Z. Phys. B 90, 277-281 (1993) Condensed Zeitsehrift far Physik B Matter Springer-Verlag 1993

The plasmon density of states of a layered electron gas H. Morawitz ~, I. Bozovic 2, V.Z. Kresin 3, G. Rietveld 4, D. van der Marel 4 IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120-6099, USA 2 Varian Research Center, 611 Hansen Way, Palo Alto, CA 94303, USA 3 Materials and Chemical Physics Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA 4 Delft University of Technology, Faculty of Applied Physics, Lorentzweg 1, 2628 CJ Delft, The Netherlands Received: 24 September 1992

Abstract. The plasmon density of states (DOS)p(co) of a mon side-band measured with such a local probe. The layered electron gas (LEG) is studied theoretically. It is plasmon-pole approximation replaces the response func- shown that p (co) is a linear function of frequency ~o as co tion with the density of states of the layer plasmons, goes to 0, and then increases more rapidly as co tends to which is defined by the screened, three dimensional plasma frequency cop. We also study the partial densities of state for constant K and p(~o)=(2n) -3 ~ O(~-Op(tc, qz)). (1) q~, where K and qz are the magnitude of the transverse K~ qz and perpendicular momentum transfers. A possible ex- The nature of the collective excitations in layered systems perimental probe for the plasmon DOS is discussed. has been studied extensively [5, 6] theoretically, because of continuing interest in superlattices and naturally layered materials such as graphite and the transition metal I. Introduction chalcogenides (TaS2, TeS2, etc.). The outline of this paper is 'as follows: In Sect. 2 we The new superconducting cuprates [1-4] with high su- rederive the layer plasmon dispersion relations [6] using perconducting transition temperatures T~ are all charac- the random phase approximation (RPA). In Sect. 3 we terized by a layered structure of conducting (CuO2) calculate the partial density of states, keeping either of planes. We propose to use the layered electron gas (LEG) the variables K (the absolute value of the LEG plasmon model [5] to describe their collective charge excitation wavevector parallel to the conducting planes) or qz (the (layer plasmons) and to calculate their density of states LEG plasmon wavevector perpendicular to the planes) (DOS) as a function of frequency in this paper. The struc- constant. In Sect. 4 we derive a simple form of the total ture of the plasmon spectrum in layered, conducting sys- plasmon DOS, based on an approximate form of the LEG tems differs in a striking way from those of the isotropic dispersion relations. In Sect. 5 we calculate both ana- (3 D) conductors. In the latter case, the plasmon branch lytically and numerically the exact layer plasmon density is described by a dispersion relation co (q)=oJ 0 +aq 2, of states and discuss the various frequency regimes whereas a layered conductor is characterized by a highly ~o/cop< l, = 1 and > 1. Finally - in Sect. 6 - discuss the anisotropic plasmon band D 0c, qz) without a gap at tc = 0 implications of these results for the shake-up spectra of except for the single branch for which qz = 0 (to is a two- core levels in a layered compound. dimensional wave-vector in the plane of the sheets and qz has been chosen perpendicular to the layers). That 2. Plasmon dispersion relations produces noticeable changes in the spectrum of collective We assume that the carriers are localized in N(N~) modes of these materials. In experimental probes, such independent layers, spaced a distance D apart. The dis- as absorption of light, the dielectric response is subject persion relation for two-dimensional carriers is assumed to strong momentum selection rules. As a result only the to be given by the effective mass approximation: ek = long-wavelength dielectric response is probed in such ex- k 2/2 m*. Electrons on different sheets are coupled by the periments. In local experimental probes, such as the sud- Coulomb interaction. We can write an electronic Ham- den creation of a core hole by X-ray photo-electron spec- iltonian H el. troscopy, one can, in principle, couple to the full spectrum of plasmons. This shows up as a side-band of the core- H ~1-= ~, eKIA~,+ A,~l hole excitation spectrum due to plasmon shake-up. It is Ir conceivable that the plasmon-pole approximation can be + Z Z V~~ + + ,,"z A,,+ " - ,," z" A,,I" A,~,z used for the calculation of the spectral shape of the plas- K,,,,,,~" l,t" (2) 278 where t~, te' and K" are two-dimensional momentum var- the background dielectric constant of the medium due to iables, and l, l' label different layers, At,. l, A k,z+ are carrier all other high frequency excitations. (A typical value is annihilation and creation operators and the interaction v~ 0.05). With these definitions, and solving for the roots matrix element V~ (l, l') is the two-dimensional Fourier of e (te, qz; co) = 0, we obtain with (8) transform of the Coulomb interaction between electrons on layers l and l'. For an infinite system V(I,l')= F(y,z) [ cop ] Vq-P). 0=1 v~y [COZp_(VFy)2]I/2 1 . (10) 2 7re 2 V~~ ') =-- e -~Dll-r I (3) From this equation we solve the layer plasmon dispersion ~Mte relation F2(Y'Z) (11) If we make a Fourier transformation of the variable c~ v~ Y2-~ v~+ 2F(y,z)/y" r = l- l' we obtain Alternatively we can express this as a dispersion relation q~0(te;q~)= ~. V(O)(r)exp(iq~Dr) for O as a function of te and q~. When written in this r = --ao form the dispersion relation reads 2 7~e2 --- F(teD, q~D), (4) Op (n, qz) CM te 1/2 where we use the dimensionless function F (the 'layer 2 1 .~-VFte 1+ [ 2F(xD'q~D)] form factor'), introduced by Fetter [5b] in the hydro- 4F(xD, q~D) dynamic treatment of the layer plasmon dispersion rB x A r B K sinh teD F (te D, q~D ) - cosh teD - cos q~D " (5) For later use it is convenient to express F as a function of COp and y using (10), and also to express z in terms of The study of the screening properties and collective modes F and y, by inverting (5). The resulting expressions are requires the knowledge of the fully screened interaction found to be V(1, l'). Within the random phase approximation, the v~yV~2p-(vFy) 2 latter obeys the equation F(mp,y) - co, - Vco - (v y)2 " V(I,I')= V~ ") (12) sinh ( y ) ~'~. +H~176 Z V~ V(l",l'), (6) z(Ogp, y)=arccos (cosh (y) ~/ l" where the single layer polarizability H ~ is given by It should be noted here that (11) goes beyond the earlier RPA results as it used the exact form of the single sheet 2 f(8~+q)--f(ek) (7) polarizability )~0 (9) and not its expansion valid for H~ =A - ~ gk+q--ek -ico co ~ VFX only. On the other hand, we have not included the effects of the non-local exchange and correlation part Here A is the area of one layer and f (e,~) is the Fermi of the electron-electron interactions. Some exploration of function. The layer plasmons are defined as the zeroes of the inclusion of such effects by a local field correction the layer dielectric function for the in-plane electron-electron interaction has been recently made in [8]. We note that the general conse- e (te, q~; co) = 1 - ~0 (te, q~)Re {H(~ (8) quence of such treatments as the GRPA (generalized ran- dom phase approximation) is to reduce the strength of In the frequency regime co > vete, the real part of the the electron-electron interactions. polarizability Z0 of a single sheet is [5, 7]

