Conditions for the Existence of Backward Surface Magnetostatic Waves in a Ferrite–Insulator–Metal Structure V

TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Conditions for the existence of backward surface magnetostatic waves in a ferrite–insulator–metal structure V. I. Zubkov and V. I. Shcheglov

Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino ͑Submitted December 10, 1996; resubmitted March 13, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 1–7 ͑July 12, 1998͒ A theoretical analysis is made of the dispersion of surface magnetostatic waves ͑SMSWs͒ at different frequencies propagating in ferrite–insulator–metal ͑FIM͒ structures magnetized by a uniform magnetic ﬁeld. It is established that depending on the ratio between the thicknesses of the ferrite and insulating layers in FIM structures, backward SMSWs exist in different frequency ranges and have different wave numbers and directions of propagation. The conditions for which backward SMSWs may be observed directly in FIM structures are determined. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00107-4͔

Studies of the dispersion of forward and backward sur- with varying ⍀H , whereas the angle ␸ is considered to be a face magnetostatic waves ͑SMSWs͒ in ferrite–insulator– free parameter which is defined by the antenna exciting the metal ͑FIM͒ structures are of great interest because of their SMSW (␸ϭ␸0) ͑Refs. 1–6͒. Below we determine the rela- possible applications for analog processing of information in tionship between the frequency range of existence, the wave the microwave range ͑see Ref. 1 and the literature cited numbers, and directions of propagation of backward SM- therein͒. The numerous studies on the dispersion of SMSWs SWs, and we show that the type of wave cannot be accu- in FIM structures with specific parameters1–6 create the im- rately determined merely using the dispersion law. pression that the problem of the existence of backward SM- It is known1–3,5 that SMSWs exist in a frequency range SWs in FIM structures has been well studied. However, the between a lower limit ⍀l and an upper limit ⍀u(s,␸) and in absence of direct experimental evidence of their existence4–6 a range of angles ␸ bounded by the ‘‘cutoff’’ angles is incomprehensible. Some possible reasons for this are sug- Ϯ␸c(s). We then have ⍀u(ϱ,0)Ͻ⍀u(s,␸)Ͻ⍀u(0,0) and gested below. ␸c(s)Ͻ␸c(0), where ⍀u(ϱ,0) and ⍀u(0,0) and ␸c(ϱ) and We shall consider an infinite FIM structure in the yz ␸c(0) are the upper frequencies and cutoff angles for a plane, consisting of a ferrite film of thickness d magnetized SMSW in the ferrite film (sϭϱ) and in the ferrite–metal to saturation and an ideal conducting metal layer, separated structure (sϭ0): by an insulating layer ͑in magnetostatics, a vacuum layer͒ of thickness s ͑subsequently, this layer is simply denoted by s͒. ⍀lϭͱ⍀H͑⍀Hϩ1͒, ͑2͒ The xϭ0 plane is the surface of the ferrite film closest to the ⍀u͑ϱ,0͒ϭ⍀Hϩ0.5, ͑3͒ metal layer. The magnetizing field H0 is directed along the z 2 Ϫ1 axis. In this FIM structure an SMSW propagates with the ␸c͑ϱ͒ϭarccos͕͓⍀ϩͱ⍀ Ϫ⍀H͑⍀Hϩ1͔͒͑⍀Hϩ1͒ ͖, frequency ␻ and wave vector k, whose group velocity vg is ͑4͒ directed at the angles of ␸ and ␺ to the y axis. The dispersion relation for the SMSW in the FIM struc- ⍀u͑0,0͒ϭ⍀Hϩ1, ͑5͒ 7 ture is written in the form: 2 Ϫ1 ␸c͑0͒ϭarccosͱ͓⍀ Ϫ⍀H͑⍀Hϩ1͔͒͑⍀Hϩ1͒ . ͑6͒ ͓␤Ϫ2␮␣ coth͑␣kd͔͒ϩ͑␤ϩ2Ϫ2p␯ cos ␸͒ Figure 1 shows dispersion curves ⍀s(ky ,d) for a SMSW ϫexp͑Ϫ2ks͒ϭ0, ͑1͒ in an FIM structure with different values of s. It can be seen that there are three types of curves. First, in the frequency Ϫ1 2 2 1/2 2 2 2 where ␣ϭ͓␮ sin ␸ϩcos ␸͔ ; ␤ϭ(␯ Ϫ␮ ϩ␮)cos ␸ range ⍀u(ϱ,0)Ͻ⍀Ͻ⍀u(0,0) the functions ⍀s(kyd) for 2 2 Ϫ1 2 2 Ϫ1 Ϫ␮Ϫ1; ␮ϭ1ϩ⍀H(⍀HϪ⍀ ) ; ␯ϭ⍀(⍀HϪ⍀ ) ; SMSWs in an FIM structure with small s (0Ͻsрsb1) have ⍀ϭ␻(4␲͉␥͉M )Ϫ1; ⍀ ϭH (4␲M )Ϫ1;4␲M is the a maximum at kdϭk d and tend to ⍀ (ϱ,0) for kd ϱ 0 H 0 0 0 max,s u → saturation magnetization of the ferrite film, ␥ is the electron ͑curves 2 and 3͒. Each forward SMSW with kϽkmax,s corre- gyromagnetic ratio, and pϭϮ1. Only SMSWs propagating sponds to a backward SMSW with k in kmax,sϽkϽϱ. Second, in the xϭ0 plane of the FIM structure (pϭ1) can be both in the frequency range ⍀lϽ⍀Ͻ⍀u(ϱ,0) the function 1–3 forward and backward waves. ⍀s(kyd) for an SMSW in a FIM structure with large s The dispersion relation ͑1͒ is considered to be the dis- (sуsb1) has a maximum and a minimum ͑curves 4–7͒.A persion law ⍀s(k) for given parameters ␸, d, s, and ⍀H and backward SMSW exists for sb1ϽsϽsb2 and in the frequency is used to determine the type of SMSW ͑forward or range ␦⍀(s,0)Ӷ⍀l . The forward SMSWs have wave num- 1–5 backward͒. The form of the dispersion law ⍀s(k) depends bers k in the ranges 0ϽkϽkmax,s and kmin,sϽkϽϱ while the on the ratio between d and s and does not vary qualitatively backward SMSWs have wave numbers in the range

FIG. 1. Dispersion curves ⍀s(kyd) for SMSWs in an 4 FIM structure with different s for ⍀Hϭ0.25 ͑sb1Ϸ 3d and 1 sb2Ϸ2d͒. Curves: 1—sϭ0; 2—sϭ 3d; 3—sϭd; 4—sϭ1.467d; 5—sϭ1.6d; 6—sϭ1.733d; 7—sϭ1.867d; 8—sϭ2d; 9—sϭ4d, and 10—sϭϱ. For curves 2 and 6 the points kmax,sЈ , kmin,sЈ , ⍀u(s,0), and ␦⍀s are indicated.

kmax,sϽkϽkmin,s ͑curves 4–7͒. Third, the functions ⍀s(kyd) points. If these are numbered according to increasing ky , for SMSWs in an FIM structure with sb2ϽsϽϱ have a point forward SMSWs exist at the first and third points with back- of inflection ͑curve 8͒, and only forward SMSWs could exist ward SMSWs at the second. in such structures. It can be seen from the curves kz(ky) ͑Fig. 2͒ that in FIM However, relation ͑1͒ defines the dispersion surface structures with different s, backward SMSWs in different ⍀s(ky ,kz) ͑ky and kz are the projections of the SMSW wave frequency ranges propagate in different ranges of variation of vector k on the y and x axes͒ and the dispersion law ⍀s(ky) the angle ␸ and have different wave numbers. is its intersection with the plane kzϭ0 and does not give We shall analyze SMSWs in FIM structures with small s complete information, which is obtained by additionally ͑the line of direction intersects the kz(ky) curves at two studying its intersections with the planes ⍀iϭconst(ky0kz), points͒. In the frequency range between ⍀u(ϱ,0) and 8,9 i.e., the curves kz(ky) ͑as for an SMSW in a ferrite film ͒. ⍀u(0,0) backward SMSWs exist for angles ␸ between Figure 2 gives the curves kz(ky) for an SMSW wave with the Ϫ␸c(0) and ϩ␸c(0), since the curves kz(ky) resemble non- frequencies ⍀ϭ0.608, 0.695, 0.745, and 0.775 in an FIM canonical ellipses lying at kyϾ0 ͑dotted curve 2͒ and the line structure with different s. In the ferrite film and the ferrite– of direction can only intersect these for ͉␸0͉Ͻ͉␸c(0)͉. In the metal structure the curves kz(ky) resemble canonical frequency range between ⍀l and ⍀u(ϱ,0) backward SM- 2,8,9 hyperbolas ͑curves 1 and 4͒. We denote these as qH(ϱ) SWs exist in the range of angles ͉␸c(ϱ)͉Ͻ͉␸0͉Ͻ͉␸c(0)͉ for sϭϱ and qH(0) for sϭ0. The direction of propagation since the curves kz(ky) are strictly increasing functions ͑solid of the SMSWs on the k y0kz plane is given by a straight line curves 2 and 3͒ situated between qH(0) and qH(ϱ), and the emerging from the origin at the angle ␸0 ͑subsequently line of direction can only intersect them in this range of called the line of direction; when discussing Fig. 2 we shall angles. The curves kz(ky) at the point with the lowest value draw this line mentally͒. For ͉␸0͉Ͻ͉␸c,s͉ this line intersects of ky ͑ky,inf,s for kzϭ0͒ are close to qH(0); ky,inf,s increases qH(ϱ) and qH(0) at a single point at which the projection with increasing s. of the group velocity on the direction of the phase velocity We shall now analyze SMSWs in an FIM structure with shows that the SMSWs in the ferrite film and in the ferrite– large s ͑the line of direction intersects the curves kz(ky)at metal structure are forward waves. In the FIM structure the three points͒ where two forward and one backward SMSW line of direction intersects the curves kz(ky) at two or three exist. In an FIM structure with sb1ϽsϽsb2 the curves kz(ky) Tech. Phys. Lett. 24 (7), July 1998 V. I. Zubkov and V. I. Shcheglov 501

FIG. 2. Curves of kz(ky) for SMSWs at frequencies ⍀ϭ0.695 ͑solid curves͒, 0.745 ͑dashed curves͒, and 0.775 ͑dotted curves͒ in an FIM structure with different s for ⍀Hϭ0.25. Solid and dotted curves: 1—sϭ0; 2—sϭd; 3—sϭ2d; 4—sϭ4d; 5—sϭϱ. Dashed curves: 5 1—sϭ0; 2—sϭ1.533d; 3—sϭ 3d; 4—sϭ4d. At frequencies ⍀ϭ0.695 and 0.745 curves 5 differ little from curves 4.At⍀ϭ0.775 curves 3–5 do not exist.

either consist of two parts, one resembling an ellipse, the these difficulties are known.5,6 The group velocity of back- other resembling qH(ϱ) ͑dashed curve 3͒ or they are con- ward SMSWs with ͉␸c(ϱ)͉Ͻ͉ϱ0͉Ͻ͉␸c(0)͉ is in fact lower tinuous curves, having a maximum ͑for kyϭky,max͒ and a than that of the forward SMSWs. These backward SMSWs minimum ͑for kyϭky,min͒͑dashed curve 2͒. We then find may be observed if ͉␸0͉Ϸ͉␸c(ϱ)͉ holds and narrow-band ky,maxϽky,minϽky,inf,ϱ for the curve qH(ϱ). A backward converters are used. SMSW and a second forward SMSW exist for angles ␸0 In our opinion, the only reason backward SMSWs were between 0° and Ϯ␸c(ϱ) in the frequency range ␦⍀(s,␸), not observed in Refs. 4 and 5 is because broad-band convert- lying between ⍀l and ⍀u(ϱ,0); for sуsb1 this is around ers were used. In Ref. 6 backward SMSWs could have been ⍀u(ϱ,0). In an FIM structure with sb2ϽsϽϱ the curves excited but their role was not identified because this was not kz(ky) are increasing functions exhibiting weakly defined the purpose of the study. convexity and concavity ͑curve 3͒. A backward SMSW and a second forward SMSW exist in a narrow range of angles This work was supported by the Russian Fund for Fun- damental Research ͑Grant No. 96-02-17283a͒. ͉␸c0͉Ͻ͉␸0͉Ͻ͉␸c(ϱ)͉, where ␸c0 is the angle of inclination of the line passing through the origin and the tip of the concavity. Their frequency range of existence ␦⍀(s,␸) lies be- 1 W. S. Ishak, Proc. IEEE 76, 171 ͑1988͒. 2 G. A. Vugal’ter, and I. A. Gilinski , Izv. Vyssh. Uchebn. Zaved. Radiofiz. tween ⍀l and ⍀u(ϱ,0) ͑for s ϱ, around ⍀l͒. → 1,4–6 32͑10͒, 1187 ͑1989͒. On analyzing familiar experiments ͒ concerned ex- 3 Yu. I. Bespyatykh, A. V. Vashkovski, and V. I. Zubkov, Radiotekh. clusively with the study of SMSWs propagating only for Elektron. 20, 1003 ͑1975͒. 4 O. S. Esikov, N. A. Toloknov, and Yu. K. Fetisov, Radiotekh. Elektron. ␸0ϭ0, we see that the dispersion curves for the forward 25, 128 ͑1980͒. SMSWs in an FIM structure show good agreement with the 5 A. B. Valyavski, A. V. Vashkovski, A. V. Stal’makhov, and V. A. theoretical ones. No backward SMSWs are observed; this is Tyulyukin, Radiotekh. Elektron. 33, 1820 ͑1988͒. attributed to their high propagation losses ͑as a result of their 6 V. I. Zubkov, E´ . G. Lokk, and V. I. Shcheglov, Radiotekh. Elektron. 34, low group velocity͒. 1381 ͑1989͒. 7 ´ However, the group velocity of backward SMSWs with A. V. Vashkovski, V. I. Zubkov, E. G. Lokk, and V. I. Shcheglov, Zh. Tekh. Fiz. 65͑8͒,78͑1995͓͒Tech. Phys. 40, 790 ͑1995͔͒. ␸0ϭ0 is comparable with that of the forward SMSWs ͑Fig. 8 V. I. Zubkov, E´ . G. Lokk, and V. I. Shcheglov, Radiotekh. Elektron. 35, 1͒. These backward SMSWs may be observed by using 1617 ͑1990͒. 9 ´ narrow-band converters to excite SMSWs with high wave A. V. Vashkovski, V. I. Zubkov, E. G. Lokk, and V. I. Shcheglov, Radiotekh. Elektron. 40, 950 1995 . numbers ͑Fig. 2͒ and then the only difficulties involved are ͑ ͒ measurements for small s, although ways of surmounting Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Films of a-Si:H doped with erbium from the metalorganic compound Er„HFA…3*DME, emitting at 1.54 ␮m V. B. Voronkov, V. G. Golubev, N. I. Gorshkov, A. V. Medvedev, A. B. Pevtsov, D. N. Suglobov, and N. A. Feoktistov

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg; ‘‘V. G. Khlopin Radium Institute’’ Scientific-Industrial Organization, St. Petersburg ͑Submitted December 22, 1997; resubmitted February 12, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 8–13 ͑July 12, 1998͒ For the first time standard low-temperature (Ͻ300 °C) plasma-enhanced chemical vapor deposition technology has been used to obtain a-Si͑Er͒:H films emitting at 1.54 ␮m at room temperature. The fluorine-containing metalorganic compound Er͑HFA͒3*DME, exhibiting enhanced volatility and fairly good thermal stability, was used for the first time as the Er source. The establishment of photoconduction in the synthesized samples indicates that they are of satisfactory electronic quality and potentially useful for developing light-emitting diodes at 1.54 ␮m. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00207-9͔

Materials based on erbium-doped silicon are attracting as a passivator for broken bonds,9 which helps to suppress interest primarily because of their potential usefulness for nonradiative recombination centers. In addition, the high developing silicon optoelectronic devices emitting in the 1.5 electronegativity of fluorine compared with nitrogen and ␮m range which coincides with the range of minimum opti- oxygen results in a stronger ligand field surrounding the Er3ϩ cal losses in fiber-optic communication lines. Films of ion. erbium-doped amorphous hydrogenated silicon ͑a-Si:H͒ ex- The metalorganic compounds most widely used to intro- hibit an enhanced photoluminescence intensity at 1.54 ␮m duce rare-earth elements into semiconducting materials are relative to crystalline silicon together with weak thermal ␤-diketonate complexes having the composition ML3, where quenching.1,2 Two basic methods are used to introduce Er M is a rare-earth metal, Lϭ͑CH3͒3CC͑O͒CHC͑O͒C͑CH3͒3, into a-Si:H: 1͒ ion implantation3 and 2͒ magnetron sputter- 10,11 Lϭ͑CH3͒3CC͑O͒CHC͑O͒͑C3F7͒, and so on. However, ing of a mosaic erbium–silicon or erbium–graphite target in unsolvated -diketonates of rare-earth elements are 1,4 ␤ a silane atmosphere. In both cases, the synthesized films coordination-unsaturated compounds inclined to hydrolysis demonstrated photoluminescence in the 1.54 ␮m range both and thermolysis. In order to avoid this, the coordination va- at liquid-nitrogen and room temperatures. A disadvantage of cancies must be blocked with suitable auxiliary ligands. this method is the high concentration of intrinsic defects in Here, erbium was introduced from the complex the films, which hinders their subsequent use for the devel- Er͑HFA͒*DME ͑Fig. 1͒, specially synthesized using a opment of optoelectronics devices. In Refs. 5 and 6 Si͑Er͒ 3 technique described in Ref. 12, where films were synthesized using plasma enhanced chemical va- HFAϭCF C͑O͒CHC͑O͒CF ͑hexafluoracetylacetone͒ and por deposition ͑PE CVD͒, a modified plasma-chemical vapor 3 3 DMEϭCH OCH CH OCH ͑1.2-dimethoxyethane͒. The deposition technology which is usually used to obtain a-Si:H 3 2 2 3 auxiliary ligand in this compound is selected so that it expels films. Erbium was introduced into the film from metalor- coordinated water molecules from the complex which could ganic compounds during the PE CVD process at cause high-temperature hydrolysis and which is also easily Tϭ430 °C under cyclotron resonance conditions, which ensured epitaxial Si growth. In this case, the photolumines- eliminated from the complex, without contaminating the cence at 1.54 ␮m was recorded at liquid-helium tempera- growing film with impurity phases. In Ref. 13 the use of such tures. compounds was demonstrated for the chemical vapor depo- * The aim of the present study was to obtain a-Si͑Er͒:H sition of MF3 films ͑MϭNd, Eu, Er͒. The Er͑HFA͒3 DME films using standard low-temperature (Ͻ300 °C) PE CVD complex is converted to the gas phase at fairly low tempera- technology7 usually used to produce individual layers and ture (120– 140 °C). At temperatures of 320– 340 °C the multilayer ‘‘device-quality’’ a-Si:H structures for solar cells. metalorganic compound undergoes gas-phase thermal disso- 13 Particular attention was paid to the choice of metalor- ciation with the formation of an ErF3 film. ganic compound used to introduce the Er3ϩ ions into the film Amorphous a-Si͑Er͒:H films were grown by rf dissocia- in a chemical environment suitable for converting erbium tion of an argon–silane mixture in a glow discharge plasma. into an optically active center. It has been established4–6 that The parameters of the plasma-chemical process were as fol- O, N, and F impurities help to increase the 1.54 ␮m emission lows: silane content in argon,10%; pressure, 0.1–0.2 Torr; intensity in erbium-doped silicon. Moreover, a fluoride frequency, 17 MHz; rf power, 0.03– 0.1 W/cm2; substrate neighborhood for the erbium leads to smaller lattice distor- temperature, 200– 250 °C; and gas mixture flux, 5–10 sccm. tions compared with an oxygen one.8 We also note that like The erbium was introduced by sublimation of hydrogen, fluorine is widely used in amorphous silicon films Er͑HFA͒3*DME powder. A 0.1 g weighed portion was placed

width at half maximum Ϸ7 meV͒ centered at ␭ϭ1.54 ␮m, 4 4 which is attributed to the I13/2 I15/2 intracenter transition of the Er3ϩ ion. At room temperature→ the line shows appreciable broadening in the short-wavelength direction and its intensity drops approximately fourfold. Measurements of the photoluminescence intensity as a function of the excitation power showed that this dependence is linear up to Ϸ200 mW, and then the signal saturates. The second, broad band near 1 ␮m reflects the well-known ‘‘tail–tail’’ transitions in undoped a-Si:H ͑Ref. 4͒. However, the elongation of this line in the long-wavelength direction may evidence the presence of an appreciable concentration of oxygen in the * 14 FIG. 1. Er͑HFA͒3 DME complex: samples. As was to be expected, this line undergoes strong a—auxiliary ligand DMEϭCH3OCH2CH2OCH3; thermal quenching caused by a thermally activated transition b—principal ligand HFAϭCF C͑O͒CHC͑O͒CF . 3 3 of nonequilibrium carriers to nonradiative states. To sum up, standard low-temperature (Ͻ300 °C) PE in a stainless steel boat positioned approximately 4 cm from CVD technology has been used to obtain a-Si͑Er͒:H films the discharge gap where the substrate with the growing emitting in the 1.54 ␮m range at room temperature. A a-Si:H film was placed. The rate of sublimation of the met- fluorine-containing metalorganic compound Er͑HFA͒3*DME alorganic compound was varied by heating the boat from 25 possessing enhanced volatility and fairly good thermal sta- to 200°. Fused quartz and crystalline silicon were employed bility was used for the first time as an Er source. This com- as the substrates. The thickness of the films was determined pound had been used previously13 to obtain films of pure in situ by laser interferometry and was 2000–3000 Å for erbium fluoride. The observation of photoconduction in the different samples. The films were annealed at Tϭ300 °C at synthesized samples indicates that they are of satisfactory normal pressure in a pure nitrogen flux ͑50 sccm͒ for 15 min. electronic quality and potentially useful for the development The concentration of impurities in the film was deter- of 1.54 ␮m light-emitting diodes. mined by SIMS and was 1019 cmϪ3 for erbium, 2ϫ1020 cmϪ3 for fluorine, and 1020 cmϪ3 for oxygen, with a This work was supported by the International Scientific- fairly uniform distribution over sample thickness. The con- Technical Program ‘‘Physics of Solid-State Nanostructures’’ ductivity of the films was at most 10Ϫ10 ScmϪ1 and the ͑Project No. 96-1012͒. photoconductivity was around 10Ϫ7 ScmϪ1. The photoluminescence of the films was measured using 1 M. S. Bresler, O. V. Gusev, V. Kh. Kudoyarova, A. N. Kuznetsov, P. E. an automated KSVU-23 system. The detector was a cooled Pak, E. I. Terukov, I. N. Yassievich, B. P. Zakharchenya, W. Fuhs, and germanium photodiode. The photoluminescence was excited A. Sturm, Appl. Phys. Lett. 67, 3599 ͑1995͒. 2 using the 4880 Å Arϩ laser line at 40 mW. E. I. Terukov, V. Kh. Kudoyarova, M. M. Mezdrogina, V. G. Golubev, A. Sturm, and W. Fuhs, Fiz. Tekh. Poluprovodn. 30, 820 ͑1996͓͒Semi- Photoluminescence spectra at room and liquid-nitrogen conductors 30, 440 ͑1996͔͒. temperatures are shown in Fig. 2. At low temperature the 3 J. H. Shin, R. Serna, G. H. van den Hoven, A. Polman, W. G. J. M. van spectrum exhibits two bands. The first is a narrow line ͑full Sark, and A. M. Vredenberg, Appl. Phys. Lett. 68, 997 ͑1996͒. 4 A. R. Zanatta, L. A. O. Nunes, and L. R. Tessler, Appl. Phys. Lett. 70, 511 ͑1997͒. 5 J. L. Rogers, P. S. Andry, W. J. Varhue, E. Adams, M. Lavoie, and P. B. Klein, J. Appl. Phys. 78, 6241 ͑1995͒. 6 P. S. Andry, W. J. Varhue, F. Lapido, K. Ahmed, P. B. Klein, P. Hengen- hold, and J. Hunter, J. Appl. Phys. 80, 551 ͑1996͒. 7 The Physics of Hydrogenated Amorphous Silicon ͑Topics in Applied Physics, Vols. 55–56͒, edited by J. D. Joannopoulos and G. Lucovsky ͑Springer-Verlag, Berlin, 1984; Mir, Moscow, 1987͒, Part 1, p. 363. 8 W.-X. Ni, K. B. Joelsson, C.-X. Du, I. A. Buyanova, G. Pozina, W. M. Chen, G. V. Hansson, B. Monemar, J. Cardenas, and B. G. Svensson, Appl. Phys. Lett. 70, 3383 ͑1997͒. 9 A. Madan, in The Physics of Hydrogenated Amorphous Silicon ͑Topics in Applied Physics, Vols. 55–56͒, edited by J. D. Joannopoulos and G. Lucovsky ͑Springer-Verlag, Berlin, 1984; Mir, Moscow, 1987͒, Part. 1, p. 310. 10 M. Morse, B. Zheng, J. Palm, X. Duan, and L. C. Kimerling, Mater. Res. Soc. Symp. Proc. 422,41͑1996͒. 11 A. J. Kenyon, P. F. Trwoga, M. Federighi, and C. W. Pitt, Mater. Res. Soc. Symp. Proc. 358, 117 ͑1995͒. 12 N. P. Grebenshchikov, G. V. Sidorenko, and D. N. Suglobov, Ra- diokhimiya 32͑3͒,14͑1990͒. 13 N. I. Gorshkov, D. N. Suglobov, and G. V. Sidorenko, Radiokhimiya 37͑3͒, 196 ͑1995͒. 14 R. Street, Adv. Phys. 30, 593 ͑1981͒. FIG. 2. Photoluminescence spectra of a-Si͑Er͒:H films at liquid-nitrogen ͑1͒ and room ͑2͒ temperatures. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Formation of a cathode crater in a low-voltage cold-cathode vacuum arc M. K. Marakhtanov and A. M. Marakhtanov

N. E. Bauman State Technical University, Moscow ͑Submitted October 7, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 14–19 ͑July 12, 1998͒ The cathode crater in a low-voltage cold-cathode vacuum arc is planar. The area of the crater and the density of the current ﬂowing through it depend on the thermal and electrical properties of the cathode metal. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00307-3͔

Experiments were carried out to establish the dimensions the cathode spot. The quantity of metal removed by the arc and shape of the craters left by a low-voltage cold-cathode depends on the electric charge passing through the crater. vacuum arc. The density of the current flowing through the Thus, the volume of a planar crater should increase over time crater was determined for Zn, Al, Cu, Fe, Ti, and Cr cath- T, mainly as a result of an increase in its area ϳD2.We odes at voltages of 16.7–36 V and arc currents of 42–121 A, measure the diameter D left by the arc at the end of the depending on the material. The experiments were carried out using the Bulat-6 industrial vacuum-arc evaporation system at an argon pressure of 0.04 Pa in the vacuum chamber. The cathodes were in the shape of truncated cones with base di- ameters of 56 and 60 mm and a height of 40 mm. The arc burns on the surface of the smaller base, over which the cathode spot moves causing cathode sputtering. The opposite base was water-cooled and the inner wall of the vacuum chamber served as the anode. The entire sputtered surface of the spent cathode was covered with craters. The outline of the crater was bounded by folds of solidified metal. Two groups of craters whose sizes differ by a factor of ten are clearly distinguishable. The large cathode craters 1 are distributed fairly sparsely while the small cathode craters 2 completely cover the sputtered surface ͑Fig. 1͒. For the measurements we used a NEOPHOT 21 microscope, Carl Zeiss, Jena ͑crater depth and ϫ180 pho- tographs͒, an MIM-7 metallurgical microscope ͑depth and diameter of craters͒, and an MBS-9 laboratory microscope ͑diameter and distance between craters͒. Each dimension quoted here is the average of 25–50 measurements. The diameter D of the cathode crater is assumed to be 0.5(DmaxϩDmin). The crater depth h was determined by focusing the microscope objective first onto the tips of the folds bounding the crater ͑Fig. 1a͒ and then onto the crater floor ͑Fig. 1b͒. The crater depth was taken to be the difference between the readings of the micrometer used to move the objective. The accuracy of the measurements of h is Ϯ0.0025 mm. The true crater depth is less than h because the latter was measured from the tips of the folds, which protrude above the surface of the cathode. The average crater dimensions were Dϭ0.498, 0.264, 0.173, 0.183, 0.169, and 0.087 mm and hϭ0.043, 0.017, 0.0185, 0.0157, 0.0082, and 0.0059 mm for Zn, Al, Cu, Fe, Ti, and Cr cathodes, respectively. The average cathode crater is planar since the ratio FIG. 1. Photomicrographs of the surface of an aluminum cathode, arc volt- h/Dϭ0.048– 0.107 is obtained for these metals. age 36 V, arc current 52 A; 1—cathode crater, 2—small cathode crater; It was previously established1 that an arc moves over the objective focused onto the tips of the metal folds surrounding the crater 1 cathode by replacing old craters with new ones, the average ͑a͒; objective focused onto the bottom of the cathode crater 1 ͑b͒; the black Ϫ6 Ϫ3 spots on the photographs correspond to protrusions at the cathode surface or crater lifetime being finite at Tϭ10 –10 s. It is also its unfocused image; the white spots correspond to indentations or the bot- known2 that the entire arc current flows through the crater of tom of the crater and small craters.

FIG. 2. Diagram of ‘‘switching-thermal circle’’ for cold metal cathodes: with high thermal conductivity ␭Ͼ100 ͑a͒ and low thermal conductivity, ␭Ͻ100 ͑b͒; the diameter D of the cathode crater and the distance R between neighboring craters vary, obeying an explicit dependence related to the thermal and electrical characteristics of the metal cathode.

period T. Thus, the ratio jϭI/D2 is equal to the minimum craters are positioned arbitrarily, separated by the same dis- current density in the cathode crater if we neglect any varia- tance R, we obtain a ‘‘switching-thermal circle’’ system tion of the arc current I with time. ͑Fig. 2͒. Here the cathode craters are shown as small circles The current density j for which a crater can still exist is of diameter D. We assume that after the arc has been 0.19ϫ109, 0.75ϫ109, 3.51ϫ109, 3.61ϫ109, 3.33ϫ109, quenched at the hot point of a disappearing crater, it enters and 12.29ϫ109 A/m2, for Zn, Al, Cu, Fe, Ti, and Cr, respec- the switching part of the period T. During the switching tively. These values agree with the current densities of 3.4 time, the arc systematically reappears and is quenched in ϫ109, 3.2ϫ109, and 1.5ϫ109 A/cm2, which were observed several small cathode craters whose position gradually in experiments using Ag, Cu, and Mo exploding wires, moves from the hot point of the disappearing crater to the respectively.3 cold boundary of the circle R. Here, at the cold surface of the The maximum temperature of the cathode crater barely cathode, the crater again acquires the normal size D. exceeds the boiling point Tb of the metal. Assuming that the The displacement of the crater over the cathode is current density of the thermionic emission from a boiling caused by the thermal and electrical properties of the cath- 6 2 Ϫ1 Ϫ1 crater is jeϭ1.2ϫ10 Tb exp͕Ϫe␸/kTb͖, then we have je ode. For Zn, Al, and Cu for which ␭Ͼ100 W•m •K , the Ϫ6 2 7 2 ϭ1.3ϫ10 A/m for a Zn cathode and jeϭ3.9ϫ10 A/m dimensions D and R decrease as the cathode melting point for a Ti cathode. Here Zn has the boiling point T f , K and the thermal conductivity ␭ increase ͑Fig. 2a͒. For Tbϭ1179 K and work function ␸ϭ4.24 V, while Ti has Fe, Ti, and Cr we ﬁnd ␭Ͻ100, and the dimensions D and R Tbϭ3560 K and ␸ϭ3.95 V. Given the obvious inequality decrease as the ratio of the thermal and electrical conductivi- Ϫ1 jeӶ j, the contribution of the thermionic emission from the ties ␭/␴,W•⍀•K , increases ͑Fig. 2b͒. cathode to the arc current can be neglected for all these met- Almost the entire mass of the cathode may ﬂow into the als. arc via a few large cathode craters. For example, the average 2 The average distance between the craters is Rϭ1.69, crater on Ti has the mass Mϭ␳•D •hϭ4540 1.13, 0.69, 1.90, 1.27, and 0.84 mm for Zn, Al, Cu, Fe, Ti, ϫ(0.169ϫ10Ϫ3)2ϫ0.0082ϫ10Ϫ3ϭ0.11ϫ10Ϫ8 kg, where and Cr, respectively. The ratio of this distance to the crater ␳ϭ4540 kg/m3 is the Ti density. The average rate of sput- diameter is R/Dϭ3.4– 10.4 for Zn–Fe, respectively. If the tering of titanium in the arc was determined as Sϭ4.42 506 Tech. Phys. Lett. 24 (7), July 1998 M. K. Marakhtanov and A. M. Marakhtanov

ϫ10Ϫ6 kg/s. If the rate S is constant in time, the mass M also on the thermal and electrical conductivities of the cath- leaves the average crater in the time tϭM/Sϭ2.5ϫ10Ϫ4 s, ode metal. which is approximately equal to the crater lifetime T ͑see above͒. 1 I. G. Kesaev, Cathode Processes in an Electric Arc ͓in Russian͔, Nauka, Moscow, p. 6, 144. The physical quantities used were taken from Ref. 3. 2 To sum up, in a low-voltage cold-cathode vacuum arc F. H. Webb, H. H. Hilton, P. H. Levine, and A. V. Tollestrup, Exploding Wires, Vol. 2, edited by W. G. Chace and H. K. Moore ͑Plenum Press, the cathode crater has a planar profile with h/DՇ0.1. The New York, 1962͒,p.37–77. 3 crater disappears when the density of the current flowing A. P. Babichev, N. A. Babushkina, A. M. Bratkovski et al.,inHandbook through its cross section is (0.2– 12)ϫ109 A/m2 ͑Zn–Cr͒. of Physical Quantities, edited by I. S. Grigor’ev and E. Z. Melikhova ͓in Russian ,Eńergoatomizdat, Moscow 1991 , 1232 pp. The average crater diameter and the average distance be- ͔ ͑ ͒ tween neighboring craters depend on the melting point, and Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Spontaneous formation of arrays of three-dimensional islands in epitaxial layers ´ V. G. Dubrovski ,G.E. Cirlin, D. A. Bauman, V. V. Kozachek, and V. V. Mareev Institute for Analytical Instrumentation, Russian Academy of Sciences, St. Petersburg; Institute of High-Performance Computations and Databases, St. Petersburg ͑Submitted December 19, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 20–26 ͑July 12, 1998͒ An analysis is made of a theoretical model of the formation of three-dimensional nanometer-size islands in molecular beam epitaxy. The kinetics of the self-organization processes are described using a lattice gas model of the adsorbate with self-consistent allowance for lateral interactions in the activation energies of the diffusion processes. It is shown that at below-critical temperatures in a certain range of thicknesses, decay of the spatially uniform state gives rise to arrays of three-dimensional nano-islands which do not participate in the coalescence process after growth has ceased. The average size of the islands, their geometric proﬁle, and the spatial ordering depend strongly on the kinetic parameters of the model. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00407-8͔

Studies of nanostructure formation processes in molecu- ␪t͑R,t͒ lar beam epitaxy MBE and its modiﬁcations are of consid- pl͑R,t͒ϭ ϱ , lу1. ͑2͒ ͑ ͒ 1ϩ͚ ␪ ͑R,t͒ erable interest from the fundamental viewpoint as well as lϭ2 l for practical applications in microelectronics and The three-dimensional surface of the epitaxial ﬁlm is de- 1 optoelectronics. One of the most promising methods of ob- scribed by the function taining nanostructures is their direct formation as a result of surface self-organization effects in MBE growth. Requiring ϱ no pretreatment or aftergrowth surface treatment, this h͑R,t͒ϭ lpt͑R,t͒. ͑3͒ l͚ϭ0 method can produce systems of three-dimensional, dislocation-free, nanometer-size islands ͑quantum dots͒ at a The system of nonlinear rate equations proposed in Ref. 8 for 2 high surface density. With a suitable choice of growth con- ␪t(R,t) has the general form ditions, a reduction in the spread of island sizes as well as 3 ␪l͑R,t͒ץ their spatial ordering are observed experimentally. Despite ϭADϩMϩP, ͑4͒ tץ the considerable success achieved in the equilibrium theory of nanostructure formation in heteroepitaxial systems,4,5 Monte Carlo simulation,6,7 and continuum models of diffu- where the operators AD, M, and P describe adsorption– sion instabilities in MBE,6 the kinetic theory of self- desorption processes, two-dimensional diffusion within layer organization effects in MBE has not been adequately devel- l, and interlayer transitions in the three-dimensional lattice. oped. In particular, very few microscopic models are We subsequently use the following explicit representations available to examine the relationship between the growth of the kinetic operators to make self-consistent allowance for kinetics and the surface morphology, or the temperature de- the interaction between lattice gas particles pendence of the characteristics of these nanostructures. ␪ Here we attempt to construct such a model by general- l ␣ ADϭ␣lJ␴͑1Ϫ␪l͒␪lϪ1Ϫ ␣ exp͑Ϫ␾l ␪l͒, ͑5͒ izing the model of the dynamics of a nonideal three- tl dimensional adsorbate.8 The proposed model is used to make a numerical analysis of self-organization effects in an epitax- 1 Mϭ ͑␪Ј exp͑Ϫ␾D␪Ј͒͑1Ϫ␪ ͒␪ Ϫ␪ ial layer, which result in the formation of ordered arrays of D ͚ l l l l lϪ1 l ztl RЈ three-dimensional nano-islands. D The lattice gas model is used to describe the growth ϫexp͑Ϫ␾l ␪l͒͑1Ϫ␪lЈ͒␪lЈϪ1͒, ͑6͒ kinetics. The occupation numbers of the site (l,R) in layer l with the discrete coordinate R in the substrate plane n (R) ␪lЈϪ1 ␪lЈϩ1 l Pϭ exp͑Ϫ␾ P ␪Ј ͒ϩ are 0 or 1. The occupation number averaged over all micro- ͚ ͫͩ t lϪ1 lϪ1 t RЈ lϪ1,l lϩ1,l scopic states at time t gives the probability of observing an adatom at the site (l,R)␪ (R,t)ϭ n (R) . The statistically P l ͗ l ͘t ϫexp͑Ϫ␾ ␪Ј ͒ ͑1Ϫ␪ ͒␪ Ϫ␪ averaged probability that the local height of the surface at lϩ1 lϩ1 ͪ l lϪ1 l point R is l monolayers at time t is given by 1Ϫ 1Ϫ P ͑ ␪lЈϪ1͒␪lЈϪ2 ͑ ␪lЈϩ1͒␪lЈ ϫexp͑Ϫ␾l ␪l͒ ϩ , 1Ϫ␪1͑R,t͒ ͩ t t ͪͬ l,lϪ1 l,lϩ1 p0͑R,t͒ϭ ϱ , ͑1͒ 1ϩ͚lϭ2␪l͑R,t͒ ͑7͒

