Primordial Pairing and Binding of Superheavy Charged Particles in the Early Universe¦ V

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JETP Letters, Vol. 77, No. 7, 2003, pp. 335Ð338. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 403Ð406. Original English Text Copyright © 2003 by Dubrovich, Khlopov.

Primordial Pairing and Binding of Superheavy Charged Particles in the Early Universe¦ V. K. Dubrovich*, 1 and M. Yu. Khlopov**, ***, **** * Special Astrophysical Observatory, Russian Academy of Sciences, St. Petersburg, 196140 Russia 1 e-mail:[email protected] ** Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia *** Moscow Engineering Physics Institute, Moscow, 115409 Russia **** Physics Department, Università degli studi “La Sapienza,” I 00185 Roma, Italy Received July 10, 2002; in ﬁnal form, February 18, 2003

Primordial superheavy particles, considered as the source of Ultra High Energy Cosmic Rays (UHECR) and produced in local processes in the early Universe, should bear some strictly or approximately conserved charge to be sufficiently stable to survive to the present time. Charge conservation dictates that they be produced in pairs, and the estimated separation of particle and antiparticle in such a pair is shown to be in some cases much smaller than the average separation determined by the averaged number density of considered particles. If the new U(1) charge is the source of a long-range field similar to the electromagnetic field, the particle and antiparticle, possessing that charge, can form a primordial bound system with an annihilation timescale, which can satisfy the conditions assumed for this type of UHECR source. These conditions severely constrain the possible properties of the considered particles. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 98.70.Sa; 98.80.Cq

The origin of cosmic rays with energies exceeding If the considered particles are absolutely stable, the the GZK cutoff energy [1] is widely discussed, and one source of UHECRs is related with their annihilation in of the popular approaches is related to decays or anni- the Galaxy. But their averaged number density, con- hilation in the Galaxy of primordial superheavy parti- strained by the upper limit on their total density, is so cles [2Ð5] (see [5] for review and references therein). low that a strongly inhomogeneous distribution is The mass of such particles is assumed to be higher than needed to enhance the effect of annihilation to the level the reheating temperature of the inflationary Universe, desired to explain the origin of UHECR by this mecha- so it is assumed that the particles are created in some nism. nonequilibrium processes, such as inflation decay [6] at In the present note, we offer a new approach to the the stage of preheating after inflation [7]. solution of the latter problem. If superheavy particles possess new U(1) gauge charge, related to the hidden The problems related to this approach are as fol- sector of particle theory, they are created in pairs. The lows. If the source of ultrahigh-energy cosmic rays Coulomb-like attraction (mediated by the massless (UHECR) is related with particle decay in the Galaxy, U(1) gauge boson) between particles and antiparticles the timescale of this decay necessary to reproduce the in these pairs can lead to their primordial binding, so UHECR data needs a special nontrivial explanation. that the annihilation in the bound system provides the Indeed, the relic unstable particle should survive to the mechanism for UHECR origin. present time and, having mass m on the order of Note, first of all, that in quantum theory particle sta- 1014 GeV or larger, should have the lifetime τ, exceed- bility reflects the conservation law, which according to ing the age of the Universe. On the other hand, even if Noether’s theorem is related with the existence of a particle decay is due to gravitational interaction and its conserved charge possessed by the particle under con- τ 4 probability is on the order of 1/ = (m/mPl) m, where sideration. Charge conservation implies that a particle 19 mPl = 10 GeV is the Planck mass, the estimated life- should be created together with its antiparticle. This time would be many orders of magnitude smaller. This means that, being stable, the considered superheavy implies some specific additional suppression factors in particles should bear a conserved charge, and such the probability of decay, which need a rather nontrivial charged particles should be created in pairs with their physical realization [2, 5], e.g., in the model of cryptons antiparticles at the stage of preheating. [8] (see [9] for review). Being created in the local process of inflation field decay, the pair is localized within the cosmological ¦This article was submitted by the authors in English. horizon in the period of creation. If the momentum dis-

336 DUBROVICH, KHLOPOV tribution of created particles peaks below p ~ mc, they If the considered charge is the source of a long- do not spread beyond the proper region of their original range field similar to the electromagnetic field, which localization, being in the period of creation l ~ c/H, can bind a particle and antiparticle into an atomlike sys- where the Hubble constant H at the preheating stage is tem analogous to positronium, it may have important ≤ ≤ in the range Hr H Hend. Here, Hend is the Hubble con- practical implications for the UHECR problem. The stant at the end of inflation and Hr is the Hubble con- annihilation timescale of such a bound system can pro- stant in the period of reheating. For relativistic pairs, vide a rate of UHE particle sources corresponding to the region of localization is determined by the size of UHECR data. the cosmological horizon in the period of their derela- tivization. In the course of successive expansion, the A pair of a particle and antiparticle with opposite distance l between particles and antiparticles grows gauge charges forms a bound system when, in the course of expansion, the absolute magnitude of the with the scale factor, so that, after reheating at the tem- α perature T, it is equal to (here and further, if not indi- potential energy of the pair V = y/l exceeds the kinetic 2 cated otherwise, we use the units ប = c = k = 1) energy of the particle’s relative motion Tk = p /2m. The mechanism is similar to that proposed in [12] for the m 1/2 1 lT()= ------Pl- --- . (1) binding of magnetic monopoleÐantimonopole pairs. It H T is not a recombination one. The binding of two oppositely charged particles is caused just by their Cou- The averaged number density of superheavy parti- lomb-like attraction, once it exceeds the kinetic energy cles n is constrained by the upper limit on their modern of their relative motion. density. If we take their maximum possible contribu- Ω tion in units of critical density, X, not to exceed 0.3, In case plasma interactions do not heat superheavy the modern cosmological average number density particles created with relative momentum p ≤ mc in the ≥ should be period corresponding to Hubble constant H Hs, their 14 Ω initial separation, being on the order of Ð2210 GeV 3 n = 410× ------X T . m 0.3 p lH()= ------, (3) This corresponds at the temperature T to a mean dis- mH Ð1/3 tance (ls ~ n ) equal to 1/3 1/3 experiences only the effect of general expansion, pro- 7m 0.3 1 l ≈ 1.6× 10 ------. (2) portional to the inverse first power of the scale factor, s 14 Ω 10 GeV X T while the initial kinetic energy decreases as the square of the scale factor. Thus, the binding condition is ful- 1/3 1/3 6m 0.3 (l ≈ 4.6× 10 ------cm filled in the period corresponding to the Hubble con- s 14 Ω 10 GeV X stant Hc, determined by the equation in ordinary units). H 1/2 p3 One finds that superheavy nonrelativistic particles ------= ------2 , (4) Hc 2m α H created just after the end of inflation, when H ~ Hend ~ y 1013 GeV, are separated from their antiparticles by dis- where H is the Hubble constant in the period of particle tances more than 4 orders of magnitude smaller than the α average distance between these pairs. On the other creation and y is the “running constant” of the long hand, if the nonequilibrium processes of superheavy range U(1) interaction possessed by the superheavy particle creation (such as decay of inflation) take place particles. If the local process of pair creation does not at the end of the preheating stage and the reheating tem- involve nonzero orbital momentum, due to primordial perature is as low as it is constrained from the effects of pairing the bound system is formed in the state with gravitino decays on 6Li abundance (T < 4 × 106 GeV zero orbital momentum. The size of the bound system r strongly depends on the initial momentum distribution [10, 11]), the primordial separation of pairs given by ≤ Eq. (1) can exceed the value given by Eq. (2). This and for p mc is equal to means that the separation between particles and anti- 4 α p y particles can be determined in this case by their aver- l ==------2------, (5) aged density, if they were created at c α 3 2 β2 2 ym H m 14 2/3 Ω 2/3 ≤ ∼ Ð15 10 GeV X HHs 10 mPl ------where m 0.3 Ω 2/3 α 4 2 ymH ∼ 10 ------X GeV. β = ------. (6) 0.3 p2

JETP LETTERS Vol. 77 No. 7 2003 PRIMORDIAL PAIRING AND BINDING OF SUPERHEAVY CHARGED PARTICLES 337

The annihilation timescale of this bound system can be results of the classical problem of massive particles estimated from the annihilation rate given by with opposite electric charges falling down the center due to radiation in the bound system. The correspond- α2 Ψ 2()σv ∼ Ð3 y ing timescale is given by (see [14] for details) wann = ()0 ann lc ------2Cy, (7) m 3 2 l m τ = ------c ------. (12) where the “Coulomb” factor Cy arises similarly to the 64π α2 case of a pair of electrically charged particle and anti- y particle. For the relative velocity v/c Ӷ 1, it is given by Using Eq. (11) and the condition l(H) ≥ l , one obtains πα β s [13] Cy = 2 yc/v. Finally, taking v/c ~ , one obtains for this case the restriction for the annihilation timescale Ω Ω 5 14 9 X tU Ð10X 10 GeV 10 5 r = ------≤ 310× ------. (13) τ ∼ 1 p m 1 4 X 0.3τ 0.3 m ann ------------------= ------. (8) X 8πα5 mc H m πmβ5 y The gauge U(1) nature of the charge possessed by ≥ ≥ For Hend H Hs, the annihilation timescale is superheavy particles assumes the existence of massless U(1) gauge bosons (y photons) mediating this interac- 5 10 τ × 191 p tion. Since the considered superheavy particles are the ann = 210------α- ------lightest particles bearing this charge and they are not in 50 y mc (9) thermodynamic equilibrium, one can expect that there 104 GeV 5m 4 should be no thermal background of y photons and that × ------s, H 14 their nonequilibrium fluxes can not significantly heat 10 GeV the superheavy particles. α 14 where p ~ mc, y = 1/50, and m = 10 GeV in the range The situation changes drastically if the superheavy Ð26 × 19 Ω 10/3 particles possess not only new U(1) charge but also from 10 s up to 2 10 (0.3/ X) s. The size of the bound system is given by some ordinary (weak, strong, or electric) charge. Due to this charge, superheavy particles interact with the equi- 4 4 2 librium relativistic plasma (with the number density × Ð7p m 10 GeV lc = 510------------cm, (10) n ~ T3) and, for the mass of particles m ≤ α2m , the rate mc 1014 GeV H Pl of heating nσv∆E ~ α2T3/m is sufficiently high to bring × Ð10 ≤ Ω ≤ × Ð7 × ranging for 2 10 X/0.3 1 from 7 10 to 6 the particles into thermal equilibrium with this plasma. 10Ð3 cm. Here, α is the running constant of the considered (weak, strong, or electromagnetic) interaction. Provided that the primordial abundance of superheavy particles created at the preheating stage corre- Plasma heating causes the thermal motion of super- Ω ≤ 2 sponds to the appropriate modern density X 0.3 and ≤ m heavy particles. At T m------, their mean free that the annihilation timescale exceeds the age of the α2m × 17 Pl Universe tU = 4 10 s, owing to the strong depen- path relative to scattering with plasma exceeds the free dence on the initial momentum p, the magnitude rX = thermal motion path, so it is not diffusion but free Ω t motion with thermal velocity v that leads to complete X U × Ð10 T ------τ----- can reach the value rX = 2 10 , which was loss of initial pairing, since v t formally exceeds l at 0.3 X T s found in [2] to fit the UHECR data on superheavy par- Ω 2/3 14 5/3 ticle decay in the halo of our Galaxy. ≤ Ð10 X 10 GeV T 10 mPl------------. ≤ 0.3 m For late particle production (i.e., at H Hs), the ≤ binding condition can retain the form (4) if l(H) ls. In While plasma heating keeps superheavy particles in ≥ ≥ the opposite case, when l(H) ls, the primordial pairing thermal equilibrium, the binding condition V Tkin can- is lost and even zero-orbital-momentum particles and not take place. At T < TN (where N = e, QCD, w, respec- antiparticles originating from different pairs form, in tively, and Te ~ 100 keV for electrically charged parti- ≈ general, bound systems with nonzero orbital momen- cles; TQCD ~ 300 MeV for colored particles and Tw tum. The size of the bound system is in this case 20 GeV for weakly interacting particles; see [14] for obtained from the binding condition for the initial sep- details), the plasma heating is suppressed and super- aration, determined by Eq. (2), and is equal to heavy particles go out of thermal equilibrium. 1015 m 2/30.3 2/3p 2m In the course of successive expansion, the binding l ≈ ------. (11) c α 14 Ω  condition is formally reached at Tc, given by 2 ymPl 10 GeV X mc H Ω 1/3 14 1/3 The lifetime of the bound system can be reasonably α × Ð8X 10 GeV T c = T N y310------------. (14) estimated in this case with the use of the well-known 0.3 m