Re{H~ (te; co)} =)-~o,[ [CO~_(Vvte)2],/2--1.] (9) 3. Plasmon partial density of states Let us first address the question of the LEG plasmon In order to keep the expressions simple we introduce frequency dependence for co ~ 1. In this region (see Fig. 1) dimensionless variables for the in-plane and out-of-plane y~ 1 holds also, and the hyperbolic functions may be wave-vector components te and qz and the LEG plas- expanded in a Taylor series for small y. The LEG plasmon mon frequency co_, by defining y-=teD, z=-qzD, dispersion relations then reduce to the much simpler form cop (y, z) ~ Op (y, z)f0 o with ~o2 = 2 vZ/(DrB) and VF = (y, z) = 2 OF/(D(20) (rB/2D) 1/2. Here h2e~ is the effec- y2 rB = ~177"e (13) tive Bohr radius, m* the effective carrier mass and c g is [ + 1 -- cos z] [2 + VFI.gy2 .'1 2 + 1 -cosz)] 279

1.5 i r i // / /

1.0

t.O i ',/,t ~- I //Z p(qD,w) 0.5

0 i I i I r I i 0 1 2 3 KD

0.30 I i o 0.5 1 .o y=KD

-- 0.2 0.25 --.-- 0.4 --- 0.6 ...... 0.8 0.20 .... 1,0 1.2 8 0.15 l

o.10 - I rl i !

dlnp 0.05 i, !!, l\iAjl dlnw 4 i .4.~_//~ ! ~ 0 0 0.5 1.0 c a,,/w 0

Fig. la-d. a Layer plasmon dispersion for v 2= 0.05 and (from top to bottom) z/~ =0, 0.2, 0.4, 0.6, 0.8, 1. b Partial DOS with z fixed i O'o'O'O'rO'e4"O4"e'O" at (from bottom to top) z/z~ = 0, 0.2, 0.4, 0.6, 0.8, 1. c Partial DOS with y fixed at (from left to right) y = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2. d Total DOS (solid), and exponent ,8 in p = co ~ versus frequency 0.5 1.0 (open circles) W/WO