1063-7850/98/24(7)/3/$15.00 507 © 1998 American Institute of Physics 508 Tech. Phys. Lett. 24 (7), July 1998 Dubrovski et al. where ␪lЈϵ␪l(RЈ), ␪lϵ␪l(R), the time dependence is omit- ␪l͑x,y,tϭ0͒ϭ␪l͑0͒ϩ␥ min͑1Ϫ␪l͑0͒,␪l͑0͒͒ ted to simplify the notation, and ␪ϵ1 and t0,1ϭϱ in the absence of particle exchange between the epitaxial layer and 2␲x 2␲y ϫsin ␻x sin ␻y . ͑8͒ the substrate. Summation is performed over all nearest ͩ L ͪ ͩ L ͪ neighbors in the two-dimensional lattice. In Eqs. ͑5͒–͑7͒ J is the particle flux density to the surface, ␴ is the area of the adsorption cell, ␣l is the coefficient of adsorption at the sur- The lattice period L in the plane of the substrate, the face of the lϪ1 layer, ta , tD , t and t are the char- l l lϪ1,l l,lϪ1 value of ␪l(0), and the perturbation amplitude ␥ were var- acteristic lifetimes of an absorbed particle in layer l, diffu- ied. The perturbation frequencies ␻xϭ␻y for given L were sion in layer l, lϪ1 l ͑upward ‘‘jumping’’͒ and l lϪ1 → → taken to be equal to the critical frequency at which the ͑downward ‘‘sliding’’͒ interlayer transitions for zero coating, ground state with ¯h(0)ϭconst becomes unstable in the linear and z is the coordination number of the two-dimensional approximation at below-critical temperature (␾DϾ4) ͑Refs. a D P * lattice. The binding energies ␾l , ␾l , and ␾l expressed in 8 and 9͒. The instability of the spatially periodic solution is units of kBT in self-consistent form allow for the influence of caused by anomalous diffusion in the spinoidal region10 lateral interactions at desorption activation barriers, two- where particle attraction predominates over the usual diffu- dimensional intralayer hopping diffusion, and interlayer 8,9 sion because the diffusion flux is directed along the density transitions. The Langmuir factors (1Ϫ␪l(R))␪lϪ1(R) im- gradient of the lattice gas. Under these assumptions, the sys- ply vacancy forbiddenness and overhanging layers, which is tem of equations 4 – 7 for (R,t) was integrated numeri- 7 ͑ ͒ ͑ ͒ ␪l a normal assumption in the theory of MBE growth. Expres- cally at each time step with periodic boundary conditions. sions ͑5͒–͑7͒ are given for a homogeneous planar substrate. Formulas ͑1͒–͑3͒ were then used to calculate the evolution A numerical analysis was made for a cubic mesh assum- of the surface morphology. ing a fixed quantity of material at the surface ADϭ0, i.e., The results of the calculations show that for ␾Dу4.5, after the end of growth in the absence of desorption Jϭ0, P D * ͑ regardless of the value of ␾ , the time ratio t↑↓/t , and the a * * * tl ϭϱ͒. An analysis was made of the case of heteroepitaxy parameters of the first layer in the range of thickness between where the kinetic constants and interaction parameters were 2 and 4 ML, the application of a weak critical-frequency D D assumed to be the same in all layers from lϭ2: t1 Þt2 perturbation causes the epitaxial layer to decay into a system D D P P ϭt ϭ ...ϵt ; t Þt ϭt ϭt ϭ ...ϵt↑↓ ; ␾ Þ␾ 3 * 12 21 23 32 * 1 2 of three-dimensional nano-islands. In this case, the geometry ϭ␾Pϭ ...ϵ␾P ; ␾DÞ␾Dϭ␾Dϭ...ϵ␾D. The initial 3 * 1 2 3 * of the islands, their distribution over the surface of the sub- ¯ state ␪l(0) with uniformly filled layers and thickness h(0), strate, and the spatial ordering depend strongly on the critical D D constant over the surface of the substrate and equal to the parameters. By varying the characteristics t1 , t12 , ␾1 , and P quantity of deposited material in one monolayer, was per- ␾1 , we can model the formation of arrays of islands by the turbed by a biperiodic perturbation given by Volmer–Weber mechanism and by the Stranski–Krastanow

FIG. 1. Surface morphology with an average thickness of D P D 2 ML, ␾ ϭ5, ␾ ϭ3, t↑↓/t ϭ800, and time tϭ0.5t↑↓ . * * * * * The numbers on the axes are in nanometers. Tech. Phys. Lett. 24 (7), July 1998 Dubrovski et al. 509

FIG. 2. As in Fig. 1 for tϭ2t↑↓ ͑quasi-steady-state surface * structure͒.

mechanism.3,5 Figures 1 and 2 show typical dynamic patterns flux, and also to make systematic allowance for the lattice of the formation of three-dimensional islands with the aver- mismatch in the microscopic growth model. age thickness ¯h(0)ϭ2 ML. At the initial stage of decay, the The authors are grateful to the scientific program ‘‘Phys- surface morphology resembles nonlinear density oscillations. ics of Solid-State Nanostructures’’ for partially financing this This is superseded by the formation of islands with clearly work. defined boundaries. The surface structure shown in Fig. 2 is The authors also thank N. N. Ledentsov and V. A. stable for times exceeding its characteristic formation time in Shchukin for useful discussions of various aspects of this order of magnitude. Thus, we can talk of the formation of an work. ensemble of islands which do not take part in the coalescence process for a long time.5 It should be noted that as the temperature increases (4р␾Dр4.5) no islands with clearly * 1 defined boundaries are formed. A spatially ordered struc- Y. Arakava and A. Yariv, IEEE J. Quantum Electron. QE-22, 1887 ͑1986͒. ture clearly does not form at above-critical temperatures 2 P. M. Petroff and G. Medeiros-Ribeiro, Mater. Res. Bull. 21,50͑1996͒. D (␾ Ͻ4). 3 G. E. Cirlin, V. N. Petrov, A. O. Golubok, S. Ya. Tipissev, V. G. * To sum up, the proposed model can in principle describe Dubrovskii, N. N. Ledentsov, and D. Bimberg, Surf. Sci. 377–379, 895 the dynamics of surface transformation for a fixed quantity ͑1997͒. 4 J. Tersoff and R. M. Tromp, Phys. Rev. Lett. 70, 2782 ͑1993͒. of deposited material. The results show that in many cases, 5 V. A. Shchukin, N. N. Ledentsov, P. S. Kop’ev, and D. Bimberg, Phys. the influence of lateral interactions on the activation energy Rev. Lett. 75, 2968 ͑1995͒. of elementary diffusion processes at the surface can give rise 6 M. Rost, P. Smilauer, and J. Krug, Surf. Sci. 369, 393 ͑1996͒. 7 A. L. Barabasi, Appl. Phys. Lett. 70, 2565 ͑1997͒. to stable arrays of three-dimensional nano-islands. The kinet- 8 G. V. Dubrovski, and V. V. Kozachek, Zh. Tekh. Fiz. 65͑4͒, 124 ͑1995͒ ics of the diffusion processes and the interaction constants ͓Tech. Phys. 40, 359 ͑1995͔͒. strongly influence the surface morphology. However, the 9 V. G. Dubrovskii, G. E. Cirlin, D. A. Bauman, V. V. Kozachek, and V. V. Mareev, Vacuum, in press. theory must be generalized to describe surface self- 10 organization effects directly during the MBE growth process, S. A. Kukushkin and A. V. Osipov, Prog. Surf. Sci. 51,1͑1996͒. to allow for adsorption and desorption, a two-component Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Energy dissipation of mechanical oscillators induced by an electric ﬁeld applied to the surface of an oscillating body N. A. Vishnyakova, M. L. Gorodetski , V. P. Mitrofanov, and K. V. Tokmakov M. V. Lomonosov State University, Moscow ͑Submitted April 16, 1997; resubmitted December 22, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 27–33 ͑July 12, 1998͒ Various factors inﬂuencing the damping of the torsional oscillations of a pendulum are discussed. Damping caused by dissipative surface diffusion processes or charge transfer of surface states is considered to be the most probable. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00507-2͔

The development of antennas to record gravitational ra- of the pendulum and the lower electrode, the oscillations diation from cosmic sources has stimulated the fabrication of exhibited increased damping. This effect occurred when the pendulum-type high-Q mechanical oscillators. It has been pendulum was situated in vacuum and in a room atmosphere. noted in various experimental studies that one of the sources In the second case, the damping was appreciably stronger. of dissipation limiting the Q-factor of these systems is the We stress that these are not losses caused by gas friction action of external electric fields on the oscillating body.1,2 since the difference between the losses with and without the The aim of the present paper is to study the damping of the field was determined. Figure 2a gives the insertion losses mechanical oscillations of an oscillator when an electric field Ϫ1 Qe as a function of the voltage U applied between the body is created between the conducting surface of the oscillating of the pendulum and the lower electrode, separated by a gap body and the surface of a special electrode. of 0.5 mm. The lower electrode was made of p-type silicon Most of the experiments were carried out using the pen- with a resistivity of 20 ⍀•cm. The curves were obtained for dulum shown schematically in Fig. 1. An aluminum disk 6 different gas environments. Curve 1 was plotted using the cm in diameter and weighing around 100 g was suspended results of measurements made in a dry nitrogen atmosphere. by means of three 100 ␮m diameter tungsten wires. The Within measurement error, this agrees with the similar de- resonant frequency of the torsional oscillations of this pen- pendence measured in vacuum at a pressure of 10Ϫ2 Torr. dulum was around 1 Hz. The lower surface of the disk had Curve 2 was obtained in a nitrogen atmosphere with a small radial protrusions at 2 mm intervals. An electric field was quantity of water vapor while curve 3 was obtained in a generated between the body of the pendulum and an elec- room atmosphere with around 70% relative humidity. The trode positioned beneath the disk, parallel to it. The elec- hysteresis clearly observed in the moist atmosphere showed trodes were planar disks made of aluminum, brass, and sili- up less clearly in vacuum. Note that the character and mag- con. The gap between the electrode and the pendulum could nitude of the effect did not depend on the polarity of the be varied between 0.2 and 2 mm. Torsional oscillations of applied voltage. It is also interesting to note that if an ac the pendulum were excited by a resonant force generated by electric field with a frequency of 300 Hz or higher was ap- applying a periodic electric voltage to the auxiliary electrode plied between the pendulum and the lower plate, no addi- and were recorded using a capacitive detector. The insertion losses were determined as the difference between the recip- tional damping was observed within measurement error. It rocal Q factor of the pendulum oscillations measured with was also established that the introduced damping does not and without the electric field. The measurements were made remain constant after the electric field is switched off. The in a special chamber in which either a controlled atmosphere nature of this change also depended on the state of the am- or a vacuum was created. bient atmosphere. Similar effects were observed for alumi- The design of the pendulum and the electrode used to num or brass lower electrodes. In this case, the effect also generate the field was dictated by the need to create a high increased strongly with increasing atmospheric humidity, but electric field strength in the gap, without introducing any this increase was between three and five times less than that appreciable extra rigidity in the oscillatory system. This for the silicon electrode under similar conditions. Ϫ1 eliminated any dissipation of the pendulum mechanical en- Figure 2b gives the insertion losses Qe caused by the ergy as a result of Joule losses to the resistance in the voltage electric field as a function of the applied voltage when the supply circuit, induced by an electric current flowing in this conducting coating of the pendulum and the electrode were circuit as a result of a change in the capacitance between the made of aluminum. The gap between the pendulum and the field-generating electrode and the oscillating body. It also electrode was 2 mm. The pendulum was suspended by quartz eliminated any dissipation as a result of coupling between threads and thus had a high intrinsic Q factor in the absence the oscillator and the electrode, induced by the electric field.2 of an electric field.1 The measurements were made in When an electric voltage was applied between the body vacuum at a pressure of 10Ϫ4 Torr. The experimental depen-

FIG. 1. Schematic of pendulum and circuit used to apply the electric field. dence was accurately approximated by a quadratic function ͑shown by the dashed curve in Fig. 2b͒. Another interesting effect deserves attention. In the experiments carried out at atmospheric pressure, additional damping of the pendulum was also observed in the absence of an external electric field, for example, if the field was FIG. 2. Electric-field-induced losses versus voltage applied between pendu- applied for some time and then switched off or when the lum and electrode: a—when the electrode plate is made of silicon. The humidity of the ambient atmosphere changed drastically, measurements were made for various compositions of the ambient gas: 1— In general, the damping observed for these pendulums dry nitrogen, 2—nitrogen containing a small amount of water vapor, 3—air on application of an electric field can be attributed to various with Ϸ70% humidity; b—for an aluminum-coated pendulum and electrode at a pressure of Ϸ10Ϫ4 Torr. factors. For instance, the ponderomotive force acting on the pendulum alters the tension in the suspension wires. How- ever, our experimental investigations showed that any change in the tension of the suspension wires in the absence and the electrode between which the field is applied. It is of a field barely influences the Q factor of the pendulum. The known that the electrophysical state of the surface is deter- absence of electric discharges in the interelectrode gap, mined to a considerable extent by the composition of the which could also introduce additional losses in the pendulum surrounding atmosphere and may vary substantially as a re- oscillations, was monitored by measuring the noise in the sult of the adsorption of water molecules, for example.3 It voltage supply circuit. Discharges were accompanied by ex- may be supposed that a possible mechanism for dissipation cess noise. The oscillations of the pendulum in an electric of the oscillator energy is surface diffusion of adsorbed at- field are accompanied by induced currents on the lower elec- oms caused by spatial variation of the electric field. In par- trode and the adjacent conducting surfaces. The damping ticular, water molecules, having a high static dipole moment, caused by these induced currents depends on the bulk resis- can migrate under the action of a field gradient. In measure- tivity of the materials. Calculations showed that for these ments of the electric-field-induced damping of the pendulum materials ͑silicon, aluminum, and brass͒ this damping is oscillations made in vacuum, diffusion processes at the sur- much less than that observed in these experiments. In addi- face make a smaller contribution to the dissipation. In this tion, these processes should also be observed when an ac case, however, another loss mechanism may be observed as a voltage is applied between the pendulum and the electrode, result of the presence of localized surface states. The electric but in this case no additional damping of the pendulum os- field applied to the surface varies under the pendulum oscil- cillations was observed. lations, causing carriers to be transferred from the bulk to An analysis of these results suggests that the dissipation surface states and back, or from some surface states to oth- introduced into the mechanical oscillator by an electric field ers. depends on the state of the surfaces of the oscillating body Depending on the magnitude of the applied voltage, the 512 Tech. Phys. Lett. 24 (7), July 1998 Vishnyakova et al. thickness of the oxide layer, and the composition of the am- switching off the external field is associated with the electric bient gas atmosphere, charge may be transferred to surface fields formed during adsorption but this loss mechanism re- states under the action of the electric field by different mains unclear. Studies of the energy dissipation of mechani- mechanisms: diffusion, tunneling, and hopping. The impor- cal oscillations induced by an electric field may provide ad- tant thing is that all these processes are dissipative in these ditional information on the surface properties of a solid. systems and are caused by oscillations of a body and are thus accompanied by scattering of the oscillator elastic energy. The authors would like to thank V. B. Braginski,S.P. Note that surface diffusion processes are observed when the Vyatchanin, and G. S. Plotnikov for useful discussions and surfaces are examined with a scanning tunneling microscope valuable comments. where a strong electric field is created between the tip and This work was supported by the Russian State Commit- the surface.4 One of the quantities characterizing these sur- tee for Higher Education ͑Grant No. 9508.054͒ and the Na- face processes is their relaxation time, which depends on the tional Science Foundation, USA ͑Grant NPHY-9503642͒. surface structure. For processes associated with slow surface 1 states the relaxation time may be between tenths of a second V. B. Braginsky, V. P. Mitrofanov, and K. V. Tokmakov, Phys. Lett. A 5 218, 164 ͑1996͒. and minutes. When the electric field varies with a frequency 2 V. B. Braginski, V. P. Mitrofanov, and V. I. Panov, Low-Dissipation around 1 Hz, which corresponds to the oscillation frequency Systems ͓in Russian͔, Nauka, Moscow ͑1981͒, 144 pp. 3 of the pendulum, the surface states undergo relaxation. When F. F. Vol’kenshten, Electronic Processes at the Surface of Semiconduc- the external electric field varies at higher frequencies, the tors in Chemisorption, ͓in Russian͔, Nauka, Moscow ͑1987͒, 432 pp. 4 J. Mendez, J. Gommez-Herrero, J. I. Pascual et al., J. Vac. Sci. Technol. carriers do not have time to be transferred to other surface A 2, 1145 ͑1996͒. states and thus the dissipation caused by these processes de- 5 V. F. Kiselev and O. V. Krylov, Electronic Effects in Adsorption and creases. This can explain the absence of any mechanical re- Catalysis at Semiconductors and Insulators ͓in Russian͔, Nauka, Moscow action of the pendulum to an rf field. We can merely postu- ͑1979͒, 236 pp. late that the change in the pendulum Q factor observed after Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Impact excitation of squeezed vibrational packets in molecules A. V. Belousov, V. A. Kovarski , and O. B. Prepelitsa Institute of Applied Physics, Academy of Sciences of Moldavia, Kishinev ͑Submitted October 16, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 34–38 ͑July 12, 1998͒ An analysis is made of vibrational packets created in molecules by collisions with fast protons. Some characteristic features of the Raman light scattering spectrum are discussed for a molecule with squeezed vibrational states. It is suggested that squeezed vibrational states in the ground electronic state of a molecule may be used for isotope separation. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00607-7͔

Squeezed states of oscillators for an electromagnetic v2 vt m ␮ Ϫ2 a field ͑squeezed light͒ and for squeezed vibrations ͑squeezed F͑t͒ϭ cosh ; ␭ϭ , ͑2͒ 2 2R0 maϩmb phonons͒ have recently attracted considerable attention.1,2 A characteristic feature of the excitation of squeezed vibra- x is the deviation of the vibrational coordinate from the equi- tional states is that the rates of chemical reactions may in- librium position, ␮ is the reduced oscillator mass of the im- 3,4 crease abruptly and exponentially ͑laser femtochemistry͒. pinging particle, and ma and mb are the atomic masses in a Most studies usually examine the preparation of squeezed diatomic molecule. vibrational states in excited molecular states, for example, by The Schro¨dinger time-dependent equation for an oscilla- excitation of a molecular system with ultrashort laser pulses. tor, taking account of W(x,t), may be expressed in the form However, the creation of squeezed vibrational wave packets in the ground electronic state would allow a new class of ϩϱ t ␺͑x,t͒ϭ dx1 dt1G0͑xt͉x1t1͒W͑x1 ,t1͒␺͑x1 ,t1͒. processes to be studied, which assume a super-Poisson dis- ͵Ϫϱ ͵Ϫϱ tribution of vibrational occupation numbers. The simplest ͑3͒ example could be the spectrum of Raman light scattering by a molecule in the ground electronic state. Here G0(xt͉x1t1) is the Green’s function of the oscillator. Here we propose a new method of preparing squeezed We assume that the time behavior of the interaction potential vibrational states in the ground electronic state of a mol- W(x,t) in the active region tϳ␶0 may be sufficiently accu- ecule, based on the excitation of molecular vibrations by rately approximated by a Gaussian pulse collisions between molecules and fast protons. The impact v2 ␮ 2 2 time ␶0 in these collisions is estimated using the formula F͑t͒ϭ exp͑Ϫt /4␶ ͒. ͑4͒ 2 0 ␶0ϭR0 /v, where R0 is the characteristic distance at which impact takes place, of the order of 1 Å ͑Ref. 5͒, v is the 7 We use the condition of collisional nonadiabaticity, i.e., the proton velocity: for vϳ10 cm/s we have ␶0ϳ1 fs. For vi- approximation of short times t1 . We replace ␺(x1 ,t1) on the brations with the energy ប␻ϳ0.1 eV this implies that a right-hand side of Eq. ͑2͒ by ␺(x ,0). The wave function wave packet can be excited from ten vibrational states, i.e., 1 ␺(x1,0) is an eigenfunction of the Hamiltonian: ␻␶0Ӷ1. This case of inelastic collisions is the opposite of the case of slow adiabatic collisions for which ␻␶0ӷ1, and HϭHoscϩW͑x1 ,0͒. ͑5͒ is well described using the familiar Landau–Teller theory.6 It is natural to consider molecular vibrational states accurately Initially, the oscillator is assumed to be unexcited (ប␻ described by the harmonic model and having fairly long life- ӷkt) so that times. Such molecules may include many of those studied in Ϫ9 1 1 ͑xϪ␰͒2 laser isotope separation where ␶Vϳ10 s, ប␻ϳ0.1 eV ͑see ␺͑x1,0͒ϭ 1/4 exp Ϫ 2 , Ref. 7, for example͒. ␲ ͱ␦2 ͫ 2␦ ͬ To illustrate the fundamental relationships governing the excitation of squeezed vibrational packets, we shall use a v2 v2 2 ␭ 2 ប 2 2 ␮ ␭ simple one-dimensional model of the interaction potential of ␰ϭ 2 ; ␦ ϭ ; ⍀ ϭ␻ ϩ 2 . ͑6͒ 2d␻ ␮⍀ 2R mab a heavy classical particle with a molecule, which was pro- 0 posed in Refs. 8 and 9: The integrals in Eq. ͑6͒ can be calculated directly by using the explicit form of the Green’s function of the oscil- x 1 lator and the expression for the pulse ͑3͒. To simplify the W͑x,t͒ϭϪF͑t͒ 1ϩ␭ ϩ ␭2x2 . ͑1͒ ͩ R 2 ͪ calculations, we replace the narrow Gaussian pulse profile 0 (␶0 0) with a delta function of t, which appreciably sim- Here we have plifies→ the calculations. We give the result of integrating in

1063-7850/98/24(7)/2/$15.00 513 © 1998 American Institute of Physics 514 Tech. Phys. Lett. 24 (7), July 1998 Belousov et al. formula ͑6͒ using the accurate expression for the Green’s ¯nϭ͓exp(ប␻/kT)Ϫ1͔Ϫ1, i.e., asymmetry is observed. The function of the oscillator and the expression for ␺(x,0) ͑5͒, anti-Stokes component in the absence of squeezed states is which yields usually lower than the Stokes component. For coherent and 2 2 2 ͉␺͑x,t͉͒ ϭN exp͓Ϫ͑xϪx0͑t͒͒ /␴ ͑t͔͒. ͑7͒ squeezed states the intensities Ia and Is are equal, so that ୐ϳ1. A more accurate analysis of the variation of the pa- Here the following notation is introduced: rameter ୐ for various velocities of the proton beam incident 2 2 2 2 x0͑t͒ϭ␰ cos ␻t; ␴ ͑t͒ϭ␦ ͑cos ␻tϩ␩ sin ␻t͒, on the molecules to excite squeezed states will allow the squeezing parameter to be determined experimentally. Since 2 2 2 2 1/2 ␩ϭ⍀/␻ϭ͓1ϩ␮v ␭ /2R0mab␻ ͔ , ͑8͒ the average occupation numbers vary with time ͑because of N is the normalization constant. The probability density ob- the time dependence of the dispersion of the squeezed vibra- tained for the squeezed state is characterized by the squeez- tion͒, the ratio ୐ϭIa /Is will also vary with time. In this case, ing parameter ␩ which, in the particular case where the qua- unlike the method of two successive ͑short optical͒ pulses, dratic correction ␭2 in the expansion of the potential ͑1͒ is comprising a preparatory optical pulse and a readout optical neglected, gives ␩ϭ1 and formula ͑7͒ describes a coherent pulse, the squeezing parameter can be determined by the packet. Thus, for ␩Ͼ1 expression ͑7͒ describes a squeezed simpler technique described above and, which is particularly wave packet with a super-Poisson distribution and the dis- important, for vibrations in the ground electronic state. This persion ␴2(t) greater than that of the coherent state ␦2. circumstance may be used in laser isotope separation since Note that expression ͑7͒ may be obtained by accurately the amplitudes of the squeezed vibrational states are fairly solving the problem with the potential ͑1͒ if this potential is large and differ appreciably for molecular isotopes. used as the Po¨schl–Teller potential in the Schro¨dinger problem for the motion of a classical oscillator in this potential.10 In this approach it is convenient to use a secondary quantization basis where the squeezed state is represented as two 1 V. P. Bykov, Usp. Fiz. Nauk 161͑10͒, 145 ͑1991͓͒Sov. Phys. Usp. 34, successive transformations of shear and rotation, acting on 910 ͑1991͔͒. 2 ´ the vacuum state. The problem with the Po¨schl–Teller po- A. V. Vinogradov and J. Janskzy, Zh. Eksp. Teor. Fiz. 100, 386 ͑1991͒ ͓Sov. Phys. JETP 73, 211 ͑1991͔͒. tential is solved using a familiar procedure where the con- 3 ´ V. A. Kovarski, Zh. Eksp. Teor. Fiz. 110, 1216 ͑1996͓͒JETP 83, 670 stants in the solution of this equation are related to the ͑1996͔͒. 4 above-barrier reflection coefficient, as suggested by L. P. V. A. Kovarski, Pis’ma Zh. Tekh. Fiz. 20͑24͒,59͑1994͓͒Tech. Phys. Pitaevski .10 The accuracy of this solution is the same as the Lett. 20, 999 ͑1994͔͒. 5 A. A. Radtsig and B. N. Smirnov, Handbook of Atomic and Molecular solution ͑7͒ with fairly steep functions F(t). It is easily con- Physics ͓in Russian͔, Atomizdat, Moscow ͑1980͒, 240 pp. firmed that only this constraint on the function F(t) gives a 6 L. Landau and E. Teller, Phys. Z. Sowjetunion 10,34͑1936͒. class of super-Poisson solutions of the form ͑7͒. We have 7 V. S. Letokhov, Nonlinear Selective Photoprocesses in Atoms and Mol- ecules ͓in Russian͔, Nauka, Moscow ͑1983͒, 408 pp. confined ourselves to the simplest calculations of the wave 8 K. V. Stupochenko, S. A. Losev, and A. I. Osipov, Relaxation in Shock packet for the steep potential F(t). Waves ͑Springer-Verlag, Berlin, 1967͓͒Russ. original, Nauka, Moscow, Experimental observations of a squeezed vibrational 1965, 482 pp.͔. 9 packet are primarily based on the fact that the occupation B. F. Gordiets, A. I. Osipov, and L. A. Shelepin, Kinetic Processes in Gases and Molecular Lasers ͓in Russian͔, Moscow ͑1980͒, 510 pp. numbers of the oscillator obey a super-Poisson distribution. 10 A. I. Baz’, Ya. B. Zel’dovich, and A. M. Perelomov, Scattering, Reactions In analyses of the Raman spectrum of light scattering by a and Decay in Nonrelativistic Quantum Mechanics, transl. of 1st Russ. ed. molecule, the ratio of the intensities of the blue Ia ͑Israel Program for Scientific Translations, Jerusalem, 1966͓͒Russ. original, 2nd ed., Nauka, Moscow, 1971͔, 544 pp. ͑anti-Stokes͒ and red Is ͑Stokes͒ satellites of the scattered ¯ ¯ light is determined by the condition ୐ϭIa /IsХn/(nϩ1), Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Inﬂuence of the type of cathode on the dynamic characteristics of ionization waves A. S. Aref’ev and Yu. A. Yudaev

Ryazan State Radio Engineering Academy ͑Submitted October 31, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 39–42 ͑July 12, 1998͒ The inﬂuence of different types of cathodes on the dynamic characteristics of ionization waves is determined experimentally. It is established that when the discharge is initiated by negative pulses, the maximum wave propagation velocity is observed for a cold cathode while the maximum peaking of the output pulse leading edge is observed for a heated cathode. The mechanism for this effect is discussed. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00707-1͔

Studies of the influence of different types of cathodes on the ionization wave was determined using the formula the dynamic characteristics of wave breakdown of gas gaps are of considerable theoretical and practical interest. An VϭL2 /͑t2Ϫt1Ϫ͑L1ϩL3͒/c͒, analysis of the scientific literature1,2 does not answer the question as to how the emission from the surface of the elec- where t1 and t2 are the times of appearance of the signal at trode influences the dynamic characteristics of the ionization the first and second current detectors, respectively, c is the waves ͑displacement velocity, formation time, current pulse velocity of light, and L1 , L2 , and L3 are distances as shown peaking, and so on͒. Moreover, many publications assert that in Fig. 1. the type of cathode does not influence the properties of the Figures 2a and 2b give the propagation velocity of the ionization waves. However, our experimental investigations ionization wave as a function of the neon pressure for the have shown that thermionic emission from the cathode plays prototype with different types of cathodes. The discharge some role in the formation of ‘‘slow’’ ionization waves when was ignited by negative pulses from the cathode side with a the propagation velocity lies in the range 106 –107 m/s. The rate of rise of 5ϫ1011 V/s. apparatus used for the experimental investigations is shown A comparative analysis of these curves reveals that the schematically in Fig. 1. The distance between the ends of the ionization wave propagates at a higher velocity with a cold electrodes was 270 mm and the diameter was 30 mm. To cathode than with a heated one. This behavior may be attrib- achieve uniform experimental conditions, the investigations uted to the following factors. The propagation velocity of the were carried out using a single prototype where an indirectly ionization wave is influenced by two interrelated processes, heated oxide cathode 4 functioned as the cold and the heated impact and photoionization. The relative influence of these cathode. factors on the ion formation mechanism depends on the filler The measurements of the wave propagation velocity gas concentration. At low pressure, both processes play a were made assuming that the current-recording detectors 5 negligible role and the ionization wave has a low velocity. A are separated by the distances L1 and L3 from the volume rise in pressure increases the role of photoionization and im- where the plasma forms ͑Fig. 1͒, so that the propagation pact ionization. The maximum propagation velocity ͑Fig. 2͒ velocity of the high-voltage pulse in these sections was taken corresponds to the optimum conditions for the formation of equal to the velocity of light and the propagation velocity of charged particles.

FIG. 1. Prototype to investigate dynamic characteristics of ionization waves: 1—envelope, 2—screen, 3—anode, 4— cathode, 5—current shunts, 6—high-voltage cable, 7— adjusting device, 8—insulator, 9—slit for visual observations, and 10—cathode heater leads.

FIG. 2. Propagation velocity of ionization wave versus neon pressure for various types of cathodes: a—heated, b— cold. Negative control pulses dU/dtϭ5ϫ1011 V/s: 1—13 kV; 2—11 kV; 3—9 kV.

As the amplitude of the initiating pulses increases, the Changes in the cathode geometry ͑length, diameter, surface maximum of the wave propagation velocity shifts toward area, configuration͒ did not alter the temporal and energy higher pressures for both the cold and heated cathodes. An characteristics of the ionization waves. increase in the amplitude of the voltage pulse applied to the Formation of the discharge by positive pulses from the gap leads to an increase in the energy acquired by the elec- anode side showed that the dynamic characteristics of the trons over the mean free path but at the same time, reduces ionization waves do not depend on the nature of the second the probability of the gas atoms being ionized, which causes electrode, except for the leading edge duration of the current a drop in the wave propagation velocity in the gap. Only an pulse and the voltage recorded in the cathode circuit. With a increase in pressure can reduce the mean free path for which heated cathode the output pulse showed stronger peaking the electron ionization probability will have a maximum. than for the cold cathode, although the propagation velocity With a heated cathode it is considerably easier for elec- was the same in both cases. Changes in the amplitude and trons to escape from the surface of the electrode, so that the slope of the applied voltage pulses altered the quantitative plasma forms more rapidly than with a cold cathode, when a results, without changing the general behavior of the pro- longer time is required to produce the same concentration of cesses. charged particles. The presence of a heated cathode makes Using these phenomena in plasma current switches the plasma formation process more vigorous but the charged could improve their speed by several factors and increase the particles begin to form at a lower electrode voltage, making rate of rise of the anode current.3 the wave propagation velocity lower than that for a cold 1 ´ cathode ͑Fig. 2͒. E. I. Asinovski, L. M. Vasilyak, and V. V. Markovets, Teplofiz. Vys. The type of cathode also influences the shape of the Temp. 21, 577 ͑1983͒. 2 current pulse. With a heated cathode, the output pulse shows A. N. Lagar’kov and I. M. Rutkevich, Ionization Waves in Electrical Breakdown of Gases, Springer-Verlag, New York ͑1994͓͒Russian orig.͔ stronger peaking at a lower propagation velocity. Nauka, Moscow ͑1989͒, 206 pp. When the rate of rise of the initiating voltage was in- 3 A. S. Arefiev and Yu. A. Yudaev, in Proceedings of the Twelfth Interna- creased to 5ϫ1012 V/s, the maximum propagation velocity tional Conference on Gas Discharges and their Applications, Greifswald, shifted to higher pressure and the differences in the propaga- Germany, 1977, Vol. 2, p. 800. tion velocity of the ionization waves progressively decrease. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Synergic model of the superplastic deformation of materials A. K. Emaletdinov

Servis Technological Institute, Ufa ͑Submitted October 30, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 43–47 ͑July 12, 1998͒ A study is made of a synergic model of superplastic deformation of ultraﬁne-grained materials at high temperatures as a manifestation of collective modes of motion and self-organization in a system of stimulated grain-boundary slip. The evolution of grain-boundary slip is accompanied by the formation of a wavefront which separates two steady states of the system and leads to the appearance of the experimentally observed ‘‘running neck’’ at the surface of the sample. The synergic model is used to explain the scaling effect of superplasticity. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00807-6͔

The superplasticity of inorganic materials is of major when grain-boundary slip becomes a cooperative process in- practical signiﬁcance and is observed as a result of the de- corporating the grain-boundary slip of many grains. The or- velopment of a stable ultraﬁne-grained structure and defor- der parameters determining the behavior of the grain system mation in a certain range of temperature and strain rate ͑usu- in superplastic deformation will be collective, long- ally for grain sizes dsӍ10– 15 ␮m, TsӍ0.5– 0.6Tm , wavelength modes of grain-boundary slip. The controlling ␧˙ pϭ10Ϫ3 –10Ϫ1 sϪ1͒͑Refs. 1–3͒. It has now been convinc- parameters will be the strain rate ␧˙ p and the temperature T. ␶ ingly established that three deformation micromechanisms g ␶D are involved simultaneously in superplastic deformation: The defect interaction diagram has the form ␳1 ␳2␳3 → ␶ grain-boundary slip, intragranular dislocation glide, and dif- 0 where ␶ is the nucleation time for a lattice dislocation at the fusion creep.1–4 Grain-boundary slip occurs in two forms:5 g boundary and ␶ is the dissociation time for a lattice dislo- intrinsic slip at the velocity V and slip stimulated by lattice D 0 cation at grain-boundary defects in the boundary.4 We write slip at the velocity V , where V ӷV . It was established in s s 0 the basic equations for the dislocation kinetics in the relax- Ref. 6 that in superplasticity, collective modes appear in the ation time approximation in the form motion of ensembles of grains. The aim of the present study is to analyze the synergic model of superplastic deformation as a manifestation of grain-boundary, self-oscillating, dissipative structures. The production of entropy in superplastic deformation is related to the rate of irreversible deformations: dissipative processes and their contribution to the total deformation. In the optimum range of strain rates ͑region II͒, the contribu- tions of grain-boundary slip ␥g , intragranular dislocation glide ␥D , and dislocation creep ␥ p are respectively given 1,4 by ␥gտ0.8, ␥DՇ0.2, ␥ p 0. From the point of view of nonequilibrium thermodynamics→ and synergetics,7 the appearance of a temperature–rate range for the onset of superplasticity is caused by a change in the dissipative process controlling the deformation: in region I the main processes are isolated dislocation glide and creep, in region II grain- boundary processes predominate, and in region III multiple dislocation glide is the dominant process. On the basis of the physical model of superplastic deformation proposed in Ref. 4, we shall analyze a polycrystalline material consisting of equiaxial grains of average size d, deformed at the optimum rate ␧˙ p ͑region II͒. The deformation is accomplished by grain-boundary defects of density ␳1 and lattice dislocations of density ␳2 . The action of the dislocation glide produces stimulated grain-boundary slip with FIG. 1. Diagram showing the evolution of micromechanisms responsible for superplastic deformation: ␳ —density of grain-boundary defects, the bulk density ␳ ͑Fig. 1͒. In superplastic deformation col- 1 3 ␳2—density of lattice dislocations, and ␳3—density of introduced grain- lective modes appear in the motion of ensembles of grains boundary defects.

FIG. 2. Density of grain-boundary defects ␳3 /␳c1 and strain rate ˙ p ˙ ␧ /␧0 as functions of distance x/⌬H in traveling neck.