JETP LETTERS Vol. 77 No. 7 2003 338 DUBROVICH, KHLOPOV

However, for electrically charged particles, the binding (M.Yu.Kh.) expresses his gratitude to IHES and LUTH does not in fact take place to the present time, since one (Observatory Paris-Meudon) for hospitality, and both ≤ gets from Eq. (14) Tc 1 K. Bound systems of hadronic authors are grateful for support in their visit to College and weakly interacting superheavy particles can form, de France (Paris, April 2002). We are grateful to D. Far- ≈ respectively, at Tc ~ 0.3 eV and Tc 20 eV, but even for gion and S. Sarkar for useful comments. weakly interacting particles the size of such bound systems approaches half a meter (30 m for hadronic parti- REFERENCES cles!). This leads to an extremely long annihilation timescale of these bound systems, which cannot fit 1. K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zat- UHECR data. This makes it impossible to realize the sepin and V. A. Kuzmin, Pis’ma Zh. Éksp. Teor. Fiz. 4, considered mechanism of UHECR origin if the super- 114 (1966) [JETP Lett. 4, 78 (1966)]. heavy U(1) charged particles share ordinary weak, 2. V. Berezinsky, M. Kachelriess, and A. Vilenkin, Phys. strong, or electromagnetic interactions. Rev. Lett. 79, 4302 (1997). Disruption of primordial bound systems in their col- 3. V. A. Kuzmin and V. A. Rubakov, Yad. Fiz. 61, 1122 lisions by tidal forces in the Galaxy reduces their con- (1998) [Phys. At. Nucl. 61, 1028 (1998)]. centration in the regions of enhanced density. Such spa- 4. D. Fargion, B. Mele, and A. Salis, astro-ph/9710029; Astrophys. J. 517, 725 (1999); Preprint No. 1179/97, tial distribution, specific for these UHECR sources, INFN (1997); T. J. Weiler, Astropart. Phys. 11, 303 makes it possible to distinguish them from other possi- (1999). ble mechanisms [4, 9, 15] in the future AUGER and 5. H. Ziaeepour, Gravit. Cosmol. Suppl. 6, 128 (2000). EUSO experiments. 6. D. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D 59, The crucial physical condition for the formation of 023501 (1999). primordial bound systems of superheavy particles is the 7. L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. existence of new strictly conserved local U(1) gauge Rev. D 56, 3258 (1997). symmetry, ascribed to the hidden sector of particle the- 8. J. Ellis, G. Gelmini, J. Lopes, et al., Nucl. Phys. B 373, ory. Such symmetry can arise in the extended variants 399 (1992); S. Sarkar, Nucl. Phys. A, Proc. Suppl. 28, of GUT models (see, e.g., [10] for review), in heterotic 405 (1992). string phenomenology (see [13] and references 9. S. Sarkar and R. Toldra, Nucl. Phys. B 621, 495 (2002). therein), and in D-brane phenomenology [16]. Note 10. M. Yu. Khlopov, Cosmoparticle Physics (World Sci., that in such models the strictly conserved SU(2) sym- Singapore, 1999). metry can also arise in the hidden sector, which leads to a nontrivial mechanism of primordial binding of super- 11. Yu. L. Levitan, E. V. Sedelnikov, I. M. Sobol, and M. Yu. Khlopov, Yad. Fiz. 57, 1466 (1994) [Phys. At. heavy particles due to macroscopic size SU(2) confine- Nucl. 57, 1393 (1994)]. ment, as was the case for “tetons” [17]. 12. V. K. Dubrovich, Gravit. Cosmol. Suppl. 8, 122 (2002). The proposed mechanism offers a link between the 13. M. Yu. Khlopov and K. I. Shibaev, Gravit. Cosmol. observed UHECRs and the predictions of particle the- Suppl. 8, 45 (2002). ory, which cannot be tested by any other means and on 14. V. K. Dubrovich, D. Fargion, and M. Yu. Khlopov, Astro- which the analysis of primordial pairing and binding part. Phys. (2003) (in press). can put severe constraints. If viable, the considered 15. V. Berezinsky and A. A. Mikhailov, Phys. Lett. B 449, mechanism makes UHECR a unique source of detailed 237 (1999). information on the possible properties of the hidden 16. G. Aldazabal, L. E. Ibanez, F. Quevedo, and sector of particle theory and on the physics of the very A. M. Uranga, J. High Energy Phys. 0008, 002 (2000); early Universe. L. F. Alday and G. Aldazabal, J. High Energy Phys. The work was partially performed in the framework 0205, 022 (2002). of State Contract 40.022.1.1.1106 and supported in part 17. L. B. Okun’, Pis’ma Zh. Éksp. Teor. Fiz. 31, 156 (1980) by the Russian Foundation for Basic Research, grant [JETP Lett. 31, 144 (1980)]; M. Yu. Khlopov, Yad. Fiz. nos. 02-02-17490 and UR.02.01.026. One of us 32, 1600 (1980) [Sov. J. Nucl. Phys. 32, 828 (1980)].

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Evolution of the Cosmological Constant in Effective Gravity¦ G. E. Volovik Low-Temperature Laboratory, Helsinki University of Technology, FIN-02015 HUT, Finland Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, 117334 Russia e-mail: [email protected].ﬁ Received February 20, 2003; in ﬁnal form, March 5, 2003

In contrast to the phenomenon of nulliﬁcation of the cosmological constant in equilibrium vacuum, which is the general property of any quantum vacuum, there are many options in modifying the Einstein equation to allow the cosmological constant to evolve in a nonequilibrium vacuum. An attempt is made to extend the Ein- stein equation in the direction suggested by the condensed matter analogy of the quantum vacuum. Different scenarios are found depending on the behavior of and the relation between the relaxation parameters involved, some of these scenarios having been discussed in the literature. One of them reproduces the scenario in which the effective cosmological constant emerges as a constant of integration. The second one describes the situation when, after the cosmological phase transition, the cosmological constant drops from zero to a negative value; this scenario describes the relaxation from this big negative value back to zero and then to a small positive value. In the third example, the relaxation time is not a constant but depends on matter; this scenario demonstrates that vacuum energy (or its fraction) can play the role of cold dark matter. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 04.20.Cv; 98.80.Es

1. INTRODUCTION This is the first message from condensed matter to the physics of the quantum vacuum: One should not It is clear that the pressure of the vacuum in our Uni- worry about the huge vacuum energy, the trans-Planck- verse is very close to zero as compared to the Planck ian physics with its degrees of freedom will do the job energy scale, and thus the experimental cosmological of the cancellation of the vacuum energy without any constant is close to zero. However, if at this almost zero fine tuning and irrespective of the details of the trans- pressure one starts calculating the vacuum energy by Planckian physics. There are other messages that are summing all the positive and negative energy states, also rather general and do not depend much on the one obtains a huge vacuum energy that by about 120 details of trans-Planckian physics. For example, if the orders of magnitude exceeds the experimental limit. cosmological constant is zero above the cosmological This is the main cosmological constant problem [1Ð4]. phase transition, it will become zero below the transition after some transient period. Exactly the same “paradox” occurs in any quantum liquid (or in any other condensed matter) at zero pres- Thus, from the quantum-liquid analogue of the sure. The experimental energy of the ground state of, quantum vacuum, it follows that the cosmological con- say, the quantum liquid at zero pressure is zero. On the stant is not a constant but is an evolving physical other hand, if one starts calculating the vacuum energy parameter, and our goal is to find the laws of its evolu- by summing the energies of all the positive and negative tion. In contrast to the phenomenon of the cancellation energy modes up to the corresponding Planck (Debye) of the cosmological constant in the equilibrium vac- scale, one obtains a huge energy. However, there is no uum, which is the general property of any quantum vac- real paradox in quantum liquids, since if one adds all uum, there are many options in modifying the Einstein the trans-Planckian (microscopic, atomic) modes one equation to allow the cosmological constant to evolve. immediately obtains the zero value, irrespective of the However, condensed matter physics teaches us that we details of the microscopic physics [5]. The fully micro- must avoid the discussion of the microscopic models of scopic consideration restores the GibbsÐDuhem rela- the quantum vacuum [6] and use instead the general tion ⑀ = Ðp between the energy (the relevant thermody- phenomenological approach. That is why we do not fol- namic potential) and the pressure of the quantum liquid low the traditional way of description in terms of, say, at T = 0, which ensures that the energy of the vacuum the scalar field, which mediates the decay of the dark state ⑀ = 0 if the external pressure is zero. energy [7], and instead present an attempt at a purely phenomenological description by introducing dissipa- ¦This article was submitted by the author in English. tive terms directly into the Einstein equation.