For z ~ 1 and co ~ 1, we obtain the acoustic plasmon dis- Clearly for (Dp ~ 1 the second term is much smaller than persion relation: the first term and can be neglected, hence (11) has co proportional to y for all values of the wavevector per- cop (y, z) pendicular to the planes (except for the special point =y~v~+(1--cosz)-~(2+v~(1--cosz)) -1 (14) z = 0). As the integral over y is two-dimensional, a direct consequence of this is that the partial DOS for each fixed For z < 1 and ~ ~ 1, and retaining only leading orders in value of z is proportional to co in this limit. In Fig. 1 we z and y (11) becomes: plot a set of dispersion curves for a number of z-values for v 2 = 0.05, which is a typical value. Also indicated for y2 2 _ 2 2 (15) the same set of z-values is the partial DOS obtained by COp (y,z)-- VFy -~ y2+z2- integrating numerically over y, while keeping z fixed. It is clear from this plot, that for each value of z the partial It is not transparent from this form that the dispersion DOS rises linearly with frequency for co below about 0.4. is still acoustic for small co. We rewrite the formula in A singular point is reached at z = 0, where the dispersion the following form sets in at co = 1. As the total DOS is obtained by inte- y2 y2 grating over z between - ~ and 7r one expects to obtain a linear dependency on co for the plasmon DOS. This is co~(Y'Z) =7 (1 + v~ (Y~+ zb) - co~ ~--z (16) not so obvious if one considers instead the partial DOS 280

for fixed y, while integrating over all z. This is indicated where Ym corresponds to the value of y for which cop = co in Fig. lc. One clearly sees the 1-dimensional nature of on the lowest branch (cos (z)= -1) of the dispersion this partial DOS reflected in the divergencies at the points curves for fixed z. With (11) we obtain where the energy corresponds to the lower and higher branches (z = ___ n and z = 0 respectively). sinh (ym) ]2 1 + cosh (y.,) vFy,,2 2 , § - (.02. (23) 4. Plasmon DOS using an approximation 2 sinh (Ym) v~+ to the dispersion formula Ym 1 +cosh(ym)

We first derive the DOS using a simplified dispersion We now make a transformation of variables x-y/co in formula, which nevertheless possesses the most relevant the integration of (22), and define x,,,=y,,Jco. We will physical properties. If v~ is of the order of 0.05 as usual, be interested in the behaviour for co ~ 0, where the above one may neglect the first term in (11). This only decreases expression reduces to the group velocity somewhat, without affecting the quasi- acoustic nature of the dispersion relation. We also replace 2 1 + VF2 (24) the second term in (11) with yF/2, which corresponds to Xm'=4 1 +4@ (1 +V~)" neglecting some higher order corrections. We further- more replace the sinh(y) and cosh(y) in F with their By combining (5), (11) and (12) we calculate dz/dcop and limiting values for small y, which is a good approximation express it as a function of x and co for small frequencies. For larger frequencies it is still reasonable and only results in a pile-up of states near the bulk plasma frequency. The result is d~p ,o,x- 2o3 ( ~co 2/dE) ( ~ El ~ cos z) sin z y2 co~ (y, z) = y2+2( 1 -cos(z))" (17) x sinh (x co) - oa 2 (1 - v2 x2) 3/2 sin z" (25) We will first calculate the number of states per unit vol- ume with co~ (y, z) < co using the expression: With the help of (12) we express sin z as a function of x and co N(co) = (2 re)-3 S d3k (Dp( y, Z) < 69 sin 2 z = 1

=(2nD)-3 i dz l d2Y" (18) -- 7~ r y,z) • r @xco l//l _ v2 x2 The integral over y is equal to the area z~y2 enclosed by the contour of constant y for given z and co. Hence we which, in the limit co ~ 1 becomes rewrite (17) as sinz~[l_[l+ ~ 1 1 y2 _ 2 (1 - cos (z)) co2 (19) ]12 ( /1 -- 1~222 1)]211/2" 1 --o9 2