˙ ϭAϪ / ϩ / Ϫk ϩD 2 , D1 /D2Ӷ1. Converting to slow variables in Eqs. ͑1͒,weob- ␳1 ␳1 ␶g ␳2 ␶D ␳2␳3 1ٌ ␳1 1/2 tain in the ﬁrst approximation „t/␶0ϭ␶, ␰ϭx/(D2␶0) , ˙ ␳ϭ␳3 /␳ … ␳2ϭϪ␳2 /␶Dϩ␳1 /␶g , *

˙ 2 2 2 ␳˙ ϭ ␳ Ϫ␳ /␶ ϩk␳ ␳ Ϫc␳2ϩD ٌ2␳ , ͑1͒ ␳ϭ1ϩA␳/͑b0ϩd0␳ ␳͒Ϫ␣0␳Ϫc␳ ␳ ϩٌ ␳. ͑2͒ 3 ͑ * 3͒ 0 2 3 2 3 * * where ␶0 is the annihilation time for grain-boundary defects Equation ͑2͒ for the slow amplitude of the cooperative at sinks ͑such as grain boundary joins, and so on͒, A is the grain-boundary slip has three established steady-state points. rate of nucleation of dislocations, and k and c are normaliza- Physically this means that the system is stable at the mini- tion coefficients. Here it is assumed that the defect distribu- mum and maximum densities of grain-boundary defects and tion exhibits substantial spatial nonuniformity, described by is unstable for ␳c2 . That is to say, a front or ‘‘neck’’ appears introducing the ‘‘diffusion term.’’ Within a single grain we in the system, separating the two steady states ␳c1 and ␳c3 . find ␳1 , ␳3ϭconst and gradients occur only when consider- Solving Eq. ͑2͒ in terms of the self-similar variable ␰ϭx ing neighboring grains. The rapid transfer of active deforma- ϪVt, where V is the velocity of the front, with the boundary tion from some boundaries to neighboring ones ͑Fig. 1͒ is conditions ␳ , ␰ Ϯϱ, ␳Јϭ0, ␳ϭ␳ , we obtain an im- c2 → ␰ c1 accompanied by the formation of density gradients ␳1 and plicit expression for the defect distribution ␳ in the form ␳3 , and the ‘‘diffusion’’ coefficients of the defect flux den- ␰ϭ͐d␳/F(␳,␣0 ,d0 ,␳ ,c,k), where F(z) is a given func- 2 2 * sities will be given by D1Շd /␶D and D2Շd /␶g . The ini- tion. In the first approximation we have tial conditions are (␳10 ,0,0). We shall analyze the system using bifurcation theory 8 ␳͑␰͒Շ␳c1ϩ͑␳c3Ϫ␳c1͒/͓1ϩexp͑2␣0͑␳c3Ϫ␳c1͒͒␰/V͔. methods and Poincare´ maps. A search for the steady-state ͑3͒ points of the system and their analysis using the linearized system of equations ͑1͒ revealed that the following steady The width of the front is determined in the first approxima- states exist: two saddle-type points (A␶0 ,A␶0␶D /␶g,0) and tion by ⌬H ␶ D (␳ Ϫ␳ )/␳ (D ␶ )1/2. Estimates of V 2 Ӎ g 2 c3 c1 * 1 0 (␳s1 ,␳s2 ,␳s3) for AϾ␶g /K␶D␶0 and the unstable focus point and ⌬H for superplastic deformation are of the order (␳ ,␳ ,␳ ) for ␳ Ͼ␳ . An analysis of the solutions of Ϫ1 Ϫ1 F1 F2 F3 * c* VӍ10 cm/s and ⌬HӍ10 cm, which is similar to the the kinetic equations ͑1͒—space-time dissipative experimental data.1–3 A numerical solution of the system ͑1͒ structures—may be sought in the form of expansions as a is shown in Fig. 2. Fourier series in terms of plane waves ␳ϳ͚k exp(ptϩikx). Dissipative structures can only form when the dimen- The characteristic equation of the system ͑1͒ has two types of sions of the nonequilibrium system ͑the sample͒ L exceed solutions: 1͒ a spatially homogeneous solution and 2͒ a pe- certain critical values.7 For superplasticity, the minimum di- riodic, self-oscillating solution ͑traveling neck regime1–3͒. mension Lc is determined by the condition for the appear- Periodic solutions are generated according to the type of ance of collective modes and self-organization in the system Hopf bifurcation. Thus, there exists a critical density of ac- of stimulated grain-boundary slip and has the form tivated grain-boundary slip boundaries when a self- oscillating regime is established in the system 1/2 1/2 Lcϭ͑F1͑d0 , f Ј͑␳c1͒,gЈ͑␳c3͒,V,␶g͒͒ տ͑⌬H•V•␶g͒ , 2 2 2 2 ␳c ϭ͑k␶DA ␶0/␶g͒͑1ϩz͒/͑3ϩz͒, zϭ2␶g /Ak␶0␶D . * where f (z) and g(z) are the right-hand sides of the system The type of wave regime is analyzed using the method ͑1͒, and F1(z) is a given function. An order-of-magnitude 7 2 3 of multiscale expansions in terms of the small parameter estimate of Lc gives 10 –10 ␮m. In Ref. 6, a departure Tech. Phys. Lett. 24 (7), July 1998 A. K. Emaletdinov 519

5 from superplastic deformation was in fact observed when the O. A. Kabyshev, V. V. Astanin, and R. Z. Valiev, Dokl. Akad. Nauk sample dimension was of the order of 60 ␮m. SSSR 245, 1356 ͑1979͓͒Sov. Phys. Dokl. 24, 289 ͑1979͔͒. 6 V. V. Astanin, Fiz. Met. Metalloved. 79͑3͒, 166 ͑1995͒. 7 1 G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Sys- O. A. Kabyshev, Plasticity and Superplasticity of Metals ͓in Russian͔, Metallurgiya, Moscow ͑1975͒, 280 pp. tems ͑Wiley, New York, 1977; Mir, Moscow 1979, 512 pp.͒. 2 8 A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Ma er, Quali- M. V. Grabski, Structural Superplasticity of Metals ͓in Russian͔, Metal- lurgiya, Moscow ͑1975͒, 280 pp. tative Theory of Second-Order Dynamic Systems ͓in Russian͔, Nauka, 3 A. S. Tikhonov, Superplasticity of Metals and Alloys ͓in Russian͔, Nauka, Moscow ͑1966͒, 326 pp. Moscow ͑1978͒, 141 pp. 4 O. A. Kabyshev, R. Z. Valiev, and A. K. Emaletdinov, Dokl. Akad. Nauk SSSR 279, 369 ͑1984͓͒Sov. Phys. Dokl. 29, 967 ͑1984͔͒. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Resolution of optically addressed liquid-crystal spatial light modulators V. F. Nazvanov and D. I. Kovalenko

Saratov State University ͑Submitted April 18, 1997; resubmitted February 25, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 48–53 ͑July 12, 1998͒ Results are presented of calculations of the modulation transfer function of optically addressed, liquid-crystal, spatial light modulators based on layered photoconductor–liquid crystal structures, including surface-plasmon light modulators. Allowance is made for diffusive spreading of carriers in the photoconductor and the propagation length of surface plasmons in the layered structures. © 1998 American Institute of Physics. ͓S1063-7850͑98͒00907-0͔

Several models have been proposed to understand the liquid crystal, ␧1 is the parallel permittivity of the liquid constraints on the resolution of optically addressed spatial crystal, and ␧2 is its perpendicular permittivity. light modulators.1–11 It has been shown that in optically ad- In their derivation of formula ͑1͒ the authors of Refs. 8, dressed spatial light modulators based on photodetector– 12, and 16 neglected diffusive spreading of carriers in the liquid crystal layered structures, the resolution is primarily photoconductor layer, although they did not deny the impor- determined by the thicknesses and dielectric constants of the tance of this effect. photoconductor and liquid crystal layers. Theoretical and ex- Diffusive spreading ͑and/or drift in the presence of an perimental investigations1,4–6,10,13 have also shown that, in electric field͒ of photocarriers in the photoconductor may be addition to the thickness of the liquid crystal layer, the reso- taken into account directly7 in calculations of the modulation lution is also influenced by the orientation of the molecules transfer function by multiplying the ratio SN /S0 by the in the liquid crystal layer and by its dielectric anisotropy. frequency-contrast characteristic of the photoconductor.17,18 Allowance should also be made for diffusive spreading In fact, the use of a photoconductor layer as one of the ͑and/or drift͒ of photoexcited carriers in the photoconductor main elements in optically addressed light modulators is which drastically reduces the resolution of spatial light largely attributable to the capacity of a photoconductor with modulators. Direct allowance for the carrier diffusion length a specific contrast to produce an image of the exciting radia- in the photoconductor in calculations of the resolution of tion, as occurs in iconics for photographic materials.17 optically addressed liquid-crystal spatial light modulators Calculations18 show that the distribution of the nonequilib- was apparently proposed for the first time by one of the rium carrier concentration in a photoconductor obeys the present authors in Ref. 7. same cosinusoidal distribution as the excitation function. Here we present an expanded version of Ref. 7 in which However, the amplitude of the variation in the concentration we calculate the modulation transfer function of optically differs from that of the excitation function by the factor addressed liquid-crystal spatial light modulators. In addition, T(␯,LD), which depends on the spatial frequency ␯ and the 14,15 in the surface plasmon calculations we also allow for the diffusion wavelength LD . Consequently, in the photocon- propagation length of the surface plasmons. It should be ductor the excitation function is converted into the spatial noted that a similar7 but slightly different version was also distribution function of the concentration with the frequency- proposed in Ref. 11 to allow for the diffusion spreading contrast characteristic length of the carriers in the photosensor part of optically 2 2 2 addressed, liquid-crystal spatial light modulators. T͑␯,LD͒ϭ1/͑1ϩ4␲ ␯ LD͒. ͑2͒ An expression was derived in Ref. 12 for the modulation transfer function of a light modulator formed by a two-layer In the presence of an electric field in the photoconductor, structure consisting of an isotropic dielectric ͑photoconduc- the carrier displacement length in the field19 should be taken tor͒ and a layer of dielectric anisotropic material, assuming a instead of LD . In general, for a photoconductor layer in an sinusoidal charge distribution at the interface between the optically addressed spatial light modulator, this length de- two layers. For a photoconductor–liquid crystal structure the pends on the voltage V across the photoconductor layer, the modulation transfer function depends mainly on the ratio intensity of the writing light, the degree of modulation m, 8,12,16 7 SN /S0 , given by and also the spatial frequency. Figure 1 gives results of calculating the modulation SN 1 ␧PS /dPSϩ␧1 /dLC transfer function of an optically addressed spatial light ϭ , S modulator comprising a bismuth silicate–liquid crystal struc- 0 ␲␯ ␧PS coth͑␲␯dPS͒ϩͱ␧1␧2 coth͑␲␯ͱ␧2 /␧1dLC͒ ͑1͒ ture from Ref. 16, neglecting and allowing for the diffusion length ͑curves 1 and 2, respectively͒. This clearly shows that where d PS is the thickness of the photosensor, ␧ PS is the LD influences the resolution of the optically addressed spatial permittivity of the photosensor, dLC is the thickness of the light modulator. Using the experimental data from Ref. 16,

FIG. 2. Modulation transfer function of an optically addressed spatial light FIG. 1. Modulation transfer function of an optically addressed spatial light modulator based on an amorphous silicon–liquid crystal structure14 with modulator based on a bismuth silicate–liquid crystal structure16 with a LDϭ5 ␮m and a modulation coefﬁcient of 1: 1—LSPϭ0; 2—LSP modulation coefﬁcient of 1: 1—LDϭ0; 2—LDϭ8 ␮m. ϭ5 ␮m; 3—LSPϭ10 ␮m; 4—LSPϭ20 ␮m.

we estimated the carrier spreading length in bismuth silicate 14 to be 13.6 ␮m, which agrees with the real value.20 curves of the modulation transfer function to calculate the The resolution of surface-plasmon liquid-crystal light lengths LSP for the two directions of propagation of the sur- modulators is also extremely interesting.14,15,21 In this type of face plasmons studied in Ref. 14 and we obtained ͑for our spatial light modulator, in addition to allowing for the diffu- assumed carrier spreading length of 2 ␮m in a layer of amor- sive spreading of carriers in the photoconductor ͑correspond- phous silicon͒ 8.2 ␮m ͑for ␯ϭ25 line/mm͒ and 16.9 ␮m ͑for ing to the frequency-contrast characteristic of the photosen- ␯ϭ13 line/mm͒, respectively. sor in an optically addressed spatial light modulator͒,itis To conclude, direct allowance for the diffusive spreading also necessary to take into account21 the blurring of the ‘‘pic- of the carriers in the photosensor part of an optically ad- ture’’ written in the photoconductor as a result of the finite dressed, liquid-crystal spatial light modulator can be made by introducing the frequency-contrast characteristic of the propagation length L PS of the plasmons in the photoconductor–liquid crystal layered structure. It is easy to photoconductor into the expression for the modulation trans- imagine that the system for reading the image from an opti- fer function of the spatial light modulator. For surface- cally addressed spatial light modulator comprising a plasmon, optically addressed, liquid-crystal spatial light photoconductor–liquid crystal structure with surface plas- modulators, the propagation length of the surface plasmons mon excitation14,15 is the second component of a general is taken into account by introducing the frequency-contrast system for reproduction of an object, while the first is a characteristic of the plasmon image reading system at the system for writing the object in the photoconductor and thus same time as that of the photosensor. in the photoconductor–liquid crystal structure. Then, in accordance with iconics,17 for the entire surface-plasmon, optically addressed liquid-crystal spatial light modulator sys- 1 A. A. Vasil’ev, Trudy Fiz. Inst. Akad. Nauk SSSR 126,3͑1981͒. tem, we can introduce the frequency-contrast characteristic 2 U. Efron et al., Opt. Eng. 22, 682 ͑1983͒. determined by simply multiplying the characteristics of the 3 S. S. Ignatosyan, V. P. Simonov, and B. M. Stepanov, Opt. Mekh. Promst. system components, and specifically the frequency-contrast No. 1, 7 ͑1986͓͒Sov. J. Opt. Technol. 53,6͑1986͔͒. 4 A. A. Vasil’ev, Yu. K. Gruzevich, S. N. Levov et al., Preprint No. 1 ͓in characteristic Russian͔, Lebedev Physics Institute, Academy of Sciences of the USSR, T ͑␯͒ϭT͑␯,L ͒T͑␯,L ͒, ͑3͒ Moscow ͑1986͒. 0 D SP 5 A. A. Vasil’ev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators ͓in Russian͔, Radio i Svyaz’, Moscow ͑1987͒, 320 pp. where T(␯,LSP) is similar to Eq. ͑2͒, where the diffusion ͑or 6 Yu. D. Dumarevski, N. F. Kovtonyuk, and A. I. Savin, Image Conversion spreading͒ length LD should be replaced by the surface plas- in Semiconductor–Insulator Structures ͓in Russian͔, Nauka, Moscow mon propagation length. Thus, to allow directly for both the ͑1987͒, 176 pp. diffusion spreading length and the surface plasmon propaga- 7 V. F. Nazvanov and A. V. Novikov, Abstracts of Papers presented at the tion length in calculations of the modulation transfer func- First All-Union Seminar ‘‘Optics of Liquid Crystals,’’ Leningrad, 1987 ͓in Russian͔, p 189. tion of a surface-plasmon, optically addressed liquid-crystal 8 S. S. Ignatosyan, Opt. Mekh. Promst. No. 4, 4 ͑1988͓͒Sov. J. Opt. Tech- spatial light modulator, expression ͑1͒ for SN /S0 must be nol. 55, 198 ͑1988͔͒. 9 multiplied by T0(␯) as given by Eq. ͑3͒. I. V. Fedorov, V. Yu. Zlatov, A. O. Radkevich, and A. V. Khlestov, Vopr. Figure 2 gives results of calculations of the modulation Radioe´lektroniki Ser. Obshch. Vopr. Radioe´lektron. ͓in Russian͔, No. 10, pp 51–59 ͑1990͒. transfer function of a surface-plasmon, optically addressed, 10 N. F. Kovtonyuk and E. N. Sal’nikov, Photosensitive MIS Devices for liquid-crystal spatial light modulator with the layer param- Image Conversion ͓in Russian͔, Radio i Svyaz’, Moscow ͑1990͒, 160 pp. eters taken from Ref. 14, allowing not only for the diffusion 11 P. R. Barbier, L. Wang, and G. Moddel, Opt. Eng. 33, 1322 ͑1994͒. 12 length but also for the surface plasmon propagation length. It W. R. Roach and E. D. Frans, IEEE Trans. Electron Devices ED-21, 8–453 ͑1954͒. is easy to see that LSP limits the resolution of a surface- 13 V. G. Chigrinov et al., Mol. Cryst. Liq. Cryst. 55, 193 ͑1979͒. plasmon spatial light modulator. We used experimental 14 M. E. Gardwell and E. M. Yeatman, Appl. Opt. 31, 3880 ͑1992͒. 522 Tech. Phys. Lett. 24 (7), July 1998 V. F. Nazvanov and D. I. Kovalenko

15 V. F. Nazvanov, O. A. Afonin, and A. I. Grebennikov, Quantum Electron. 20 M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photosensitive Elec- 25, 1028 ͑1995͒. trooptic Media in Holography and Optical Information Processing ͓in 16 P. Aubourg et al., Appl. Opt. 21, 3706 ͑1982͒. Russian͔, Nauka, Leningrad ͑1983͒,p.88. 17 21 K. V. Vendrovski and A. I. Vetsman, Iconics ͓in Russian͔, Nauka, V. F. Nazvanov, Scientific-Technical Report on ‘‘Plasmon–Polariton’’, Moscow ͑1968͒,p.97. Scientific Research Program No. 01.9.10.03393, Scientific-Research Insti- 18 N. P. Aban’shin, D. I. Bilenko, and V. A. Lodgauz, Izv. Vyssh. Uchebn. tute of Mechanics and Physics, Saratov, on behalf of Saratov State Uni- Zaved. Fiz. 10, 143 ͑1971͒. versity ͑1992͓͒in Russian͔. 19 S. M. Ryvkin, Photoelectric Effects in Semiconductors ͑Consultants Bureau, New York, 1964͓͒Russ. original, Fizmatgiz, Moscow, 1963͔. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Reduction in the critical conditions for instability of a highly charged droplet moving relative to a medium S. O. Shiryaeva, V. A. Koromyslov, and O. A. Grigor’ev

Yaroslavl State University ͑Submitted September 19, 1997; resubmitted February 10, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 54–57 ͑July 12, 1998͒ It is shown that as the ﬂow velocity of an ideal liquid ﬂowing around a charged perfectly conducting droplet of ideal liquid increases, the critical self-charge of the droplet at which instability occurs decreases rapidly. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01007-6͔

⌿1ץ ␰ץ ⌿1ץ ␰ 1ץ Charged droplets moving relative to some medium are rϭRϩ␰: ϩ ϭ ; rץ ␪ץ ␪ץ t r2ץ -encountered in a wide range of problems in technical phys ics, geophysics, and technology.1–3 However, whereas a considerable number of investigations have examined the ⌿2ץ ␰ץ breakup of free-falling droplets in the atmosphere,4 problems ϭ ; rץ tץ involving the laws governing the buildup of instability in droplets with respect to their self-charge and with respect to ⌿2ץ ⌿1ץ ;a tangential discontinuity in the velocity field at the free Ϫ␳ ϩ␳ Ϫ␳ ٌ͑⌿ ͒2ϪP ϩP ϭ0 t 1 1 Q ␴ץ t 2ץ 1 surface of the droplet have not yet been addressed. Since the critical conditions for the instability of capillary waves at a ⌽ϭconst; charged liquid surface in the presence of a tangential velocity 5 field do not depend on viscosity, we shall examine models ␴ P ͑␰͒ϭϪ ͑2ϩ⌬ ͒␰; of ideal liquids to simplify the following reasoning. ␴ R2 ⍀ We shall assume that an ideal incompressible dielectric 2 2 medium having the density ␳1 and permittivity ␧ moves at a Q Q P ϭϪ ␰ϩ constant velocity U relative to a spherical droplet of radius Q 2␲␧R4 4␲␧R4 R, comprising an ideal, perfectly conducting liquid of den- 1 sity ␳2 , carrying the charge Q, where ␴ is the surface ten- ϫ ͑nϩ1͒Yn͑␮͒ ␰Yn͑␮͒d␮, sion at the interface of the media. We shall find the critical ͚n ͵Ϫ1 conditions for instability of capillary vibrations of the droplet under these conditions. where ⌬⍀ is the angular component of the Laplace operator We shall solve the problem in a spherical coordinate in spherical coordinates. system with its origin at the center of the droplet using a We shall seek the solution of this problem in the linear linear approximation with respect to a perturbation ␰(␪,t)of approximation with respect to the surface perturbation ␰ ͑as the free surface of the droplet caused by thermal capillary was done for the similar problem of the Helmholtz waves and having peak values of ϳ10Ϫ8 cm. The equation instability6͒ in the form for the free surface of the droplet has the form r(␪,t)ϭR ϩ␰(␪,t). Ϫ͑nϩ1͒ ⌿1͑r,t͒ϭ␸ϩ͚ Anr Y n͑␮͒exp͑St͒; The expression for the velocity field of the irrotational n liquid flow around the unperturbed droplet has the form6 n R3 ⌿2͑r,t͒ϭ͚ Bnr Yn͑␮͒exp͑St͒; V͑r͒ϭϪ ͓3n͑U n͒ϪU͔ϩU. ͑1͒ n 2r3 •

We shall assume that the wave motion in the droplet and ␰͑r,t͒ϭ͚ ZnYn͑␮͒exp͑St͒. in the surrounding medium is irrotational, with the harmonic n velocity potentials ⌿1 ,⌿2 , and the electrostatic potential ⌽ Yn are spherical functions, An , Bn , and Zn are first-order satisfying the problem coefficients, ␸ is the potential of the liquid velocity field ͑1͒, 2 -⌿ ϭ0; ͑iϭ1;2͒; ⌬⌽ϭ0; of zeroth order, and the quadratic term ϳ(ٌ⌿1) in the dy⌬ i namic boundary conditions at the free surface of the liquid is retained because it contains as a term. r ϱ: ٌ⌿1ϭU; ⌽ 0; ␸ → → This problem can easily be solved by conventional 5 ,rϭ0: ٌ⌿2ϭ0; methods. In terms of dimensionless variables where Rϭ1

␴ϭ1, and ␳2ϭ1, we obtain an infinite system of homoge- from Eq. ͑3͒ that as the velocity of the flow around the drop- neous algebraic equations for the amplitudes of the capillary let increases, the critical Rayleigh parameter WϭW for the * vibrations of the droplet: onset of instability decreases rapidly: ␳ 1 W ϭ4Ϫ␳U2M . 2 * 2 ␳U KnZnϪ2Ϫ␳USLnZnϪ1ϩ ϩ ͭͩ͑nϩ1͒ nͪ This observation justifies renewed efforts to construct a physical model for the initiation of a lightning discharge 2 2 ϫS Ϫ␳U Mnϩ͑nϪ1͓͒nϩ2ϪW͔ based on the idea of ignition of a corona discharge near a ͮ free-falling large melting hailstone in a thundercloud8–10 2 ϫZnϪ␳USInZnϩ1ϩ␳U JnZnϩ2ϭ0, ͑2͒ consistent with the real conditions obtained in a thundercloud ͑using measured values of the charges on hailstones, 2 Q 9␣n␣nϪ1 9␤n␣nϩ1 the intracloud electric field, and the falling velocity of the Wϵ ; M ϵ ϩ ; 4␲␧ n 2n 2͑nϩ2͒ hailstones͒. To conclude, we note that the ideal liquid approximation 9␣ ␣ ͑9nϩ6͒␣ n nϪ1 n used for the analysis does not restrict the generality of the Knϵ ; Lnϵ ; 2n 2n͑nϩ1͒ result since the critical conditions for instability of a droplet with respect to its self-charge do not depend on the viscosity ͑9nϩ12͒␤n 9␤n␤nϩ1 I ϵ ; J ϵ ; ␳ϵ␳ /␳ ; of the liquid. n 2͑nϩ1͒͑nϩ2͒ n 2͑nϩ2͒ 1 2

1 n͑nϪ1͒ ͑nϩ1͒͑nϩ2͒ S. I. Shevchenko, A. I. Grigor’ev, and S. O. Shiryaeva, Nauch. Prib. 1͑4͒, 3 ͑1991͒. ␣nϵ ; ␤nϵ . ͱ͑2nϪ1͒͑2nϩ1͒ ͱ͑2nϩ1͒͑2nϩ3͒ 2 A. I. Grigor’ev, Yu. A. Syshchikov, and S. O. Shiryaeva, Zh. Prikl. Khim. 62, 2020 ͑1989͒. A necessary and sufficient condition for the existence of 3 A. I. Grigor’ev and S. O. Shiryaeva, Izv. Ross. Akad. Nauk Ser. Mekh. a solution of this system is that the determinant composed of Zhidk. Gaza No. 3, 3 ͑1994͒. 4 the coefficients of the unknown amplitudes Z should be A. L. Gonor and V. Ya. Rivkind, Itogi Nauki i Tekhniki. Ser. Mekhanika n Zhidkosti i Gaza Vol. 17 ͓in Russian͔, VINITI, Moscow ͑1982͒, pp. 98– zero. This constraint yields a dispersion equation for the 159. problem which, neglecting mode interaction, has the simple 5 S. O. Shiryaeva and A. I. Grigor’ev, Methods of Calculating Critical form Conditions for Electrohydrodynamic Instabilities ͓in Russian͔, Yaroslavl State University Press, Yaroslavl ͑1996͒,60pp. n Ϫ1 6L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. ͑Pergamon 2 2 ␳ S ϭ͓n͑nϪ1͓͒WϪnϩ2͔Ϫn␳U M ͔ ϩ1 . Press, Oxford, 1987͓͒Russ. original, 3rd ed, Nauka, Moscow, 1986, n n ͩ ͑nϩ1͒ ͪ 733 pp.͔. ͑3͒ 7 A. I. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 61͑3͒,19͑1991͓͒Sov. The droplet becomes unstable when S2 passes through Phys. Tech. Phys. 36, 258 ͑1991͔͒. 2 8 V. A. Dyachuk and V. M. Muchnik, Dokl. Akad. Nauk SSSR 248,60 zero and becomes negative. If this condition is satisfied, the ͑1979͒. amplitude of the dominant mode begins to increase exponen- 9 A. I. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 59͑5͒,6͑1989͓͒Sov. Phys. Tech. Phys. 34, 502 ͑1989͔͒. tially with time, which leads to the successive excitation of 10 the amplitudes of higher-order modes3 and the droplet de- A. I. Grigor’ev and S. O. Shiryaeva, Physica Scripta 54, 660 ͑1996͒. cays according to the law described in Ref. 7. It can be seen Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Charge properties of aluminum oxide layers synthesized by molecular layering S. G. Sazonov, Z. N. Zuluev, V. E. Drozd, and I. O. Nikiforova

Physical Research Institute, St. Petersburg State University ͑Submitted January 21, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 58–63 ͑July 12, 1998͒ An investigation was made of the charge and conducting properties of layers of insulator obtained by molecular layering in Si–Al2O3–Al structures. No charge trapping in the insulator is observed at 77 K when depleting voltages are applied to the structure. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01107-0͔

As the variety of semiconductors used in microelectron- permittivity of the Al2O3 layers decreases with increasing Ts ics increases, there is a need to obtain high-quality insulating from 7.0 to 5.0. 0 layers at relatively low temperatures to prevent degradation For all the samples, the initial fixed charge QFB in the of the semiconductor surface. Layers of Al2O3 synthesized insulator was negative and decreased as the synthesis tem- by molecular layering are potentially useful from this point perature increased. The capacitance–voltage characteristics of view.1,2 exhibit an injection type of hysteresis. The aim of the present paper is to make a complex study The experimental dependences of the charge QFB of the electrophysical properties of metal–insulator– trapped during the action of the field, the integrated density semiconductor ͑MIS͒ structures with an Al2O3 insulator ob- of surface states, and the hysteresis of the capacitance– tained by molecular layering, using the capacitance–voltage voltage characteristic as a function of the polarizing field and current–voltage characteristics. E ϭV /d ͑V is the voltage applied to the metal electrode ´ p p p For the experiments we used KEF-5 n-Si ͑111͒ wafers for 1 min͒ are complicated for Si–Al2O3–Al structures with and KDB-10 p-Si(100) wafers. Silicon was chosen as the different parameters. substrate because the characteristics of a silicon–insulator With a positive gate voltage, the negative charge in the interface depend only weakly on the synthesis temperature of insulator of n-Si structures increases with increasing V p the insulator. In addition, the pretreatment stage of the sub- ͑Fig. 1a͒ as a result of trapping of electrons injected from the strate surface has been developed most thoroughly for sili- semiconductor into the insulator. con. As a result, the MIS structure had a fairly high-quality Since structurally different materials may be formed at interface ͑the surface state density was different temperatures Ts , the observed dependence of QFB 11 Ϫ2 Ϫ1 ϳ5ϫ10 cm eV ͒ which allowed us to study the elec- on Ts for V pϾ0 may be attributed to an increase in the trophysical properties of the insulator and their dependence barrier for the electrons at the semiconductor–insulator con- 3 on the synthesis conditions. tact as Ts increases. The samples were synthesized in the temperature range The behavior of QFB(Ep) for E pϽ0 is qualitatively dif- Tsϭ150– 310 °C using trimethyl aluminum and nitric oxide ferent ͑the polarity of the trapped charge is reversed as the 2 (NO2). Aluminum contacts of 0.24 mm were deposited by field intensity increases͒ and depends strongly on Ts . The thermal deposition in vacuum. nonmonotonic behavior of QFB(Ep) may be attributed to bi- The spread of the insulator thickness measured using an polar injection of carriers in the Al2O3. LE´ F-3 ellipsometer over a 60 mm diameter substrate was Annealing the samples immediately after deposition of less than 2%. The results of an investigation of the transmis- the insulator reduced the trapped charge. sion spectra of the films show that the band gap does not The dependence of QFB on the polarizing strength E p exceed 6 eV. The composition of the synthesized films from differs qualitatively at 300 and 77 K because for E pϽ0 and ESCA data corresponds to Al2O3 stoichiometry ͑the atomic Tϭ77 K the value of QFB varies very little down to ratio of the elements is Al/Oϭ0.7͒. Electron diffraction E pϳϪ7 MV/cm, after which the initial negative charge analyses revealed that the films are amorphous. The shows a slight decrease ͑Fig. 1b͒. The absence of a negative electrical strength of a layer of thickness dϭ100 nm was charge buildup in the n-type Si structure ͑as at Tϭ300 K in 7.5 MV/cm. strong fields E pϽ0͒ can be attributed to the temperature de- The current–voltage characteristics of these structures pendence of the level of electron injection from the metal were linear when plotted as log J vs E1/2. The depth of a electrode into the insulator. As a result, at 77 K the injection Poole–Frenkel center is 0.4 eV but the conduction mecha- current is so small that over the polarization time, the elec- nism cannot be identified specifically because of the small trons captured by traps do not have time to create any appre- range of measurable current ͑three or four orders of magni- ciable charge. However, the low concentration of holes ͑mi- tude͒. Layers of Al2O3 with Tsϭ150 °C have the highest nority carriers͒ and the longer times taken to form the resistivity and breakdown field strength. The low-frequency inversion layer in n-Si at this temperature, have the result

FIG. 1. Trapped charge density versus field strength in n –Si–Al2O3–Al structure at Tϭ300 K ͑a͒ and 77 K ͑b͒. T : 1—150 °C; 2—200 °C; s FIG. 2. Trapped charge density versus field strength ͑a͒ and polarization 3—240 °C; 4—310 °C. 0 time ͑b͒ in p –Si–Al2O3–Al structure: 1—Tϭ77 K, QFBϭϪ1 11 Ϫ2 0 11 Ϫ2 ϫ10 cm ; 2—Tϭ300 K, QFBϭϪ6ϫ10 cm ; 3—Tϭ77 K, V pϭ30 V. that no positive charge builds up for V pϽ0. This factor also explains the absence of electron trapping in the p-Si structure at positive voltages ͑Fig. 2a͒. An increase in the polar- gating insulating heterostructures obtained by molecular lay- ization time leads to a buildup of charge in the insulator ͑Fig. ering ͑insulating layers imbedded in the bulk of the Al2O3͒ 2b͒. At negative gate voltages, positive charge accumulates since the charge properties of these structures will only be in this structure as a result of trapping of holes injected from determined by the characteristics of the embedded layers and the semiconductor into the insulator and at V pϾ0 and their interfaces. Tϭ300 K, the negative charge increases as a result of injection of electrons from the metal. 1 V. B. Aleskovski and V. E. Drozd, Acta Polytech. Scand. 195, 155 ͑1990͒. Thus, although traps are present in the bulk of the insu- 2 V. E. Drozd, A. P. Baraban, and I. O. Nikiforova, Appl. Surf. Sci. 83, 583 lator, at fairly low temperature no charge trapping is ob- ͑1994͒. served in the insulator over a fairly long period ͑tens of min- 3 S. G. Sazonov, V. E. Drozd, Z. N. Zuluev, and O. E. Nikiforova, in utes͒ at depleting polarizing voltages ͑up to breakdown Proceedings of the International Scientific-Technical Conference ‘‘Dielectrics-97,’’ St. Petersburg, 1997 ͓in Russian͔,p.66–68. levels͒ as a result of the low level of carrier injection into the insulator. This opens up additional possibilities for investi- Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Diffraction of low-energy electrons at a rippled surface S. A. Knyazev and V. E. Korsukov

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg ͑Submitted November 28, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 64–69 ͑July 12, 1998͒ An analysis is made of a kinematic model of the scattering of low-energy electrons at a one- dimensional periodic toothed structure simulating a rippled surface. Calculations of the intensity proﬁles of the diffraction peaks are made for periodic systems of identical symmetrical triangular teeth with a variable number of atoms on each generatrix of the tooth and various angles of inclination of the generatrices to the horizontal. The main effect observed as a result of scattering at a periodic toothed structure involves broadening or splitting of the diffraction peaks compared with the diffraction patterns from a smooth chain of atoms. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01207-5͔

In low-energy electron diffraction analyses of the trans- This gives rise to the problem of diffraction of low-energy formation of the surface structure of Ge͑111͒ and mica dur- electrons at a periodic surface structure. The aim of the ing stretching and bending of the crystals, we observed present study is to determine how the transition from a broadening, which was quasireversible with loading, and smooth to a rippled surface influences the intensity profile of splitting of the peaks in the diffraction patterns.1,2 It was the diffraction peaks on the low-energy diffraction patterns. shown by high-energy electron diffraction and scanning tun- For a rippled surface the diffraction conditions in the neling microscopy1,3 that under deformation, the single- direction of the grooves are the same as those for a planar crystal surface is transformed to the polycrystalline and surface. Thus, diffraction at a rippled surface—the intensity nanocrystalline state. This is accompanied by the appearance of the electron scattering at this structure—may be simulated of a rough surface relief ͑protrusions and indentations of by the scattering at a one-dimensional atomic chain having varying scale are observed, ranging from atomic terraces to the same profile as the rippled surface and the angular dis- extended hillocks and indentations with depths between 10 tribution of the intensity need only be considered in the di- and 1000 nm͒. Thus, the process of mechanical loading of rection of this chain. The intensity of the diffraction peaks crystals may be viewed as involving the formation of shal- was calculated using a kinematic approximation for a one- low sections with different orientations to one another, and dimensional periodic toothed structure with triangular sym- as one variant, the formation of a rippled surface structure. metric teeth which gives

exp i͑2Na␲/␭͑cos ␸ sin͑␪͒ϩsin ␸͑cos͑␪͒ϩ1͒͒͒Ϫ1 Iϭ ͯ exp i͑2a␲/␭͑cos ␸ sin͑␪͒ϩsin ␸͑cos͑␪͒ϩ1͒͒͒Ϫ1 exp i͑2͑NϪ1͒a␲/␭͑cos ␸ sin͑␪͒Ϫsin ␸͑cos͑␪͒ϩ1͒͒͒Ϫ1 2 ϩ Ϫ1 exp i͑2a␲/␭͑cos ␸ sin͑␪͒Ϫsin ␸͑cos͑␪͒ϩ1͒͒͒Ϫ1 ͯ ϫ͉͑sin͑2M͑NϪ1͒a␲/␭ cos ␸ sin͑␪͒͒/͑sin 2͑NϪ1͒a␲/␭ cos ␸ sin͑␪͉͒͒2.

The number of atoms N on one generatrix is the number of the transition to structures with more teeth substantially scattering atoms, M is the number of teeth, a is the distance changes the diffraction pattern. For the ﬁrst structure, this between neighboring atoms, ␸ is the angle of inclination of pattern is a set of isolated peaks and for the second, it com- the generatrix to the horizontal, and ␪ is the scattering angle. prises a set of peaks split into various sub-peaks. Figure 1 shows how the angular dependence of the in- In order to explain the nature of the diffraction peaks tensity of the diffraction peaks I(␪) changes in the transition observed for scattering at a toothed structure, we constructed from a smooth horizontal chain of atoms ͑Fig. 1a͒ to a a series of I(␪) graphs for various angles of inclination of toothed structure with few atoms per generatrix (Nϭ4) and the tooth generatrices. For a small angle ␸(1°) the diffrac- a comparatively large number of teeth (Mϭ20) ͑Fig. 1b͒, tion pattern for Nϭ4 and Mϭ20 contains peaks consistent and then to a structure with many atoms per generatrix with the diffraction from a smooth horizontal chain. An in- (Nϭ20) and few teeth (Mϭ4) ͑Fig. 1c͒. It can be seen that crease in the number of atoms per generatrix to 20 signiﬁ-

FIG. 1. Angular distribution of the intensity for diffraction

at a one-dimensional chain, Erϭ120 eV, 1—electron beam; a—horizontal atomic chain MϫMϭ80; b—toothed structure, ␸ϭ6, Nϭ4, Mϭ20; c—toothed structure, ␸ϭ6, Nϭ20, Mϭ4.

cantly transforms the diffraction pattern: the peaks become lar reﬂection from the generatrices becomes more clearly de- split, their intensity drops appreciably, and the diffraction ﬁned. pattern varies abruptly with the electron energy. As a result A series of I(␪) graphs was constructed as a function of of changing from ␸ϭ6° to ␸ϭ13° ͑Nϭ4, Mϭ20͒, the the number of teeth. Figure 2 gives the diffraction pattern structure of the diffraction pattern becomes sparser, the an- obtained for diffraction at one, two, and twenty teeth gular position of the diffractions peaks shifts, these peaks ͑Nϭ4, ␸ϭ6°, Epϭ120 eV͒. All the curves were normal- separate into individual groups, and their intensity varies rap- ized relative to the structure with one tooth. The results idly with E p . As the number of atoms per tooth generatrix clearly show that the diffraction structure is formed by scat- increases ͑Nϭ20, Mϭ4͒, the peak corresponding to specu- tering at one tooth and increasing the number of teeth in the Tech. Phys. Lett. 24 (7), July 1998 S. A. Knyazev and V. E. Korsukov 529

son between the angular position of these peaks and the position of the diffraction peaks from a smooth chain of atoms positioned at angles Ϫ␸ and ϩ␸ to the horizontal ͑in accordance with the generatrices of the tooth͒ showed that the angular position of the peaks for diffraction at one tooth may be ascribed to the ͑00͒, ͑10͒, and ͑01͒ peaks obtained by diffraction from two smooth atomic chains inclined to the horizontal at angles similar to the generatrices of the tooth. This set of calculations shows that even a simple one- dimensional kinematic model of electron scattering at a toothed surface predicts an appreciable transformation of the low-energy diffraction patterns from a rippled surface compared with those for a smooth surface. These changes mainly involve broadening and splitting of the diffraction peaks. For FIG. 2. Angular distribution of intensity for various numbers of teeth. All toothed structures containing a considerable number of at- the curves are normalized to one tooth. 1—one tooth, 2—two teeth, and oms per generatrix, the angular broadening of the peaks can 3—twenty teeth. be estimated as the angle of inclination of the tooth generatrix to the horizontal. These results can be used to estimate the parameters of rippled structures formed as a result of the periodic structure to twenty merely reduces the angular deformation of single crystals from the changes in the low- width of the diffraction peaks. Similar results were obtained energy electron diffraction patterns from the loaded surface. for other values of N and ␸.