340 VOLOVIK

2. EINSTEIN EQUATION which distort the effective metric field gµν and thus play Standard formulation. Let us start with the nondis- the role of the gravitational field. As distinct from the Λ sipative equation for gravity—the Einstein equation. It T µν term, which is of the zeroth order in gradients of is obtained from the action curv the metric, the T µν term is of the second order in gra- SS= ++SΛ S , (1) E M dients of gµν. The higher order gradient terms also nat- where SM is the matter action; urally appear in induced gravity. In the induced gravity, the free gravitational field is 1 4 ᏾ SE = Ð------π -∫d x Ðg (2) absent, since there is no gravity in the absence of the 16 G quantum vacuum. Thus, the total energyÐmomentum is the Einstein curvature action; and tensor comes from the original (bare) fermionic degrees Λ of freedom. That is why all the contributions to the 4 energyÐmomentum tensor are obtained by the variation SΛ = Ð------π -∫d x Ð,g (3) 8 G of the total fermionic action over gµν: where Λ is the cosmological constant [8]. Variation δ µν total 2 S 1 ρ M over the metric g gives T µν ==------µν- Ð------Gµν ++Λgµν T µν. (9) Ðgδg 8πG 1 M ------()Gµν Ð Λgµν = T µν, (4) µν 8πG According to the variational principle, δS/δg = 0, the where total energyÐmomentum tensor is zero, which gives rise to the Einstein equation in the form of Eq. (7). Since 1 total Gµν = Rµν Ð ---᏾gµν (5) T µν = 0, it satisfies the conventional and covariant 2 total ν total ν conservation laws, T µν, = 0 and T µ ν = 0, and thus M ; is the Einstein tensor and T µν is the energyÐmomentum serves as the covariant and localized energyÐmomen- tensor for matter. This form of the Einstein equation tum tensor of matter and gravity. implies that the matter fields on the right-hand side of In this respect, there is not much difference between Eq. (4) serve as the source of the gravitational field, Λ different contributions to the energyÐmomentum ten- while the term belongs to the gravity. sor: all come from the original fermions. However, in Moving the Λ-term to the right-hand side of the Ein- the low-energy corner, where the gradient expansion stein equation changes the meaning of the cosmologi- for the effective action works, one can distinguish cal constant. The Λ-term becomes the energyÐmomen- between different contributions: (i) some part of the tum tensor of the vacuum, which in addition to the mat- M ter is the source of the gravitational field [9]: energyÐmomentum tensor (T µν ) comes from the excited fermions—quasiparticles—which in the effec- 1 M Λ Λ Λ ------Gµν ===T µν + T µν, T µν ρΛgµν ------gµν.(6) tive theory form the matter. The other parts come from 8πG 8πG the fermions forming the vacuum—the Dirac sea. The contribution from the vacuum fermions contains (ii) the Here, ρΛ is the vacuum energy density and pΛ = ÐρΛ is the vacuum pressure. zeroth-order term in the gradients of gµν; this is the energyÐmomentum tensor of the homogeneous vac- Einstein equation in induced gravity. In the uum—the Λ term. Of course, the whole Dirac sea cannot induced gravity introduced by Sakharov [10], the grav- be sensitive to the change of the effective infrared fields ity is the elasticity of the vacuum, say, fermionic vac- gµν: only small infrared perturbations of the vacuum, uum, and the action for the gravitational field is induced by the vacuum fluctuations of the fermionic matter which we are interested in, are described by these effec- fields. This kind of effective gravity, emerges in quan- tive fields; (iii) the stress tensor of the inhomogeneous tum liquids [5]. In the induced gravity, the Einstein ten- distortion of the vacuum state, which plays the role of curv sor must also be moved to the matter side, i.e., to the gravity; the second-order term T µν in the stress tensor right-hand side: represents the curvature term in the Einstein equation. M Λ curv 0 = T µν ++T µν T µν , (7) The same occurs in induced QED [11], where the electromagnetic field is induced by the vacuum fluctu- where ations of the same fermionic field. The total electric current curv 1 T µν = Ð------Gµν (8) π int Maxwell 8 G µ δS δS δS j ==------+ ------δ - has the meaning of the energyÐmomentum tensor pro- δAµ δAµ Aµ (10) duced by deformations of the fermionic vacuum. It µ µ describes such elastic deformations of the vacuum, = jcharged particles +,jfield

JETP LETTERS Vol. 77 No. 7 2003 EVOLUTION OF THE COSMOLOGICAL CONSTANT IN EFFECTIVE GRAVITY 341 is produced by excited fermions (the first term) and by flat Universe at T = 0; i.e., a quiescent flat Universe fermions in the quantum vacuum (the second term). without matter is not gravitating. The electric current in the second term is produced by such elastic deformations of the fermionic vacuum that play the role of the electromagnetic field, and the lowest 3. MODIFICATION OF EINSTEIN EQUATION order term in the effective action describing such dis- AND RELAXATION OF THE VACUUM ENERGY tortion of the vacuum state is the induced Maxwell Dissipation in Einstein equation. The Einstein action equation does not allow us to obtain the time dependence of the cosmological constant, because of the Maxwell 3 Ðg µν ν S = dtxd ------F Fµν, (11) ∫ 16πα Bianchi identities Gµ; ν = 0 and covariant conservation νM law for matter fields (quasiparticles) T µ ν = 0, which which is of the second order in gradients of Aµ. Accord- ; ∂ Λ ing to the variational principle, δS/δAµ = 0, the total together lead to µ = 0. But they allow us to obtain the electric current produced by the vacuum and excited value of the cosmological constant in different static fermions is zero. This means that the system is always Universes, such as the Einstein closed Universe [8], locally electroneutral. This ensures that the homoge- where the cosmological constant is obtained as a func- neous vacuum state without excitations has zero elec- tion of the curvature and matter density. To describe the tric charge, i.e., our quantum vacuum is electrically evolution of the cosmological constant, the relaxation neutral. The same occurs with the hypercharge, weak term must be added. charge, and color charge of the vacuum: they are zero The dissipative term in the Einstein equation can be in the absence of matter and fields. introduced in the same way as in two-fluid hydrody- In the traditional approach, the cosmological con- namics [12], which serves as the nonrelativistic ana- stant is fixed, and it serves as the source for the metric logue of the self-consistent treatment of the dynamics field: in other words, the input in the Einstein equation of the vacuum (the superfluid component of the liquid) is the cosmological constant, the output is de Sitter and matter (the normal component of the liquid) [5]: expansion, if matter is absent. In effective gravity, M Λ curv diss where the gravitational field, the matter fields, and the T µν ++T µν T µν +T µν = 0, (12) cosmological “constant” emerge simultaneously in the diss low-energy corner, one cannot say that one of these where T µν is the dissipative part of the total energyÐ fields, is primary and serves as a source for the other momentum tensor. In contrast to the conventional dissi- fields, thus governing their behavior. The cosmological pation of the matter, such as viscosity and thermal con- constant, as one of the players, adjusts to the evolving ductivity of the cosmic fluid, this term is not a part of matter and gravity in a self-regulating way. In particu- M T µν . It describes the dissipative back reaction of the M lar, in the absence of matter (T µν = 0), the nondistorted vacuum, which does not influence the matter conserva- curv νM vacuum (T µν = 0) acquires zero cosmological con- tion law T µ; ν . The condensed-matter example of such stant, since, according to the “gravi-neutrality” condi- a relaxation of the variables describing the fermionic M vacuum is provided by the dynamic equation for the tion Eq. (7), it follows from equations T µν = 0 and order parameter in superconductors—the time-depen- curv Λ T µν = 0 that T µν = 0. In this approach, the input is the dent GinzburgÐLandau equation, which contains the vacuum configuration (in the given example, there is no relaxation term (see, e.g., the book [13]). matter and the vacuum is homogeneous), the output is Let us consider how the relaxation occurs by the the vacuum energy. In contrast to the traditional example of the spatially flat Robertson–Walker metric approach, here the gravitational field and matter serve 2 2 2 2 as a source of the induced cosmological constant. ds = dt Ð a ()t dr . (13) This conclusion is supported by the effective gravity The Ricci tensor and scalar and the Einstein tensor are and effective QED, which emerge in quantum liquids or any other condensed matter system of the special uni- ∂2a ∂2a ()∂ a 2 0 t i δi t t versality class [5]. The nullification of the vacuum R0 ==Ð3------; R j Ð j ------2+ ------2 , (14) energy in the equilibrium homogeneous vacuum state a a a of the system also follows from the variational principle, or more generally from the GibbsÐDuhem relation ∂2a ()∂ a 2 ᏾ t t applied to the equilibrium vacuum state of the fermi- = Ð6 ------+ ------2 , (15) onic system if it is isolated from the environment. In the a a absence of the environment, one has pΛ = 0, while from 2 the GibbsÐDuhem relation ρΛ = ÐpΛ at T = 0 it follows ()∂ 0 0 1᏾ ta ≡ 2 Λ G0 ==R0 Ð3------2 3H , (16) that ρΛ = 0. This corresponds to T µν = 0 for a quiescent 2 a