After integration over z we obtain Here we note, that in this limit sin z is a function of x and does not depend on co. For vF--*0 the above ex- 1 a) 2 pression reduces to N(co)--2nD3 1_o)2" (20) sin z~-x ]//1 - 2. (28) Differentiating once gives us the density of states: We are now ready to write down the expression for the 1 co density of states in the limit where co ~ 1. As this implies (21) P(CO)=rcD 3 (1_o32)2" y ~ 1, we also substitute cosh (y) and sinh (y) with their limiting behaviour for small y. Combining (22), (24), (25) and (27) we obtain 5. Plasmon DOS using the full dispersion formula co xm x3 dx 1. (29) We now calculate the DOS per unit volume using the P (co) 2 rc2D 3 ~ (1 - v 2 x2) 3/2 sin z relation p(co)= V-' ~. fi(cop(y,z)--co) As the integrand and x m are independent of co we see y,z that the plasmon DOS is proportional to co for co ~ 1. We can check this expression by choosing ve= O, as dis- 2 }m dz - (2 gD) 3 ~ 2 nydy, (22) cussed in the preceding section. From (24) we see that 0 ,~Wp o),y x m = 2. Expression (29) can now be calculated analyti- 281 callyand becomes, using (28) the core-lineand deconvolutiontechniques it mightbe possibleto determinethe plasmonDOS with such an 4(.o i /'/2 experiment. P (a)):~D-5o ~l//i_u du- 7rD"a) (30) Anotherpossibility is, thatthe plasmonDOS shows up as the inverselife-time of an electron-holepair, due whichis equalto thelimiting behaviour for smalleo that to electron-electroninteractions. This possibilitymerits, we alreadyderived in (21). in ourview, further study as a possiblemicroscopic route Finallywe comparethese analytical results to com- to themarginal Fermi liquid model [9]. putationsmade by integratingthe plasmondispersion relationof (11) numerically.In Fig. ld we displaythe total plasmonDOS p(co), as well as the function 7. Conclusions oo/p,d p/da), which corresponds to the exponentfl in p -~co ~. Also thenumerical results show that the density We calculatedthe plasmon density of statesof a layered of statesdepends linearly on oo for smallfrequencies in electrongas. It is shown,that this DOS is proportional this numericalcalculation. Interestingly there is only a to frequencyco for sufficientlysmall co. This resultcan weakdiscontinuity in the first derivativeof the DOS at in principlebe verified with core-level spectroscopy, where f2p/f2o = 1. In realitya cutoffof theplasmon DOS must theshake-up spectrum corresponds to thedensity of states be foundat largervalues of thein-plane wavevector, due of collectiveexcitations. to non-parabolicdispersion of theelectronic bands near theBrillouin-zone boundary. These effects have not been References includedin this study.The DOS basedon the approxi- mationfor thedispersion relations (21), has a singularity 1. Bednorz,G., Mueller,K.A. : Z. Phys.333, 23 (1986) atg2p/f20 = 1, dueto thefact that with this approximation 2. Wu et al.: Phys.Rev. Lett. 58, 405 (1987) the plasmondispersion saturates for largevalues of the 3. Maeda,H., Tanaka,Y., Fukutomi,M., Asano,T.: Jpn.J. Appl. in-planewavevector. Phys.Lett. 87, 2761 (1988) 4. Sheng,Z.Z., Herman, A.M.: Nature332, 138 (1988) 5. Visscher,P.B., Falicov,L.M.: Phys.Rev. B3, 2541(1971); Fetter,A.L.: Ann. Phys.(N.Y.) 88, 1 (1974);Grecu, D.: Phys. 6. Discussion Rev.B8, 1958(1973); Das Sarma, S., Quinn,J.J. :ibid.25, 7603 (1982);Tselis, A.C., Quinn,J.J.: ibid.29, 3318(1984); Jain, In the abovesections we haveshown that, under condi- J.K., Allen,P.B.: Phys. Rev. Lett. 54, 2437(1985); Giuliani, tionswhere the experimentalprobe couples to theplas- G.F.,Quinn, J.J.: ibid.51,919; Hawrylak, P., Wu,J.-W., Quinn, monDOS, oneshould observe this as a linearfrequency J.J.: Phys.Rev. B32, 4272 (1985); Hawrylak, P., Eliasson,G., Quinn,J.J.:Phys. Rev. B37, 10187 (1988) dependencyof theresponse function. One suchpossible 6. Kresin,V.Z., Morawitz,H.: Phys.Lett. A145,368 (1990); probewould be core level spectroscopyof a 'probing" Morawitz,H., Kresin,V.Z.: In: Electronicproperties of high atomembedded in a LEG. The plasmonDOS should T~ superconductorsand relatedcompounds, Kuzmany, H., thenshow up as the shake-upspectrum of thecore line. Mehring,M., Fink, J. (eds.).Springer Series of SolidState A necessaryrequirement would be to havea verynarrow Sciences,Vol. 99, p. 376.Berlin, Heidelberg, New York: Springer core-line,with an intrinsicline-width much smaller that 1990;Kresin, V.Z., Morawitz, H.: Phys.Rev. B43, 2691 (1991) 7. Kresin,V.Z., Morawitz, H.: Phys.Rev. B37, 7854 (1988) thewidth of theplasmon side-band. Usually the intrinsic 8. Griffin,A., Pindor,A.J.: Phys.Rev. B39, 11503 (1989) widthof a coreline is of theorder of oneeV dueto life- 9. Varma,C.M., Littlewood,P.B., Schmitt-Rink, S., Abrahams, timeeffects, about equal to thescreened plasmon energy E., Ruckenstein,A.E.: Phys.Rev. Lett. 63, 1996(1989); 64, of the high T~cuprates (1 eV). With a properchoice of 497(E)(1990)

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