In order to identify the nature of the diffraction peaks 1 formed by scattering at one tooth, we calculated I( ) for V. E. Korsukov, S. A. Knyazev, A. S. Luk’yanenko et al., Fiz. Tverd. ␪ Tela ͑St. Petersburg͒ 38, 113 ͑1996͓͒Phys. Solid State 38,60͑1996͔͒. various numbers of atoms per generatrix. The calculations 2 S. A. Knyazev, V. E. Korsukov, and B. A. Obidov, Fiz. Tverd. Tela showed that changing N from two to four appreciably alters ͑St. Petersburg͒ 36, 1315 ͑1994͓͒Phys. Solid State 36, 718 ͑1994͔͒. 3 the diffraction pattern. An increase in the number of atoms S. N. Zhurkov, V. E. Korsukov, A. S. Luk’yanenko, B. A. Obidov, and on the generatrix (Nϭ10– 20– 50) leads to the formation of A. P. Smirnov, JETP Lett. 51, 370 ͑1990͒. a stable angular distribution of diffraction peaks. A compari- Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Criteria for the existence of generalized surface spin waves S. V. Tarasenko

A. A. Galkin Physicotechnical Institute, Donetsk ͑Submitted April 17, 1997; resubmitted February 25, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 70–75 ͑July 12, 1998͒ A criterion for the existence of generalized surface spin waves is proposed on the basis of an analysis of the dipole-exchange surface spin dynamics of a tangentially magnetized, semibounded, easy-axis antiferromagnet. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01307-X͔

For a normally magnetized, semibounded, easy-axis fer- tions should be supplemented by the corresponding boundary romagnet ͑antiferromagnet͒ it was shown in Refs. 1 and 2 conditions. If the spins at the surface of the easy-axis anti- that hybridization of the magnetic dipole and exchange ferromagnet are free ͑m˜ ,˜l are small deviations of the vectors mechanisms for dispersion of the magnetic vibrations in a m and l from their equilibrium orientation͒, the system of bounded crystal may result in the formation of a dipole- boundary conditions which must be satisfied by a dipole- exchange, generalized surface ͑quasi-surface͒ spin wave, exchange surface spin wave localized at the interface be- propagating along the boundary of the magnet, which is not tween the magnetic and nonmagnetic media (xϭ0) may be achieved in the pure-exchange or in the magnetostatic ap- represented in the form proximation. So far however, the criterion which the dipole- l˜ץ ˜mץ exchange dynamics of an unbounded magnet must satisfy for a generalized surface spin wave to form at its boundary re- ϭ ϭ0; B nϭBn; ͓H n͔ϭ͓Hn͔; x m mץ xץ mains unclear. In the present paper an analysis is made of the conditions ˜ ˜ for the formation of dipole-exchange surface spin waves in a ͉m͉, ␾m͉l͉ 0 for x Ϫϱ; → → semibounded easy-axis antiferromagnet (Cr2O3) and it is shown for the first time that the criterion for the existence of ␾ 0 for x ϱ, ͑2͒ → → a generalized surface spin wave may be the presence of sections of maximum negative curvature on the isofrequency where Bm , Hm , ␾m ͑B,H,␾͒ are respectively the magnetic surface of the normal spin vibrations of an unbounded mag- induction vector, the magnetic field vector, and the magne- net and their specific spatial orientation relative to the normal tostatic potential in a magnetic ͑nonmagnetic͒ medium. It is to the surface of the magnet and a certain direction of propa- well-known that in this model of an easy-axis antiferromag- gation. By way of example, an analysis is made of a two- net, neglecting the finite dimensions of a real magnetic sublattice ͑M1,2 is the sublattice magnetization, ͉M1͉ϭ͉M2͉ sample, the antiferromagnetic resonance spectrum consists of ϭM 0͒ model of an easy-axis antiferromagnet ͑0Z is the easy two branches which, according to their mode of excitation by axis͒. In this case, the density of the thermodynamic poten- the microwave field h, can be divided into a quasi- tial W of the easy-axis antiferromagnet may be represented ferromagnetic mode (␻F), linearly excited and with hЌH, in terms of the vectors of ferromagnetism m and antiferro- and a quasi-antiferromagnetic mode (␻AF) linearly related to magnetism l in the form hʈH. For simplicity and conciseness, we shall subsequently assume that the external field H is such that ␦ ␣ ␤ Wϭ m2ϩ ͑ l͒2Ϫ l2ϪM͑HϩH ͒, 2 2 “ 2 z m ␻Fӷ␻AF . ͑3͒

M1ϩM2 M1ϪM2 mϭ , lϭ , ͑1͒ In this case, the indirect coupling of these modes of an un- 2M 0 2M 0 bounded crystal via the magnetic-dipole interaction field can where ␦, ␣, and ␤Ͼ0 are the constants of homogeneous be neglected and the surface dipole-exchange spin dynamics exchange, inhomogeneous exchange, and uniaxial anisot- of a semibounded easy-axis antiferromagnetic can be inves- ropy, respectively, HʈOY is the external magnetic field and tigated merely in terms of the quasi-antiferromagnetic mode ˜ ˜ Hm is the field induced by magnetic-dipole interaction. The of the spin-wave excitation spectrum (͉l x,z͉,͉my͉). Then the dipole exchange dynamics of this model of a magnetic me- corresponding characteristic equation determining the wave dium is known to be described by a closed system of equa- vector component of the magnetic vibrations k(kx) normal to tions consisting of the Landau–Lifshitz equations for the the surface, as a function of the experimentally determined vectors m and l and the magnetostatics equations. Assuming frequencies ␻, kЌ , and ␺ ͑cos ␺ϭky /kЌ , kЌ is the wave vec- that the easy-axis antiferromagnetic medium occupies the tor component of the spin vibrations tangential to the inter- half-space xϽ0(nʈOXЌH), this system of dynamic equa- face of the media͒, may be expressed in the form

ﬁed. In the regions of existence of dipole-exchange generalized surface spin waves ͑region III͒ and evanescent surface 2 2 spin waves ͑region IV͒, the conditions q1ϭ(q2)* and 2 2 q1Ͼ0, q2Ͻ0, respectively, hold ͑‘‘*’’ indicates complex conjugation͒. The condition for nontrivial solvability of the system ͑2͒, ͑4͒, and ͑5͒ determines the implicit form of the dispersion law for the exchange-dipole surface spin waves propagating along the free boundary of an easy-axis antiferromagnet with HЌnЌl. In the long-wavelength limit with the additional constraint c2␥Ӷ˜R, this dispersion relation can be obtained in the explicit form

␻2ϭ␻˜ 2ϩc2͑k2 ϩ␥͑k ͒͒; ␥͑k 0͒ 0; 0 Ќ Ќ Ќ→ → ˜R ␥͑k ͒ϭ2k ϩk2 Ќ Ќͱc2 Ќ

2 FIG. 1. Possible modes of dipole-exchange spin-wave excitations in a ˜R ˜R 1 ˜R Ϫ k ϩk2 ϩ Ϫ . ͑6͒ bounded easy-axis antiferromagnet with nʈOX, HʈOY, lʈOZ as a function ͩͱ Ќͱc2 Ќ 4c2 2 ͱc2ͪ of the frequency ␻ and the wave vector component tangent to the magnetic surface k : f ϭ␻˜ 2ϩ˜Rϩc2k2 , f ϭ␻˜ 2Ϯ2ͱ˜Rck ; ck ϭͱ˜R. Ќ 1 0 Ќ Ϯ 0 Ќ * Comparing formula ͑6͒ with the regions of existence of the possible types of exchange-dipole modes of spin-wave excitations shown in Fig. 1, it can be concluded that the surface 2 ˜ 2 ␻ Ϫ␻0 spin wave ͑6͒ for kЌӶk is a generalized surface spin wave. k4ϩAk2ϩBϭ0; Aϭ2k2 Ϫ ; * x x Ќ c2 Now for fixed ␺ formula ͑4͒ is used to analyze the change in the curvature of the isofrequency surface K(␪)(cos ␪ 2 2 ˜ 2 ˜ 2 2 2 2 c kЌϩRϪ␻ ϩ␻0 ϭkx /k,k ϭkxϩkЌ) of the quasi-antiferromagnetic mode of Bϭk2 ; Ќͩ c2 ͪ the spectrum of normal spin vibrations of an unbounded antiferromagnet ͑4͒ as a function of the external parameter ␻. ˜ 2˜ ˜ 2 2 ˜ 2 2 Rϭ␻0⑀; ␻0ϭ␻0ϩ⑀c kЌ , ͑4͒ For this purpose an analysis is made of the shape of the curve obtained by intersection of the isofrequency surface with the ˜ 2˜ ˜ 2 where Rϭ␻ ⑀; here ⑀ϭ(4␲M0 /HE)cos ␺ and 0 plane in which the vectors n and kЌ lie. This analysis shows ␻2Ϸg2(2H H ϪH H2/2H ); c2ϭg22H (1ϪH2/ 0 E A A E E that for ␻2Ͻ␻2ϩ˜R(2ϩ˜⑀), ␪ϭ0,␲ in the direction of (2H )2)␣M , g is the gyromagnetic ratio, H and H are 0 E 0 E A propagation of the generalized surface spin wave being stud- the fields of intersublattice exchange and uniaxial anisotropy, ied, sections with maximum negative curvature K(␪) form respectively, and H ӷH . Since relation ͑4͒ is a biquadratic E A on this curve: equation for kx , it can be confirmed that all possible modes of dipole-exchange linear magnetic vibrations propagating in K͑␪ϭ0,␲͒Ͻ0. ͑7͒ the plane of an easy-axis antiferromagnetic film in the limit ͑3͒ are bipartial spin-wave excitations. Thus, the amplitude It is easily confirmed that for the dipole-exchange general- structure of the small vibrations of the magnetostatic poten- ized surface spin waves studied in Refs. 1 and 2, a similar relation is found between the conditions for the existence of tial ␾m ͑for an easy-axis antiferromagnet͒ along the normal to the interface is given in the form a generalized surface spin wave and the sections of maximum negative curvature on the curve obtained by intersec- ␾mϭA1 exp͑i␻tϪq1x͒ϩA2 exp͑i␻tϪq2x͒, tion of the isofrequency surface of the normal spin vibration 2 2 of an unbounded magnet with the plane in which the vectors q ϵϪkx , ͑5͒ n and kЌ lie. Thus, this criterion can be applied to specify the where A1,2 are arbitrary constants determined from the conditions required for the formation of a generalized surface boundary conditions spin wave of the dipole-exchange or elasto-exchange type by Formulas ͑4͒ and ͑5͒ can be used to classify the possible calculating the dipole-exchange ͑elasto-exchange͒ magnon modes propagating along the surface of an easy-axis antifer- spectrum of an unbounded magnet. romagnet with nʈOX as a function of their conditions of We have so far considered the case with HyÞ0 in which localization near the interface between the magnetic and non- condition ͑3͒ is satisfied between the antiferromagnetic reso- magnetic media, which are determined from Eq. ͑4͒ by the nance frequencies of an unbounded ferromagnet. We shall 2 character of the roots q j ( jϭ1,2) on the plane of the external now assume that the condition opposite to ͑3͒ is satisfied parameters ␻ and kЌ for given ␺ ͑see Fig. 1͒. It follows from between the quasi-ferromagnetic and quasi-antiferromag- Eqs. ͑4͒ and ͑5͒ that in the region of existence of dipole- netic branches of the antiferromagnetic resonance spectrum exchange surface spin waves ͑region I͒ or volume spin of this magnet, which is possible for HʈOX(lʈOZ). The cal- waves ͑region II͒ on the plane of the parameters ␻, kЌ , and culations show that if the magnetic-dipole mechanism of in- 2 2 ␺, the inequalities q1,2Ͼ0orq1,2Ͻ0, respectively, are satis- teraction between the quasi-ferromagnetic and quasi- 532 Tech. Phys. Lett. 24 (7), July 1998 S. V. Tarasenko antiferromagnetic modes of the spectrum of an easy-axis spectrum of normal bulk magnons of an unbounded magnet antiferromagnet is neglected for the same relative orienta- is localized, criterion ͑7͒ is the condition for the existence of tions of the vectors n,l,kЌ , the dipole-exchange generalized generalized surface spin waves or an evanescent generalized surface spin wave determined above is as before the result of surface spin wave, respectively. the normal quasi-antiferromagnetic mode of an unbounded For a metallized magnetic surface or completely an- crystal being localized near the surface of the magnet. Now, chored spins dipole-exchange generalized surface spin waves however, its spectrum ͑6͒ will lie at a higher frequency than of this type do not occur. that of the quasi-ferromagnetic mode of antiferromagnetic resonance and hence the surface spin wave ͑6͒ will now be 1 R. E. De Wames and T. J. Wolfram, J. Appl. Phys. 41, 987 ͑1970͒. 2 an evanescent generalized spin wave since it generates quasi- B. A. Ivanov, V. F. Lapchenko, and A. L. Sukstanski, Fiz. Tverd. Tela ferromagnetic bulk spin waves as it propagates. Thus, de- ͑Leningrad͒ 27, 173 ͑1985͓͒Sov. Phys. Solid State 27, 101 ͑1985͔͒. pending on whether the low- or high-frequency mode of the Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Use of magnetic ﬁeld screening by high-temperature superconducting ﬁlms to switch microwave signals T. M. Burbaev, S. I. Krasnosvobodtsev, V. A. Kurbatov, N. P. Malakshinov, V. S. Nozdrin, and N. A. Penin

P. N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow ͑Submitted September 18, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 76–81 ͑July 12, 1998͒ The efficiency with which YBCO films screen an alternating magnetic field near the superconducting transition was measured. In the decimeter range measurements were made of the characteristics of a switch whose operating principle was based on the change in the screening of an alternating magnetic field by a superconducting transition. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01407-4͔

The use of a superconducting transition to switch micro- Our prototype switch consisted of a strip line fitted with wave signals usually implies a device with thin supercon- magnetic coupling loops which was screened by a high- ducting bridges whose normal-state resistance is sufficiently temperature superconducting film. The switch is shown sche- high to ensure that the switching element has a high quality matically in Fig. 1 which gives a block diagram of the mea- factor and to facilitate current control of the switch.1 Never- suring apparatus. The device was designed to be inserted in a theless, the normal-state resistance of the bridges is generally portable helium Dewar. The temperature of the sample was less than 1000 ⍀, which leads to appreciable (у20%) active measured with a thermocouple. Measurements of the trans- losses. This limits the switchable power and thus the range of mission coefficient KTϭ(S1 /S2) and the switching coeffi- application of these switches. However, there is another cient Ksϭ(KTN /KTS) were made in the frequency range method of using a superconducting transition to switch sig- 0.01–300 MHz using SMV6.1 and SMV8.5 measuring de- nals which can almost eliminate losses in the switching ele- tectors. Here S1 and S2 are the signals on entry to and at the ment over a fairly wide frequency range; this involves using exit from the switching device and KTN and KTS are the the screening properties of a superconducting plate. In this transmission coefficients in the normal and superconducting case, the switching coefficient will be determined by the dif- state of the high-temperature superconducting film, respec- ference in the screening properties in the superconducting tively. The signal generator was either the built-in calibration and normal states. In the first case, the screening is at a oscillator of the measuring detector or, when higher powers maximum while in the second, it is determined by the con- were required, a G4-37A generator. ductivity and by the signal frequency. Figure 1 shows the system used to measure the switch- We first note that the screening of an electromagnetic ing coefficient with the switch controlled by current pulses. wave incident normally on a superconducting plate will be For measurements of the transmission coefficient, the signal efficient in the normal state, even for very thin films with from the measuring detector 4 was fed directly to the Y input dϽ1000 Å. Here, the condition for weak absorption is of an automatic plotter 7. Measurements of the transmission dӶ(2/z ␴ ), where z ϭ377 ⍀ and ␴ is the bulk conduc- 0 0 0 0 coefficient in the absence of the superconducting film K and tivity of the sample at low frequencies. This condition cannot 0 with the coupling loop screened by copper foil of the same be satisfied for superconductors. area as the high-temperature superconducting film K at For the screening of an alternating magnetic field di- Cu frequencies 10 MHz, at which the depth of penetration of rected perpendicular to the plane of a superconducting plate, у the situation is different. the copper is substantially less than the thickness of the foil, showed that the maximum switching coefficient K The transmission of the magnetic field by the sample in Smax the normal state is determined by the relation between the ϭ(K0 /KCu) is 12 dB. This value is determined by the design sample thickness and the depth of penetration characteristics and is mainly limited by ‘‘leakage’’ of mag- 1/2 d pϭ(2/␮␮0␻␴0) . This difference is attributed to the netic field beneath the screen through the insulating sub- boundary conditions at the surface where, in this case, the strate, which is sandwiched between the superconducting magnetic field has an antinode while the electric field has a film and the coupling loop. 2 node. The condition dӶd p yields the inequality f We investigated YBCO films 1500 Å thick fabricated by 2 2 Ϫ1 3 Ӷ(␲d z0␧0) , which determines the frequency range laser deposition on strontium titanate substrates. In the fre- where the superconducting sample in the normal state will be quency range used the electrical properties of the substrate almost transparent to the alternating magnetic field. For our influenced the results negligibly, no difference being ob- high-temperature superconducting films an estimate gives f served in the transmission coefficient with the film in the Ӷ1.7ϫ1013 Hz. normal state and without the film.

FIG. 1. Schematic of apparatus: 1—rf generator, 2—switching device, a—superconducting film with current contacts b, c—thermocouple, 3— control current pulse generator, 4—measuring detector, 5—oscilloscope, FIG. 3. Switching coefficient versus temperature at 100 MHz for control 6—dc microvoltmeter, 7—automatic xy plotter, and 8—synchronous pulse currents: 1—1.6 A, 2—0.8 A, 3—0.4 A, 4—0.2 A, 5—0.08 A. The dashed voltmeter. curve gives the temperature dependence of the transmission coefficient.

Figure 2 gives temperature dependences of the relative transmission coefficient measured at different frequencies. Our ac measurements of the temperature dependence of The absolute value of KT increases with the square of the frequency and at 100 MHz in the open ͑normal͒ state of the the electrical conductivity of the films revealed no significant superconducting screen is Ϫ30 dB. Also plotted is the mea- differences in the curves obtained at different measurement sured dependence of the dc resistance of the film. frequencies, which could explain the magnetic field screen- It can be seen that as the measurement frequency in- ing results. We thus postulate that the observed frequency creases, the onset of screening, caused by the appearance of shift effect is attributable to the response time required for the superconducting phase with decreasing temperature, is the alternating magnetic field to polarize the thermally ex- shifted toward higher temperatures. At the same time, the cited vortices which form in quasi-two-dimensional 5 reduction in the dc resistance with decreasing temperature superconductors and combine to form pairs rotating in op- begins at temperatures higher than that at which magnetic posite directions at temperatures slightly below the supercon- field screening appears. ducting transition. The dependences observed for different types of films Figure 3 gives the temperature dependence of the exhibit different behavior of the screening curves at different switching coefficient measured at a frequency of 100 MHz frequencies. For YBCO films the slope of the magnetic field for a YBCO film switch. Also plotted is the temperature transmission increases with decreasing frequency, whereas dependence of the transmission coefficient at this frequency. for BiSrCaCuO films and single crystals, the dependences A comparison of the curves shows that the control current are almost independent of frequency—the curves are shifted required for complete switching is 1.6 A at an operating in temperature without any significant change in profile.4 temperature of 87 K. Smaller control currents do not give complete switching because of the finite width of the superconducting transition and the extremely large dimensions of the film, 10ϫ10 mm. At some expense in switching speed, the control signal power can be reduced in principle by controlling the temperature of the superconducting element. To assess the operating capability of the switch at elevated signal powers, we used an inverse switching circuit in which the signal was fed to the feedback loop and the coefficient of transmission to the main line was measured. This ensured the maximum magnetic flux through the superconducting film and a low absorbed power in the matched load connected to the main line. Measurements made at 400 Hz revealed no substantial change in the temperature dependence of the transmission coefficient when the signal power was raised to 2 W, except for a small temperature shift of the curve, not exceeding 1 K, which was evidently caused by the sample being heated by the power dissipated in the load. FIG. 2. Temperature dependences of the transmission coefficient at frequencies: 1—1 MHz, 2—30 MHz, 3—100 MHz, 4—300 MHz, 5—dc resistance These results show that the screening of an alternating of film. magnetic field by superconducting films may be of interest Tech. Phys. Lett. 24 (7), July 1998 Burbaev et al. 535 for the practical development of superconducting switches 2 P. Grosse, Free Electrons in Solids, Mir, Moscow ͑1982͒. for elevated-power microwave signals. 3 A. I. Golovashkin, E. V. Ekimov, S. I. Krasnosvobodtsev et al., Physica C 162–164, 715 ͑1989͒. This work was supported by the ‘‘Superconductivity’’ 4 T. M. Burbaev, G. A. Kalyuzhnaya, V. A. Kurbatov, V. S. Nozdrin, and program, Project No. 96081. N. A. Penin, Fiz. Tverd. Tela ͑St. Petersburg͒ 39, 228 ͑1997͓͒Phys. Solid State 39, 200 ͑1997͔͒. 5 V. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. 1 M. M. Gadukov, A. B. Kozyrev, V. N. Osadchi, and V. F. Vratskikh, Rev. Lett. 40, 783 ͑1978͒. Pis’ma Zh. Tekh. Fiz. 20͑18͒,86͑1994͓͒Tech. Phys. Lett. 20, 761 ͑1994͔͒. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Theory of the thermoelastic generation of mechanical vibrations in internally stressed solids by laser radiation K. L. Muratikov

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg ͑Submitted February 25, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 82–88 ͑July 12, 1998͒ An investigation was made of the behavior of transient deformations in solids with internal stresses exposed to time-modulated laser radiation. The nonlinear theory of thermoelasticity is used to propose a model for the excitation of mechanical stresses which takes account of the stress dependence of the thermoelastic coupling parameter. An expression is derived for the electric signal recorded by a piezoelement when mechanical vibrations are generated in a uniformly deformed sample. The results are compared with existing experimental data and show qualitative agreement. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01507-9͔

Pikץ -An interesting line of research into the thermoelastic ef ϭ␳ ⌬u¨ , ͑2͒ x 0 iץ -fect concerns the possible use of this effect to record me chanical stresses in solids.1–5 Various experimental data have k 1,4,5 2,3,6 ,xm)tkm is the Pioli–Kirchhoff tensorץ/ uiץ)now been obtained for metals and ceramics as confir- where Pikϭ mation of this possibility. However, the mechanism respon- tkm is the stress tensor which is related to the internal ; ukmץ/Wץsible for the influence of mechanical stresses on the results of energy density W of the solid by tkmϭ 1 -xm) is the deץ/ ulץxkץ/ ulץ xk ϩץ/ umץ xm ϩץ/ ukץ)laser thermoelastic measurements has not been adequately ukmϭ 2 clarified. In Ref. 4, a model was proposed for the formation formation tensor, and ␳0 is the density of the solid in the of the thermoelastic signal, where its dependence on the me- initial state. chanical stresses was mainly attributed to the dependence of The energy density of the deformed solid may be ex- the thermophysical parameters of the material on these pressed as the sum WϭW1ϩW2 ͑where W1 is the mechani- stresses. In Ref. 6, however, it was shown that in ceramics a cal energy density and W2 is the energy density associated strong dependence of the thermoelastic signal on the residual with the thermoelastic deformations͒. Here it is assumed that stresses may be observed, even in the absence of any appre- the solid is initially isotropic and its mechanical energy com- 11 ciable changes in their thermophysical properties. ponent is determined by the Murnaghan model. For the Thus, the present paper proposes a model for the forma- thermoelastic energy density we shall assume by analogy tion of the thermoelastic signal generated by laser radiation with Ref. 7 that the thermoelastic coupling coefficient de- in solids which is capable of explaining these characteristics. pends on the deformation tensor and this dependence is lin- An important difference is that this model assumes that the ear. We shall also assume that the strains ⌬uik created as a thermoelastic coupling coefficient depends on the mechani- result of the action of laser radiation on the object are small. cal stresses. Such a dependence was noted previously for the Under these conditions, the thermoelastic energy density coefficient of thermal expansion7 and for the elastic may be expressed in the following form, to within linear 8 modulus. Since the coefficient of thermoelastic coupling in terms in ⌬uik isotropic solids is the product of these quantities, it is impor- W ϭϪ␥ ͑␦ ϩ␤U ͒⌬u ͑TϪT ͒, ͑3͒ tant to include this dependence in the thermoelastic coupling 2 0 ik ik ik 0 coefficient. where ␥0 is the thermoelastic coupling coefficient for the The generation of mechanical vibrations by laser radia- nondeformed solid, ␤ is a coefficient which determines the tion in mechanically stressed solids is analyzed in terms of dependence of the thermoelastic coupling on the initial 9 nonlinear mechanics with initial deformations. We shall as- strain, and T0 is the ambient temperature. sume that the initial deformations are not small. Thus, we Note that for ␤ϭ0 equality ͑3͒ reduces to the usual ex- shall assume that the displacement vector of points in the pression for the thermoelastic energy density of an isotropic solid is given in the form solid. Expressions ͑2͒ and ͑3͒ can be used to obtain the equation of motion for the displacement vector of the particles of u͑r͒ϭrϩU͑r͒ϩ⌬u͑r͒, ͑1͒ a solid in which mechanical vibrations are excited by laser radiation as a result of the thermoelastic effect. Here we where U͑r͒ describes the initial deformation and ⌬u͑r͒ is the confine our analysis to a uniformly deformed solid with the displacement of the solid particles caused by the thermoelas- components of the initial deformation vector defined as (i) (i) tic deformations under action of the laser radiation. UiϭA xi ͑A are constants characterizing uniform defor- The equation of motion for the elements of the solid in mation in different directions͒. We shall also assume that the nonlinear mechanics10 can be expressed in the form displacement vector ⌬u and the temperature fluctuations ⌬T

tions generated by the laser radiation, the boundary condition on the upper surface of the sample may be considered to be deﬁned at the free surface. At the interface between the object and the piezoelement, we use the condition for continuity of the normal component of the stress vector at the interface zϭl. In addition, to simplify the problem, we adopt the assumption made in nonlinear mechanics that the boundary conditions at a deformed surface may be replaced by those at an undeformed surface.12 The requirement for continuity of the normal component at the sample–piezoelement interface allows us to ﬁnd the signal recorded by the piezoelement. This involves determin-

ing the component of the displacement vector ⌬u3(z,t) ͑Refs. 4 and 5͒. Assuming that the exciting laser radiation is FIG. 1. Geometric configuration of sample and piezoelement: 1—sample harmonically modulated, we can express this component in and 2—piezoelement. the form ⌬u(z,t)ϭ⌬u(z,␻)ei␻t ͑where ␻ is the angular modulation frequency͒. Then, using Eq. ͑4͒ and the boundary conditions specified for ⌬u (z,␻) we obtain the following in the solid are small quantities and we shall use these to 3 result: linearize the equations of motion. Without going into the details of the calculations, we give the final result for the vector components ⌬ui . Using expressions ͑2͒ and ͑3͒,we ͑0͒ Ϫ␴l obtain the following equations U3 e ⌬u3͑z,␻͒ϭϪ cos Qz T cos Ql⌬ץ u⌬2ץ u⌬2ץ ͑i͒ i ͑i͒ k ͑i͒ ¨ f k ϩhk ϭg ϩ␳0⌬ui , ͑4͒ xi ϩ ͑3͒ ϩץ xiץxkץ xkץxkץ ␥0͑1 A ͒͑1 ␤U33͒ ͑0͒ ϩ ͑3͒ ͑3͒ ⌬Tsϩ␴U3 ͑1ϩA͑i͒͒2 ͫ f ϩh ͬ ͑i͒ ͑0͒ 3 3 where f k ϭ tkk ϩ ͑bϩnUiiϩnUkk͒ , ͫ 2 ͬ ͑0͒ Ϫ␴z ϫsin Q͑zϪl͒ϩU3 e , ͑5͒ bϭ2␮ϩ͑2mϪn͒Upp , h͑i͒ϭ͑1ϩA͑i͒͒͑1ϩA͑k͒͒ k (0) (3) (3) (3) 2 where U3 ϭϪ͓␥0␴(1ϩA )(1ϩ␤U33)/(f3 ϩh3 )␴ 1 ϩ␳ ␻2͔⌬T , ␴ϭ(1ϩi)ͱ␻/2୐, ୐ is the thermal diffusivity ϫ c ϩ2mU ϩ ͑bϩnU ϩnU ͒ , 0 s ͫ ii kk 2 ii kk ͬ of the sample, Qϭ ␳ ␻2/ f (3)ϩh(3), and ⌬T is the ampli- ͱ 0 3 3 s tude of the temperature fluctuations of the sample surface. g͑i͒ϭ␥ ͑1ϩA͑i͒͒͑1ϩ␤U ͒, 0 ii The mechanical and electrical characteristics of the pi- 2 (0) 13 ciiϭKϪ 3␮ϩ2(lϪmϩ n/2)U ppϩ(2mϪn)Uii , tkk are the ezoelement are related by well-known equations. Then, us- components of the initial deformation tensor, K and ␮ are the ing the condition for continuity of the normal stress compo- bulk modulus and the shear modulus of the material, and l, nent at the sample–piezoelement interface, we obtain the n, and m are the Murnaghan constants. following expression for the voltage V(␻) of the electric Note that on the right-hand sides of the expressions for signal at the output of the piezodetector: (i) (i) (i) f k , hk , cii , and g , summation is not performed over the repeated indices. Equation ͑4͒ can be used to determine the strain in a solid provided that the temperature distribution ͑T͒ u3⌬ץ created by the exciting laser radiation is known. Here we ␧ l1 V͑␻͒ϭϪ ͑f͑3͒ϩh͑3͒͒ , ͑6͒ ͯ zץ shall assume that the appearance of internal stresses in an ͑ET͒ ͑ST͒ ͑T͒2 3 3 C ␧ ϩe zϭl object does not cause appreciable changes in its thermophysical parameters. A similar situation has been observed 6 experimentally in various ceramics. We shall also assume (ET) (ST) (T) that the surface of the sample is uniformly illuminated by where C , ␧ , and e are the piezoelectric character- laser radiation and the radiation is harmonically time- istics determined as in Ref. 13. modulated. Expression ͑6͒ can be used to determine the piezoelectric In addition to the temperature distribution, the boundary signal under fairly general conditions. However, the analysis conditions must also be defined to solve Eq. ͑4͒, these being is confined here to the case where the modulation frequency determined by the method of recording the varying deforma- of the exciting radiation is not too high, so that the condition tions in the object. Here we consider the case where these QlӶ1 is satisfied. We shall also assume that the sample is deformations are recorded by a piezoelement connected to fairly thick in thermal terms, i.e., ␴lӶ1. Then, using expres- the sample ͑see Fig. 1͒. To determine the varying deforma- sion ͑6͒, the signal from the piezodetector is obtained as 538 Tech. Phys. Lett. 24 (7), July 1998 K. L. Muratikov

͑T͒ 3/2 2 ͑3͒ ␧ ୐␳0 ␻ l1 ␥0͑1ϩA ͒͑1ϩ␤U33͒⌬Ts V͑␻͒ϭi 2 3/2 . ͑7͒ C͑ET͒␧͑ST͒ϩe͑T͒ 4 ͑1ϩA͑3͒͒ Kϩ ␮ ϩ͓t͑0͒ϩ͑1ϩA͑3͒͒͑2lU ϩ͑4mϩn͒U ͔͒ ͭ ͩ 3 ͪ 33 pp 33 ͮ

Expression ͑7͒ can be used to analyze some general laws recorded near the ends of cracks in titanium.3,6 These governing the behavior of the piezoelectric signal from ob- changes are usually also a few tens of percent. jects with internal stresses. Here the analysis is confined to To sum up, this theory qualitatively explains the main physical rather than geometric nonlinearities,9,12 i.e., we shall characteristics of the generation of mechanical vibrations by assume A(3)Ӷ1. According to the results presented in Ref. 7, laser radiation in internally stressed solids. However, before the coefficient ␤ is positive. Thus, the action of tensile quantitative agreement can be achieved between theory and stresses helps to intensify the thermoelastic coupling in the experiment, this model must be further developed to account sample while compressive stresses tend to reduce this cou- more comprehensively for the behavior observed in these pling. experimental investigations. Experiments to excite mechanical vibrations in loaded titanium rods using laser radiation were described in Ref. 5. In accordance with expression ͑7͒, an increase in the piezo- 1 electric signal was recorded when mechanical vibrations M. Kasai and T. Sawada, Photoacoustic and Photothermal Phenomena II, Springer Series in Optical Sciences Vol. 33 ͑Springer-Verlag, Berlin, were excited by laser radiation in zones of tensile stresses 1990͒,p.33. while a reduction in this signal was observed in zones of 2R. M. Burbelo, A. L. Gulyaev, L. I. Robur, M. K. Zhabitenko, B. A. compressive stresses. The influence of the internal stresses Atamanenko, and Ya. A. Kryl, J. Phys. ͑Paris͒, Colloq. 4, Colloq. 7, 311 ͑1994͒. on the piezoelectric signal can be estimated using data on the 3 7 H. Zhang, S. Gissinger, G. Weides, and U. Netzelmann, J. Phys. ͑Paris͒, coefficient ␤ for metals. Thus, for the conditions reported in Colloq. 4, Colloq. 7, 603 ͑1994͒. Ref. 5, expression ͑7͒ shows that the stress dependence of the 4 M. Qian, J. Acoust. 14͑2͒,97͑1995͒. thermoelastic coupling coefficient may change the piezoelec- 5 R. M. Burbelo and M. K. Zhabitenko, Progress in Natural Science ͑Taylor tric signal by 10%. and Francis, Washington, 1996͒, Supplement 6, p. 720–723. 6 K. L. Muratikov, A. L. Glazov, D. N. Rouz, D. E. Dumar, and G. Kh. This value is slightly lower than that obtained in Ref. 5. Kva, Pis’ma Zh. Tekh. Fiz. 23͑3͒,44͑1997͓͒Tech. Phys. Lett. 23, 188 However, it should be borne in mind that for most metals the ͑1997͔͒. Murnaghan constants are negative. Then, in accordance with 7 R. I. Garber and I. A. Gindin, Fiz. Tverd. Tela ͑Leningrad͒ 3, 176 ͑1961͒ expression ͑7͒, the dependence of the piezoelectric signal on ͓Sov. Phys. Solid State 3, 127 ͑1961͔͒. 8 T. Bateman, W. P. Mason, and H. J. McSkimin, J. Appl. Phys. 32͑5͒, 928 the stresses, caused by mechanical nonlinearities, will be ͑1961͒. similar to the dependence of the thermoelastic coupling co- 9 A. N. Guz’, Prikl. Mekh. 6͑2͒,3͑1970͒. 10 T. Tokuoka, and M. Saito, J. Acoust. Soc. Am. 45͑5͒, 1241 ͑1969͒. efficient. Under these conditions, the total change in the pi- 11 ezoelectric signal will be slightly greater than that caused A. I. Lur’e, Nonlinear Theory of Elasticity, North-Holland, Amsterdam ͑1990͓͒Russ. orig. Nauka, Moscow ͑1980͒, 512 pp.͔. only by the change in the thermoelastic coupling coefficient. 12 V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Unfortunately it is difficult to obtain a more detailed estimate Graylock Press, Rochester, N.Y. ͑1953͓͒Russ. orig. Gostekhizdat, Mos- cow ͑1948͒, 211 pp.͔. of the piezoelectric signal because no data on the Murnaghan 13 constant are available for titanium. Substantial changes in the W. Jackson and N. M. Amer, J. Appl. Phys. 51, 3343 ͑1980͒. piezoelectric signal as a result of internal stresses were also Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Anomalously high conductivity in a thin polyphthalidylidene biphenylene film V. A. Zakrevski , A. N. Ionov, and A. N. Lachinov A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg; Institute of Molecular and Crystal Physics, Russian Academy of Sciences, Ufa ͑Submitted April 6, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 89–94 ͑July 12, 1998͒ An anomalously high conductivity was observed for the first time in a thin film of polyphthalidylidene biphenylene inserted between two metal electrodes with no electric field applied but exposed to the action of a small uniaxial mechanical pressure. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01607-3͔

In recent years, various reports of high conductivity in The aim of the present study was specifically to obtain polymer films have appeared ͑see Ref. 1 and the literature new experimental data on the conditions for the formation of cited therein͒, and these unexpected results have attracted the a highly conducting state and to discuss possible reasons for attention of researchers. However, no generally accepted ex- the appearance of conducting channels in the film. Experi- planation has yet been provided for these observations espe- ments were first carried out to study the properties of thin cially because of the ͑in our opinion͒ inadequate treatment of polyphthalidylidene biphenylene films exposed to the action the possible consequences which may result from the differ- of uniaxial pressure in the absence of an external electric ent experimental conditions. Thus, in the introductory part of field. this study we shall merely discuss some of the results ob- The polymer sample was a PPB film ϳ1 ␮m thick. The tained so far. electrodes were made of tin and had a diameter of 15 mm. It was established in Ref. 2 that a thin (dХ12-␮m) film The PPB film was deposited from a cyclohexane solution 1͒ of poly͑3,3Љ-phthalidylidene-4,4Љ-biphenylylene͒͑PPB͒, directly onto the polished surface of one of the tin electrodes one of a promising class of polyphthalidylidene arylene by centrifuging. The film was then placed in a drying cabinet polymers, becomes highly conducting in a relatively low- for 60 min at 100 °C to remove the cyclohexanone solvent. intensity electric field (EϽ103 V/cm) if a small 5 The second tin electrode was lightly clamped to the polymer (PϽ10 Pa) uniaxial mechanical pressure is applied to the film. metal electrodes. It was shown in Ref. 3 that if the metal The formation of the conducting state in this sandwich electrodes were in the superconducting state, the conductiv- structure was recorded as follows. The electrodes were con- ity of this sandwich structure was so high that the instru- nected to a digital voltmeter with a fairly high input resis- ments recorded effectively zero resistance. This effect was tance (R Ͼ108 ⍀). In the insulating state, when the resis- also observed for oxidized polypropylene1,4 and polyimide v tance of the sandwich structure R (R Ͼ1014 ⍀) exceeded films.5 s s the input resistance of the voltmeter R , the voltmeter dis- Key parameters in these polymer film switching effects v play recorded a small fluctuating noise voltage, no greater are the electric field strength and the electric current at which than a few millivolts, caused by external induction in the the conductivity of the material changes abruptly. These pa- voltmeter input circuit. As the pressure was raised to the rameters can determine the mechanism responsible for the 4 highly conducting state in the polymer. For example, switch- threshold level (ϳ10 Pa), the structure changed from insu- ing of comparatively thick polyimide films (dϳ12 ␮m) lating to conducting, as a result of which Rs became lower sandwiched between low-melting electrodes is accompanied than Rv . The signal induced at the input to the voltmeter by metallic dendrite intergrowth through the polymer film in then disappeared and the voltmeter displayed zero voltage. an electric field close to breakdown (Eϳ106 V/cm) if there An attempt was then made to measure the resistance of is no special limitation on the breakdown current between the film. The sandwich structure in which the conducting the electrodes.6 In this case, the polymer will clearly be state was only produced by external mechanical pressure was shunted by the metal bridge. It was shown in Refs. 4 and 5 placed in a helium cryostat. After the conducting state had that polymer films 3–5 ␮m thick are switched to the highly been attained at room temperature and during cooling of the conducting state in a relatively low-intensity electric field helium cryostat to liquid nitrogen temperature, the metal (EϽ103 V/cm). When the breakdown current was limited to electrodes of the sandwich structure were shorted to elimi- 10–100 ␮A, no solid metallic dendrite formed, but the pres- nate induction of static electricity from external uncontrol- ence of electrode material in the form of isolated micropar- lable sources of electromagnetic radiation. At temperatures ticles in the polymer matrix could not be ruled out. In this between 4.2 and 300 K the resistance of this sandwich struc- case, the anomalously high conductivity could be attributed ture remained constant at ϳ0.04 ⍀. A current of 100 ␮A, to a conducting channel with a modified polymer whose set by the large load resistance, was selected to measure the structure differed from that of the initial polymer. temperature dependence of the resistance below 4.2 K and