JETP LETTERS Vol. 77 No. 7 2003 342 VOLOVIK

∂2 equations must be supplemented by the covariant con- j j 1 j j2 a G ==R Ð ---᏾δ δ H + 2------t - servation law to prevent the creation of matter: i i 2 i i a (17) δ j()2 ú ∂ M 3 M 3 = i 3H + 2H . a ()ρ a = p a . (24) ∂a In the lowest order of the gradient expansion, the diss Let us consider the simplest case, when the relax- dissipative part T µν of the stress tensor describing the ation occurs only in the pressure sector, i.e., τ = 0. We relaxation of Λ must be proportional to the first time 1 derivative. assume also that the ordinary matter is cold, i.e., its pressure pM = 0, which gives ρM ∝ aÐ3. Then one finds Cosmological constant as integration constant. Λ τ two classes of solutions: (i) = const and (ii) H = 1/3 2. Let us start with the following guess: The first one corresponds to the conventional expansion with constant Λ term and cold matter, so let us discuss diss ∂ρΛ T µν = τΛ------gµν. (18) the second solution, H = 1/3τ . ∂t 2 Relaxation after cosmological phase transition. This noncovariant term implies the existence of a pre- τ ≡ τ Λ ferred reference frame, which is a natural ingredient of In the simplest case, when 2 = const, the term and the trans-Planckian physics, where general covariance the energy density of matter ρM exponentially relax to is violated. The modified Einstein equations in the 1/3τ2 and to 0, respectively: absence of matter are correspondingly 1 1 M 2 ΛτΛú H ==----- , Λ()t ------8Ð πGρ (),t 3H = + Λ , (19) 3τ 3τ2 (25) 2 M 3H + 2Hú = Λτ+ ΛΛú . (20) ρ ()t t ------= exp Ð-- . ρM τ This gives the constant Hubble parameter H = const, ()0 i.e., the exponential de Sitter expansion or contraction. The cosmological “constant” Λ relaxes to the value Such a solution describes the behavior after the cosmo- determined by the expansion rate: logical phase transition. According to the condensed- matter example of the phase transition, the cosmologi- Λ 2 ()Λ 2 t cal “constant” is (almost) zero before the transition, ()t = 3H + ()0 Ð 3H expÐ----- . (21) τΛ while after the transition it drops to a negative value and then relaxes back to zero [5]. Equation (25) corre- This is consistent with the Bianchi identity, which sponds to the latter stage, but it demonstrates that, in its Λ ∂ Λ τ Λú relaxation after the phase transition, crosses zero and requires that t( + Λ ) = 0. Actually, this situation finally becomes a small positive constant determined corresponds to the well-known case when the cosmo- by the relaxation parameter τ, which governs the expo- logical constant arises as an integration constant (see Λ nential de Sitter expansion. reviews [1, 4]). Here it is the integration constant 0 = Λ τ ∂ Λ + Λ t . Such a scenario emerges because the dissi- Dark energy as dark matter. Let us now allow τ to pative term in Eq. (18) is proportional to gµν. vary. Usually the relaxation and dissipation are deter- Model with two relaxation parameters. Since mined by matter (quasiparticles). The term that violates diss the Lorentz symmetry or the general covariance must T µν is a tensor, the general description of the vacuum contain the Planck scale EPlanck, since it must disappear relaxation requires introduction of several relaxation at infinite Planck energy. The lowest order term, which ប τ times. This also violates the Lorentz invariance, but we contains the EPlanck in the denominator, is / ~ have already assumed that the dissipation of the vac- 2 T /EPlanck, where T is the characteristic temperature or uum variables due to trans-Planckian physics implies energy of matter. In the case of radiation, it can be writ- the existence of the preferred reference frame. In isotro- ten in terms of the radiation density: pic space, we have only two relaxation times: in the energy and pressure sectors. In the presence of matter, 1 M one has ------8= παGρ , (26) 3τ2 2 ΛτΛú π ρM 3H = ++1 8 G , (22) where α is a dimensionless parameter. If Eq. (26) can 3H2 + 2Hú = Λτ+ Λú Ð 8πGpM. (23) be applied to the cold baryonic matter too, then the 2 solution of class (ii) becomes again H = 1/3τ, but now Since the covariant conservation law for matter does τ depends on the matter field. This solution gives the not follow now from the Bianchi identities, these two standard power law for the expansion of the cold flat

JETP LETTERS Vol. 77 No. 7 2003 EVOLUTION OF THE COSMOLOGICAL CONSTANT IN EFFECTIVE GRAVITY 343 Λ τ τ Universe and the relation between and the baryonic single relaxation parameter. The first example ( 1 = 2) matter density ρM: reproduces the well-known scenario in which the effective cosmological constant emerges as a constant of 2/3 M 4 τ τ at∝ ,8πGρ = ------, integration. The second example ( 1 = 0 and 2 = const) 3αt2 describes the situation that occurs if, after the cosmo- (27) logical phase transition, Λ acquires a large negative 2 M H ==-----, Λα()Ð 1 8πGρ . value: Λ relaxes back to zero and then to a small posi- 3t τ tive value. The third example, when 2 is determined by ρ 2 π In terms of the densities normalized to c = 3H /8 G the baryonic matter density, demonstrates that the vac- (the critical density corresponding to the flat Universe uum energy (or its fraction) can play the role of cold Ω ρ ρ dark matter. in the absence of the vacuum energy), Λ = Λ/ c and ΩM ρM ρ These examples are too simple to describe the real = / c one has evolution of the present Universe and are actually M 1 α Ð 1 excluded by observations [2]. A general consideration Ω ==--- , ΩΛ ------. (28) α α with two relaxation functions is needed. In this general case, it corresponds to the varying in time of the param- Since the effective vacuum pressure in Eq. (23) is pΛ ∝ ρ eter wQ = pQ/ Q describing the equation of state of the Ð(Λ + τΛú ) = 0, in this solution the dark energy behaves Λ τ Λú Λ τ Λú quintessence with wQ(t) = Ð( + 2 )/( + 1 ). The as cold dark matter. Thus, the vacuum energy can serve recent observational bounds on w can be found, for as the origin of the nonbaryonic dark matter. Q example, in [14]. I thank A. Achúcarro, T.W.B. Kibble, I.I. Kogan, 4. CONCLUSIONS L.B. Okun, A.K. Rajantie, and A.A. Starobinsky for In the effective gravity, the equilibrium time-inde- fruitful discussions. This work was supported by ESF pendent vacuum state without matter is nongravitating; COSLAB Programme and by the Russian Foundation i.e., its relevant vacuum energy, which is responsible for Basic Research. for gravity, is zero. In a nonequilibrium situation, the cosmological constant is nonzero, but it is an evolving REFERENCES parameter rather than a constant. The process of relaxation of the cosmological constant, when the vacuum is 1. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). disturbed and out of equilibrium, requires some modi- 2. V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9, 373 fication of the Einstein equation violating the Bianchi (2000). identities to allow the cosmological constant to vary. In 3. P. J. E. Peebles and B. Ratra, astro-ph/0207347. contrast to the phenomenon of nullification of the cos- 4. T. Padmanabhan, hep-th/0212290. mological constant in the equilibrium vacuum, which is 5. G. E. Volovik, The Universe in a Helium Droplet (Clar- the general property of any quantum vacuum and does endon Press, Oxford, 2003). not depend on its structure or on the details of the trans- 6. R. B. Laughlin, gr-qc/0302028. Planckian physics, the deviations from general relativ- 7. U. Alam, V. Sahni, and A. A. Starobinsky, astro- ity can occur in many different ways, since there are ph/0302302. many routes from the low-energy effective theory to the 8. A. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. 1, 142 high-energy “microscopic” theory of the quantum vac- (1917). uum. However, it seems reasonable that such modifica- 9. M. Bronstein, Phys. Z. Sowjetunion 3, 73 (1933). tion can be written in the general phenomenological 10. A. D. Sakharov, Dokl. Akad. Nauk SSSR 177, 70 (1968) way, as, for example, the dissipative terms are intro- [Sov. Phys. Dokl. 12, 1040 (1968)]. duced in the hydrodynamic theory. Here we suggested Λ 11. Ya. B. Zel’dovich, Pis’ma Zh. Éksp. Teor. Fiz. 6, 922 to describe the evolution of the term by two phenom- (1967) [JETP Lett. 6, 345 (1967)]. enological parameters (or functions)—the relaxation 12. I. M. Khalatnikov, An Introduction to the Theory of times. The corresponding dissipative terms in the stress Superfluidity (Nauka, Moscow, 1965; Benjamin, New tensor of the quantum vacuum are determined by trans- York, 1965). Planckian physics and do not obey the general covari- 13. N. B. Kopnin, Theory of Nonequilibrium Superconduc- ance. tivity (Clarendon Press, Oxford, 2001). We discussed here the simplest examples of the 14. D. Pogosyan, J. R. Bond, and C. R. Contaldi, astro- relaxation of the vacuum to equilibrium, described by a ph/0301310.

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JETP Letters, Vol. 77, No. 7, 2003, pp. 344Ð348. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 412Ð416. Original Russian Text Copyright © 2003 by Gornov, Ber, Gurov, Lapushkin, Morokhov, Pechkurov, Poroshin, Sandukovsky, Tel’kushev, Chernyshev.

Spectroscopy of the 5H Superheavy Hydrogen Isotope M. G. Gornov1, †, M. N. Ber1, Yu. B. Gurov1, *, S. V. Lapushkin1, P. V. Morokhov1, V. A. Pechkurov1, N. O. Poroshin1, V. G. Sandukovsky2, M. V. Tel’kushev1, and B. A. Chernyshev1 1 Moscow Engineering Physics Institute, Kashirskoe sh. 31, Moscow, 115409 Russia * e-mail: [email protected] 2 Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russia Received June 24, 2002; in ﬁnal form, February 12, 2003

The formation of the 5H superheavy hydrogen isotope was experimentally sought in the reactions induced by stopped πÐ mesons absorbed by 9Be nuclei. Peaks in missing-mass spectra were observed in two reaction channels, 9Be(πÐ, pt)X and 9Be(πÐ, dd)X, and were attributed to the 5H resonance states. The lowest state has param- ± Γ ± eters Er = 5.5 0.2 MeV and = 5.4 0.5 MeV [Er is the resonance energy measured from the (triton + two neutrons) threshold]. Therefore, 5H is bound more weakly than 4H. Excited states of 5H were also observed. All ± Γ ± ± Γ ± three resonance levels (E1r = 10.6 0.3 MeV, 1r = 6.8 0.5 MeV; E2r = 18.5 0.4 MeV, 2r = 4.8 1.3 MeV; ± Γ ± E3r = 26.7 0.4 MeV, 3r = 3.6 1.3 MeV) can decay into ﬁve free nucleons. © 2003 MAIK “Nauka/Interpe- riodica”. PACS numbers: 21.10.-k; 25.80.Hp