FIG. 1. Resistance of an Sn–PPB–Sn structure versus temperature under an external uniaxial pressure of ϳ104 Pa.

the voltage drop was measured with the voltmeter at the between the electrodes, as is found in our case, where the sandwich structure. thickness of the film was less than 1 ␮m. Figure 1 gives the temperature dependence of the resis- Pressure must be applied to the electrodes to achieve tance of a tin–PPB–tin sandwich structure. It is known that reliable mechanical contact between the metal and the poly- tin becomes superconducting for TcХ3.7 K. Thus, at tem- mer ͑or the contact area must be increased͒. This condition peratures TϽ3.7 K the resistance in the electrodes can be must be satisfied for efficient transport of electrons from the neglected and all the measured resistance will be determined metal to the polymer. exclusively by the conducting polymer channel. It can be At present, there is no rigorous theoretical model which seen from the figure that for Tр3.7 K the resistance drops could explain the high conductivity in polyphthalidylidene sharply and for TϽ3.6 K is less than 0.001 ⍀, which is the biphenylene. A possible mechanism for electron transport resolution limit of this apparatus. could involve resonant ͑activation-free͒ tunneling between The highly conducting state in the sandwich structure is nearest localized states. These low-energy states form chains obtained under experimental conditions when there is no with linear dimensions of order 1 ␮m. However, in this ap- doubt that no electrode material is present in the polymer proach the conductivity in the chain may be very high but film. Thus, the highly conducting state is an intrinsic physi- still finite. cal property of the metal–PPB film–metal sandwich structure. The highly conducting state presupposes that there is a high concentration of free carriers with a fairly high mobil- 1͒In Ref. 2 this polymer is called polydiphenylenephthalide. The present ity. In this case, the carriers are injected into the polymer name complies with the international classification—poly͑3,3Љ- from the electrodes. phthalidylidene-4,4Љ-biphenylylene͒—and more accurately describes the Electrification of polymers on contact with metals polymer structure. ͑charge transport across the interface͒ is a well-known phenomenon studied by various authors.7–9 It is assumed that electrons entering the polymer are trapped by deep local 9 1 states situated near the Fermi level of the metal. The deep V. M. Arkhangorodski, A. N. Ionov, V. M. Tuchkevich, and I. S. levels are associated with molecular fragments ͑atomic Shlimak, JETP Lett. 51,67͑1990͒. 2 A. N. Lachinov, A. Yu. Zherebov, and V. M. Kornilov, JETP Lett. 52, 103 groups͒ with a high positive electron affinity, such as end ͑1990͒. groups, side radicals, or impurities. According to estimates 3 A. N. Ionov, A. N. Lachinov, M. Rivkin, and V. M. Tuchkevich, Solid made in Ref. 9, these local deep levels are separated from the State Commun. 82, 609 ͑1992͒. 4 vacuum level by approximately 4 eV. This value is similar to A. N. Ionov and V. M. Tuchkevich, Pis’ma Zh. Tekh. Fiz. 16͑16͒,90 ͑1990͓͒Sov. Tech. Phys. Lett. 16, 638 ͑1990͔͒. the work function of a polycrystalline metal whose surface is 5 A. M. El’yashevich, A. N. Ionov, M. M. Rivkin, and V. M. Tuchkevich, coated with a layer of adsorbate ͑more accurately, it is simi- Fiz. Tverd. Tela ͑St. Petersburg͒ 34, 3457 ͑1992͓͒Sov. Phys. Solid State lar to the lowest work function of the spotty surface of a 34, 1850 ͑1992͔͒. 6 metal from which contaminants have not been removed͒ and A. M. El’yashevich, A. N. Ionov, V. M. Tuchkevich et al., Pis’ma Zh. Tekh. Fiz. 23͑7͒,8͑1997͓͒Tech. Phys. Lett. 23, 538 ͑1997͔͒. as a result, conditions are created for the injection of elec- 7 D. K. Davis, J. Phys. D 2, 1533 ͑1969͒. trons into the polymer. Fairly high concentrations of centers 8 T. J. Fabish, H. M. Saltsburg, and M. L. Hair, J. Appl. Phys. 47, 930 ͑1976͒. with a high electron affinity are evidently only achieved in 9 small volumes. This is why the anomalously high conductiv- C. B. Duke and N. J. Fabish, Phys. Rev. Lett. 37, 1075 ͑1976͒. ity being discussed can only be observed with short distances Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Stochastically induced hysteresis in optical carrier generation Yu. V. Gudyma

Chernovtsy State University ͑Submitted March 18, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 1–4 ͑July 26, 1998͒ A study is made of a mechanism for the occurrence of a noise-induced kinetic transition in a thin amorphous semiconductor wafer. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01707-8͔

The appearance of new steady states in nonequilibrium or nonradiative͒. The form of the absorption coefficient ͑1͒ open systems in response to external multiplicative noise is with allowance for Eq. ͑2͒ ensures positive feedback in one of the most dramatic effects in self-organizing systems.1 Eq. ͑3͒. However, the number of real physical systems in which this It is convenient to introduce the dimensionless variables effect occurs is fairly limited and it is therefore relevant to Ϫ1 Ϫ1 search for such objects ͑see Ref. 2 for one of the most recent ␩ϭbn, ␪ϭb at, ⌬ϭ͑ប␻ϪE0͒/E0 , examples͒. Here we propose a model for the occurrence of a Ϫ1 2 2 Ϫ1 noise-induced kinetic transition in a thin amorphous semi- ␤ϭa b ␯BE0J͑ប␻͒ . conductor wafer. Experimental and theoretical investigations show that in Then, in accordance with Eqs. ͑3͒ and ͑1͒, we have most amorphous semiconductors the fundamental absorption 3 d edge may be described by a simple power law. For some ␩ 2 2 ϭ␤͑⌬ϩ␩͒ Ϫ␩ . ͑4͒ semiconductors in the energy range above the exponential d␪ tail ͑GeTe, As2Te3,As2Se3,orAs2S3) or for amorphous silicon which has no exponential absorption edge, the spectral Equation ͑4͒ has two steady-state solutions, unstable and dependence of the absorption coefficient is described by the stable, which correspond to a soft regime of bistability in the formula nonequilibrium carrier distribution. Note that in the model of direct allowed transitions the corresponding generation- 2 ␣͑␻͒ϭB͑ប␻ϪE0͒ /ប␻, ͑1͒ recombination equation may have three roots near the absorption threshold. A similar situation was considered for where the values of the parameters B and E0 are determined crystalline semiconductors in Ref. 5. The indeterminate loss experimentally. This expression is similar to that for the ab- of coherence of the light is described by the process ␤(t) sorption coefficient for indirect transitions in crystalline ϭ␤ϩ␴␰(t), where the external noise ␰(t) is characterized 4 semiconductors. The renormalized value of the band gap by very fast fluctuations compared with the characteristic evolution time of the system ␶ϭb/a ͑quasi-white noise͒. * Eg ϭEg͑1Ϫbn͒ ͑2͒ In terms of generalized functions, Gaussian white noise is the derivative of a Wiener process so that Eq. ͑4͒ is trans- implies that the light beam is of sufficiently high intensity for formed to give the Stratonovich stochastic differential equa- the lower states of the energy troughs to be filled more rap- tion which may be made to correspond to the Fokker–Planck idly than they decay. equation which determines the evolution of the transition The influence of interelectron interaction on the electron probability p(␩,␪͉␩Ј,␪Ј) energy spectrum in a nondegenerate semiconductor has been studied on many occasions. The simplest phenomenological ץ p͑␩,␪͉␩Ј͒ץ form of this dependence has been selected here. ϭϪ ͓␤͑⌬ϩ␩͒2Ϫ␩2 ␩ץ ␪ץ It is also implicit that the light intensity is so high that the concentrations of photogenerated electrons and holes ϩ␴2͑⌬ϩ␩͒3͔p͑␩,␪͉␩Ј͒ considerably exceed their equilibrium values ͑which implies 2 2 2 ץ ␤ nϷp). If the sample thickness is small compared with ␴ ϩ ͕⌬ϩ␩͖4p͑␩,␪͉␩Ј͒. ͑5͒ ␩2ץ Bប␻)Ϫ1, we can write the equation for the generation- 2) recombination kinetics in the form The phenomena described by Eq. ͑5͒ take place on two dn time scales: the fast time scale is associated with inverse ϭ␯␣JϪan2, ͑3͒ dt relaxation to the local minimum after the perturbation, while the slow scale is associated with a transition from the meta- where ␯ is the quantum yield, J is the photon flux density, stable minimum to the global minimum. The steady-state and a is the coefficient of interband recombination ͑radiative distribution in the stationary process has the form

2 Ϫ1 ps͑␩͒ϭN͕␤͑⌬ϩ␩͒ ͖ which has three solutions ͑two stable and one unstable͒. For ␴ 0 there are only two of these solutions and they are 2 ␩ → 2 2 Ϫ4 exactly the same as the steady-state solutions of the determi- ϫexp Ϫ 2 ͓␤͕⌬ϩU͖ ϪU ͔͓⌬ϩU͔ dU . ͩ ␴ ͵0 ͪ nate equation ͑4͒. Even unusually fast completely random ﬂuctuations of the control parameters substantially change ͑6͒ the macroscopic behavior of the nonlinear dynamic system The constant N is determined from the normalization ͑3͒: they give rise to additional steady states. Under the ac- condition. The solution ͑6͒ may be assigned a potential form tion of fast external noise, the system is converted from a since its maxima correspond to stable steady states and its soft to a hard ͑hysteresis͒ regime with a multivalued distri- minima correspond to unstable ones. We shall use the bution of optically generated carriers in the thin semicon- ‘‘probability’’ potential, after writing the steady-state prob- ducting wafer. Estimates show that a laser radiation intensity ability density in the form of 100 W/cm is required for the experimental observation of light absorption hysteresis in these materials for wafers of ϭ Ϫ 2 ps N exp͓ 2V͑␩͒/␴ ͔, ͑7͒ the order of 100–1000 Å thick and in this case, the paramet- where ric ﬂuctuations are less than 0.01–0.1 of the value.

␩ V͑␩͒ϭϪ ͓␤͕⌬ϩU͖2ϪU2͔͓⌬ϩU͔Ϫ4dU ͫ͵0 1 W. Horsthemke and R. Lefever, Noise-Induced Transitions ͑Springer- Verlag, Berlin, 1984; Mir, Moscow 1987, 400 pp.͒. 2 2 ␴ 2 R. E. Kunz and E. Scholl, Z. Phys. B 99, 185 ͑1996͒. Ϫ ln͓͑⌬ϩ␩͒ ͔ . ͑8͒ 3 2 ͬ N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials ͑Clarendon Press, Oxford, 1971; Mir, Moscow, 1974, 472 pp.͒. 4 A. L. Semenov, Zh. Eksp. Teor. Fiz. 111, 2147 ͑1997͓͒JETP 84, 1171 The extrema of the steady-state probability density are ͑1997͔͒. easily obtained from the equation 5 V. A. Kochelap, L. Yu. Mel’nikov, and V. N. Sokolov, Fiz. Tekh. Polu- provodn. 16, 1167 ͑1982͓͒Sov. Phys. Semicond. 16, 746 ͑1982͔͒. ͑␤Ϫ␴2⌬͒⌬2ϩ͑2␤Ϫ3␴2⌬͒⌬␩ϩ͑␤Ϫ1Ϫ3␴2⌬͒␩2Ϫ␴2␩3ϭ0, ͑9͒ Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

A new method of reconstructing spectra A. M. Egiazaryan and Kh. V. Kotandzhyan

Institute of Applied Physics Problems, Armenian Academy of Sciences ͑Submitted July 28, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 5–9 ͑July 26, 1998͒ A theoretical method is proposed for accurate reconstruction of the spectrum using bounded sets of discrete values of the spectrum intensities. The method is based on a well known measurement theorem from optics. This method was used to solve the corresponding integral equation to eliminate instrumental distortions and to accurately reconstruct the spectra using the appropriate discrete values. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01807-2͔

INTRODUCTION of the train amplitude is described by the function f (t). The ¨ Fourier transform F(␻) will then be the wave amplitude in Methods of obtaining x-ray spectra, Mossbauer spectra, 2 and also theoretical methods of analyzing experimental data the frequency space and J(␻)ϭ͉E(␻)͉ its spectral inten- have been dealt with fairly extensively in the literature.1–4 sity. Since spectral lines reflect the distribution of electrons over It then follows that J(␻) is a function of bounded spec- the energy levels of crystal atoms and molecules, the highest tral width and hence can be reconstructed from sets of its possible resolution is required. In analyses of experimental discrete values. We used this idea for a complete reconstruc- data, curves are approximated by summing power series, tion of Mo¨ssbauer spectra recorded at our Institute. Figure 1 with the result that the experimental points are smoothed. shows the experimental curve and the theoretical curve re- However, certain difficult problems must be overcome to constructed using formula ͑1͒. In this case the shortest inter- improve the resolution. The aim of the present paper is to val between the experimental points was taken as 1/2B. propose a theoretical method of accurately reconstructing the In the usual methods of treating experimental data ͑such spectrum from bounded sets of discrete values of the spec- as the power series method͒, the experimental curves are trum intensities. The method is based on the measurement 5 approximated by certain analytic functions. Depending on theorem well known in optics. We propose a formula for the accuracy of the approximation, only some of the experi- complete reconstruction of the spectra assuming that the mental points lie on this analytic curve and these points are wave trains have a finite duration. This formula was applied smoothed. When the spectrum is reconstructed by the to solve the corresponding integral equation to eliminate in- method described above, all the experimental points lie ex- strumental distortions when these are described by a dispersion distribution. We obtained a formula for accurate recon- actly on the curve g(x), as can be seen from Fig. 1. struction of the spectra and present an example of suitably processed experimental curves.

THEORETICAL BASIS OF THE METHOD It is known that for a specific class of functions, known as functions of bounded spectral width, the functions can be completely reconstructed from sets of discrete values if the constraint is imposed that the interval between these discrete values does not exceed a certain value. By functions g(x)of bounded spectral width, we have in mind functions whose Fourier transforms are only nonzero in a finite region of space 2B. This statement is known as the Kotel’nikov theorem, according to which ϱ k sin ␲͓2BxϪk͔ g͑x͒ϭ ͚ g , ͑1͒ kϭϪϱ ͩ2Bͪ ␲͓2BxϪk͔ i.e., the function g(x) can be completely reconstructed from its corresponding discrete values. Regardless of their physical nature, the wave paths have FIG. 1. Experimental curve of the Mo¨ssbauer spectrum ͑circles͒ and the a finite duration 2B. Let us assume that the time dependence corresponding theoretical reconstruction ͑triangles͒.

To eliminate instrumental distortions and reconstruct the 1 1 1 Ϫ ϽPϽ accurate form of the spectrum I(␻), we need to solve the 2 2 integral equation rect Pϭ 1 1 0 PϾ , PϽϪ , ϱ dx 2 2 J͑␻͒ϭ I͑x͒ 2 , ͑2͒ ͵Ϫϱ ␻Ϫx ␲⍀ 1ϩ ͫ ͩ ⍀ ͪ ͬ the Fourier transform of the dispersion distribution has no zeros.6 From formula ͑2͒ we obtain where ⍀ is the half-width of the instrumental distortion ϱ curve. ϱ ϱ Bearing in mind formula ͑1͒, we apply the convolution I͑w͒ϭ ͚ ͵ dq͵ dP rect P exp͓i2␲qw theorem for Fourier transformation to Eq. ͑2͒. Since we have kϭϪϱ Ϫϱ Ϫϱ k sin ␲͓2B␻Ϫk͔ ϱ ϩ2␲⍀͉q͉ϩ12␲Pk͔J ␦͑2BPϩq͒, ͑4͒ ϭ rect P exp͓Ϫ12␲P͑2B␻Ϫk͔͒dp, ͩ2Bͪ ␲͓2B␻Ϫk͔ ͵Ϫϱ ͑3͒ where ␦(x) is the Dirac delta function. where After various transformations, formula ͑4͒ becomes

ϱ k ͑8␲BwϪ4␲kϩ8␲⍀B͒exp 2␲⍀B sin͑2␲BwϪ␲k͒Ϫ8␲⍀B I͑w͒ϭ ͚ J 2 2 . ͑5͒ kϭϪϱ ͩ2Bͪͫ ͑4␲BwϪ2␲k͒ ϩ͑4␲⍀B͒ ͬ

It is easily observed that when ⍀ϭ0, i.e., when the de- The authors would like to thank Academician A. R. vice does not distort the spectrum, the expansion ͑5͒ is iden- Mkrtchyan for valuable discussions. tical to expansion ͑1͒. Thus, expansion ͑5͒ can be used to reconstruct the exact form of the spectrum I(w) from the experimentally determined corresponding discrete values. With the proposed method, we can accurately describe 1 M. A. Blokhin, Physics of X-Rays ͓in Russian͔, Gostekhizdat, Moscow the experimental curves by analytic expressions and thereby ͑1957͒. 2 M. A. Blokhin, Methods of X-ray Spectroscopic Research, Pergamon avoid smoothing of the experimental points. Press, Oxford ͑1965͓͒Russ. orig. Fizmatgiz, Moscow ͑1959͔͒. When the proposed method is used in spectroscopy, the 3 Z. G. Pinsker, X-Ray Crystal Optics ͓in Russian͔, Nauka, Moscow ͑1982͒. problem of reducing the goniometer pitch does not arise, 4 R. G. Gabrielyan and Kh. V. Kotandjian, Phys. Status Solidi B 151, 655 since the reconstruction is equivalent to using goniometers ͑1989͒. 5 R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography ͑Aca- with inﬁnitely small pitch. demic Press, New York, 1971; Mir, Moscow, 1973͒. The proposed method is saves a lot of time compared 6 A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and with the conventional experimental methods of recording Series, Vols. 1–3 ͑Gordon and Breach, New York, 1986, 1986, 1989͒ Russ. original, Vols, 1–3, Nauka, Moscow, 1981, 1983, 1986 . spectra and the distortions introduced by the apparatus can ͓ ͔ be eliminated to produce accurate spectra. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Nonlinear stage of instability evolution in a monostable active medium A. A. Pukhov

Joint Institute of High Temperatures, Scientiﬁc-Research Center of Applied Problems in Electrodynamics, Russian Academy of Sciences, Moscow ͑Submitted January 8, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 10–15 ͑July 26, 1998͒ A theoretical analysis is made of the nonlinear stage of instability evolution in a monostable active medium described by a ‘‘reaction–diffusion’’ type equation. The boundary of that region of the medium in which instability develops propagates at constant velocity. Group-theoretical analysis of the problem yields an analytic expression for the propagation velocity of this boundary. The results may be important for the analysis of instability evolution in a wide range of active media. © 1998 American Institute of Physics. ͓S1063-7850͑98͒01907-7͔

A wide range of highly nonequilibrium physical systems ␪(x,t)Ͼ␪2 . However, for a wide range of monostable active ͑active media͒ are described by a nonlinear ‘‘reaction– media numerical simulation has shown that despite the time- diffusion’’ equation dependent behavior of the temperature ␪(x,t) in the hot region, the front propagates at a constant velocity u, as shown ␪ץ ϭ⌬␪ϩ f ͑␪͒, ͑1͒ in Fig. 2 ͑Refs. 4–6͒. Thus, the distribution ␪(x,t) in the t cold tail of the front has the form ␪(x,t)ϭ␪(xϪut) and in aץ 2z, and ␪ may be the tem- system moving together with the wave, asymptotically (␪ץ/2ץ2yϩץ/2ץ2xϩץ/2ץwhere ⌬ϭ perature of the medium, the reagent concentration, the elec- Х␪1) satisfies the equation tric field, and so on, depending on the nature of the dissipa- d2␪ d␪ tive process ͑see Ref. 1 and the literature cited therein͒.To 2 ϩu ϩf͑␪͒ϭ0. ͑2͒ be specific, let us take ␪ to be the temperature. The nonlinear dx dx stage of instability evolution in these systems is completely In fairly general form the nonlinear source function f (␪) determined by the form of the source function f (␪). For may be given as5 example, for bistable systems ͑Fig. 1, curve 1͒ the transition between the two stable states of the medium ␪(x,y,z,t) f ͑␪͒ϭa␪m͑␪nϪb͒, ͑3͒ ϭ␪ and ␪(x,y,z,t)ϭ␪ is accomplished by the propagation 1 3 n of a switching autowave. In the one-dimensional case, the where m, n, a, b are arbitrary positive values and bϭ␪2 . asymptotic behavior of this wave ␪(x,t)ϭ␪(xϪut) is char- The absence of a self-similar solution means that the velocity acterized by its constant propagation velocity u ͑Ref. 1͒. u cannot be determined as an eigenvalue of Eq. ͑2͒ with the Methods for approximate calculation of the wave velocity boundary conditions d␪/dxϭ0atxϭϮϱ using standard and its dependence on the parameters of the problem have methods from the theory of finite-amplitude autowave 1–3 7,8 now been studied in fairly great detail.2,3 propagation. in this case, group-theoretical concepts can In many situations, however, the system loses its bista- be applied to find the value of u. We assume that the expo- bility ͑such as when the dissipation depends strongly on tem- nents m and n are fixed and that a and b are control param- perature, the differential conductivity increases abruptly, and eters of the problem. Equations ͑2͒ and ͑3͒ are then invariant so on1–3͒. The qualitative form of the dependence f (␪) for with respect to the transformation group of the variables this case is shown in Fig. 1 ͑curve 2͒. When the temperature xϭLϪpxЈ, ␪ϭLq␪Ј, uϭL puЈ, ␪ exceeds the threshold ␪2 in a certain region of the medium, the system undergoes unbounded self-heating. This circum- aϭL2pϩ͑1ϪmϪn͒qaЈ, bϭLnqbЈ, ͑4͒ stance is responsible for the particular instability evolution in a monostable medium. The nonlinear wavefront ␪(x,y,z,t) which comprise a group of expansions with the scale factor separating the ‘‘hot’’ and ‘‘cold’’ regions is constructed as L. The exponents of the scale factors of the expansion are follows. The temperature ␪ in the cold region is equal to ␪1 , determined by the condition of invariance with respect to the whereas that in the hot region increases without bound. Thus, transformation ͑4͒, given by ͑2͒ and ͑3͒, so that the new the evolution of instability in this case is the result of two ͑primed͒ variables satisfy the same equations ͑2͒ and ͑3͒. parallel processes: expulsion of the cold region by the hot This has the result that the expansion group ͑4͒ contains free and a rapid increase in the temperature of the hot region. parameters L, p, and q which may have arbitrary values. We shall first consider the one-dimensional case (D It is clear from physical reasoning that the velocity u in ϭ1). Equation ͑1͒ has no self-similar solution of the travel- Eqs. ͑2͒ and ͑3͒ is a function only of a and b: uϭF(a,b). ing autowave type ␪(x,t)ϭ␪(xϪut) because of the un- This relation should be invariant with respect to the group bounded increase in the temperature of the hot region transformation ͑4͒, i.e., uЈϭF(aЈ,bЈ) ͑Ref. 9͒. We shall

cases, the self-similarity is approximate: ␪(r,t)Х␪(rϪut) and for rӷ1 we obtain for the wavefront velocity dr/dt ϭuϪ(DϪ1)/r ͑Refs. 1 and 2͒. The curvature of the wavefront slows the wave which yields a logarithmic correction to the uniform propagation of the front rХutϪuϪ1(D Ϫ1)ln t. Thus, a group-theoretical analysis of Eqs. ͑1͒ and ͑3͒ can be used to obtain the functional dependence of u on the control parameters for any dimensions of the problem. To conclude, we note that these characteristics of the evolution of instability in a monostable active medium show similarities to so-called ‘‘blowup regimes.’’ These regimes are observed in problems involving nonlinear active media where the speciﬁc heat and heat release depend strongly on temperature,11 which are described by

␪ץ ␪k ϭٌ͑␪lٌ␪͒ϩ f ͑␪͒. ͑7͒ tץ

By applying the procedure described above to Eqs. ͑3͒ and FIG. 1. Characteristic dependence of the nonlinear source function f (␪)on ͑7͒, we obtain for the wavefront velocity ␪in the bistable ͑curve 1͒ and monostable ͑curve 2͒ cases.

u ϰ a1/2b͑mϩnϩlϪ2kϪ1͒/2n. ͑8͒ seek a solution in the form u ϰ arbs ͑Ref. 10͒, where the exponents r and s are to be determined. We then have However, when using this result, one should bear in mind u/arbsϭL p͑2rϪ1͒ϩq[nsϩ͑1ϪmϪn͒r]uЈ/aЈrbЈs, ͑5͒ that the ‘‘blowup regime’’ may exhibit an explosive buildup of instability when the temperature of the medium goes to and from the condition of invariance u/arbsϭu /a rb s we Ј Ј Ј infinity within a finite time (mϩnϪkϪ1Ͼ0) ͑Ref. 11͒. obtain p(2rϪ1)ϩq͓nsϩ(1ϪmϪn)r͔ϭ0. On account of the arbitrariness of p and q in the transformation ͑4͒,we The author is grateful to N. A. Buznikov for useful dis- obtain single-valued expressions for the exponents: rϭ1/2, cussions of the results. sϭ(mϩnϪ1)/2n. Thus, for the velocity u we have This work was supported by the program ‘‘Topical Di- rections in the Physics of Condensed Media’’ Project No. u ϰ a1/2b͑mϩnϪ1͒/2n. ͑6͒ ͑ 96083͒ and by the Russian Fund for Fundamental Research The coefficient of proportionality of the order of unity in Eq. ͑Project No. 96-02-18949͒. ͑6͒ cannot be obtained using group concepts. Numerical calculations4,5 or additional reasoning6 are required to determine this. In the symmetric two-dimensional (Dϭ2, rϭ(x2 2 1/2 2 2 2 1/2 1 ϩy ) ) and three-dimensional (Dϭ3, rϭ(x ϩy ϩz ) ) V. A. Vasil’v, Yu. M. Romanovski, and V. G. Yakhno, Autowave Processes ͓in Russian͔, Nauka, Moscow ͑1987͒, 240 pp. 2 A. Yu. Loskutov and A. S. Mikhalov, Foundations of Synergetics, 2nd ed., Springer, New York ͑1994–1996͓͒Russ. orig., Nauka, Moscow ͑1990͒, 272 pp.͔. 3 A. Vl. Gurevich and R. G. Mints, Usp. Fiz. Nauk 142͑1͒,61͑1984͓͒Sov. Phys. Usp. 27,19͑1984͔͒. 4 Yu. M. L’vovski, Pis’ma Zh. Tekh. Fiz. 15͑15͒,39͑1989͓͒Sov. Tech. Phys. Lett. 15, 596 ͑1989͔͒. 5 S. V. Petrovski, Zh. Tekh. Fiz. 64͑8͒,1͑1994͓͒Tech. Phys. 39, 747 ͑1994͔͒. 6 A. A. Pukhov, Supercond. Sci. Technol. 10, 547 ͑1997͒. 7 N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics ͑Reidel, Dordrecht, 1985͓͒Russ. original, Nauka, Moscow, 1983, 280 pp.͔. 8 G. W. Bluman and S. Kumei, Symmetries and Differential Equations ͑Springer-Verlag, New York, 1989͒. 9 L. Dresner, Similarity Solutions of Nonlinear Partial Differential Equa- tions ͑Pitman, London, 1983͒. 10 L. Dresner, IEEE Trans. Magn. 21, 392 ͑1985͒. 11 A. A. Samarski, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhalov, Blow-up in Quasilinear Parabolic Equations, W. de Gruyter, FIG. 2. Schematic showing the evolution of the wavefront profile in a New York ͑1995͓͒Russ. orig., Nauka, Moscow ͑1987͒, 477 pp.͔. monostable active medium.4–6 The curves are separated by different time intervals. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Inﬂuence of the speciﬁc input energy on the characteristics of a nuclear-pumped Ar/Xe laser M. V. Bokhovko, A. P. Budnik, I. V. Dobrovol’skaya, V. N. Kononov, and O. E. Kononov

Russian State Science Center—Physics and Power Institute, Obninsk ͑Submitted January 14, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 16–20 ͑July 26, 1998͒ Results are presented of experiments and model calculations on the pumping of a 1.73 ␮m Ar/Xe laser with 235U fission fragments over a wide range of specific powers. An analysis is made of the mechanism for quenching of the lasing at high specific input energies. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02007-2͔

It has been suggested that the He/Ar/Xe mixture for zones of the reactor. The peak flux density of the thermal which the best power characteristics ͑efficiency, output neutrons on the surface of the cell was between 2.4ϫ1016 power͒ have been achieved so far, should be used as the and 2ϫ1017 cmϪ2 sϪ1, the maximum thermal neutron flu- active medium in a power prototype of a large laser facility ence was 5ϫ1013 cmϪ2, and the specific pump power for the pumped by pulses from a nuclear reactor.1 In this case, the fission fragments was 3 kW cmϪ3. The maximum laser ra- specific pump energy of the uranium-235 fission fragments diation energy recorded experimentally was 0.49 J and cor- in the laser unit will be around 1 kJ lϪ1 and the postulated responded to a specific output energy of 0.9 J lϪ1 and a spe- laser output power may be a few tens of kilojoules, depend- cific power of 5.5 kW lϪ1. ing on the maximum specific output energy of the active The kinetic processes in an Ar/Xe laser pumped by fis- medium. The specific output energy of an Ar/Xe laser sion fragments were modeled mathematically to identify the achieved so far is ϳ1JlϪ1and is determined by the quench- mechanism for quenching of the lasing. The kinetic model ing of the lasing as the specific input energy increases. This was constructed using data given in Refs. 9 and 10 and took characteristic of a laser utilizing the 5d –6p transitions of the into account 46 components involved in 389 reactions and Xe atom has been investigated experimentally and theoreti- radiative transitions. The modeling assumed that the electron cally by various authors.2–4 However, the energy overload- energy distribution function deviated from Maxwellian, the ing mechanism of this laser and possible methods of sur- change in the temperature of the medium was calculated dur- mounting this problem have not yet been fully clarified.5,6 ing the pump pulse, and the temperature dependence of the Here we report the first investigation of the energy char- reaction rate constants and the stimulated emission cross sec- acteristics of a cw Ar/Xe laser carried out in order to study tion was taken into account. The pump pulse power, and the the mechanisms for quenching the lasing over a wide range parameters of the cavity and the medium were defined ac- of specific nuclear pump powers ͑between 0.3 and cording to the experimental conditions. 3kWcmϪ3͒. The kinetic processes in an Ar/Xe laser are Figure 1 gives oscilloscope traces of the pump and 1.73 modeled numerically to interpret the experimental results. ␮m laser radiation pulses obtained in experiments with three The experiments were carried out using the BARS-6 specific input energies, and also shows the results of the two-region pulsed reactor, which can deliver a thermal neu- mathematical modeling. For an input energy of 0.1 J cmϪ3 tron flux density of up to 2ϫ1017 cmϪ2 sϪ1 on the surface of energy overloading of the Ar/Xe laser shows up as a limita- the laser cell with a neutron pulse duration of 200 ␮s. The tion of the experimentally observed output power at around 3 inner surface of the laser cell was coated with a thin layer of kW when the input energy reaches ϳ20 mJ cmϪ3. In this uranium-235 with a total weight of 1 g. The 550 cm3 laser case, the laser efficiency in terms of deposited energy is cell had antireflection-coated optical windows and was filled 1.5%. The output power and efficiency calculated using the with a 200:1 Ar/Xe mixture at a pressure of 380 Torr. The kinetic model are twice the experimental values and no ap- cavity was formed by two dielectric mirrors. The 1.73 ␮m preciable energy overloading occurs. An increase in the spe- laser radiation was directed into an IMO-2N calorimeter lo- cific input energy to 0.17 J cmϪ3 substantially shortens the cated within the reactor confinement and some was fed into laser pulse ͑from 300 to 80 ␮s͒, approximately halves the the experimental hall where it was recorded using germa- peak power, and reduces the efficiency to 5ϫ10Ϫ2%. The nium photodetectors and a digital storage oscilloscope.7 The calculated results also reveal energy overloading in the form thermal neutron pulse was recorded using vacuum fission of quenching of the lasing and its duration shows good chambers8 and the fluence was measured by the activation of agreement with that observed experimentally. No power gold foils. These data were used to calculate the specific limitation is observed in this case and the efficiency is re- input energy and the pump power for the uranium-235 fis- duced approximately threefold, to 0.8%. When the specific sion fragments. The experiments were carried out with the input energy is increased further to 0.7 J cmϪ3, this trend is laser cell in three different positions relative to the active still observed. Thus, in broad terms, the experimentally ob-

pump power quenching of the lasing is caused by collisional processes involving electrons and not by an increase in the temperature of the gaseous medium. In this case, the rate of filling of Xe atoms in the upper active level drops sharply because of an increase in the electron concentration and especially their average energy, since the highly excited Xe atoms, formed as a result of dissociative recombination of ArXe ions, are predominantly ionized by electrons rather than quenched by Ar atoms accompanied by filling of the upper active level. This ultimately results in quenching of the lasing. In contrast to the proposed mechanism, the quenching of the lasing at low pump powers is attributed, as in Ref. 10, to simple mixing of the laser levels by electrons. To sum up, these results show that the maximum specific output power of a fission-fragment-pumped Ar/Xe laser may be enhanced by incorporating molecular gas additives to reduce the electron temperature and possibly also electronega- tive gases to reduce their concentration. However, a reduction in the initial temperature of the gas mixture proposed in Ref. 11 is not effective. This work was supported by the Russian Fund for Fun- damental Research, Grant No. 96-02-16922.

1 A. V. Gulevich, P. P. D’yachenko, A. V. Zrodnikov et al., At. Energy 80, 361 ͑1996͒. 2 A. M. Voinov, L. E. Dovbysh, V. N. Krivonosov et al., Dokl. Akad. Nauk SSSR 245,80͑1979͓͒Sov. Phys. Dokl. 24, 189 ͑1979͔͒. 3 W. J. Alford and G. N. Hays, J. Appl. Phys. 65, 3760 ͑1989͒. 4 W. J. Alford, G. N. Hays, M. Ohwa, and M. J. Kushner, J. Appl. Phys. 69, 1843 ͑1991͒. 5 O. V. Sereda, V. F. Tarasenko, A. V. Fedenev, and S. I. Yakovlenko, ´ FIG. 1. Oscilloscope traces of pump ͑a, W cmϪ3͒ and laser radiation pulses Kvantovaya Elektron. ͑Moscow͒ 20, 535 ͑1993͓͒Quantum Electron. 23, 459 ͑1993͔͒. ͑b, W͒ and calculated results ͑c, W͒ for three speciﬁc input energies: 1— 6 0.1 J cmϪ3, 2—0.17 J cmϪ3, and 3—0.7 J cmϪ3. A. V. Karelin, A. A. Sinyavski, and S. I. Yakovlenko, Kvant. Elektron. ͑Moscow͒ 24, 387 ͑1997͒. 7 V. I. Regushevski, O. E. Kononov, M. V. Bokhovko, and V. N. Kononov, Preprint No. 2478 ͓in Russian͔, Physics and Power Institute, Obninsk served energy overloading of a nuclear-pumped Ar/Xe laser ͑1995͒. 8 utilizing the 1.73 ␮m Xe-atom transition is fairly well repro- V. I. Regushevski, M. V. Bokhovko, V. N. Kononov et al., Preprint No. duced by the kinetic model used for the calculations. The 2531 ͓in Russian͔, Physics and Power Institute, Obninsk ͑1996͒. 9 M. Ohwa, T. J. Moratz, and M. J. Kushner, J. Appl. Phys. 66, 5131 discrepancy between the experimental and calculated thresh- ͑1989͒. old and peak power may arise from the incomplete agree- 10 J. W. Shon, M. G. Kushner, G. A. Hebner, and G. N. Hays, J. Appl. Phys. 73, 2686 ͑1993͒. ment between the real cavity parameters and those used for 11 the calculations. G. A. Hebner, IEEE J. Quantum Electron. 31, 1262 ͑1995͒. An analysis of the kinetic processes showed that at high Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Use of a microconductor with natural ferromagnetic resonance for radio-absorbing materials S. A. Baranov

T. G. Shevchenko Dnestr State University, Tiraspol ͑Submitted February 2, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 21–23 ͑July 26, 1998͒ Thin radio-absorbing screens with an attenuation of 30 dB at frequencies of 8–10 GHz were fabricated using a composite with microconductor sections. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02107-7͔

The existence of natural ferromagnetic resonance in mm͒ absorption resonances were observed in the composite amorphous microconductors in the frequency range as a result of the dipole length l being comparable to ␭eff/2, ϳ1–10GHz ͑Refs. 1–7͒ has opened up the possibility of where ␭eff is the effective wavelength of the absorbed field in developing broad-band radio-absorbing materials. The the composite ͑i.e., geometric resonance͒. present work treats the radio-absorbing properties of planar, A disadvantage of amorphous materials based on metal thin ͑no more than 2 mm thick͒ samples made of rubber alloys compared with ferrites is their high electrical conduc- ͑KLTGS grade͒ filled with sections of microconductor ͑mi- tivity. This prevents the dipole concentration in the compos- crowire dipoles͒. The radio-absorbing properties were inves- ite from being increased substantially. The samples investi- tigated in the range 8–10 GHz. Note that the best results gated contained no more than 5–8 g of microwire per 100 g were obtained for dipoles 1–3 mm long with a core diameter rubber so that the absorption can only be of dipole nature. 2rcϳ1–3␮m. It has also been established experimentally that strong The curve giving the frequency dependence of the at- inductive coupling exists between the dipoles,4 which can be tenuation is the same as the magnetic permeability curve of estimated by introducing the inductive impedance,9 the natural ferromagnetic resonance, except that its half- 2l width is greater ͑see Fig. 1͒. Measurements of the attenuation Yϳ␮␮0l ln , ͑1͒ factor were made using the method described in Ref. 8 and ͫ rc ͬ the magnetic permeabilities were calculated by a method and its ratio to the resistance R, similar to that used in Ref. 6. The maximum attenuation 2 2 exceeded 30 dB ͑a sheet of rubber without dipoles exhibits Y/Rϳrc /␦ , ͑2͒ very low attenuation, no greater than 2 dB͒. where ␦ is the skin layer depth. The investigations were made using FeBSiMnC alloys A reduction of rc compared with ␦ may increase the having natural ferromagnetic resonance in the range 8–10 absorption coefficient, since it creates the possibility of in- Hz with a half-width of ϳ1 – 1.5 GHz and a magnetic per- creasing the dipole concentration. Note that the width of the 2 meability ␮effϳ10 . In a certain range of dipole lengths ͑1–3 composite curve may be increased by using the dispersion of the magnetic susceptibility, which can extend the geometric resonance. To sum up, the following conclusions can be drawn. Microwire has been used to fabricate composites in the form of planar radio-absorbing screens less than 2 mm thick which can operate efficiently in the range 8–11 GHz of practical interest. The use of thinner microwire can improve the characteristics of the screens by increasing the density of absorbing dipoles without increasing the effective conductivity of the sample. However, the fabrication of microwire with cores thinner than 1 ␮m presents technological difficulties.