† 6 πÐ ± Γ ± Nuclei near the nucleon stability boundary are of tions Li( , p)X (Er = 11.8 0.7 MeV, = 5.6 0.9 MeV) interest due both to their unique properties, among 7 πÐ ± Γ ± and Li( , d)X (Er = 9.1 0.7 MeV, = 7.4 which the neutron halo is most brilliant [1], and to the 0.6 MeV), respectively [4]. Experimental results [3, 4] importance of the precise determination of their param- are close to each other, but the data obtained on the lith- eters for testing and developing nuclear models. Super- ium isotopes were based on a smaller statistical sample. heavy hydrogen isotopes are lightest neutron-rich resonances and, therefore, allow the simplest description by For a long time, the formation of 5H on heavy-ion theoretical models. At the same time, only superheavy beams was observed only in the 7Li(6Li, 8B)5H reaction, ≈ Γ ≈ hydrogen isotopes among all nuclei at the nucleon sta- where the resonance state with Er 5.2 MeV and bility boundary have the unfilled proton 1s shell. Thus, 4 MeV was identified [5]. A narrow 5H state with ± Γ ± the data concerning these nuclei can provide the possi- energy Er = 1.7 0.3 MeV and width = 1.9 0.4 MeV bility of a critical test of the models based on the helium was recently observed in the 1H(6He, pp)X reaction [6]. and lithium spectroscopy. Thus, the experimental spectroscopy of 5H remains The question of the existence of the 5H isotope was uncertain. The parameters of the observed states differ open for a long time. Enhancement in the proton spec- from each other by more than the measurement errors trum near the kinematic threshold of the 6Li(πÐ, p)X presented above. These discrepancies are possibly reaction was observed in [2]. However, the interpreta- caused by noticeable selectivity of the population of 5H tion of experimental data was uncertain; this enhance- levels. In any case, new experimental data are neces- ment was possibly caused by a substantial contribution sary. from the three-body reaction channels. Theoretical calculations also strongly differ from An indication of the formation of 5H in the 9Be(πÐ, each other [7Ð9]. All models predict the nucleon insta- 5 5 bility of H, but the resonance energy of the ground pt) H reaction induced by stopped pions was obtained ≈ in our previous work [3]. Attributing enhancement in state calculated in [7] ( 6 MeV) is significantly higher the missing-mass spectrum to one state, we obtained than the values of 2.5Ð3 MeV obtained in [8, 9]. In ± Γ addition, the authors of [8] predicted the existence of the resonance parameters Er = 7.4 0.7 MeV and = ± 5H excited states with resonance energies ≈4.5Ð 8 3 MeV [Er is the resonance energy measured from the (triton + two neutrons) threshold]. More recently, 7.5 MeV. we observed indications of the formation of 5H in the In this paper, we present the spectroscopic results inclusive spectra of protons and deuterons in the reac- obtained for 5H in a joint experiment carried out at the Moscow Engineering Physics Institute and Northwest- † Deceased. ern University (Evanston, Illinois). In this experiment,

SPECTROSCOPY OF THE 5H SUPERHEAVY HYDROGEN ISOTOPE 345 the formation of neutron-rich isotopes was investigated in reactions induced by stopped pions absorbed by 1p- shell nuclei. A 9Be target was again used to seek for 5H, because results could be obtained with a significantly larger statistical sample and with the noticeably better energy resolution. Moreover, 9Be is preferable over lithium targets, because the ratio between the yields of the 5H-formation, channels and the physical background in the correlation data is higher than in the inclusive data [3, 4]. MM Experiment was carried out on the low-energy pion channel (LEP) at the Los Alamos Meson Physics Facil- Fig. 1. Missing-mass spectrum for the 9Be(πÐ, tt)X reaction. ity with a multilayer semiconductor spectrometer made The solid line is the complete description; lines 1 and 2 are at the Moscow Engineering Physics Institute [10]. A the phase-space distributions for the 9Be(πÐ, tt)dn and beam of 30-MeV pions passed through a beryllium 9Be(πÐ, tt)p2n reactions, respectively. moderator and stopped in a thin target with a thickness of 24 mg/cm2. The 9Be target included less than 1% of uncontrolled impurities. The rate of pion stops was equal to 6 × 104 sÐ1. Charged particles formed after the observed near the zero missing masses is assigned to absorption of pions by nuclei were detected by two tele- the 9Be(πÐ, tt)t three-body reaction channel. The peak ° ± ∆ ≅ ± scopes set at an angle of 180 to each other. Each tele- position (EMM = 0.0 0.1 MeV) and its width ( 1.2 scope consisted of silicon detectors. Two surface-bar- 0.1 MeV) are consistent with the results obtained with rier detectors with thicknesses 100 and 400 µm were 11B. The peak is slightly broadened, because the angu- placed at the beginning of a telescope, and 3-mm-thick lar acceptance of the spectrometer causes an increase in lithium drift detectors followed them. The area of a sen- the width of a device line with decreasing mass of unde- sitive region of each detector was equal to 8 cm2. The tected reminder. The absence of time variations in the total thickness of each telescope was equal to 43 mm, spectrometer characteristics was corroborated by the which was sufficient for all long-range nuclear particles constant spectrum shape for different parts of the accu- formed in the absorption reaction to be completely mulated statistical sample. decelerated in the sensitive volume of a telescope. The spectrometer and experimental procedure were To calibrate the detectors in energy, a radioactive described in more detail in [10]. source of α particles and a precise-amplitude generator Figures 2 and 3 show the missing-mass spectra for were used. For absolute scale calibration, two-body the 9Be(πÐ, pt)X and 9Be(πÐ, dd)X reactions, respec- reactions with pion absorption and known final nuclear tively. Missing masses are measured from the sum of states were taken. The systematic error did not exceed masses of a triton and two neutrons. First, we note that 100 keV. The energy resolution of the spectrometer for there are no manifestations of 5H bound states in the the detection of single-charged ions (p, d, t) was better negative missing-mass region. The weak background in than 0.5 MeV. this region is caused by random coincidences in corre- In this work, 5H states were sought in the missing- lation measurements. At the same time, the structures, mass spectra obtained from correlation data. In this which can be attributed to the formation of resonance case, the energy resolution was determined from mea- states, are observed in the positive missing-mass region surements made on a 11B target in the same experimen- in both spectra. To separate these 5H states and to deter- tal run. When two single-charged ions are detected in mine their parameters, we used the least-squares three-body channels of the absorption reaction on this approximation in the description of measured spectra nucleus, 5Ð9He isotopes are formed. The ground states by the sum of n-body phase-space distributions (n ≥ 4) of these isotopes are reliably determined and well sep- and BreitÐWigner distributions. The angular and arated from excited levels [11, 12]. Data analysis, energy broadenings of the spectrometer, and the back- whose results were partially reported in [13, 14], indi- ground of random coincidences, were also taken into cated that the missing-mass resolution is ≤1.0 MeV, and account in the calculations. the error in the absolute scale calibration does not The missing-mass spectra for the 9Be(πÐ, pt)X and exceed 0.1 MeV. 9Be(πÐ, dd)X reactions shown in Figs. 2a and 3a, In the experiment with the 9Be target, the energy res- respectively, are satisfactorily approximated (χ2 per olution and calibration, as well as the possible time degree of freedom is equal to 1.05 and 0.94, respec- variation of these quantities, were checked by correla- tively) by four 5H resonance states and the sum of tion measurements of tt events. Figure 1 shows the n-body phase-space distributions (4 ≤ n ≤ 6). The reso- missing-mass spectrum obtained in these measure- nance parameters of the 5H isotope that were obtained ments. The triton mass is the reference point. A peak by fitting two spectra, together with their weighted

JETP LETTERS Vol. 77 No. 7 2003 346 GORNOV et al.

MM MM

Fig. 2. Missing-mass spectra for the 9Be(πÐ, pt)X reaction: (a) measured spectrum and (b) spectrum measured under Fig. 3. Missing-mass spectrum for the 9Be(πÐ, dd)X reac- the restriction Px ≤ 100 MeV/c on the momentum of unde- tion: (a) measured spectrum and (b) spectrum measured tected residual. The solid lines are the total description and under the restriction Px ≤100 MeV/c on the momentum of BreitÐWigner distributions, the numbered lines are the undetected reminder. The solid lines are the total descrip- phase-space distributions for (1) all reaction channels, tion and BreitÐWigner distributions, the numbered lines are (2) 9Be(πÐ, pt)4Hn, (3) 9Be(πÐ, pt)t2n, and (4) 9Be(πÐ, the phase-space distributions for (1) all reaction channels, pt)d3n channels. (2) 9Be(πÐ, dd)t2n, and (3) 9Be(πÐ, pt)4Hn channels. averages, are listed in the table. The Γ values are the consistent with the existence of four resonance states of FWHMs of the peaks shown in the figures. Errors of the the 5H isotope. parameters presented in the table are caused by the sta- Figures 2 and 3 also show the contributions from the tistical and systematic errors of measurements. multiparticle phase-space distributions to the spectra. The spectral shapes and the relations between yields The comparison of these distributions with the mea- of resonances are noticeably different for the 9Be(πÐ, sured spectra shows that the observed features cannot 9 πÐ be reproduced without resonance states. Since two pt)X and Be( , dd)X reactions (Figs. 2a, 3a). This dif- high-excited states are less noticeable, two approxima- ference possibly indicates that the mechanisms of these tions of spectra with three resonance states sequentially reactions are different. Nevertheless, the parameters of excluding 18.5- and 26.7-MeV levels were tested using the BreitÐWigner distributions for both channels coin- the χ2 criterion. Both hypotheses can be rejected at a cide to within the measurement errors. Therefore, the 10% significance level. We also note that a satisfactory results are reliable. approximation cannot be achieved without four-body The stability of the spectroscopic data obtained for phase-space distributions with a dineutron 2n and (or) a 4 5H can be additionally tested by using the fact that H resonance in the final state. quasifree processes, where the nucleons of residual The ground-state resonance energy of the 5H iso- nucleus are not directly involved in the reactions, con- tope, which was determined in this work, agrees with tribute significantly to the three-body channels of pion- the experimental results obtained in [5] and theoretical absorption reaction. With the aim of relative enrich- calculations made in [8] but is higher (∆E ~ 2Ð3 MeV) ment of the measured spectra with such events, we than the experimental results from [6] and theoretical posed restriction on the momentum of residual nucleus calculations [9, 10]. In our opinion, the discrepancy (Px < 100 MeV/ c). This limit certainly does not exceed between the experimental data are too large. It is worth the expected value for the Fermi momentum of the noting that the binding energy of the 5H isotope, cluster in a nucleus. The missing-mass spectra obtained according to our measurements, is much less than for with such a restriction are shown in Figs. 2b and 3b. the 4H isotope. This conclusion is valid even with the These spectra were approximated by the BreitÐWigner inclusion of uncertainty in the measurements of the res- distributions with parameters presented in the table. onance energy of 4H, which varies from ~2 MeV [15, The χ2 values per degree of freedom are equal to 1.2 16] to ~3.4 MeV [17]. This result is inconsistent with and 1.1 for pt and dd events, respectively, which are the conclusion made in [6] that the binding energy of

JETP LETTERS Vol. 77 No. 7 2003 SPECTROSCOPY OF THE 5H SUPERHEAVY HYDROGEN ISOTOPE 347 Resonance parameters of the 5H isotope Reaction channel Weighted average 9Be(πÐ, pt)5H 9Be(πÐ, dd)5H Γ Γ Γ Er, MeV , MeV Er, MeV , MeV Er, MeV , MeV 5.2 ± 0.3 5.5 ± 0.5 6.1 ± 0.4 4.5 ± 1.2 5.5 ± 0.2 5.4 ± 0.5 10.4 ± 0.3 7.4 ± 0.6 11.4 ± 0.7 5 ± 1 10.6 ± 0.3 6.8 ± 0.5 18.7 ± 0.5 3.9 ± 2.0 18.3 ± 0.5 5.5 ± 1.7 18.5 ± 0.4 4.8 ± 1.3 26.8 ± 0.4 3.0 ± 1.4 26.5 ± 1.0 6 ± 3 26.7 ± 0.4 3.6 ± 1.3

the superheavy hydrogen isotopes as a function of the excitation energy Ex = 35.7 and 34 MeV, respectively, number of neutrons behaves similarly to the binding are, possibly, the isobar analogs for the level with E2r = energy of heavy helium isotopes: bonding is stronger 18.0 MeV observed in our experiment. for an even number of neutrons because of the neutron In conclusion, we note that the continuation of coupling. studying superheavy hydrogen isotopes is of current The energy determined here for the ground state of interest. The solution to the problem of the effect of the 5H is supported by the following arguments. First, the structure of nuclei involved in a reaction on the popula- data are based on a larger statistical sample than in tion of the 5H levels can possibly resolve contradictions other experiments. Second, our experiment covered a between the results of different works. However, a wide missing-mass interval, which minimized the search for the heavier isotopes such as 6H and 7H is of phase-space effects; in particular, this conclusion is particular interest. At present, we continue to analyze illustrated by the spectra determined with a restriction the possible formation of these isotopes in the 9Be(πÐ, on the residual momentum. Third, the phenomenologi- pd)X and 9Be(πÐ, pp)X reactions. cal consideration of the three-body channel yields for the reactions induced by the absorption of stopped This work was supported by the Russian Foundation pions shows that the ground states of the residual nuclei for Basic Research (project no. 00-02-17511), the pro- are more populated than the excited states [13]. At the gram “Universities of Russia” (project no. UR.02.01.007), 5 and by the U.S. Civilian Research and Development same time, the formation of other states in the H iso- Foundation for the Independent States of the Former tope can possibly be suppressed in the reaction chan- Soviet Union (grant no. MO-011-0). nels under investigation due to the effects of nuclear structure of 9Be. An important result of our measurements is the REFERENCES observation of several excited levels of the 5H isotope. 1. I. Tanihata, J. Phys. G 22, 157 (1996). To date, experimental information about excited states 2. K. K. Seth, in Proceedings of 4th International Confer- of nuclei near the nucleon stability boundary is very ence on Nuclei Far from Stability (CERN, Helsingor, limited and is absent for 5H. At the same time, theory 1981), No. 81-90, p. 655. predicts the existence of, at least, two excited states 3. M. G. Gornov, Yu. B. Gurov, V. P. Koptev, et al., Pis’ma with resonance energies ~4.5Ð7.5 MeV [8]. The excita- Zh. Éksp. Teor. Fiz. 45, 205 (1987) [JETP Lett. 45, 252 tion energies of these states (~2Ð5 MeV) are close to (1987)]. our value for the ﬁrst excited state. The peak observed 4. A. I. Amelin, M. G. Gornov, Yu. B. Gurov, et al., Pis’ma in the experiment is possibly the superposition of two Zh. Éksp. Teor. Fiz. 51, 607 (1990) [JETP Lett. 51, 688 levels. However, this assumption is inconsistent with (1990)]. the fact that its parameters are identical in two reaction 5. D. V. Aleksandrov, E. Y. Nikol’skii, B. G. Novatskii, channels. et al., in Proceedings of the International Conference on Exotic Nuclei and Atomic Masses (ENAM-95), Arles, We emphasize that the resonance energies of excited France, 1995 (Frontiers, Gifsur-Yvette, France, 1995), states of 5H exceed the energy necessary for the decay p. 329. into ﬁve nucleons. Excitations of this system of free 6. A. A. Korsheninnikov, M. S. Golovkov, I. Tanihata, nucleons are high and reach ~18 MeV (or 3.6 MeV per et al., Phys. Rev. Lett. 87, 092501-1 (2001). nucleon). The origin and the formation mechanism of 7. A. M. Gorbatov, V. L. Skopich, P. Yu. Nikishov, et al., such states are unclear. Compilations of light-nuclei Yad. Fiz. 50, 1551 (1989) [Sov. J. Nucl. Phys. 50, 962 spectroscopy [11, 12] show that the same high excita- (1989)]. tions above the decay threshold were observed only for 8. B. Shul’gina, B. V. Danilin, L. V. Grigorenko, et al., the 5He and 5Li isotopes. The levels in 5He and 5Li with Phys. Rev. C 62, 014312 (2000).

JETP LETTERS Vol. 77 No. 7 2003 348 GORNOV et al.

9. P. Descouvemont and A. Kharbach, Phys. Rev. C 63, 15. D. V. Aleksandrov, E. Yu. Nikol’skiœ, B. G. Novitskiœ, 027001 (2001). et al., Pis’ma Zh. Éksp. Teor. Fiz. 62, 18 (1995) [JETP 10. M. G. Gornov, Yu. B. Gurov, P. V. Morokhov, et al., Nucl. Lett. 62, 18 (1995)]. Instrum. Methods Phys. Res. A 446, 461 (2000). 16. M. G. Gornov, Yu. B. Gurov, S. V. Lapushkin, et al., in 11. F. Ajzenberg-Selove, Nucl. Phys. A 490, 1 (1988). LI Meeting on Nuclear Spectroscopy and Atomic 12. D. R. Tilley, C. M. Cheves, J. L. Godwin, et al., Nucl. Nucleus Structure (Sarov, 2001), p. 142. Phys. A 708, 3 (2002). 17. D. R. Tilley, H. R. Weller, and G. M. Hale, Nucl. Phys. A 13. M. G. Gornov, Yu. B. Gurov, S. V. Lapushkin, et al., Izv. 541, 1 (1992). Ross. Akad. Nauk, Ser. Fiz. 62, 1781 (1998). 14. M. G. Gornov, Yu. B. Gurov, S. V. Lapushkin, et al., Phys. Rev. Lett. 81, 4325 (1998). Translated by R. Tyapaev

JETP LETTERS Vol. 77 No. 7 2003

J/ψ cc Production in e+eÐ and Hadronic Interactions¦ A. B. Kaidalov Institute of Theoretical and Experimental Physics, Moscow, 117259 Russia e-mail: [email protected] Received March 5, 2003

Predictions of the nonperturbative quark gluon strings model, based on the 1/N expansion in QCD and the string picture of interactions for production of states containing heavy quarks, are considered. Relations between fragmentation functions for different states are used to predict the fragmentation function of c quark to J/ψ mesons. The resulting cross section for J/ψ production in e+eÐ annihilation is in good agreement with recent the Belle result. It is argued that the associated production of cc states with open charm should give a substantial contribution to the production of these states in hadronic interactions at very high energies. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 13.66.Bc; 12.38.Qk

Investigation of heavy quarkonia production at high on this approach (the quark gluon strings model energies provides important information on QCD (QGSM) [8]) has been successfully applied to produc- dynamics in an interesting region of intermediate dis- tion of different hadrons at high energies. It has been tances from 1/mQ to r , where mQ is the heavy quark also used for description of inclusive spectra of hadrons QQ containing heavy (c, b) and light quarks [9Ð11]. In mass and rQQ is the radius of a heavy quarkonia state. QGSM, the fragmentation functions, which describe For c and b quarks, this is the region 0.05 fm < r < 1 fm. transitions of strings to hadrons in many cases, can be In this region both perturbative and nonperturbative predicted theoretically [8, 12] and are expressed in effects can be important. Production of J/ψ mesons is terms of intercepts of the corresponding Regge trajec- studied experimentally in e+eÐ annihilation, γp, hp, hA, tories. We will show that the model naturally leads to and AA collisions. Analysis of hadronic interactions the cross section of J/ψ production in e+eÐ annihilation shows that the simplest perturbative approach (color consistent with the Belle result. Estimate of the contri- singlet model) [1] does not reproduce experimental bution of the same mechanism in hadronic interactions data [2]. This observation leads to an introduction of the indicates that it can be important at energies s = color octet mechanism [3] of heavy quarkonia produc- 2 tion. In this approach, a set of nonperturbative matrix 10 GeV. elements is introduced, which is determined from a ﬁt Let us ﬁrst discuss heavy quarkonia production in to data. A characteristic prediction of this approach is a e+eÐ collisions. In these reactions, a cc pair is produced large transverse polarization of J/ψ, and ψ' at large directly by a virtual photon. However, the probability of transverse momenta [4] is not supported by the Teva- transition of such a state at high energies (far above the tron data [5]. threshold of charm production) to J/ψ is very small. A new mystery in the problem of heavy quarkonia The simplest diagram of QCD perturbation theory production has been added by recent result of the Belle (Fig. 1a) corresponds to a transition to a white cc state collaboration [6] on the large production of J/ψ mesons with relative momentum characteristic to J/ψ by emis- with charmed hadrons. The observed cross section at sion of two hard gluons. This cross section is sup- s = 10.6 GeV is an order of magnitude larger than theoretical predictions [7] based on perturbative QCD. It is interesting that at this energy the associated production of J/ψ with a cc pair is the dominant mechanism of J/ψ production [6]. e+ c c e+ c c c In this paper, a nonperturbative approach, based on Ð c Ð the 1/N expansion in QCD and the string picture of par- e c e c ticle production, is used for description of heavy g quarkonia production at high energies. A model based

+ Ð ¦ This article was submitted by the author in English. Fig. 1. Diagrams for J/ψ production in e e annihilation.