1 S. A. Baranov, V. I. Berzhanski, S. K. Zotov, V. L. Kokoz, V. S. Larin, and A. V. Torkunov, Fiz. Met. Metalloved. 67͑1͒,73͑1989͒. 2 S. A. Baranov, S. K. Zotov, V. S. Larin, and A. V. Torkunov, Fiz. Met. FIG. 1. Dispersion of the magnetic permeability of an amorphous microwire Metalloved. No. 12, 172 ͑1991͒. ͑left-hand ordinate͒ and attenuation factor for this amorphous microwire as 3 S. A. Baranov, S. K. Zotov, V. S. Larin, and A. V. Torkunov, in Proceed- a function of the ﬁeld frequency ͑right-hand ordinate͒. The relative measure- ings of Young Scientists’ Conference, Faculty of Physics, Lvov University, ment error was less than 10% for the frequency and less than 20% for the L’vov, 1991 ͓in Russian͔, p. 5–7, Deposited Paper No. 763–UK91, magnetic permeability. The spread of the attenuation factor measurements 30.04.91. was Ϯ5 dB. 4 S. A. Baranov, Vestn. Pridnestrov. Univ. 1͑2͒, 126 ͑1994͒.

5 L. G. Gazyan and L. M. Suslov, Radiotekhnika No. 7, 92 ͑1988͒. 8 Electromagnetic Wave Absorbers for Chamber Shielding, State Standard 6 V. I. Ponomarenko, V. Zh. Berzhanski, I. V. Dzedolik, V. L. Kokoz, GOST R500, Nov. 1992 ͓in Russian͔,p.1–15. 9 Yu. M. Vasil’ev, and A. V. Torkunov, Izv. Vyssh. Uchebn. Zaved. P. L. Kalantarov and L. A. Tsetlin, Calculation of Inductances ͓in Rus- Radioe´lektron. 32͑3͒,38͑1989͒. sian͔,E´nergoatomizdat, Moscow ͑1986͒,p.92–99. 7 V. N. Berzhanski, I. G. Gazyan, V. L. Kokoz, and D. N. Vladimirov, Pis’ma Zh. Tekh. Fiz. 16͑12͒,14͑1990͓͒Sov. Tech. Phys. Lett. 16, 467 ͑1990͔͒. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Thermal ﬂicker noise in dissipative pre-melting processes in crystalline materials L. A. Bityutskaya and G. D. Seleznev

Voronezh State University ͑Submitted January 12, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 24–27 ͑July 26, 1998͒ An analysis is made of thermal fluctuations on the premelting exothermics of materials with different types of chemical bond ͑KCl, Ge, Sb, and Cu͒. Statistical and spectral parameters of the thermal fluctuations are introduced. It is shown that for these materials under certain conditions the thermal fluctuations may be identified as two-level thermal flicker noise. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02207-1͔

Flicker noise, known for more than half a century, has the instrumental noise, which was 0.13 ͑see Table I͒. This recently attracted an upsurge of attention among means that the observed instability can be considered as a researchers1–4 because of the increasing interest being shown manifestation of the dynamic nature of the transient in problems of irreversibility, nonlinearity, and self- processes.9 organization in which the fundamental role of flicker noise is The isothermal fluctuations were investigated by digital becoming increasingly evident. Initially recorded as electri- thermal analysis with the data processed by a special pro- cal noise in electronic and semiconductor devices ͑p – n gram using the Welch periodogram method.10 junctions, transistors, metal–semiconductor contacts͒,3,4 It should be noted that in spectral analyses of random flicker noise was later observed in an extremely diverse processes4,10 the results are represented as frequency depen- range of processes, ranging from fluctuations of the mem- dences of the spectral power density S. The spectral power brane potential of a living cell to music.1,2 density is the energy characteristic of a random processes In Refs. 5–8 a special method of digital differential ther- and in the calculations, the expression for S should contain mal analysis in a dynamic mode at heating rates greater than the square of the amplitudes of its spectral components. 1 K/min with the pass band of the recorded signals control- However, the thermal fluctuations considered here are essen- lable up to 1 Hz was used to observe transient dissipative tially a thermal energy release process, and in this case their processes which appeared as sharp-edged thermal pulses spectral density, denoted as S*, should contain an expression against whose background were observed low-frequency linear in the absolute values of the amplitude. fluctuations of the temperature ⌬T. These fluctuations are For all the materials, the spectrum consisted of two sec- preserved and amplified by being held isothermally in the tions A and B approximated by straight lines for which the excitation region. The nature and parameters of the observed absolute values of the slopes satisfy ␣AϾ1, ␣BϽ1 and the noise has not been studied. critical frequency at which the slope changes is f cr ͑see Fig. The aim of the present paper is to make a spectral analy- 1 and Table I͒. We draw attention to the fact that the ratio of sis of the thermal fluctuations of the premelting isotherms for the parameters ␣B /␣A for Cu and Sb metals is close to 0.3, crystalline materials with different types of chemical bonds: whereas for the semiconductor Ge and ionic crystal KCl it is Cu, Sb, Ge, and KCl. The analysis was made using data close to 0.5. The presence of two sections in the spectrum obtained in Refs. 5–8, consisting of a set of readings from a suggests that the physical process generating the thermal differential thermocouple used to record the temperature dif- fluctuations has two levels of the same nature. ference between the sample and a standard, expressed in mil- A linear log–log dependence of the spectral power den- lidegrees Kelvin. The characteristic isothermal holding time sity on frequency is typical of the fluctuation process known was 30 min and the number of readings 1000–1500. The as flicker noise.4 The existence of two approximately recti- variance of the macroscopic thermal fluctuations for this linear sections A and B observed on the spectra of all the groups of materials considerably exceeded the variance of materials studied suggests that the thermal fluctuations in

TABLE I.

Spectral Isothermal characteristics holding

Material temperature, °C Variance ␣A ␣B ␣B /␣A f cr ,Hz Cu 1047 3.56 1.25 0.38 0.30 0.085 Ge 949 2.94 1.02 0.5 0.50 0.056 Sb 584 0.78 1.4 0.42 0.30 0.042 KCl 762 2.47 1.1 0.52 0.47 0.115

the thermal fluctuations, the relationship ␣AϽ␣B was found in this case. Since the characteristics that identify isothermal thermal fluctuations at the premelting stage as flicker noise are independent of the type of chemical bond, this indicates that the phenomenon is universal. At the same time, the parameters of these fluctuations introduced—the variance, the spectral density characteristics ␣A and ␣B , and the critical frequency f cr—while preserving the general features, have their own characteristic values under given conditions and thus may serve as typical indications of the particular material.

1 S. F. Timashev, Ross. Khim. Zh. 61͑3͒,17͑1997͒. 2 Yu. L. Klimontovich, Statistical Theory of Open Systems ͓in Russian͔, TOO ‘‘Yanus’’, Moscow ͑1995͒, p. 624. 3 G. N. Bochkov and Yu. E. Kuzovlev, Usp. Fiz. Nauk 141͑1͒, 151 ͑1983͒ ͓Sov. Phys. Usp. 26, 829 ͑1983͔͒. 4 M. J. Buckingham, Noise in Electronic Devices and Systems ͑Halsted Press, New York, 1983; Mir, Moscow, 1986͒. 5 L. A. Bityutskaya and E. S. Mashkina, Pis’ma Zh. Tekh. Fiz. 21͑17͒,85 ͑1995͓͒Tech. Phys. Lett. 21, 720 ͑1995͔͒. 6 L. A. Bityutskaya and E. S. Mashkina, Pis’ma Zh. Tekh. Fiz. 21͑18͒,8 FIG. 1. Fluctuations of the heat release during isothermal premelting of ͑1995͓͒Tech. Phys. Lett. 21, 728 ͑1995͔͒. 7 L. A. Bityutskaya and E. S. Mashkina, Pis’ma Zh. Tekh. Fiz. 21 20 ,30 copper: a—thermogram during holding at 1047 °C, b—relative spectral ͑ ͒ ͑1995͓͒Tech. Phys. Lett. 21, 828 ͑1995͔͒. power density of the fluctuations log (S*/S* ) versus the relative fre- 10 max 8 L. A. Bityutskaya and E. S. Mashkina, Pis’ma Zh. Tekh. Fiz. 21͑24͒,90 quency log ( f / f ), where f and S* are maxima. 10 max max max ͑1995͓͒Tech. Phys. Lett. 21, 1032 ͑1995͔͒. 9 ´ ´ S. E. Shnol’, E. V. Pozharski et al., Ross. Khim. Zh. 61͑3͒,30͑1997͒. S. 30–36. regions of excitation at the premelting stage of crystalline 10 S. L. Marple, Jr., Digital Spectral Analysis with Applications ͑Prentice- Hall, Englewood Cliffs, N. J., 1987; Mir, Moscow, 1990͒. materials may be identified as two-level thermal flicker 11 noise. Two levels of flicker noise were also observed in ex- R. F. Voss and J. Clarke, Phys. Rev. Lett. 13, 556 ͑1976͒. periments to study electric current fluctuations11 but unlike Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Distribution laws for microplastic deformations V. V. Ostashev and O. D. Shevchenko

Pskov Polytechnic Institute ͑Submitted January 23, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 28–30 ͑July 26, 1998͒ Calculations and construction of bispectra are used to show that the deviation of the probability density of microplastic deformations from a normal distribution law is a measure of the evolution of synchronization effects and a measure of the completeness of the relaxation processes determining the plasticity of the material. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02307-6͔

The oscillating nature of plastic deformations at local zation was estimated by calculating the bispectra ͑see Fig. 1͒. structural levels may be described by the parameters of latent The bispectrum reflects the Fourier transformation of the oscillatory processes ͑amplitude, frequency, phase͒, while centered third-order correlation function the interaction of deformation defects on different scale lev- 1 2 ϱ els is represented by the superposition and overlap of random B͑ f 1 , f 2͒ϭ R͑␶1,␶2͒ quantities with different distribution laws.1 The probability ͩ 2␲ͪ ͵͵Ϫϱ density distribution of microplastic deformations under the ϫexp͓Ϫi͑ f ␶ ϩ f ␶ ͔͒d␶ d␶ , conditions of a factor experiment depends on the load param- 1 1 2 2 1 2 eters. For samples having maximum strength characteristics, where R(␶1 ,␶2)ϭ͗x(t)x(tϩ␶1)x(tϩ␶2)͘ is a third-order this distribution is close to normal over the entire range of correlation function averaged over the ensemble. For a pro- deformations in translational and rotational modes. For cess with a normal distribution law the third moment ͑asym- samples exhibiting maximum plasticity an estimate of the metry͒ is zero. In this case, we have ͗x3(t)͘ϭR(0,0)ϭ0 and distribution law using the Pearson criterion suggests that the thus B(1,2)ϭ0. The bispectrum therefore shows how the hypothesis put forward in Ref. 1 is unreliable. The distribu- deviations of the process from Gaussian are ‘‘expanded’’ tion begins to deviate from normal for the rotational modes over frequency, i.e., occur as a result of various frequencies ␻z and then embraces the translational modes ␥xy and ␧xx . obtained under phase matching conditions f 1ϩ f 2ϩ f 3ϭ0. We conjecture that random microdeformations obey a In general, the model of a polycrystalline deformable normal law as long as they predominate among the many material at the statistical interaction stage is a system for factors responsible for inhomogeneity of the microplastic de- which each level may be represented by linear and nonlinear formation field. One such factor at the wave interaction level components, and thus the distribution law of the microplastic is synchronization, which reduces the dispersion of the group deformations approaches normal to the extent that linearity is velocity of the deformation defects and thus causes the evo- present at a given structural level. Quite clearly, the devia- lution of macrorotations.2,3 tion from normal may be defined as a measure of the evolu- The measure of the nonlinearity of the interactions of tion of synchronization processes and as a measure of the microplastic deformations and the contribution of synchroni- relaxation processes. Accordingly, a deformable material shows a specific combination of strength and plasticity characteristics.

1 V. V. Ostashev and O. D. Shevchenko, Abstracts of Papers presented at the 32nd Seminar ‘‘Topical Strength Problems,’’ St. Petersburg, 1996 ͓in Russian͔, pp. 35–36. 2 Yu. I. Meshcheryakov, New Methods in the Physics and Mechanics of Deformable Solids ͓in Russian͔, Tomsk ͑1990͒, pp. 33–43. 3 C. L. Nikias and M. R. Raghuver, Proc. IEEE 75, 869 ͑1987͒.

FIG. 1. Calculated bispectra for samples: a—with maximum strength and Translated by R. M. Durham b—with maximum plasticity.

Time evolution of three-crystal x-ray spectra during ﬂuorination of a Pt/LaF3 /Si structure A. A. Nefedov, S. E. Sbitnev, and S. S. Fanchenko

‘‘Kurchatov Institute’’ Russian Science Center, Moscow ͑Submitted January 8, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 31–34 ͑July 26, 1998͒ A qualitative change was observed in the three-crystal x-ray spectra during fluorination of a Pt/LaF3 /Si structure. This may be attributed to smoothing of the surface of the platinum film, most likely as a result of the formation of platinum tetrafluoride which may undergo partial volatilization or partial dissociation with the sublimation of platinum. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02407-0͔

A metal–ionic conductor–semiconductor structure with interface should be three-phase which was achieved by suc- a solid LaF3 electrolyte as the below-gate layer is a promis- cessive annealing in air at 350 °C. The sample was placed in ing sensor to determine concentrations of toxic gases such as a hermetically sealed chamber which had gas inlet and outlet fluorine and hydrogen fluoride in the atmosphere.1,2 The tubes, and x-ray transparent windows. The chamber was at- crystalline perfection of the interfaces and the quantitative tached to the sample holder of a standard three-crystal spec- characteristics of the surface roughness of these structures trometer, after which one tube was connected to a pump may be determined by analyzing spectra3,4 obtained by three- which created a vacuum of 20 Torr in the chamber and the crystal x-ray spectroscopy ͑TXS͒.5 This technique involves second tube was connected to a chamber containing crystal- measuring the intensity of an x-ray flux reflected succes- line XeF2 powder. This substance is highly volatile at room sively by all three crystals ͑a crystal monochromator, the temperature and has a partial pressure of gaseous xenon di- sample, and a crystal analyzer͒ as a function of the angle of fluoride of around 2.5 Torr. Xenon difluoride has similar rotation of the crystal analyzer ␪3 for a fixed angle of devia- reaction properties to fluorine and this suggests that the fluo- tion ␣ of the crystal from the exact Bragg angle. The rocking rination of platinum by xenon difluoride with the formation curve usually consists of three peaks: a principal peak, a of barely volatile platinum tetrafluoride is similar to the re- pseudopeak, and a diffuse peak. The principal peak is caused action with pure fluorine. The presence of water vapor in the by diffraction reflection from the crystal being studied, the chamber ͑the humidity was not monitored in this experi- pseudopeak characterizes the diffraction reflection curve ment͒ may substantially accelerate the formation of platinum from the crystal monochromator, and the diffuse peak is as- tetrafluoride, even at room temperature. The use of XeF2 signed to scattering of x-rays from defects in the crystal instead of pure fluorine considerably simplified the apparatus structure of the sample. and as a result we were able to measure the TXS spectra in Three-crystal x-ray spectroscopy was used in Ref. 6 to situ during fluorination of the Pt/LaF3 /Si structure. study the characteristics of a Pt/LaF3 /Si sensor in air. It was A series of TXS experiments was carried out at a fixed shown that the thickness of the LaF3 /Si interface at which angle of deviation of the sample ␣ϭ100Љ from the exact distortion of the silicon crystal structure was observed did Bragg angle ͑for silicon ␪Bϭ14.22°͒ at intervals of around not exceed 0.5 nm and the characteristic roughness of the 10 min, after XeF2 was admitted to the chamber. The mea- platinum film on the surface of the lanthanum trifluoride was surements were made using CuK␣ radiation at the wave- of the same order as the average thickness of this film, which length ␭ϭ1.54 Å. The experimental results are plotted in was estimated as 40 nm. The possibility of carrying out TXS Fig. 1. It can be seen that the diffuse peak ͑3͒ decreases and experiments in a controlled gaseous medium7 makes this almost completely disappears with time while the principal method extremely promising for studies of the kinetics of peak ͑1͒ increases in amplitude and becomes increasingly gas–surface interaction processes and especially for refining narrow. It should be noted that whereas this peak is almost the mechanisms of the electrochemical reactions which take indistinguishable from the background of the diffuse peak on place at the three-phase interface of a metal–ionic the initial curves, for fairly large angles of deviation of the conductor–semiconductor sensor with an active gas medium. sample ␣ this principal peak is clearly discernible, as was Here we present the first results of a TXS study of a confirmed in another series of experiments.6 Nevertheless, Pt/LaF3 /Si structure during fluorination. this substantial change in the profile of the principal peak is Films of LaF3 were grown on the ͑111͒ surface of a accompanied by no change in its area, which indicates that silicon single crystal by high-vacuum thermal vapor-phase no changes take place at the silicon–lanthanum trifluoride deposition at a substrate temperature of 550 °C and the plati- interface ͑the thickness of this interface is of the order of 0.5 num film was deposited by magnetron sputtering in an argon nm͒. The high intensity of the diffuse peak may be attributed plasma. For a sensor with high sensitivity the gas/Pt/LaF3 to scattering of x-rays at inhomogeneous of the LaF3 /Si in-

FIG. 1. Intensity of x-ray ﬂux reﬂected successively from

three crystals as a function of the angle of rotation ␪3 of the crystal analyzer obtained in situ during ﬂuorination: 1— principal peak, 2—pseudopeak, 3—diffuse peak ͑angle of deﬂection of sample ␣ϭ100Љ, t—time after admission of XeF2 to chamber, the curves were taken at 10 min inter- vals͒.

terface induced by stresses in the LaF3 which were either the TXS spectra obtained in another series of measurements created by the annealing of the platinum film or as a result of for LaF3 /Si structures without the upper platinum layer both the buildup of nonuniform electric charge at the three-phase before and after fluorination under similar conditions ͑these gas/Pt/LaF3 interface. The high intensity of the pseudopeak spectra showed little variation in the course of fluorination͒. compared with the principal one is caused by small-angle scattering of the wave incident on the inhomogeneous platinum film.4 Thus, the decrease in the diffuse peak and the 1 S. Krause, W. Moritz, and I. Grohmann, Sens. Mater. 9, 191 ͑1992͒. 2 narrowing of the principal peak may indicate that the rough W. Moritz, S. Krause, A. A. Vasiliev et al., Sens. Mater. 24–25, 194 ͑1995͒. platinum surface becomes smoothed to specular during the 3 A. M. Afanas’ev, M. V. Koval’chuk, E´ . F. Lobanovich et al., Kristal- experiment, which lasted around 3 h. This smoothing of the lografiya 26,28͑1981͓͒Sov. J. Crystallogr. 26,13͑1981͔͒. platinum surface is most likely caused by the formation of 4 P. A. Aleksandrov, A. M. Afanas’ev, and M. K. Melkonyan, Kristal- platinum tetrafluoride, which may partially volatilize and lografiya 26, 1275 ͑1981͓͒Sov. J. Crystallogr. 26, 725 ͑1981͔͒. 5 A. Iida and H. Kohra, Phys. Status Solidi A 51, 533 ͑1979͒. partially dissociate with the sublimation of platinum. Mea- 6 A. A. Nefedov, S. S. Fanchenko, I. A. Shipov et al., Poverkhnost’ No 3, surements of the TXS spectra after prolonged ͑two weeks͒ ͑in press͒͑1998͒. 7 holding in air showed that the spectral profile does not un- A. A. Nefedov, A. E. Rzhanov, V. I. Filippov et al., Pis’ma Zh. Tekh. Fiz. dergo any further change, which indicates that this process is 14͑5͒, 453 ͑1988͓͒Sov. Tech. Phys. Lett. 14, 203 ͑1988͔͒. irreversible. The form of these spectra is almost the same as Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Phase instability and nonlinear effects in a mechanically synthesized FeB nanocrystalline alloy V. A. Tsurin and V. A. Barinov

Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg ͑Submitted January 13, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 35–40 ͑July 26, 1998͒ Results are presented of investigations of the processes taking place during the mechanical formation of FeB alloy. It was observed that there are critical rates of introduction of damage and defect concentrations above which the system becomes unstable and two branches of nonequilibrium steady states appear. During grinding, transitions of a cyclic nature are observed between these states. It is noted that the evolution of the system is determined by the dynamics of the ﬂuctuations and in this sense, the mechanical synthesis reaction may be unpredictable. It is established that the powder grinding process and mechanical formation of an alloy from atomic components cannot be treated on a purely mechanical or mechanochemical basis. They must be viewed as part of the general problem of nonlinear dynamic systems far from thermodynamic equilibrium. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02507-5͔

In mechanical alloy formation, the system continuously model calculations and to reconstruct the density function absorbs energy, some of which is dissipated in irreversible P(H).3 processes. Nonthermal processes also take place, such as The progress of mechanochemical reactions in a mixture chemical or topological disordering as a result of successive of elemental Fe and B components is determined by the 4 substituting collisions. The frequency of these elementary composition and by the mechanical treatment conditions. events is proportional to the rate of introduction of damage Here, we investigate for the first time the alloy formation and is substantially higher than that under thermal equilib- process in a mixture of iron and boron powders having the rium conditions. In consequence, systems exposed to me- nominal composition Feϩ60 at.% B after high-speed treatment for 45 h with the ratio Aϭ13 ͑13 HST͒. The results of chanical action may be treated as dissipative systems far the model calculations of the Mo¨ssbauer spectra are pre- from equilibrium, the theory of which has attracted particular sented in Table I. The parameters obtained are similar to 1,2 Systems far from equi- attention over the last two decades. those given in Refs. 5 and 6 for nanocrystalline iron. The librium may undergo nonequilibrium phase transitions ͑i.e., 57 ratio of the subspectra S1 /S2ϭ1 indicates that Fe resonant transitions between nonequilibrium states͒ when the control- nuclei localized along grain boundaries make a significant ling parameters pass through certain critical values. Here we contribution. For this area ratio, the density of the boundaries investigate the processes of synthesis and breakup of alloys may be 1018–1021 cmϪ3. It is known that the dispersion pro- in a FeB system as a function of the energy saturation of the cess has an appreciable influence on the mobility of the at- alloy formation process and study how nonequilibrium states oms. It was shown in Ref. 7 that the coefficient of intra- form in this system. granular diffusion D of boron in iron at 110 °C is Dϭ2.6 A mixture of Fe and B powder was treated mechanically ϫ10Ϫ19 m2/s. In the nanocrystalline state, the diffusion co- in a Pulverizette centrifugal planetary mill. The degree of efficient increases to Dϭ2.6ϫ1015 m2/s and is comparable action was varied according to the ratio of the mass of the with the value of D along the grain boundaries. Thus, an analysis of the nuclear gamma ray spectrum suggests that the grinding balls (mb) to the powder mass (m p), Aϭmb /mp and also by the speed of rotation of the mill and the treatment time. Two standard regimes were used to form an alloy TABLE I. Calculated experimental spectrum of mechanically synthesized of the FeB powder. In the high-speed treatment regime Feϩ60 at.% B alloy after high-speed treatment for 45 h in a planetary mill ͑HST͒, the speeds of rotation of the mill platform and the ͑13 HST͒. container were 450 and 850 rpm, respectively, while for the Parameters of nuclear low-speed treatment ͑LST͒ these speeds were 200 and 380 gamma spectrum Subspectrum 1 Subspectrum 2 rpm, respectively. The structural phase state of the FeB pow- I.S., mm/s 0.292 0.266 der was investigated by Mo¨ssbauer spectroscopy at various G, mm/s 0.38 1.46 stages of the mechanical treatment. Nuclear gamma ray spec- H, kOe 235 233 tra of the FeB powder were obtained at 300 K using a 57Co S, % 48 52 source in a Cr matrix using a spectrometer and a resonance I.S.—isomeric shift relative to Fe; G—width of 1.6 lines of the nuclear scintillation detector. A computer program was used for the gamma spectrum; H—hyperfine field; S—relative area of the subspectra.

FIG. 1. Average hyperﬁne ﬁeld ͗Hhf͘ of mechanically synthesized nanocrystalline Feϩ60 at.% B alloy samples as a function of the grinding time in the 50 LST regime. The arrow indicates the bifurcation point B ͑splitting͒ where the branch splits into two nonequilibrium states. The dashed curve shows the transitions between these two states.

initial state of the synthesized Feϩ60 at.% B alloy is charac- caused by the formation of lattice defects. An increase in the terized by high diffusion mobility of the boron atoms. The treatment time of the nanocrystalline alloy in the less energy- absence of ␣-Fe lines in the spectrum and the presence of a stressed regime ͑50 LST͒ has the result that the rate of defect sextet with a field corresponding to Fe67B33 indicates that an formation is higher than their rate of relaxation because of appreciable fraction of the boron in the sample is in the free the reduced local heating temperature of the sample. This state, forming no chemical bond with Fe atoms. leads to intensive mixing of atoms of different species and In order to investigate phase transitions and the boron abruptly increases the rate of mass transfer. In the initial dissolution process, the initial synthesized Feϩ60 at.% B nanocrystalline alloy and also after a short grinding time powder was then subjected to further low-speed mechanical (Ͼ110 h) a single state is established ͑Fe66B33 solid solu- treatment with Aϭ50 ͑50 LST͒. This regime was used to tion͒ exhibiting asymptotic stability, since in this region the reduce the local heating temperature of the powder particles system is capable of suppressing internal fluctuations or ex- and the formation of Fe2B and FeB intermetallides. The efficiency of the alloy formation process, phase transitions, and the concentrations of the components in the phases at the different stages of mechanical treatment may be assessed from the dependence of ͗Hhf͘ on the average boron concentration in the sample.8 Figure 1 gives the average hyperfine field ͗Hhf͘ obtained by reconstructing the function P(H) from the experimental nuclear gamma ray spectra of the samples plotted as a function of the grinding time. It can be seen that the dependence is nonmonotonic. After a certain time ͑110 h͒ the value of ͗Hhf͘ increases abruptly as a result of the decomposition of the initial synthesized nanocrystalline Feϩ33 at.% B solid solution with the release of elemental Fe and B. An increase in the mechanical treatment time causes further dissolution of the boron with the formation of a near-equatomic phase in the amorphous state. Treatment above 400 h causes a phase transition in Fe50B50 regions to form an Fe66B33 solid solution with the release of B and Fe, i.e., the process observed at the initial stages of grinding is repeated. Figure 2 gives a three-dimensional representation of the distribution function of the hyperfine field P(H)asa function of the mechanical treatment time for the 50 LST regime. The redistribution of the peaks corresponding to the FIG. 2. Three-dimensional representation of the hyperfine field distribution Fe2B(Fe66B33), Fe, FeB (Fe50B50) concentrations illustrates the nonequilibrium phase transitions in the system. function P(H) as a function of the mechanical treatment time t in the planetary mill in the 50 LST regime. The redistribution of the peaks of the We postulate that the increase in the internal energy with corresponding concentrations illustrates the phase transitions between the increasing deformation and increasing work expended is two nonequilibrium states. 558 Tech. Phys. Lett. 24 (7), July 1998 V. A. Tsurin and V. A. Barinov ternal perturbations. For this reason, this branch of states is presence of oxides,9 the alloy formation reaction may take called the thermodynamic branch.1 After a certain critical place by different methods. concentration of lattice defects is reached the crystal structure of the treated material becomes unstable, since the fluc- 1 G. Nicolis and I. Prigogine, Self Organization in Nonequilibrium Systems tuations or small external perturbations are not suppressed. ͑Wiley, New York, 1977͒. The system departs from the standard state and is converted 2 A. S. Balankin, Synergetics of a Deformable Solid ͓in Russian͔,MO to a state far from thermodynamic equilibrium. The instabil- SSSR, Moscow ͑1991͒, 358 pp. 3 V. I. Nikolaev and V. S. Rusakov, Mo¨ssbauer Investigations of Ferrites ity in the nonlinear region far from equilibrium is observed ͓in Russian͔, Moscow State University Press, Moscow ͑1985͒, pp. 21–44. as splitting ͑bifurcation, indicated by the arrow in Fig. 1͒ of 4 J. S. Benjamin, Met. Sci. Forum 88–90,1͑1992͒. the thermodynamic branch into two branches of nonequilib- 5 U. Herr, J. Jing, R. Birringer, U. Gonser, and H. Gleiter, Appl. Phys. Lett. rium steady states. The solution is determined randomly at 50, 472 ͑1987͒. 6 W. Schlump and H. Grewe, Techn. Mitteil. Krupp 47͑2͒,69͑1989͒. the instant of instability via the fluctuation dynamics, and in 7 H. J. Hofler, R. S. Averback, and H. Gleiter, Philos. Mag. Lett. 68„2…,99 this sense is unpredictable. The stabilization of some particu- ͑1993͒. 8 C. L. Chein and K. M. Unruh, Phys. Rev. B 25, 5790 ͑1982͒. lar fluctuation determines the evolution of the system. This 9 means that even as a result of small perturbations caused, for J. Balogh, L. Bujdoso, G. Faigel et al., Nanostruct. Mater. 2,11͑1993͒. example, by some change in the treatment regime or the Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Phase transition waves in strong electric ﬁelds N. I. Kuskova

Institute of Pulsed Processes and Engineering, National Academy of Sciences of Ukraine, Nikolaev ͑Submitted January 13, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 41–44 ͑July 26, 1998͒ A description is given of a semiphenomenological theory which yields an expression for the velocity of a phase transition wave in a condensed medium exposed to a strong electric ﬁeld. The proposed theory may be applied to analyze an electron avalanche breakdown wave or a melting ͑evaporation͒ wave formed in a material under the action of an electric discharge. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02607-X͔

The action of strong electromagnetic fields on condensed istic size of the region of current density nonuniformity j is Ϫ1/2 media may result in the following types of phase transitions: equal to the width of the skin layer ␦ jϭ(0.5␮␴␻) , 1͒ melting and ͑or͒ evaporation as a result of electric where ␮ is the magnetic permeability and ␻ is the frequency breakdown of solid ͑or liquid͒ weakly conducting media and of the current fluctuation.7 The mechanism of wave displace- electrical explosion of conductors and 2͒ insulator– ment involves expulsion of the current from the skin layer in semiconductor or semiconductor–metal transitions as a re- which the electrical conductivity drops sharply when the sult of electric discharges in insulators and high-resistivity metal is heated by the high-density current. The rate of en- semiconductors, which under certain conditions propagate as ergy release is determined from the Joule heating rate waves.1–6 w(E)ϭ␴E2/␳, and formula ͑2͒ then yields an expression to An analysis will be made of phase transition waves pro- estimate the velocity of the so-called current wave3 through cesses involving the propagation of local regions of strong the conductor electromagnetic field variation where some phase transition takes place. ␴E2␦ uХ , ͑3͒ Let us determine the propagation velocity of a phase ␳⌬␧ transition wave. We write the general one-dimensional continuity equation describing the change in some physical char- where ⌬␧ is the change in the internal energy of the metal in acteristic of the medium f ͑such as the internal energy ␧, the skin layer and vϭ0 since the layer in which the current electrical conductivity ␴, carrier concentration n, or density wave propagates is cold. ␳͒ as a result of a phase transition which takes place under 2. In weakly nonuniform electrical explosions of con- the action of a strong electric field. Denoting the rate of ductors, such as the so-called fast regimes, all the character- change of the parameter f by wϭw(E)(Eis the electric istics of a cylindrical conductor in either the solid and liquid field strength͒ and the flow velocity of the medium by v,we states are uniform except for the magnetic pressure. Since the then have boiling point depends weakly on pressure, volume expansion cannot take place even with uniform heating. Thus, the f ץ f ץ Ϯv ϭw. ͑1͒ propagation of evaporation in the form of a phase transition r wave moving from the outer boundary to the axis of theץ tץ conductor is caused by the nonuniformity of the pressure p. We shall assume that the characteristic size of the region of The width of the evaporation wavefront in this case is given field uniformity in which the characteristic f changes by abruptly is ␦. By means of the change of variables zϭr Ϯut, where u is the wave propagation velocity, we obtain ␭ dP a2 ␦ pϭ , ͑4͒ w␦ cP͑0͒ dTb 2r uХ Ϯv, ͑2͒ ⌬ f where ␭ is the latent heat of evaporation, c is the specific where ⌬ f ϭ͉f maxϪf0͉, f 0 is the initial distribution. heat of the liquid metal, P(0) is the pressure on the conduc- If the flow of the medium is ‘‘opposite’’ to the phase tor axis, dP/dTb is the change in pressure and boiling point transition wave, there exists a threshold field strength E* at along the liquid–gas phase equilibrium curve, a is the radius which the phase transition wave can propagate. This thresh- of the conductor, and r is the instantaneous position of the old value E* is determined from the condition uϾ0, i.e., wavefront. w(E*)␦/⌬ f ϪvϾ0. Since the temperature of the conductor is uniform before 1. In ultrafast electrical explosions of conductors, the the onset of boiling, the change of the specific internal en- current distribution becomes highly nonuniform as a result of ergy at the front of the phase transition wave is ⌬␧ϭ␭ and the formation and explosion of the skin layer. The character- the velocity of this wave is given by

2 ␴E ␦ p hole pairs ͑as a result of tunneling, impact ionization, the uХ Ϫv. ͑5͒ ␳␭ Frenkel effect, or photoionization͒ and v is the electron velocity. 3. We shall now consider the phase transition waves To sum up, the similarity between the electrical explo- generated in the electrical breakdown of weakly conducting sion of conductors and the electrical breakdown of insulators media, as a result of melting and ͑or͒ evaporation of the and semiconductors can be attributed to their being caused material which causes gas bubbles. An electric discharge by the same mechanism: a phase transition wave at whose takes place in this gas within a time much shorter than that front the electrical conductivity changes considerably, falling needed to form the bubbles and results in the generation of a sharply in the electrical explosion of conductors and increas- plasma at the front of the phase transition wave. The width ing by several orders of magnitude in electrical breakdown. of the front ␦E is determined by the geometry of the dis- For the same geometry and times of action of the electro- charge gap and the plasma channel. In this case, the propa- magnetic field, the phase transition waves formed in con- gation velocity of the phase transition wave can be estimated densed media have common relationships and spectra of from propagation velocities—supersonic and subsonic— 2 depending on the field strength and the type of medium. ␴E ␦E uХ ϩv, ͑6͒ ␳⌬␧ where ⌬␧ is the change in the specific energy of the material. 1 V. Ya. Ushakov, Pulsed Electrical Breakdown in Liquids ͓in Russian͔, 4. The propagation of plasma channels in insulators and Tomsk State University Press, Tomsk ͑1975͒, 258 pp. 2 semiconductors at supersonic velocities exceeding the carrier N. I. Kuskova, Zh. Tekh. Fiz. 53, 924 ͑1983͓͒Sov. Phys. Tech. Phys. 28, 591 ͑1983͔͒. drift velocities indicates that the processes taking place in the 3 V. A. Burtsev, N. V. Kalinin, and A. V. Luchinski, Electrical Explosion streamer discharge are of a wave nature. The motion of the of Conductors and Applications to Electrophysical Systems ͓in Russian͔, streamer in the interelectrode gap constitutes an ionization Moscow ͑1990͒, 288 pp. 4 F. D. Bennet, Phys. Fluids 8, 1425 ͑1965͒. wave at whose front the carrier concentration changes sub- 5 N. I. Kuskova, S. I. Tkachenko, and S. V. Koval, J. Phys.: Condens. 0 1/2 stantially by ⌬nϭnmaxϪn0Хnmaxϭ(␧w /(e␮)) in a strong Matter 9, 6175 ͑1997͒. electric field ͑when the ionization rate is considerably higher 6 N. G. Basov, A. G. Molchanov, A. S. Nasibov et al., Zh. Eksp. Teor. Fiz. 8 70, 1751 ͑1976͓͒Sov. Phys. JETP 43, 912 ͑1976͔͒. than the recombination velocity͒. Expression ͑2͒ for the ve- 7 6 Yu. V. Novozhilov and Yu. A. Yappa, Electrodynamics, Mir, Moscow locity of the ionization wave then has the form ͑1981͒. 0 8 V. V. Vladimirov, V. N. Gorshkov, O. V. Konstantinov, and N. I. w ␦E uХ ϩv, ͑7͒ Kuskova, Dokl. Akad. Nauk SSSR 305, 586 ͑1989͓͒Sov. Phys. Dokl. 34, nmax 242 ͑1989͔͒. 0 where w ϭw(Emax) is the rate of generation of electron– Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Optical properties of the polarized Dirac vacuum S. G. Oganesyan

‘‘Laser Engineering’’ Scientific-Industrial Organization, Erevan ͑Submitted July 7, 1997; resubmitted January 22, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 45–47 ͑July 26, 1998͒ An investigation is made of the propagation of laser radiation through the Dirac vacuum polarized by a strong electric field. Calculations are made of the refractive index of the vacuum and the angle of rotation of the plane of polarization of the radiation. The possibility of measuring strong electric fields is assessed. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02707-4͔

The optical properties of free electron beams ͑Ref. 1, laser radiation linearly polarized along the x and y axes has Sections 13 and 48͒ can be used to study their internal struc- different refractive indices ture. We note that, according to Dirac,2 the vacuum is a set of free electrons filling all possible negative-energy levels. 2 2 nʈϭ1ϩ8␲RE , nЌϭ1ϩ28␲RE . ͑3͒ Here it is shown that the optical properties of the Dirac 0 0 vacuum can be used to study ͑especially to measure the If the laser radiation is elliptically polarized 1 , under the strengths of͒ very strong fields E ϽE ϭm2c3/eបϭ1.3 ͑ ͒ 0 cr action of the field E its axes will be turned through the ϫ1016 V/cm ͑in the opposite case E ϾE breakdown of the 0 0 cr angle vacuum occurs͒. Let us assume that the Dirac vacuum is polarized by a strong static electric field E0 and elliptically 2 polarized laser radiation passes through this region: 1 2␦ ␣ E0 z ␸ϭ arctan sin . ͑4͒ 2 ͫ ␦2Ϫ1 ͩ 9 E2 ␭ͪͬ cr E1xϭE1 sin͑␻tϪkz͒, E1yϭE2 cos͑␻tϪkz͒. ͑1͒

We shall describe this system using the Heisenberg–Euler Here z is the size of the field interaction zone, ␦ϭE2 /E1 is Lagrangian2 expanded as a series to the fourth order with the ratio of the major axes of the ellipse, and ␭ϭ2␲c/␻ is respect to the field, the wavelength of the laser radiation. Clearly, by measuring the refractive indices 3 or the angle of rotation 4 ,wecan 2 2 2 2 ͑ ͒ ͑ ͒ LЈϭR͓͑E ϪH ͒ ϩ7͑EH͒ ͔. uniquely determine the strength of the static electric field 2 E . Note that these results are also valid for a variable field Here we have Rϭ␣/360␲2E , ␣ϭe2/បc, EϭE ϩE , and 0 cr 0 1 E (t) if its characteristic fluctuation time is shorter than the HϭH . Note that this Lagrangian is suitable for describing 0 1 interaction time of the fields ϭz/c. static and variable fields if the frequency of the latter satisfies ␶ Let us assume that an electric field having the strength the condition ␻Ӷmc2/ប͑Ref. 3͒. Let us now calculate the E ϭ1.3ϫ1015 V/cm ͑or E ϭ0.1E ) is localized in a region H 0 0 crץ/LЈץE and the magnetization Iϭץ/LЈץpolarization Pϭ whose size is of the order a. Let us also suppose that the of the Dirac vacuum. Then collecting in the current j wavelength of the laser radiation is ␭ϭa/10 and the ratio of t those terms oscillating at the frequency ␻ץ/Pץϭc curl Iϩ the major axes of its polarization ellipse is ϭ1.1. In this and substituting these into the Maxwell equation, we find the ␦ case, the angle of rotation of the plane of the ellipse 4 is tensors of the permittivity and magnetic permeability of the ͑ ͒ ϭ4.2ϫ10Ϫ4 rad. Note that the angle of rotation de- Dirac vacuum, ␸ ␸ creases rapidly with decreasing field strength E0 .IfE0 2 ϭ1.3ϫ1014 V/cm ͑or E ϭ0.01E ͒, we obtain ␸ϭ4.2 ␧ijϭ͑1ϩ8␲RE0͒␦ijϩ16␲RE0iE0j , 0 cr ϫ10Ϫ6 rad. ␮ ϭ͑1Ϫ8␲RE2͒␦ ϩ56␲RE E ͑2͒ ij 0 ij 0i 0j This work was supported by the International Scientific ͑note that in the calculations we neglected the influence of and Technical Center, Grant No. A-87. the Dirac vacuum on the field E0͒. In order to simplify the following analysis, we assume that the field E0 is directed along the x axis. In this case, the equations for the x and y 1 V. M. Harutunian and S. G. Oganesyan, Phys. Rep. 270, 217 ͑1976͒. 2 projections of the field are separated V. B. Berestetski, E. M. Lifshitz, and L. P. Pitaevski, Relativistic Quan- tum Theory ͑Pergamon Press, Oxford, 1971͓͒Russ. original, Part 2, .j Nauka, Moscow, 1971, 287 pp͔ ץ 2E 4␲ץ 2E 1ץ x,y x,y x,y 3 2 Ϫ 2 2 ϭ , A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Vacuum Quantum .t Effects in Strong Fields ͓in Russian͔, Moscow, ͑1988͒, 290 ppץ t cץ z cץ 2 2 t. Obviously, the Translated by R. M. Durhamץ/ Eyץt, jyϭ14RE0ץ/ Exץwhere jxϭ4RE0

Electron energy balance in an ionization-unstable plasma in a magnetogasdynamic channel R. V. Vasil’eva, E. A. D’yakonova, A. V. Erofeev, and T. A. Lapushkina

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg ͑Submitted January 20, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 48–53 ͑July 26, 1998͒ An experimental investigation was made of the evolution of ionization instability in a model of a disk-shaped Faraday magnetogasdynamic channel connected to a shock tube in a pure rare gas ͑xenon͒ with an alkaline additive. The main components of the average electron energy balance were determined: the Joule heating power, the average rate of energy transport of the heavy component by elastic collisions, and the average rate of energy lost through ionization. It was found that the electron energy defect which includes losses not taken into account, increases abruptly at supercritical values of the magnetic induction and accounts for approximately half of the Joule heating power. It is concluded that some of the electron energy is transferred to pulsations. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02807-9͔

One of the fundamental problems of ionization-unstable described in Refs. 2–5. The duration of the flow was 400 ␮s. magnetogasdynamic ͑MGD͒ fluxes is the electron energy The experiments were carried out using xenon, the Mach balance, which has mainly been studied using a rare gas with number of the shock wave front in the shock tube was 6.9, an easily ionized alkali metal additive as the working me- and the initial pressure was 26 Torr with weak MGD inter- dium. Under the conditions of MGD channels, the degree of action. The Mach number of the flow in the disk channel was ionization of the additives is in equilibrium with the electron 2–3, the Hall parameter was 1–3, and the degree of ioniza- temperature and the selective heating of the electrons is de- tion of the gas 10Ϫ4 –10Ϫ3. Measurements were made of the termined by the fact that the Joule heating power is equal to effective plasma conductivity, the local values of the conduc- the average rate of energy transport of the heavy component tivity, the Hall parameter, and the electron densities and tem- in elastic collisions.1 In this model, all the characteristics peratures at various radii. Measurements were also made of associated with the formation of plasma inhomogeneities are the flow velocity and gas pressure. The atomic densities and reduced to the fact that the effective plasma conductivity temperatures were reconstructed by comparing the experi- ␴eff , which determines the Joule heating, is less than the mental and calculated data. Ϫ1 average value ͗␴͘. Using the relation ␴effϭ͗␴͘␤eff(␻/͗␯͘) The electron energy balance is compiled for a volume of ͑Ref. 1͒ to determine the degree of separation of the tempera- gas some distance from the entrance to the channel so that it ture of the light and heavy plasma components, we propose a is in a region of weakly varying gasdynamic parameters. It is formula similar to the Kerrebrok formula: assumed that the evolving plasma inhomogeneities do not perturb the gasdynamic flow. The experiments showed that 2 Te ␥M the average Hall current and the average perturbation of the Ϫ1 ϭ ␤ ͗␤͑͘1ϪK ͒2, ͑1͒ ͳ ͩ T ͪ ʹ 3␦ eff y azimuthal field are zero. The estimates also showed that the energy losses caused by the emission and excitation of atoms where ␥ϭc p /c␯ , M is the Mach number of the flow, and also associated with the variation of the average electron ␤eff ,͗␤͘ are the effective and the average Hall parameters, temperature are low compared with the energy dissipated in ␦ϭ1 is the inelastic loss factor, and Ky is the load factor. ionization. Under these assumptions, the electron energy bal- However, an analysis made in Ref. 1 shows that this formula ance has the following form: is not valid in all experiments. Some of the Joule energy may 3m well be converted to pulsations, and here we propose to de- 2 2 e ␴eff͑uB͒ ͑1ϪKy͒ ϭ k͗ne͗͘␯͑͗͘Te͘ϪT͒ termine this fraction of the energy. ma For this study a pure rare gas without alkali additives is 5 ⌬n used as the active medium under conditions when the ioniza- ͗ ͘ ϩ Eiϩ k͗Te͘ ϩ⌬W, tion in the MGD channel is nonequilibrium. Thus, the inelas- ͩ 2 ͪ ⌬t tic energy losses to ionization of the gas must be taken into ͑2͒ account in the electron energy balance. However, the problem still remains: does the Joule heating reduce to the aver- where Ei is the ionization potential. The ionization rate is age energy dissipated in the elastic and inelastic losses. determined from the increment of the electron density at two radii r ϭ7 cm, r ϭ10 cm, ⌬tϭ(r Ϫr )/u, ⌬n ϭ n The experiment was carried out using a disk-shaped Far- 1 2 2 1 ͗ ͘ ͗ e2͘ aday MGD channel connected to a shock tube with K 0, Ϫ͗ne ͘ϩ͗ne ͘(na Ϫna )/na . In the section r2 – r1 the y→ 1 1 2 1 1 KyϷ0.1. The apparatus and the measurement method were plasma flow parameters vary negligibly, and thus the average

FIG. 1. Joule heating power W j ͑1͒, elastic energy losses W2 ͑2͒, and energy dissipated in ionization W3 ͑3͒ for various magnetic inductions; Bcr is the critical magnetic ﬁeld.

values of the plasma parameters for this section are used to determine the elastic losses. The values of ␴eff were also FIG. 2. a—Relative electron energy defect for various magnetic fields; b— measured for this section. The average energy transport rate fluctuations of the electron density at two radii for various magnetic inductions. The numbers on the curves are the values of r in centimeters. is defined as ͗␯͘ϭna͗ce͗͘Qa͘(1ϩ͗ne͘/na͗Qi͘/͗Qa͘), where ͗Qa͘ and ͗Qi͘ are the energy transport cross sections for the atoms and ions, which are determined for ͗Te͘. The electron energy defect ⌬W includes the additional electron losses neglected in the energy balance. Figure 1 gives the the energy defect. The indicated range of critical magnetic components of the energy balance for subcritical and super- field values can be explained by the fact that the occurrence critical values of the magnetic field. The Joule heating slows of fluctuations in a fixed cross section depends on the mag- at maximum magnetic induction because the stream slows netic induction and on the time taken for the evolution of appreciably as a result of the action of the ponderomotive ionization instability. Since a longer lifetime is required for force. The elastic energy losses increase because the electron the plasma volume to attain larger distances, the lower the density is greater and because the energy transport rate in- value of the magnetic induction, the larger the radius before creases with electron temperature. The inelastic energy the onset of instability. As the instability while the plasma losses increase since the ionization rate increases with in- volume moves from r1 to r2 , the fluctuations grow. It can be creasing field. seen from Fig. 2 that an energy defect appears when the Figure 2a gives the energy defect relative to the Joule inhomogeneities form and increases as the fluctuations in the heating power. The values of ⌬W/W j for subcritical mag- electron density increase. netic fields are taken as a basis for comparison. The observed Thus, in fully developed ionization instability, the elec- difference of ⌬W/W j from zero in this region is within ex- tron energy defect caused by structuring of the plasma ac- perimental error. The relative error in the determination of counts for approximately half the Joule heating power. This ⌬W/W j when these values are compared for different mag- indicates that not only does the growth of the instability pro- netic inductions is less than 20%. The slight decrease ob- duce characteristic features in the current flow but various served at Bϭ1 T is evidently caused by a reduction in the paths of electron energy losses in pulsations probably also Joule heating. The fluctuations of the electron density shown exist. However, a more comprehensive model of ionization in Fig. 2b reveal correlations between the fluctuations and turbulence is required to determine these losses. 564 Tech. Phys. Lett. 24 (7), July 1998 Vasil’eva et al.

The authors would like to thank the Russian Fund for ͓in Russian͔, A. F. Ioffe Physicotechnical Institute Press, St. Petersburg, Fundamental Research for supporting this work Project No. 206 pp. ͑ 3 96-02-16904 . A. V. Erofeev, R. V. Vasil’eva, A. D. Zuev, T. A. Lapushkina, E. A. ͒ D’yakonova, and A. A. Markhotok, in Proceedings of the 12th Interna- tional Conference on MGD Electrical Power Generation, Yokohama, Japan, 1996, Vol. 1, pp. 74–82. 4 T. A. Lapushkina, R. V. Vasil’eva, A. V. Erofeev, and A. D. Zuev, Zh. 1 A. V. Nedospasov and V. D. Khait, Principles of the Physics of Processes Tekh. Fiz. 67͑12͒,12͑1997͓͒Tech. Phys. 42, 1382 ͑1997͔͒. ´ in Low-Temperature Plasma Devices ͓in Russian͔,Energoatomizdat, 5 T. A. Lapushkina, E. A. D’yakonova, and R. V. Vasil’eva, Pis’ma Zh. Moscow ͑1991͒, 224 pp. Tekh. Fiz. 24͑2͒,58͑1998͓͒Tech. Phys. Lett. 24,66͑1998͔͒. 2 R. V. Vasil’eva, A. L. Genkin, V. L. Goryachev et al., Nonequilibrium Rare-Gas Plasma with Nonequilibrium Ionization and MGD Generators Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Electron energy in exoelectronic emission from a ferroelectric L. M. Rabkin and V. N. Ivanov

Rostov State University ͑Submitted January 8, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 54–57 ͑July 26, 1998͒ An analysis is made of the change in the potential in the plane of a grid electrode on the surface of a ferroelectric caused by exoelectronic emission under the inﬂuence of a pulsed electric ﬁeld. The potential is calculated by means of integral equations from electrostatics. An estimate is made of the possible initial energies of electrons leaving the surface of the ferroelectric. © 1998 American Institute of Physics. ͓S1063-7850͑98͒02907-3͔

Exoelectronic emission from the surface of ferroelectrics electric. We shall confine our analysis to the limiting case of has recently been studied to gain a deeper understanding of an external electric field so strong that the spontaneous po- the physical properties of these materials and also with a larization is completely oriented in the direction of the elec- view to using these materials to develop highly efficient tric field and the dielectric properties of the ferroelectric are pulsed emitters for vacuum devices.1,2 In this last case, the determined by two polarization mechanisms, which we shall ferroelectric is exposed to a pulsed electric field. However, assume to be linear. Disregarding the existing anisotropy ͑the the mechanism responsible for the high initial energies of the difference between the permittivity measured parallel to the emitted electrons ͑up to a few kilo-electronvolts͒ has not direction of spontaneous polarization and that measured in been discussed in the literature, even though the spread of the perpendicular direction͒ we describe the dielectric by the the initial velocities is the limiting factor in the application of scalar relative permittivity ␧. these emitters. Here we examine a model to estimate the We position the origin of the Cartesian coordinate sys- order-of-magnitude energy of the emitted electrons. tem at the center of a slit of width 2a, with the z axis parallel In the steady state, surface polarizing charges are to the slit, the x axis perpendicular to the slit parallel to the screened by free charges distributed along the surface of the conducting half-planes, and the y axis running from the ferroelectric. When a fairly strong polarization-reversing ferroelectric into vacuum. The problem involves determining electric field is abruptly applied, the polarizing charges vary the potential within the slit ␾(x)(͉x͉рa,yϭ0) assuming rapidly, whereas the free charges do not vary significantly that the charge density distribution ␴(x)(͉x͉Ͻa) within the during polarization reversal since the emitted charge ac- slit is known and that the electrode potential is zero, i.e., counts for a small fraction of the polarizing charge and the ␾(x)ϭ0(͉x͉Ͼa). Since in this problem charges are only conductivity of the ferroelectric is low. A change in the total present at the planar interface of the two dielectrics, the po- surface charge alters its potential, which determines the en- tential and the electric field may be sought in a homogeneous ergy of the emitted electrons. medium with the equivalent relative permittivity ␧eϭ(␧ In the usual experimental setup to observe exoelectronic ϩ1)/2Х␧/2. It is convenient to introduce the electric field emission, a grounded conducting film electrode in the form component in the slit plane EϭEy(x) as the unknown, for of a grid is deposited on the emitting surface of a ferroelec- which electrostatics methods using Fourier transformation3 tric wafer and an external electric field which reverses the yield the singular integral equation polarization of the ferroelectric is generated by applying a ϩa pulsed voltage to a solid conducting electrode on the other Ϫ1 Ϫ1 ␲ E͑x1͒͑x1Ϫx͒ dx1ϭ␴͑x͒/͑2␧e␧0͒, ͑1͒ surface of the wafer. We shall assume that the wafer is an ͵Ϫa order of magnitude thicker than the width of the slits in the Ϫ1 conducting grid and that in calculations of the potential dis- where ␧0ϭ8.85ϫ10 F/m is the permittivity of free space. tribution in the plane of the grid, created by the charges Assuming that the function ␴(x) is even and E(x)is 4 distributed in this plane, the influence of the solid electrode odd, the solution of Eq. ͑1͒ is written as follows: can be neglected. Since the purpose of these calculations is ϩa to estimate the order-of-magnitude potential at the slit, for 2 2 1/2 Ϫ1 2 2 1/2 E͑x͒ϭϪ͑2␲␧e␧0͑a Ϫx ͒ ͒ ͑a Ϫx1͒ ␴͑x1͒ simplicity we shall consider a single slit between two con- ͵Ϫa ducting half-planes located on the surface of a semi-infinite ϫ͑x Ϫx͒Ϫ1dx . ͑2͒ ferroelectric. 1 1 The problem of the potential distribution created by un- In relations ͑1͒ and ͑2͒ the integrals are understood in the compensated surface charges formed at the slit after polar- sense of the principal value. The value of the integral is ization reversal of the ferroelectric induced by an external obtained by integrating E(x): electric field is difficult to solve in a general formulation because of the nonlinear dielectric properties of the ferro- ␾͑x͒ϭ͑2␲␧e␧0͒.

ϩa 2 2 2 2 2 1/2 ␾maxХE0a/2. ͑7͒ ␴͑x1͒ln͑͑a Ϫxx1ϩ͑͑a Ϫx ͒͑a Ϫx1͒͒ ͒/ ͵Ϫa On the basis of equalities ͑5͒ and ͑7͒, which were ob- ͑a͉xϪx ͉͒͒dx . ͑3͒ 1 1 tained for the limiting cases of strong and weak polarization- In order to estimate the potential, we assume that the reversing fields, we can predict in general that the maximum surface charge density ␴(x) is constant and we set this equal energy of the emitted electrons will increase with increasing to the spontaneous polarization Ps . We then find slit width. Note that the assumption used in the model, that 2 2 1/2 the external electric field at some distance from the slit is ␾͑x͒ϭPs /͑2␧0␧e͒͑a x ͒ . ͑4͒ uniform with the constant strength E0 , presupposes that as The maximum energy of the emitted electrons is ob- the width of the slit increases, the thickness of the ferroelec- tained using the highest value of the potential ␾max attained tric wafer and the amplitude of the polarization-reversing at the center of the slit (xϭ0) voltage pulse increase proportionately. We also postulate that the increase in this energy with increasing polarization- ␾maxϭPsa/͑2␧e␧0͒. ͑5͒ reversing field should be slower in strong fields. Taking the values for barium titanate ferroelectrics Ps To sum up, these calculations of the potential distribu- 2 Ϫ4 ϭ0.25 C/m , aϭ10 m, and ␧ϭ1000, we obtain ␾max tion at the surface of a ferroelectric in the plane of a grid ϭ2.8 kV, which is of the same order of magnitude as the electrode have yielded estimates of the possible initial ener- 1 experimental results. gies of the electrons leaving the surface of the ferroelectric as The potential distribution within the slit can also be cal- a result of exoelectronic emission under the action of a culated in the other limiting case of a weak external field pulsed electric field. when the ferroelectric can be described in the linear approximation of the scalar permittivity, ␧ӷ1. Taking the external field in the ferroelectric at some distance from the slit to be 1 perpendicular to the surface of the ferroelectric and uniform H. Gundel, H. Rige, E. J. N. Wilson et al., Ferroelectrics 100,1͑1989͒. 2 J. Asano, T. Imai, M. Okuyama, and Y. Hamakawa, Jpn. J. Appl. Phys., with the strength E0 , and using the method which was ap- Part 1 31, 3098 ͑1992͒. plied to derive relations ͑1͒ and ͑2͒,wefind 3 I. N. Sneddon, Fourier Transforms ͑McGraw-Hill, New York, 1951, re- 2 2 printed Dover, New York, 1995; IL, Moscow 1995͒. ␾͑x͒ϭ0.5E͑␧Ϫ1͒/͑␧ϩ1͒͑a Ϫx ͒, ͑6͒ 4 F. G. Tricomi, Integral Equations ͑Interscience, New York, 1957; IL, i.e., in this approximation regardless of the specific value of Moscow, 1960͒. the relative permittivity of the ferroelectric, we obtain Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Lasing in submonolayer InAs/AlGaAs structures without external optical conﬁnement B. V. Volovik, A. F. Tsatsul’nikov, N. N. Ledentsov, M. V. Maksimov, A. V. Sakharov, A. Yu. Egorov, A. E. Zhukov, A. R. Kovsh, V. M. Ustinov, P. S. Kop’ev, Zh. I. Alferov, I. E´ . Kozin, M. V. Belousov, and D. Bimberg

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg; Institute of Physics, St. Petersburg State University, Petrodvorets; Insitut fu¨r Festko¨rperphysik, Technische Universita¨t Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany ͑Submitted February 13, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 58–66 ͑July 26, 1998͒ Structures having a set of planes with submonolayer InAs inclusions in an AlGaAs matrix were fabricated and studied. Lasing was observed as a result of optical excitation. It is shown that lasing takes place via the ground state of excitons localized at InAs islands and may be achieved without external optical conﬁnement of the active region by wide-gap layers of lower refractive index. The low threshold excitation density shows that these structures may be used to develop low-threshold injection lasers in the visible range, exciton waveguides, and self-contained microcavities. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03007-9͔

INTRODUCTION EXPERIMENT Studies of growth processes in an InAs–GaAs system The structures were grown by molecular beam epitaxy have shown that the depositing submonolayer InAs coatings on semi-insulating GaAs͑100͒ substrates in a RIBER 32P leads to the spontaneous formation of an array of islands system with a solid-state As source under standard condi- around one monolayer high with a characteristic width of tions of As enrichment. The structure studied in detail here around 4 nm and small dispersion of the lateral was fabricated by growing a 0.3 ␮m thick GaAs buffer layer 1,2 dimensions. It is deduced from Refs. 3–5 that submono- on the substrate, followed by a 0.7 ␮m thick Al0.32Ga0.68As layer structures possess unique optical properties. An appre- layer, and a thin ͑10 nm͒ Al0.4Ga0.6As carrier-confinement ciable increase in the exciton binding energy was observed layer. Another Al0.32Ga0.68As ͑20 nm͒ layer was then fol- as a result of lateral quantization.3 In addition, a high lumi- lowed by growth of the active region which consisted of 20 nescence efficiency and high exciton oscillator strength were ultrathin GaAs quantum wells 1 nm thick with 0.5 ML InAs observed even for an InAs layer of ultrasmall average deposited at the center of each. The wells were separated by 4 thickness. Lifting of the momentum selection rules was also Al0.32Ga0.68As barriers 50 Å thick. A Al0.32Ga0.68As layer 5 demonstrated. Later it was shown that in II–VI structures 100 nm thick and a thin ͑10 nm͒ Al0.4Ga0.6As were then with CdSe/ZnMgSSe submonolayers the localization of ex- grown. The growth temperature was 600 °C, but during citons at islands and lifting of the momentum selection rules growth of the active region the temperature was reduced to allows lasing to be achieved via the exciton ground state in 485 °C to avoid segregation and re-evaporation of In atoms submonolayer superlattices.6 Lasing was achieved in struc- from the surface. tures without optical confinement by wide-gap layers of Luminescence was excited by an Arϩ laser (␭ lower refractive index as a result of modulation of the refrac- ϭ514.5 nm, Pϭ500 W/cm2), a pulsed nitrogen laser (␭ tive index near the exciton resonance.7 ϭ337 nm, pulse power 1 MW/cm2), and a halogen lamp So far, no similar effects have been reported in III–V whose light was passed through a monochromator. The lu- systems where the low electron mass makes the exciton lo- minescence signal was recorded using a cooled photomulti- calization energy at submonolayer islands extremely low plier. ͑less than 50 eV for InAs submonolayers in GaAs4͒.We proposed and fabricated structures with InAs submonolayer RESULTS AND DISCUSSION islands in an AlGaAs matrix in which lasing was achieved in structures without external optical confinement as a result of Figure 1 shows spectra of the photoluminescence, lumi- an increase in the exciton localization energy in the islands. nescence excitation, and optical reflection from this sample. Lasing is observed at low optical excitation densities which It can be seen that at low levels of excitation with a photon opens up extensive prospects for using this effect in visible energy higher than the Al0.32Ga0.68As band gap, the spectrum lasers to improve the optical confinement, in exciton contains a single peak which is attributed to radiative recom- waveguides, and self-contained microcavities, i.e., vertically bination of excitons localized at InAs islands. Note that this emitting lasers, where modulation of the refractive index peak is shifted appreciably in the long-wavelength direction near the exciton resonance region in the quantum dot allows relative to the energy of the optical transition predicted for a self-contained tuning of the cavity mode and the gain profile. GaAs quantum well 1 nm thick. This indicates that the exci-

FIG. 1. Photoluminescence, luminescence excitation, and optical reflection spectra of this structure. FIG. 2. a—Photoluminescence spectra for various excitation energies, indicated near the spectra. The position of the lines QD1 ͑reflection͒ and QW is taken from the optical reflection spectrum; b—photoluminescence spectra tons are strongly localized at InAs islands. The photolumi- for various excitation densities. The luminescence was excited by an Arϩ laser. The dashed lines give the representation the spectrum as two Gaussian nescence excitation spectrum has two characteristic features curves. denoted as QD1 and QW. Note that these lines exactly match the exciton characteristics observed in the reflection spectrum, although that associated with QW is substantially less A cavity of length Lϭ1 mm was cut to investigate las- broadened. The position of the QD1 and QW characteristics ing. Figure 3a gives the integrated luminescence intensity as in the luminescence excitation spectrum remains constant as a function of the excitation density. It can be seen that at the detection energy varies. We attribute these characteristics 800 W/cm2, the intensity increases sharply, evidently be- to states caused by excitons localized at InAs islands and cause of the onset of stimulated emission at a photon energy exciton states in the ultranarrow GaAs quantum well ͑QW͒. of 1.750 eV ͑Fig. 3b͒. Note that for injection pumping, the This interpretation is quite consistent with the spectral posi- threshold density of 800 W/cm2 corresponds to a threshold tion of the peaks and their half-width. The photolumines- current of ϳ200 A/cm2, which is the upper limit for the cence spectrum obtained for resonant excitation with a pho- threshold current because of the unknown fraction of carriers ton energy below the quantum-well band gap reveals two lost from the surface as a result of the surface nature of the lines, one ͑QD2, 1.729 eV͒ shifted substantially in the long- nitrogen laser excitation. Thus, structures with InAs/AlGaAs wavelength direction compared with QD1 and another peak submonolayers can be used to develop low-threshold injec- at 1.750 eV, close in energy to QD1 ͑Fig. 2a͒. Under low- tion lasers in the visible range even without external optical density nonresonant excitation, the QD1 line is less effi- confinement ͑the average difference in the Al composition ciently excited, evidently because carriers with a high mo- between the submonolayer superlattice and the confining lay- mentum in the plane of the GaAs quantum well are more ers is only 5%͒. The structure is shown schematically in the efficiently trapped at islands with a high localization energy inset to Fig. 3c. ͑QD2͒. However, at high excitation densities the QD1 line Figure 3c gives the temperature dependence of the always predominates in the spectrum ͑Fig. 2b͒. Similar ef- threshold excitation density. It can be seen that from ϳ70 K fects were also noted in submonolayer CdSe/ZnSe the threshold density increases exponentially and by approxi- 8 structures. mating the dependence using the formula PϭP0 As the excitation density is increased further, the line is ϫexp(T/T0) we can determine T0ϭ30 K. This behavior of shifted to a position close to QD1 in the reflection spectrum the threshold density can be attributed to the thermal emis- ͑Fig. 2b͒. In this case, the QD1 peak continues to dominate sion of carriers from islands. We observed lasing up to 170 in the photoluminescence spectrum and the QD1 states do K. The thermal stability of the emission may be improved by not saturate which evidences their high density. At high ex- increasing the localization energy which can be achieved by citation densities a stimulated emission line is observed from using wider-gap confining layers or by using vertically cor- the end of the cavity. related island growth.9 Tech. Phys. Lett. 24 (7), July 1998 Volovik et al. 569

matrix without external optical conﬁnement. The lasing occurs at low excitation densities, which makes these structures promising for the development of low-threshold injection lasers in the visible range, exciton waveguides, and self- contained microcavities.

This work was supported by the Russian Fund for Fun- damental Research, the Scientiﬁc Program ‘‘Physics of Solid-State Nanostructures,’’ and the Volkswagen Founda- tion. One of the authors ͑B.V.V.͒ would like to thank the Soros Foundation.

1 P. D. Wang, N. N. Ledentsov, C. M. Sotomayor Torres, P. S. Kop’ev, and V. M. Ustinov, Appl. Phys. Lett. 64, 1526 ͑1994͒. 2 V. Bressler-Hill, A. Lorke, S. Varma, P. M. Petroff, K. Pond, and W. H. Weinberg, Phys. Rev. B 50, 8479 ͑1994͒. 3 P. D. Wang, N. N. Ledentsov, C. M. Sotomayor Torres, I. N. Yassievich, A. Pakhomov, A. Yu. Egorov, P. S. Kop’ev, and V. M. Ustinov, Phys. Rev. B 50, 1604 ͑1994͒. 4 M. V. Belousov, N. N. Ledentsov, M. V. Maximov, P. D. Wang, I. N. Yassievich, N. N. Faleev, I. A. Kozin, V. M. Ustinov, P. S. Kop’ev, and C. M. Sotomayor Torres, Phys. Rev. B 51,14͑1995͒. 5 A. A. Sirenko, T. Ruf, N. N. Ledentsov, A. Yu. Egorov, P. S. Kop’ev, V. M. Ustinov, and A. E. Zhukov, Solid State Commun. 97, 169 ͑1996͒. 6 N. N. Ledentsov, I. L. Krestnikov, M. V. Maximov, S. V. Ivanov, S. L. Sorokin, P. S. Kop’ev, Zh. I. Alferov, D. Bimberg, N. N. Ledentsov, and C. M. Sotomayor Torres, Appl. Phys. Lett. 69, 1343 ͑1996͒; ibid., 70, 2677 ͑1997͒. 7 I. L. Krestnikov, S. V. Ivanov, P. S. Kop’ev, N. N. Ledentsov, M. V. Maximov, A. V. Sakharov, S. V. Sorokin, A. Rosenauer, D. Gerthsen, FIG. 3. a—Integrated photoluminescence intensity from end versus excitation density; b—photoluminescence spectra from end for various excitation C. M. Sotomayor Torres, D. Bimberg, and Zh. I. Alferov, Mater. Sci. Eng. densities shown on the spectra. The position of the QD1 and QW lines is B ͑in press͒. 8 taken from the optical reﬂection spectrum; c—temperature dependence of I. L. Krestnikov, M. V. Maksimov, S. V. Ivanov, N. N. Ledentsov, S. V. Sorokin, A. F. Tsatsul’nikov, O. G. Lyublinskaya, B. V. Volovik, P. S. threshold excitation density. The value of T0 was obtained using the ap- Kop’ev, and C. M. Sotomayor Torres, Fiz. Tekh. Poluprovodn. 31, 230 proximation PϭP0 exp(T/T0). The structure is shown in the inset. ͑1997͓͒Semiconductors 31, 127 ͑1997͔͒. 9 I. L. Krestnikov, M. V. Maximov, A. V. Sakharov, P. S. Kop’ev, Zh. I. Alferov, N. N. Ledentsov, D. Bimberg, and C. M. Sotomayor Torres, in CONCLUSIONS Proceedings of the Eighth International Conference on II–VI Compounds, Grenoble, France ͑J. Cryst. Growth, in press͒. We have demonstrated lasing in optically pumped structures formed by submonolayer InAs inclusions in an AlGaAs Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Thermionic valve effect and cathode crater rhythm in a low-voltage cold-cathode vacuum arc M. K. Marakhtanov and A. M. Marakhtanov

N. E. Bauman State Technical University, Moscow ͑Submitted November 6, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 67–72 ͑July 26, 1998͒ A vacuum arc with a cold cathode burns rhythmically. The period of the rhythm is determined by a thermionic valve effect which occurs between the hot crater and the cold cathode as a result of the brief retention of the heat in the crater by the oppositely propagating electron ﬂux. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03107-3͔

The experiments were carried out using the Bulat-6 voltage Vϭ24 V and current Iϭ67– 72 A, using black and vacuum-arc evaporation system and were described in white film ͑200 ASA͒, an exposure time of 1/30– 1/500 s, Ref. 1. and a Zenit camera without any light filter. The rhythmic nature of cathode craters can be seen from The rhythm can be identified on the oscilloscope traces a photograph of the cathode where the motion of the cathode in Figs. 1c and 1d. It has been established2 that any appear- spot is imaged as a chain of light Fig. 1a . The number of ͑ ͒ ance of a cathode spot at a new site is preceded by its dis- links in the chain is proportional to the exposure time. Each appearance at an old site. The arc current should change link consists of a cathode spot D from luminescence around s abruptly between these events, inducing voltage surges in the a crater D, and a switching path of length L and width ␦ ͑Figs. 1b and 2a͒. The path probably consists of spots of Rogowski loop. The time between the voltage minima on the smaller diameter ␦ burning around small cathode craters.1 oscilloscope traces of the loop can then be taken to be equal Since the links of the chain possess approximately the same to the period of the cathode spot rhythm. The average period Ϫ4 brightness, we can assume that the average period of the is t2ϭ1.4ϫ10 s for a Ti cathode at Iϭ92 A ͑Fig. 1c͒ and Ϫ3 rhythm t1ϭ␶/N is also the same for all the links, where ␶ is t2ϭ2.9ϫ10 s for an Al cathode at Iϭ73 A ͑Fig. 1d͒. The the frame exposure time and N is the number of links in a Rogowski loop was mounted on an 8 mm diameter conduc- Ϫ4 particular frame. The average period t1ϭ6ϫ10 s was de- tor near the cathode. The coil consisted of 270 turns on a termined from 16 photographs of a Ti cathode in an arc of ferrite ring of 16 mm diameter and 3ϫ4 mm cross section.