350 KAIDALOV 2 π pressed at high energies by a factor 4mc /s and at s = fragmentation function of a light quark to meson 10 GeV constitutes 10Ð3 of the total cc cross section [12] [7]. + ()α D 2 uc 21Ð D∗()0 + π+ D Γ ()1 Ð α*()0 s J/ψ production in association with an extra charmed D / ≡ u D 0 Ru ------+ = ------pair (Fig. 1b) does not have this suppression but con- π Γ2()α 2 D 1 Ð ρ()0 m ⊥ tains a smallness due to the production of this pair and u D (3) ()α a high threshold of the processes. At high energies, this 21Ð ρ()0 2 mπ⊥ 2()αρ()0 Ð α∗()0 mechanism can be considered as a fragmentation of × ------()1 Ð z D , ψ uu cc() to J/ . Calculation in the lowest order of QCD s0 perturbation theory [7] shows that this mechanism is α α ρ important at energies s ≥ 50 GeV but at s = 10 GeV where ρ, D*(0) are intercepts of the and D* Regge α is smaller than the mechanism of Fig. 1a by an order of trajectories. They are related to ψ(0) by the equation magnitude and is about 0.07 pb. This is in sharp contra- [14] σ ψ +0.21 ± diction with the Belle result: (J/ cc) = 0.87Ð0.19 α α α ρ()0 +2ψ()0 = D∗().0 (4) 0.17 pb. I shall use the following values for these intercepts: Note that, for states of comparatively large radius, α α α such as J/ψ and especially ψ' or χ , a nonperturbative ρ(0) = 0.5, ψ(0) = Ð2, and D*(0) = Ð0.75, in accord c α fragmentation can be important. Thus, I shall estimate with Eq. (4). The uncertainty in the value of ψ(0) dis- a fragmentation of cc() into J/ψ using the nonperturba- cussed in [9] is eliminated at present by experimental data on the inclusive spectra of charmed hadrons in tive model mentioned above. In this model, particle hadronic collisions. production is described in terms of production and fragmentation of quarkÐgluon strings. The behavior of the The gamma functions in Eq. (3) appear from Regge fragmentation functions is determined in the limit residues of the corresponding trajectories, which were z 1 from the corresponding Regge limit and is chosen in accord with dual models and are in a good α expressed in terms of Regge intercepts i(0) [8, 12]. agreement with data on widths of hadronic resonances The fragmentation function of a c quark to J/ψ in this [15]. The coupling is assumed to be universal (with an model is written in the form [12] account of SU(4) and heavy quark symmetry).

α α λ The quantities s0i entering in Eq. (3) can be easily ψ Ð ψ()0 ()Ð ψ()0 + Dc = aψz 1 Ð z , (1) calculated using the formulas and parameters of [13]

2α ()0 α α ()0 uc D∗ uu ρ()0 DD ψ where αψ is an intercept of the J/ψ Regge trajectory, () () () (0) s0 = s0 s0 ; (5) which is known from analysis of data on the spectrum of cc states and analysis of inclusive spectra of uu 2 2 DD ()2 s0 ==4mu⊥ 1 GeV ; s0 =mc⊥ + mu⊥ . (6) λ α 2 ≈ charmed particles (see below), = 2 'D∗ p⊥D 1. Thus, With mu⊥ = 0.5 GeV and mc⊥ = 1.6 GeV [13], we obtain this fragmentation function is characterized by one ()uc constant, aψ. In order to determine this constant, we will s0 = 3.57 GeV. Using these values for s0i in Eq. (3) use a relation between the fragmentation function of a 2 2 2 2 and mπ⊥ = 0.18 GeV , m ⊥ = 5 GeV , and the frag- c quark to J/ψ and the fragmentation function of a light D π+ quark to a D (D*) meson in the limit z 1. According mentation function D = 0.44 [8], we obtain the func- to the rules formulated in [12, 13], both functions have u + ()α λ ()α λ D Ð ψ()0 + Ð ψ()0 + () the same behavior on z: ()1 Ð z as z 1 and tion Du at z 1 in the form 0.011 Ð z . differ only by a kinematic factor related to the mass dif- This value is in reasonable agreement with phenomeno- ference between the J/ψ and D (D*) meson logical studies of charmed particle production in hadronic interactions in the framework of QGSM [10, 11]. 21()Ð α ()0 D uc D∗ uc ψ D s D/ ≡ u 0D The value of s0ψ in Eq. (2) can be calculated in the R ------ψ = ------. (2) uc DD ψD D s ψ c 0 same way with the substitution s0 s0 = 6.72 GeV2. Finally, we obtain from Eq. (2) The quantities s will be determined below. 0i ()α λ ψ ()Ð ψ()0 + D Dc = 0.05 1 Ð z ; z 1. (7) Now we shall ﬁnd the fragmentation function Du in the limit z 1. In this limit, it is related to the Thus, aψ = 0.05.

JETP LETTERS Vol. 77 No. 7 2003 J/Ψ cc PRODUCTION IN e+eÐ AND HADRONIC INTERACTIONS 351 ∞ At asymptotic energies s , the cross section q ψ + Ð for J/ production in e e annihilation is equal to p q 1 q q σ σ ψ D ψ = 2 cc∫Dc ()z dz; (8) D c Ð c 0 D q c Ð the factor of 2 in Eq. (8) takes into account J/ψ produc- D tion by both c and c quarks. At energy s ~ 10 GeV, p q there is an extra suppression due to phase space corrections for production of a heavy state. We estimate it by ψ γ 2 2 Fig. 2. Diagrams for J/ production in pp interactions. introducing an extra factor = 14Ð MD/Mcc into 2 Eq. (8). The distribution in Mcc is related to the z distribution. It has a maximum at M2 ≈ 0.27 s. For the shall estimate this suppression factor for the energy of energy of Belle experiment, the correction factor γ = the HERA-B experiment [18] Elab = 920 GeV. Let us 0.7. Thus, we obtain the following cross section for J/ψ denote an extra suppression factor compared to the sup- γ pression of a single cc pair by pp. For its estimation, it cc production at s = 10.6 GeV σψ = 1.2 pb. This is possible to introduce the same kinematical factors as value is in a good agreement with the Belle result [6] in e+eÐ collisions for each qÐc and cÐqq subchain. For and is much larger than the perturbative QCD predic- ψ γ ≈ tion [7]. The estimated uncertainty in the value of the J/ production at rapidity y = 0, pp 0.5. Another esti- cross section due to possible variation of quantities s0i, mate can be made by assuming that the J/ψÐDD sys- α γ ≈ mi⊥, and i(0) is about 50%. tem is produced by gluon fusion. This gives pp 0.4. Let us consider now J/ψ production in hadronic Using these estimates and taking into account that ψ interactions. In the approach based on the 1/N expan- σ σcc ≈ Ð2 pp/ pp 10 , we find that the associated production sion [16], the main diagrams for particle production ψ correspond to two-chain configurations, shown for pp of J/ with charmed hadrons constitutes at this energy interactions in Fig. 2a [8]. They can be considered as ~10%. At Tevatron energies, the role of this mechanism production and fragmentation of two qÐqq strings. It is is more important, and it can (at least partly) explain the excess of J/ψ production at Tevatron compared to the important to emphasize that production of one cc pair color singlet model. together with light quark pairs in this approach always ψ + Ð leads to an open charm production (Fig. 2b), and J/ψ in For ', associated production with cc in e e annihi- this case is produced by the OZI forbidden mechanism lation is not yet known experimentally. However, its Ð2 total inclusive yield is close to the one for J/ψ [19]. If [17]. This leads to a strong suppression (~10 ) for ψ heavy quarkonia production in hadronic collisions the probability of ' production by c-quark fragmentation is the same as for J/ψ, it will have an even stronger compared to open charm (beauty) production. To pro- ψ duce J/ψ in the chains by the OZI allowed mechanism, impact on ' production in hadronic collision, because the experimental ψ' cross section is smaller than for it is necessary to produce two cc pairs close in rapidity ψ ψ σ ' σcc ≈ × Ð3 (Fig. 2c). Though this mechanism is suppressed due to J/ : pp/ pp 1.6 10 , and associated production production of extra heavy quark pairs, it can compete at can constitute a large fraction of the ψ' production. very high energy with the mechanism of single cc -pair In conclusion, it was demonstrated that the nonper- production. Its contribution can be estimated from turbative QGSM model predicts a sizable J/ψ cc pro- charm quark fragmentation into heavy quarkonia in duction in e+eÐ annihilation at high energies, consistent e+eÐ annihilation. Consider production of a cc pair in with recent experimental results [6]. In the approach the qÐqq string of Fig. 2. In each of the qÐc and cÐqq based on the 1/N expansion in QCD, it was shown that substrings, an extra cc pair can be produced and frag- a large fraction of cc -quarkonia production in hadronic ment to a given quarkonium state. So it is possible to collisions at very high energies can be due to associated use the estimate of the fragmentation function of c ()c production with charmed hadrons. quarks given above or direct experimental data from I thank K. Boreskov and O.V. Kancheli for useful e+eÐ to determine the contribution of the corresponding discussions. I am especially grateful to M.V. Danilov diagrams to quarkonia production. This calculation is for drawing my attention to this problem and discussion rather straightforward except for a threshold suppres- of results of the Belle collaboration. sion factor. It is clear that, at the energies of fixed-target This work is supported in part by INTAS (grant experiments s = 10Ð40 GeV, there is a strong sup- no. 00-00366), NATO (grant no. PSTCLG-977275), and the Russian Foundation for Basic Research (grant pression of the production of J/ψ and extra DD pair. I nos. 00-15-96786 and 01-02-17383).

JETP LETTERS Vol. 77 No. 7 2003 352 KAIDALOV

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JETP LETTERS Vol. 77 No. 7 2003

JETP Letters, Vol. 77, No. 7, 2003, pp. 353Ð357. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 421Ð425. Original Russian Text Copyright © 2003 by Prudkovskiœ.

Spatiotemporal Self-Organization upon Two-Wave Mixing in a Photorefractive Medium P. A. Prudkovskiœ Faculty of Physics, Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] Received 2003

Nonlinear dynamics of recording volume holographic gratings upon two-wave mixing in an inertial photorefractive media were studied. It is found that the spaceÐtime dependence of the grating diffraction efﬁciency can be quasi-regular, allowing the understanding of the experimentally observed irregular behavior of the diffracted light intensity. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 42.40.Pa; 42.40.Lx; 42.70.Nq; 42.65.Hw