FIG. 1. Experimental illustration of a cathode crater rhythm: a—light chain of 42 cathode spots on the surface of a Ti cathode of diameter 56 mm, ␶ϭ1/60 s, Iϭ69 A, Vϭ24 V; b—diagram showing a link in the

chain consisting of a cathode spot Ds and a switching path L, with the average value Dsϭ2.82 mm; c— oscilloscope trace of output voltage of Rogowski loop in arc with Ti cathode, Iϭ92 A; d—the same in an arc with an Al cathode, Iϭ73 A.

file and short lifetime ͑Fig. 2a and Ref. 1͒. It is located on a cold conductor from which an unusually dense electron current flows into the crater. A thermal barrier is created between the crater and the cathode as a result of this current. Heat is transferred in the metal by the electronic and lattice conduction.3,4 For Zn, Al, and Cu ͑for which the electrical conductivity is ␴Ͼ0.25␴Cu) electronic heat transfer predominates and their real thermal conductivity is higher than ␴ ͑Ref. 4͒. In metals with low ␴ ͑Fe, Ti, and Cr͒ both mechanisms are involved in heat transfer and the real thermal conductivity increases with the ratio ␭/␴ ͑Ref. 4͒. Almost all the heat ¯q leaves the crater through the bottom 2, since its area is much greater than that of the side surface 4 ͑Fig. 2b͒. The density of the electron flux to the crater is 2 jeӍI/D because of its planar profile ͑Fig. 2a and Ref. 1͒. ¯ The flux Je moves ‘‘forcibly’’ under the action of the power supply source and transfers the heat flux ¯q back into the crater ͑Fig. 2b͒, producing a thermionic valve effect. The effect takes place over the period t when the crater area increases to the maximum D2, i.e., as long as the electron drift velocity exceeds the critical value VeуVecϭJe /(ene) ϭ0.009, 0.026, 0.259, 0.133, 0.092, and 0.307 m/s, respectively. Here we find jeϭ j ͑see Ref. 1͒, and the free electron 28 concentration in the appropriate metal is neϭ13.10ϫ10 , 18.06ϫ1028, 8.45ϫ1028, 17.0ϫ1028, 22.64ϫ1028, and 24.99ϫ1028 mϪ3 ͑Ref. 3͒. The valve retains in the crater FIG. 2. Schematic of thermionic valve in cold cathode: a—average diameter exactly that amount of energy ␧ required for the arc to burn D and depth h of cathode crater in highly conducting metals, current and arc voltage: Zn 48 Aϫ16.7 V, Al 52 Aϫ36 V, Cu 105 Aϫ24.5 V; b— for the time t. After depletion of the crater mass, switching components of valve: 1—molten metal, 2—bottom of crater, 3—valve layer of the cathode spot within the hot circle R is initiated ͑see of heated but solid metal, 4—side surface of crater, 5—cold cathode metal; Ref. 1͒. the flux ¯q of hot electrons ͑shown by the long arrows͒ transfers the kinetic The valve is not ideal and a fraction of the arc power energy of the electrons from the crater 1 to the cold cathode 5; the opposite f ϭQ /IV escapes to the cathode, where Q is the heat ¯ w w electron flux Je transfers electrical energy from the arc power supply to the crater; the thermal energy is partially ‘‘blocked’’ in the crater as a result of transferred by the cooling water. This fraction is f ϭ0.033, ¯ 0.182, 0.266, 0.412, 0.283, and 0.271 for Zn, Al, Cu, Fe, Ti, the high flux density Je . and Cr cathodes, respectively. Changes in D, t, and R are caused by the thermal con- The output impedance of the loop was 75 k⍀. ductivity ␭ and Tb ͑see Ref. 1͒. With decreasing ␭ or Tb , less The cathode crater rhythm can be determined from the heat escapes to the cathode and f ZnϽ f AlϽ f Cu .Asqde- known dimensions D and h ͑Fig. 2a and Ref. 1͒ and also creases, the heat density je which can be retained in the from the rate of cathode sputtering S which is 15.20 crater decreases and the diameter D is larger for low-melting Ϫ6 Ϫ6 Ϫ6 Ϫ6 ϫ10 , 4.05ϫ10 , 11.76ϫ10 , 11.57ϫ10 , 4.42 metals DZnϾDAlϾDCu . Growth of the crater to large D re- Ϫ6 Ϫ6 ϫ10 , and 2.27ϫ10 kg/s for Zn, Al, Cu, Fe, Ti, and Cr quires a long time ͑recall that t2AlϾt2Ti , and also the series 2 cathodes, respectively. The density ␳ of these metals is for t3). Over the time t a large area R is heated around the 7150, 2700, 8930, 7880, 4540, and 7150 kg/m3, respec- crater and a new crater appears at a larger distance R, being tively, and the average crater mass is Mϭ␳D2hϭ7.62 located on the cold metal where ␴ has a maximum ͑see ϫ10Ϫ8, 0.32ϫ10Ϫ8, 0.49ϫ10Ϫ8, 0.11ϫ10Ϫ8, and 0.032 Fig. 2 in Ref. 1͒. For Fe, Ti, and Cr these changes depend on ϫ10Ϫ8 kg, respectively. The average period of the crater ␭/␴. Ϫ4 Ϫ4 rhythm is then t3ϭM/Sϭ50.1ϫ10 , 7.9ϫ10 , 4.2 The existence of a thermionic valve is limited by ϫ10Ϫ4, 3.5ϫ10Ϫ4, 2.5ϫ10Ϫ4, and 1.1ϫ10Ϫ4 s for the ap- the time t during which the crater mass Mϭ␳V can accumu- propriate metal. Here it is assumed that S is constant and the late thermal energy ␧ which is transferred by the current j switching time of the spot can be neglected. to the crater volume V together with the electrical power Ϫ4 Thus, for titanium we have t1ϭ6ϫ10 s, t2ϭ1.4 Q. The condition for the existence of a crater Qу␧/t Ϫ4 Ϫ4 ϫ10 s, and t3ϭ2.5ϫ10 s. The similarity between the is then rewritten as jeEVу␳VC(1Ϫf )/t. We take values of t indicates that the process of crater nucleation and Eϭ j e /␴, ␳ϭnAmp , jeϭenevec ; for a Ti cathode disappearance on the cathode is regular. We shall determine we have CϭC1(T f ϪTc)ϩC f ϩC2(TbϪT f ), where C1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 the reason for this regularity. ϭ515 J kg K and C2Ӎ700 J kg K are the specific The crater has a high temperature ͑but probably below heats of the solid and liquid Ti, C f ϭ314 000 J/kg is the spe- the boiling point Tb of the metal͒ and also has a planar pro- cific heat of fusion of Ti, Tcϭ300 K, T f ϭ1881 K, and T 572 Tech. Phys. Lett. 24 (7), July 1998 M. K. Marakhtanov and A. M. Marakhtanov

ϭ3560 K are the temperature of the cold cathode, the melt- The density je obtained as a result of this estimate is close to 9 Ϫ2 ing point, and the boiling point of Ti, respectively, ␴Ӎ3.05 the experimental value jϭ3.33ϫ10 Am for Ti ͑Ref. 1͒, 5 Ϫ1 Ϫ1 ϫ10 ⍀ m at TϷT f , Aϭ48 u, tϭ(t1ϩt2ϩt3)/3 which suggests that a thermionic valve does exist, determin- ϭ(6ϩ1.4ϩ2.5)ϫ10Ϫ4/3ϭ3.3ϫ10Ϫ4 s. We rewrite this ing the rhythm t in a low-voltage cold-cathode vacuum arc. last inequality and we estimate the value of je for a titanium The physical quantities used here were taken from Ref. 5. cathode 1 M. K. Marakhtanov and A. M. Marakhtanov, Pis’ma Zh. Tekh. Fiz. m p ␴ A 24͑13͒,14͑1998͓͒Tech. Phys. Lett. 24, 504 ͑1998͔͒. jeу C͑1Ϫf ͒ 2 I. G. Kesaev, Cathode Processes in an Electric Arc ͓in Russian͔, Nauka, e Vec t Moscow p. 6. 3.05ϫ105 48 3 C. Kittel, Introduction to Solid State Physics, 5th edition ͑Wiley, New Ϫ8 York, 1976; Nauka, Moscow, 1978, 791 pp. . ϭ10 Ϫ4 ͒ 0.092 3.3ϫ10 4 V. E. Mikryukov, Thermal Conductivity and Electrical Conductivity of Metals and Alloys ͓in Russian͔, Metallurgizdat, Moscow ͑1959͒, 260 pp. 5 ϫ͓515ϫ͑1881Ϫ300͒ϩ314 000 A. P. Babichev, N. A. Babushkina, A. M. Bratkovski et al.,inHandbook of Physical Quantities, edited by I. S. Grigor’ev and E. Z. Melikhov ͓in ϩ700͑3560Ϫ1881͔͒͑1Ϫ0.283͒ Russian͔,E´nergoatomizdat, Moscow ͑1991͒, 1232 pp.

ϭ7.96ϫ109 A/m2. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Investigation of the inﬂuence of the thermal prehistory of zinc and cadmium melts on their crystallization V. D. Aleksandrov and A. A. Barannikov

Donbass State Academy of Building and Architecture ͑Submitted January 15, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 73–78 ͑July 26, 1998͒ Methods of thermal analysis were used to study how the temperature and hold time of zinc and cadmium melts and their rate of cooling inﬂuence their crystallization. It was established that zinc and cadmium melts exhibit no precrystallization supercooling. It was conﬁrmed that neither cooling at a rate between 0.001 and 8 K/s, nor heating Zn and Cd melts to temperatures 200 K above the appropriate melting point, nor isothermal ‘‘annealing’’ for up to 4 h, nor using a mass between 50 mg and 50 g resulted in supercooling during crystallization. The results are analyzed using a cluster model of the crystallization of Zn and Cd melts based on the similarity between the crystal-chemical parameters of the solid and liquid phases of these substances. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03207-8͔

Almost all simple substances show a tendency toward using a KSP-4 potentiometer witha1mVscale. In some precrystallization supercooling1–7 when their melts are cases, unshielded XA thermocouples inserted directly in the cooled below the melting point TL . However, no supercool- sample were used. ing, ⌬TϪ, during crystallization of solid zinc and cadmium Between 50 and 100 heating and cooling cycles were samples has been reported in the literature. Data presented in carried out on the same sample under the same experimental Ref. 8 on the supercooling of Cd particles of between 4 and conditions ͑at the same rate of cooling, for the same mass, 44 nm in aluminum alloys up to ϳ60 K have no relevance to and so on͒. solid materials, since particles of this size are a completely The reliability of the results was conﬁrmed by repeating different substance and form the subject of special investiga- a series of experiments many times for each sample. The tions in small-particle physics.9 error of the temperature measurements was 0.15–0.20 K/s. The lack of publications on supercooling in solid Zn and It was established that under all these experimental con- Cd is fairly surprising, considering the ready availability of ditions, for all Zn and Cd samples, no precrystallization su- these elements, their relatively low melting points ͑692.75 K percooling was observed regardless of mass, rate of cooling, for Zn and 594.26 K for Cd͒͑Ref. 10͒, and their noncorro- crucible material, atmosphere, temperature, and isothermal sive properties under normal conditions. hold time. The crystallization process exhibited equilibrium Here we describe the results of a study of the inﬂuence behavior and took place at a crystallization temperature cor- of the thermal prehistory of zinc and cadmium on their crys- responding to the melting point of the material and the su- 3–7 Ϫ tallization using methods of ballistic thermal analysis. percooling (⌬T ϭTLϪT) was zero. The samples were high-purity zinc and cadmium weigh- By way of examples, Figs. 1 and 2 show cooling ther- ing 50 mg, 0.1, 1.0, 2.0, 6.0, and 50 g with 5 or 6 test being mograms for 1 g zinc and cadmium samples, respectively, at made for each weight. The crucibles were made of alundum rates of 0.3 K/s in air. In both cases, the preheating of the and quartz. The tests were carried out in air, in a rare-gas melt was ϳ35 K above the melting point of the appropriate Ϫ5 atmosphere ͑argon͒, in a vacuum of up to ϳ10 Torr under material. It can be seen that at temperatures TL a crystalliza- conditions of dynamic pumping in a vacuum station, and in a tion plateau is observed without any initial supercooling. ‘‘static’’ vacuum (ϳ10Ϫ3 Torr) using a Stepanov ampoule. Preliminary isothermal holding of the melt for between Different rates of cooling were used: 0.001, 0.02, 0.3, 1.0, 1minand4hattemperatures 10, 20, 50, and 100 K above and 8.0 K/s. the melting point did not cause any supercooling. Thermal cycling was carried out in a specially prepared However, other substances ͑such as Bi, Sb, Sn, Se, Te, ‘‘gradient-free’’ resistance furnace over a temperature range InSb, PbCl2,H2O, and In–Sb alloys͒ always exhibited pre- covering the melting and crystallization range. A program- crystallization supercooling in our systems.3–7 For compari- mable controller allowed the thermal heating and cooling son Fig. 3 shows crystallization isotherms for bismuth ͑1g͒ cycles to be carried out in a particular regime. The lower recorded by us under similar crystallization conditions. It can limit of the thermal cycles was kept constant ͑at 50 K below be seen that bismuth exhibits the supercooling ⌬TϪ TL) while the upper limit was varied at two degree intervals ϭ21.4 K at a heating and cooling rate of ϳ0.25 K/s. between TL and superheating by 200 K above TL . The tem- An analysis of the crystallographic table of chemical perature of the samples was mainly measured using a elements11 shows that zinc and cadmium stand alone among shielded XA thermocouple and was recorded on chart paper the transition metals, and in fact among all the elements of

FIG. 1. Crystallization thermogram for zinc showing no precrystallization supercooling. FIG. 3. Crystallization thermogram for bismuth showing precrystallization supercooling. the Periodic Table. First, Zn and Cd are at the end of this series and have completely filled 3d and 4d electron shells. solidification the clusters readily combine to form a crystal Second, they are isomorphous and the only hcp modification so that crystallization takes place in equilibrium without su- with an anomalously high lattice parameter ratio (c/a percooling. That is to say, the crystallization of Zn and Cd ϭ1.8869 for Zn and 1.8859 for Cd͒ melts with the conser- takes place, as it were, at its own ‘‘seeds’’ ͑clusters͒ without vation of short-range order. Zinc and cadmium possess the a specific incubation period. Substances which undergo characteristics of covalent crystals which are formed by the structural rearrangement in the liquid state ͑such as Sb, Bi, sharing of six outer electrons with their six nearest neighbors Sn, H O, and so on͒ and possess configurations of atoms in in the layer. As a result of melting, the weaker bonds be- 2 clusters, unlike crystals,12 exhibit hysteresis effects. Rear- tween the layers are broken and these become structural units rangement of the atoms back into a crystal as a result of of the liquid phase in the form of flexible cluster conforma- cooling below T requires an incubation period in a meta- tions. The strong covalent bonds between the atoms in these L stable supercooled region. This is why the crystallization of clusters are not broken at high temperatures, as is evidenced these substances involves supercooling. by x-ray structural analyses,12 which indicate that when Zn To conclude, we postulate that the absence of supercool- and Cd melts are superheated to 100–120 K above T , the L ing for Zn and Cd was probably well known to researchers. structure of the liquid phase becomes close-packed with the The results were not published possibly for two reasons: it first coordination number ϳ11.0– 11.5 and does not undergo was either assumed that the absence of supercooling for structural rearrangement. these substances was not particularly noteworthy, i.e., a null Thus, the cluster structure of the liquid phase of Zn and result, or no explanation could be provided, particularly in Cd is similar to the structure of the solid phase, and during terms of the classical theories of nucleation and crystallization.

1 V. I. Danilov, Liquid Metals and Their Solidification ͓in Russian͔, State Scientific-Technical Publishers of Literature on Ferrous and Non-Ferrous Metallurgy, Moscow ͑1962͒, 434 pp. 2 B. Chalmers, Solidification Theory ͓in Russian͔, Metallurgiya, Moscow ͑1968͒, 288 pp. 3 V. I. Petrenko and V. D. Aleksandrov, Pis’ma Zh. Tekh. Fiz. 9͑22͒, 1354 ͑1983͓͒Tech. Phys. Lett. 9, 582 ͑1983͔͒. 4 V. D. Aleksandrov and V. I. Petrenko, Rasplavy 2͑5͒,29͑1988͒. 5 V. D. Aleksandrov and V. I. Petrenko, Rasplavy No. 3, 83 ͑1992͒. 6 V. D. Aleksandrov, Neorg. Mater. 28, 709 ͑1992͒. 7 V. D. Aleksandrov and V. I. Petrenko, Rasplavy No. 3, 85 ͑1993͒. 8 B. A. Muller and J. H. Perepezko, Metall. Trans. A 18, 1143 ͑1987͒. 9 Yu. I. Petrov, Physics of Small Particles ͓in Russian͔, Nauka, Moscow ͑1982͒ 360 pp. 10 Properties of Elements, Handbook Edition, edited by M. E. Drits ͓in Rus- sian͔, Metallurgiya, Moscow ͑1985͒, 672 pp. 11 V. D. Aleksandrov, Fiz. Met. Metalloved. 67, 1219 ͑1989͒. 12 B. I. Khrushchev, Structure of Liquid Metals ͓in Russian͔, FAN, Tashkent ͑1970͒, 112 pp. FIG. 2. Crystallization thermogram for cadmium showing no precrystallization supercooling. Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Qualitative analysis of the behavior of a Lorenz dynamic system based on geometric concepts V. V. Afanas’ev, Yu. E. Pol’ski , and V. S. Chernyavski A. N. Tupolev State Technical University, Kazan ͑Submitted January 9, 1998͒ Pis’ma Zh. Tekh. Fiz. 24, 79–83 ͑July 26, 1998͒ The behavior of a Lorenz dynamic system is analyzed qualitatively using geometric concepts. This analytic approach can be applied to analyze a broad class of systems with dynamic chaos. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03307-2͔

At present, the main method of studying complex non- where k1 , k2 , k3 , k4 , k5 , k6 , and k7 are constant coeffi- linear dynamic systems with strange attractors is mathemati- cients. Depending on whether the mapping point lies inside cal computer modeling. It is interesting to search for meth- or outside the surface ods which could yield qualitative conclusions as to the 2 2 2 behavior of a dynamic system without knowing its solution ␴k1x ϩk2y ϩbk3z Ϫxyz͑k3Ϫk2͒Ϫxy͑␴k1ϩrk2ϩk6͒ and which could predict the time when the system becomes ϩk5xzϩx͑␴k4Ϫk5r͒ϩy͑k5Ϫ␴k4͒ϩk6bzϭB stochastic. One such method is the Mel’nikov method of separatrix splitting, but this can only be applied to a limited ͑where B is a certain constant determined by the instanta- class of dynamic systems.1 We propose an analytic approach neous values of x, y, and z), the motion of the mapping which can be used to analyze the behavior of a wide range of point in the phase space relative to the family of surfaces systems with dynamic chaos. 2 2 2 Our proposed approach involves representing a system ͑k1x ϩk2y ϩk3z ͒/2ϩk4xϩk5yϩk6zϩk7ϭAϭconst of n differential equations describing a nonlinear dynamic varies qualitatively, moving in the direction of increasing A system in the form of a differential equation (BϽ0) or decreasing A (BϾ0). The intersections of these ⌽Ј͑x,a͒ϭϪ⌽ ͑x,a͒, ͑1͒ surfaces with the plane perpendicular to the X axis, projected 1 2 onto the YZ plane, have the form of quadratic curves. If we where xϭ(x1 ,x2 ,...,xn) is the vector of the phase variables choose the coefficients to be k1ϭ1/␴, k2ϭk3ϭ1/r, k4ϭk5 of the system and aϭ(a1 ,a2 ,...,aN) is the vector of the ϭk7ϭ0, and k6ϭϪ2, Eq. ͑1͒ for the Lorenz system ͑2͒ system parameters. The functions ⌽1,2 determine the sur- becomes simplified and has the form faces ⌽ ϭA and ⌽ ϭB in the phase space of the system. 1 2 2 2 2 2 2 2 The sign of B can be used to determine how the mapping ͑x /␴ϩy /rϩ͑zϪ2r͒ /r͒ЈϭϪ2br͑x /brϩy /br point of the dynamic system moves relative to the family of ϩ͑zϪr͒2/r2Ϫ1͒. ͑4͒ surfaces ⌽1ϭconst ͑in the direction corresponding to increasing or decreasing A). A set of different functions ⌽1,2 Equation ͑4͒ is valid for the Lorenz system ͑2͒ and for other can be obtained for the same dynamic system, which means types of nonlinear dynamic systems which can be reduced to that the motion of the trajectory in the phase space of the the form of Eq. ͑1͒ for the given choice of ⌽1,2 , but with a system as a whole can be assessed by means of a suitable different set of coefficients k1 ,k2 ,...,k7 . choice of ⌽1,2 . It is known from the theory of differential equations that We shall illustrate the proposed approach to investigate a a relation of the type ͑4͒ determines the region of asymptotic Lorenz system: motion of the phase trajectory of a dynamic system.3 The function ⌽1 in Eq. ͑4͒ is the generalized distance ␳ between x˙ ϭϪ␴xϩ␴y; y˙ϭϪyϩrxϪxz; z˙ϭϪbzϩxy, ͑2͒ the mapping point with the instantaneous coordinates where r, ␴, and b are the parameters of the system. Equation (x,y,z) and the point with the coordinates (0,0,2r). The ͑1͒ can be obtained for system ͑2͒ as the sum of a linear function ⌽2ϭ0 in Eq. ͑4͒ is an ellipsoid. Inside the ellipsoid combination of equations ͑2͒ with a linear combination of (⌽2Ͻ0) ␳ increases while outside (⌽2Ͼ0) ␳ decreases. Eqs. ͑2͒ multiplied by x, y, and z, respectively, as was done This implies that for any initial conditions, the asymptotic in Ref. 2, p. 92: motion of the phase trajectory, including that in the presence of strange attractors, will take place in a bounded region of 2 2 2 ͕͑k1x ϩk2y ϩk3z ͒/2ϩk4xϩk5yϩk6zϩk7͖Ј phase space. We shall estimate the boundaries of this region. From Eq. ͑4͒ we obtain the maximum ␳ ϭb2r2/(bϪ1). ϭϪ ␴k x2ϩk y2ϩbk z2Ϫxyz͑k Ϫk ͒ max ͕ 1 2 3 3 2 For bϾ1 the region is an ellipsoid of the form

Ϫxy͑␴k1ϩrk2ϩk6͒ϩk5xzϩx͑␴k4Ϫk5r͒ 2 2 2 2 2 2 x /͑ͱ␳max␴/r͒ ϩy /͑ͱ␳max͒ ϩ͑zϪ2r͒ /͑ͱ␳max͒ ϭ1 ϩy͑k5Ϫ␴k4͒ϩk6bz͖, ͑3͒ ͑5͒

phase coordinates x,y,z. Since the value of the phase coordinate z for a Lorenz system is positive because of the inequalities ͑6͒, the position of the mapping point of the system corresponds to the points M 1 and M 2 . For azϭR (y ϭ0) the points M 1 and M 2 are the same. From the time (yϭ0), the phase trajectory must undergo a transition from one state of unstable equilibrium to another over the time ⌬tрT ͑where Tϭ4␲/ͱ8␴(rϪ1)Ϫ(␴ϩr)2 is the oscillation period of the system which is determined in accordance with Ref. 4͒. Thus, the proposed approach allows us to determine analytically the onset of qualitative changes in the behavior of a dynamic system and specifically shows that as the system changes between unstable equilibrium states, the sign of the y coordinate changes first, followed by that of the x coordinate. This sequence of change in the signs of the x and y coordinates is confirmed by the experimental results.2 FIG. 1. The possibility of determining the instant that qualitative changes take place in the behavior of the system, allows us to find the specific time intervals for efficient application of with its center at the point having the coordinates (0,0,2r). It external controlling influences to the parameters of a Lorenz follows from Eq. ͑5͒ that if strange attractors appear in the dynamic system4 to ensure the desired behavior. Lorenz system, the values of the instantaneous phase coordi- The representation ͑1͒ can also be used to analyze the nates lie within boundaries defined by the inequalities behavior of other types of nonlinear dynamic systems with strange attractors ͑Ro¨ssler system, Ruelle–Takens system, Ϫbͱ␴r/͑bϪ1͒рxрbͱ␴r/͑bϪ1͒, and so on͒ although this is outside the scope of the present Ϫbrͱ1/͑bϪ1͒рyрbrͱ1/͑bϪ1͒, paper. From this analysis, we can draw the following conclu- 2rϪbrͱ1/͑bϪ1͒рzр2rϩbrͱ1/͑bϪ1͒. ͑6͒ sions: 1. The differential representation ͑1͒ is an effective If we choose the coefficients k1ϭr/␴, k2ϭk3ϭϪ1, method of studying the behavior of dynamic systems with k4ϭk5ϭk6ϭk7ϭ0, ⌽1,2 has the following form: strange attractors. ⌽ ͑x,y,z͒ϭx2r/␴Ϫy 2Ϫz2, ͑7͒ 1 2. An analysis of the functions ⌽1,2 ͑4͒ allows us to ⌽ ͑x,y,z͒ϭx2rϪy 2Ϫbz2. ͑8͒ determine the bounded volume of phase space in which the 2 phase trajectory of a Lorenz dynamic system undergoes A change in the sign of ⌽1,2 determined by the instantaneous asymptotic motion. coordinates x,y,z changes the form of the surfaces. For ⌽1,2 3. An analysis of the intersections of the ⌽1,2 surfaces ⌽1,2Ͼ0 the surfaces are parted hyperboloids, for ⌽1,2ϭ0 ͑7͒ and ͑8͒ of the Lorenz system can provide an analytic they are cones, and for ⌽1,2Ͻ0 they are unparted hyperbo- substantiation of the condition (yϭ0) for the onset of a tran- loids. For the instantaneous coordinates x,y,z the surfaces of sition between regions of unstable equilibrium states and al- ¯ constant ⌽1ϭA and ⌽2ϭB in the plane xϭxϭconst corre- lows us to determine the boundaries of the times intervals for spond to the cross sections these transitions in a dynamic system.

͑¯y 2ϩ¯z2͒/͑¯x2r/␴ϪA͒ϭ1 —circle, 1 V. V. Afanas’ev, Yu. E. Pol’ski, and V. S. Chernyavski, Pis’ma Zh. ¯y 2/͑¯x2rϪB͒ϩ¯z2/͑¯x2r/bϪB/b͒ϭ1 —ellipse. Tekh. Fiz. 23͑23͒,40͑1997͓͒Tech. Phys. Lett. 23, 917 ͑1997͔͒. 2 Strange Attractors, edited by Ya. G. Sinai and L. P. Shil’nikov ͓in Rus- When the instantaneous coordinates x,y,z vary, A and B sian͔, Mir, Moscow ͑1988͒. 3 vary as do the sizes of the circle and the ellipse, and their V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations ͑Princeton University Press, Princeton, N.J., 1960͓͒Russ. origi- relative position. For ¯xϭx we obtain a circle of radius nal, Gostekhizdat, Moscow, 1947͔ 2 2 4 V. V. Afanas’ev and Yu. E. Pol’ski , Pis’ma Zh. Tekh. Fiz. 15͑18͒,86 R͉¯xϭxϭͱy ϩz and an ellipse with the semiaxes ay͉¯xϭx 2 2 2 2 ͑1989͓͒Sov. Tech. Phys. Lett. 15, 741 ͑1989͔͒. ϭͱy ϩbz and az͉¯xϭxϭͱy /bϩz ͑see Fig. 1͒. The point M i (iϭ1,4) corresponds to the instantaneous value of the Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Electrodynamic acceleration of ultraheavy molecular ions S. K. Sekatski Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Province ͑Submitted October 16, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 84–88 ͑July 26, 1998͒ The design of an ‘‘electrodynamic accelerator’’ for ultraheavy molecular ions with masses of order 106 Da is analyzed. It is shown that these ions can be efﬁciently accelerated to energies of hundreds of kilo-electronvolts over lengths no greater than 20 cm by using a sequence of voltage pulses with amplitudes of several kilovolts and microsecond durations. This acceleration allows ultraheavy molecular ions to be efﬁciently recorded using conventional secondary electron multipliers or multichannel plates. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03407-7͔

Time-of-flight mass spectrometry is now being success- velocity of such heavy ions; for example, the velocity of an fully used in biochemical and medical research as a highly ion of mass Mϭ106 Da, accelerated to 10 kV, is only sensitive, reliable, and relatively simple method of detecting 1.4ϫ103 m/s, i.e., in 1 ␮s this ion only covers a distance of biological molecules and their fragments. This particularly 1.4 mm. The system required to implement this principle of applies to mass-spectrometric time-of-flight methods using electrodynamic acceleration is very compact and inexpen- pulsed laser desorption/ionization of complex molecules in sive, allowing heavy ions to be accelerated to potentials of specially selected matrixes ͑described in the English litera- hundreds of kilovolts over a length of 10–20 cm. ture as matrix-assisted laser desorption/ionization, or The principle of electrodynamic acceleration is illus- MALDI͒1 which can reliably detect relatively small frag- trated in Fig. 1, where the parameters are as follows: the first ments of protein molecules and oligonucleotides2 as well two accelerating gaps have length lϭ3 mm, the next five as very heavy molecules with masses up to 150 000 Da lϭ5 mm, the following eight lϭ7 mm, and the next eight ͑Ref. 3͒. It has been suggested that this method will be also lϭ9 mm. A sequence of rectangular pulses of amplitude be very useful for solving fundamental problems such as Uϭ10 kV and duration tϭ1 ␮s are applied to the even elec- unraveling the nucleotide sequence of DNA ͑Ref. 4͒ which trodes, as shown in Fig. 1 (ϩU for time t, then 0 for time t, requires, as do many other applications, the reliable detection and ϪU for time t, and so on͒. The odd electrodes are of molecules and their fragments with extremely high grounded. It is suggested that either diaphragms or grid elec- masses, up to 106 Da and higher.5 trodes highly permeable to ions may be used. The corre- However, studies of such heavy molecules present ap- sponding ion-optical effects of ion beam focusing and defo- preciable difficulties, since the velocity they acquire at the cusing are not considered ͑these clearly cannot be of any potentials of a few or even tens of kilovolts used in typical significance because the ions pass through the grids at times time-of-flight mass spectrometers is inadequate for detection when the electrode potentials are zero͒. The operation of the using conventional secondary electron multipliers and multi- accelerating system is analyzed for a pulsed source of heavy channel plates.6 The efficiency of detection of high-mass ions similar to that described in Refs. 2 and 3. ions increases as their energy increases ͑i.e., the accelerating During the voltage pulse the ion is uniformly accelerated 7,8 potential of the mass spectrometer͒. However, increasing from the initial velocity vi to viϩ1ϭviϩeUt/lM, covering a 2 this potential even to values of 30–50 kV, not to mention distance siϭvitϩeUt /2lM. A successful acceleration cycle higher values, presents major problems for compact labora- requires the condition liϩsiϽl to be satisfied, where li is the tory systems. It would seem that the types of detectors other distance between the ion and the first electrode at the instant than secondary electron multipliers or multichannel plates, of application of the voltage pulse. Then, during the time t now being actively discussed and investigated to solve this the ion propagates in the absence of the field, covering the problem, such as cryogenic detectors operating at 0.4 K and distance s0ϭviϩ1t. If the gap length satisfies lϽliϩsiϩs0 calorimetric particle detectors ͑see Ref. 9 and the literature Ͻ2l, then after the application of a voltage pulse of opposite cited therein͒ cannot completely solve this problem because polarity, the ion again begins to be accelerated and we have they are expensive and difficult to operate. liϩ1ϭliϩsiϩs0Ϫl. 6 The present note is intended to draw attention to the fact The calculations show that ions of mass M 0ϭ10 Da, that ions of such high mass can easily be accelerated ‘‘elec- entering this system with an initial energy of 10 kV, are trodynamically’’ by passing them through a set of accelerat- accelerated to a potential of 127.3 kV at the exit if l0 ͑the ing gaps having typical lengths l of the order of a few mil- distance between the ion entering the accelerator and the limeters to which are applied high-voltage pulses having input electrode at the instant of application of the first volt- amplitudes U of a few kilovolts and durations t between one age pulse͒ is between 0 and 0.85 mm, which corresponds to and a few microseconds. This is attributable to the low a time width ␦t of the order of 0.6 ␮s for the initial ion

FIG. 1. Diagram illustrating the concept of an electrodynamic accelerator for heavy molecular ions.

packet. In this case, there is a large margin for possible varia- required, the system can either be terminated after any num- tion of the initial ion energy ␦U and its mass ␦M; both these ber of accelerating gaps or the structure could be extended. parameters can vary within a percentage of the initial value Thus, there is no doubt that the electrodynamic accelera- and despite this, the incoming ion will still be accelerated tion of very heavy molecular ions is possible in principle. after passing through the accelerator. The energy acquired by the ion depends on its mass but these variations are small and In conclusion, the author would like to thank V. S. are not significant. For this accelerator, for example, the en- Letokhov for useful discussions and interest in this work. ergy varies between 126.6 keV for Mϭ1.007M 0 and 127.9 1 keV for Mϭ0.993M 0 . Note that if the probability P of de- M. Karas, D. Backmann, U. Bahr, and F. Hillenkampf, Int. J. Mass Spec- tection of such an ion is estimated using the empirical for- trom. Ion Processes 78,53͑1987͒. 2 mula derived by Geno and MacFarlane8 ͑since this formula K. Tang, S. L. Allman, and C. H. Chen, Rapid Commun. Mass Spectrom. 6, 365 ͑1992͒. was obtained for ions of far lower mass than the case 3 K. Tang, N. I. Taranenko, S. L. Allman et al., Rapid Commun. Mass 6 Mϭ10 Da considered here, its validity must be checked Spectrom. 8, 727 ͑1994͒. experimentally͒: Pϭ1Ϫexp(Ϫ␥), with ␥ϭ2.58ϫ10Ϫ7m 4 R. C. Beavis and B. T. Chait, US Patent No. 5. 228, 644. 5 ϫexp(2.31ϫ10Ϫ4v), where m is the ion mass in daltons and C. H. Chen and PerSeptive Biosystems Inc., Framingham, MA ͑private communication͒. its velocity v is in meters per second, we obtain the com- 6 M. R. Anbund and B. V. Polenov, Open Secondary Electron Multipliers pletely satisfactory value Pϭ0.56. and Their Application ͓in Russian͔,Eńergoatomizdat, Moscow ͑1981͒, 140 pp. The proposed design was not specially optimized to ob- 7 tain the highest possible values of t, U, and M but these R. J. Beuhler and L. Friedman, Int. J. Mass Spectrom. Ion Phys. 23,81 ␦ ␦ ␦ ͑1977͒. parameters may nevertheless be considered to be quite satis- 8 P. W. Geno and R. D. MacFarlane, Int. J. Mass Spectrom. Ion Processes factory. The results of a more detailed analysis of the opera- 92, 195 ͑1989͒. 9 tion of an electrodynamic accelerator, its optimization, and D. Twerenbold, J. L. Vuilleumier, D. Gerber et al., Appl. Phys. Lett. 68, also its possible usage as a mass spectrometer and mass filter 3503 ͑1996͒. will be published separately. Depending on the acceleration Translated by R. M. Durham TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998

Induced of Zeeman coherence in an ensemble of cesium atoms interacting with a two-frequency „microwave؉rf… magnetic ﬁeld N. A. Dovator

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg ͑Submitted December 30, 1997͒ Pis’ma Zh. Tekh. Fiz. 24, 89–93 ͑July 26, 1998͒ Interference between Zeeman states corresponding to the forbidden magnetic-dipole transition 2 ⌬Fϭ0, ⌬mFϭ2 is reported in connection with the simultaneous interaction of 6 S1/2 cesium atoms with a resonant microwave field and an rf field which varies at twice the Larmor frequency and is directed perpendicular to a static magnetic field H0ϭ0.2 Oe. © 1998 American Institute of Physics. ͓S1063-7850͑98͒03507-1͔

Studies of coherence effects in the interaction between The experiment was carried out as follows. Resonant, radiation and matter are now attracting increased interest, circularly polarized pump radiation directed parallel to a mainly because the coherent coupling of atomic states leads static magnetic field H0Ӎ0.2 Oe creates a nonequilibrium 2 to substantial changes in the macroscopic characteristics of a population of 6 S1/2-cesium atomic levels. Microwave radia- material. Such effects include ‘‘coherent population tion froma3cmrange microwave oscillator is passed 1 trapping’’ and ‘‘electromagnetically induced trans- through a waveguide and a rectangular horn antenna to a 2 parency.’’ It should be noted that in these studies and in glass cell containing cesium vapor ͑at room temperature͒ and others concerned with interference effects in ensembles with neon buffer gas. The orientation of the horn antenna is se- 3–5 optically oriented atoms, the coherent coupling of the lected so that the direction of the magnetic induction vector atomic states took place as a result of a two-photon process of the microwave field forms an angle of 45° with that of the in a three-level quantum system. In this case, each pair of H field. interfering states ͑between which no direct transition took 0 As a result, the microwave field had magnetic field place͒ was coupled with a third state by interaction with one strength components with the polarizations ␴ (H␴) and ␲ of two resonant electromagnetic fields belonging to the same 1 (H␲). The frequency of the microwave field was tuned to the frequency range ͑optical or microwave͒. 1 frequency of the (4.4) (3.3) hyperfine structure transition Here an experiment is described to observe coherence ↔ 2 which was monitored using the Sz signal for the optically between the ͑4.4͒ and ͑4.2͒ Zeeman levels of 6 S1/2 cesium 6 ␴ atoms ͑between which the direct magnetic-dipole transition oriented atoms. The ␴-polarized rf magnetic field (H2 ) was generated using a Helmholtz coil whose axis was perpen- ⌬Fϭ0, ⌬mFϭ2 is forbidden͒ under conditions of a three- photon coherent process induced using two variable mag- dicular to H0 . The frequency of the rf generator used to netic fields whose frequencies differ substantially ͑micro- supply the coil was close to twice the Larmor frequency wave and rf͒. f rfϭ2 f 0ϵ2␥/2␲H0 (␥/2␲ϭ350 kHz/Oe). A transverse ͑to Figure 1a shows some of the ground-state levels of the H0) probe beam, linearly polarized in the direction orthogo- cesium atoms which are coherently coupled as a result of the nal to H0 , was used to record the coherent superposition of interaction of the microwave and rf magnetic fields with the ͑4.4͒ and ͑4.2͒ states which occurred in the experiments. The ensemble of atoms. The apparatus is shown schematically in modulation of the absorption of the transverse light beam, Fig. 1b. which occurs at the frequency of the rf field under conditions

FIG. 1. a—Diagram of ground-state levels of cesium atoms assumed in the analysis of interaction between these atoms and a two-frequency (microwaveϩrf) magnetic ﬁeld; b—block diagram of apparatus: 1,2—cesium spectral lamp, 3—circular polarizer, 4—working cell, 5—Helmholtz coil, 6—rf generator, 7—microwave generator, 8—horn antenna, 9—linear polarizer, 10—band-pass ampliﬁer, and 11—photodiode.

FIG. 2. Output voltage traces from band-pass amplifier obtained by sweeping the rf field frequency with the micro- Ϫ1 wave field switched on (H1Þ0, Pmwϳ10 W) and off (H1ϭ0). The scale on the right-hand ordinate refers to the case H1ϭ0. Curves 1–3 were obtained for Uϭ20, 15, and 8 V, respectively, where U is the output voltage of the rf generator.

of a three-photon coherent process and evidences the re- Among the characteristic features of the observed Zee- quired Zeeman coherence, was detected using a photodiode man coherence signal we must include the drop in its reso- positioned at the exit window of the cell ͑see Fig. 1b͒. The nance frequency when we increased the amplitude of the rf signal from the photodiode was then fed to a band-pass am- field. This frequency shift may arise because the ␴-polarized plifier ( f bpaϭ2 f 0 , ␦bpaϭ10 kHz) and after amplitude detec- rf magnetic field used experimentally, whose frequency was tion, was recorded by an automatic plotter. close to twice the Larmor frequency, is a nonresonant per- The Zeeman coherence signal was recorded as a reso- turbation for each pair of neighboring Zeeman levels ͑with nant variation in the modulated absorption of cesium vapor ⌬Fϭ0, ⌬mFϭ1). It was shown in Ref. 7 that such a per- as the frequency of the rf field was swept. Figure 2 shows turbation should reduce the magnetic resonance frequency traces of this signal obtained for different rf magnetic fields which can be seen clearly as the ‘‘collapse’’ of the Zeeman ͑the coil constant is Kϭ2.5ϫ10Ϫ3 Oe/V). levels ͑dashed lines in Fig. 1a͒. As a result, the resonant The appearance of a coherence signal between the ͑4.4͒ frequency of the coherence signal f resϭ2 f 0 should also de- and ͑4.2͒ Zeeman levels may be explained using the model crease, as was observed in these experiments. of a three-level quantum system interacting with two mono- 1 chromatic electromagnetic fields of different frequency ͑Fig. B. D. Agap’ev, M. B. Gorny, B. G. Matisov et al., Usp. Fiz. Nauk 163,1 1a͒. According to this model, the ͑4.4͒ level is coherently ͑1993͓͒Phys. Usp. 36, 763 ͑1993͔͒. 2 S. E. Harris, Phys. Today 50„7…,36͑1997͒. coupled with the ͑4.2͒ level via the third ͑3.3͒ level as a 3 ´ O. S. Vasyutinski, N. A. Dovator, and R. A. Zhitnikov, Zh. Eksp. Teor. result of one-photon excitation of the (3.3) (4.4) hyperfine Fiz. 72, 928 ͑1977͓͒Sov. Phys. JETP 45, 484 ͑1977͔͒. ␴ ↔ 4 ´ structure transition using the H1 component of the micro- N. A. Dovator and R. A. Zhitnikov, Zh. Eksp. Teor. Fiz. 77, 505 ͑1979͒ wave field and two-photon excitation of the (3.3) (4.2) ͓Sov. Phys. JETP 50, 257 ͑1979͔͒. 5 ´ neighboring hyperfine-structure transition by absorption↔ of a E. B. Aleksandrov, A. B. Mamyrin, and Yu. S. Chidson, Zh. Eksp. Teor. ␲ Fiz. 72͑4͒, 1568 ͑1977͓͒Sov. Phys. JETP 45, 823 ͑1977͔͒. ␲-polarized microwave quantum (H1 ) and the simultaneous 6 N. M. Pomerantsev, V. M. Ryzhkov, and G. V. Skrotski , Physical Prin- ␴ emission of a ␴-polarized rf quantum (H2 ). In this case, the ciples of Quantum Magnetometry ͓in Russian͔, Nauka, Moscow ͑1972͒, conditions for excitation of the magnetic-dipole transitions 448 pp. 7 L. N. Novikov and A. G. Malyshev, JETP Lett. 15,89͑1972͒. are satisfied (⌬Fϭ1, ⌬mFϭ͉1͉) both in terms of energy and magnetic quantum number. Translated by R. M. Durham