Investigations into photorefraction, i.e., into the pro- spatiotemporal evolution is described by a set of partial cesses of light-induced change in the refractive index of differential equations, which can be written in the form a medium, have allowed one to discover many new [6Ð8] photorefractive materials in the last several decades and τ ∂Ᏹ find numerous applications for them [1, 2]. However, ------m + Ᏹ = γ E , E*, with the development of the hologram record technique ∂t 1 2 (1) and information storage in such media, it has become ∂ ∂ E2 Ᏹ∗ E1 Ᏹ clear that wave-mixing processes may be much more ------==i E1, ------i E2, complicated than was assumed previously [3]. ∂x ∂x τ ε πσ In this work, we discuss certain features of wave where m = /4 is the characteristic Maxwellian mixing in a photorefractive medium in the course of response time of the medium, ε is its dielectric constant, recording volume holographic grating by two crossing σ is conductivity, and γ = Geiϕ is the effective photore- light beams. The time dependence of the diffraction fraction coefficient, which includes a combination of efficiency of the recorded grating, as a rule, is smooth the photovoltaic and electrostatic tensor components and practically monotonic at low light intensities or in and the diffusion coefficient. media with weak photorefractive nonlinearity [4, 5]. However, an increase in the light intensity or the use of From the second and third equations, it follows that | |2 stronger nonlinearities may be accompanied by sharp the total light intensity in the system is constant: E1 + | |2 differentials in the diffraction efficiency and their sen- E2 = const = I0. Taking into account this conservation sitivity to the experimental conditions. In this work, it law, it is convenient to make the change of variables E1 ψ ψ is shown that such a jumpwise behavior of diffraction i 1 β i 2 β efficiency of a volume holographic grating is a conse- = I0e cos( /2) and E2 = I0e sin( /2). In the new quence of the formation of a regular spatiotemporal variables, system (1) takes the form structure in the course of recording grating in an inertial ∂ᐆ ψ photorefractive medium. ᐆ β Ði ------∂τ- + = sin e , Description of the model. We consider the standard (2) ∂β ∂ψ ()ψϕ model of recording holographic grating in a medium ------Ð itanβ------= Ðiᐆei + , with photorefractive response. Two coherent light ∂y ∂y beams intersect in a medium to form an interference ᐆ Ᏹ γ ψ ψ ψ pattern. Due to the diffusion or photovoltaic effect, the where the variables = 2 / I0, = 2 Ð 1, and new electric-charge carriers are rearranged in the medium to time and length scales produce an electrostatic field which modulates the τ ==t/τ , yGIx (3) refractive index through the electrooptical effect. m 0 Therefore, the state of light and medium is specified by are introduced. Therefore, the behavior of the system three complex field variables: slowly varying field depends qualitatively on a single parameter ϕ, which amplitudes of two beams E1, 2(x, t) and spatially oscil- determines the ratio between the real and imaginary lating amplitude of electrostatic field Ᏹ(x, t). In the components of the photorefraction coefficient γ. The approximation of a perfectly transparent medium, their behavior of all remaining parameters can be taken into

354 PRUDKOVSKIŒ the beam intensities in the photorefractive medium are not rearranged. In all other cases, the intensity of one beam will transfer to another (in our notation, energy transfer occurs from the first to the second beam if 0 < ϕ < π). In the case of imaginary photorefractive nonlinearity ϕ = π/2 (which corresponds, in particular, to the dif- fusional photorefraction mechanism), the system can be simplified, because there is a stable solution of the form argᐆ = πn Ð ψ = const. The rest of the set of Eqs. (2) amounts to the equation ∂2β ∂β ------+ ------= sinβ, (4) ∂y∂τ ∂τ which has the form of a modified sine-Gordon equation with the additional first derivative. The analytic solution to this equation and to the complete set of Eqs. (2) cannot be found. For this reason, we present here only numerical solutions. Results of numerical integration. The set of Eqs. (2) consists of four real equations, two of which contain only time derivatives and the other two contain only spatial derivatives. Due to this, one can easily develop a computational algorithm based on the numerical integration of the two parts of system (2), similar to the usual differential equations, to obtain the solution | ≡ by specifying the intensity ratio I2/I1 y = 0 I02/I01 = ()β ψ| ≡ ψ tan 0/2 and phase difference y = 0 0 for two light beams at the medium input as boundary conditions and the absence of electrostatic field ᐆ at τ = 0 as the initial condition. The numerical integration gave the spatiotemporal dependences for the second-beam intensity normalized to the total intensity I0, with the ψ Ð4 boundary conditions 0 = 0, I02/I0 = 10 and various phases of the photorefractive nonlinearity coefficient ϕ. For such a low initial input intensity of the second beam, its output intensity is a measure of the diffraction efficiency of the recorded grating. Figure 1a shows the three-dimensional intensity profile of the second beam for the imaginary coefficient γ (ϕ = π/2). It demonstrates the dependence of intensity on both time and length. One can see that the obtained Fig. 1. Three-dimensional intensity profile for the second- solution describes a smooth increase in the intensity of τ ary light beam I2 as a function of time and thickness y of the second beam at the output of photorefractive the photorefractive medium for (a) purely imaginary coeffi- medium with small damping oscillations near the max- cient γ of photorefractive nonlinearity (ϕ = π/2) and (b) equal real and imaginary parts of this coefficient imum. This is precisely the behavior that was consid- (ϕ = π/4). ered classical for the processes of recording holograms and photoinduced light scattering at the dawn of photorefraction [5]. account by changing the space and time scales. In par- However, in the general case, the behavior of light ticular, the variation of light intensity I0 is equivalent to intensity at the output of the photorefractive medium changing the thickness of photorefractive medium. may be much more complicated. Figure 1b shows the three-dimensional intensity map for the second beam in It is known [9, 10] that the recorded holographic the case of γ with equal real and imaginary parts (ϕ = grating for real γ is not shifted relative to the interfer- π/4). One can see that the evolution of beam intensity at ence pattern. In this case, stationary energy exchange is short lengths is almost the same as in the case of imag- impossible; after the termination of recording grating, inary γ. Recall that, on the scale adopted, the effective

JETP LETTERS Vol. 77 No. 7 2003 SPATIOTEMPORAL SELF-ORGANIZATION 355 τ τ 100 100 (a) (b) 80 80

60 60

40 40

20 20

0 0 0 50 100 150 200 0 50 100 150 200 20 (c) (d) 60 15

40 10

20 5

0 0 0 50 100 150 200 0 50 100 150 200 5 (e) y 1.0 0.9–1.0 4 0.8–0.9 0.7–0.8 3 0.6–0.7 0.5–0.6 0.4–0.5 2 0.3–0.4 0.2–0.3 0.1–0.2 1 0–0.1 0 0 0 50 100 150 200 y

τ Fig. 2. Two-dimensional intensity distribution for the second beam I2 as a function of time and thickness y of the photorefractive medium for different phases ϕ of the photorefractive nonlinearity coefficient γ: ϕ = (a) π/5, (b) π/4, (c) 3π/10, (d) 2π/5, and (e) π/2. length unit depends both on the total light intensity and dependences contain a strongly fluctuating region, on the magnitude of photorefractive nonlinearity (3). whose length increases approximately proportional to Since the laser intensity in the early studies of photore- the interaction length. fraction was too low to produce a strong light field, the Let us consider a set of spatiotemporal dependences effective thicknesses of photorefractive media were for the intensity of second beam and various values of rather small, so that almost all observed dependences phase ϕ. The corresponding dependences are shown in were smooth and monotonic. However, for the larger Fig. 2 in the form of two-dimensional brightness distri- interaction lengths (or beam intensities), the time butions (Fig. 2a corresponds to Fig. 1a, and Fig. 2d cor-

JETP LETTERS Vol. 77 No. 7 2003 356 PRUDKOVSKIŒ the other. The Maxwellian time in such crystals equals several tens of minutes and decreases with increasing light intensity because of an increase in the photoelec- tron conductivity. As the process of holographic grating evolves, the intensity of the ﬁrst beam is partly transferred to the second beam. τ To compare with the experimental data, the I2( ) dependences were extracted from the complete solution to Eqs. (2) with various effective thicknesses of the photorefractive medium.

The experimental curves for the intensity I2 of the second beam at the output of a photorefractive crystal are shown in Figs. 3a and 3b for relatively low intensities of the first beam, and the appropriate theoretical curves obtained for various effective thicknesses are shown in Figs. 3c and 3d. As was pointed out above, the length scale depends on the total light intensity; because of this, the increase in the effective thickness corresponds to the increase in the intensity of the first beam for a fixed crystal thickness. Figures 3e and 3f and Figs. 3g and 3h show, respectively, the corresponding theoretical curves for higher intensities of the first beam; I1 in Fig. 3e is higher than the intensity of the first beam in Fig. 3a by almost an order of magnitude, and I1 in Fig. 3e is one order of magnitude higher than the analogous intensity in Fig. 3b. It is seen from these graphs that the theoretical curves obtained by the numerical integration of Eqs. (2) describe not only the smooth evolution of the energy-exchange process at Fig. 3. Comparison of the experimental time dependences low intensities but also the sharp changes in the diffrac- of the intensity I2 of the second beam (a, b, e, f) with the tion efficiency of holographic grating at high light analogous theoretical dependences for different total light intensities or large thicknesses of the photorefractive intensities I0 (c, d, g, h). medium (Fig. 3e). Thus, it is shown in this work that the spatiotempo- responds to Fig. 1b). One can see that, if the photo- ral dependence of the diffraction efficiency of a holo- refraction coefficient has a nonzero real part, the whole graphic grating may have a complex quasi-regular τÐy plane can be separated into two regions divided by structure if the coefficient of photorefraction nonlinear- a near-straight line emerging from the origin of coordi- ity is complex and the pumping is strong enough nates. In the upper part adjacent to the time axis, the (Fig. 2). This is manifested experimentally in the form intensity I2 rapidly and more or less monotonically of sharp intensity differences in the intensity of dif- increases and becomes saturated at I = I . At the same fracted light (Fig. 3e). The boundary of this structure in 2 0 τ time, the two-dimensional structure formed in the the Ðy plane is a straight line emerging from the origin region adjacent to the spatial axis is rather complicated of coordinates. Its slope depends on the phase of the and has a regular quasiperiodic character. It should be coefficient of photorefractive nonlinearity. This, in noted that the slope of the line separating the regions principle, provides an original method of measuring the increases with decreasing phase ϕ. ratio between the real and complex parts of the photovoltaic tensor components. Let us now compare these results with the experimental data. This work was supported by the Russian Foundation for Basic Research (project no. 02-02-16843) within Comparison with experiment. The experimental the framework of the program “Fundamental Optics curves presented below were obtained while studying and Spectroscopy.” two-wave mixing, photoinduced light scattering, and parametric scattering of holographic type in photorefractive copper-doped lithium niobate crystals [11, 12]. REFERENCES Two light beams were brought together in a photorefractive crystal, with the intensity of one of them being 1. T. R. Volk, N. V. Razumovski, A. V. Mamaev, and approximately four orders of magnitude higher than for N. M. Rubinina, J. Opt. Soc. Am. B 13, 1457 (1996).

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2. V. A. Vysloukh, V. Kutuzov, and V. V. Shuvalov, Kvan- 8. P. A. Prudkovskiœ, O. V. Skugarevskiœ, and A. N. Penin, tovaya Élektron. (Moscow) 23, 157 (1996). Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 5, 38 (1998). 3. V. A. Vysloukh, V. Kutuzov, V. M. Petnikova, and 9. B. Ya. Zel’dovich, Kratk. Soobshch. Fiz. 5, 20 (1970). V. V. Shuvalov, Zh. Éksp. Teor. Fiz. 111, 705 (1997) 10. V. L. Vinetskiœ, N. V. Kukhtarev, S. G. Odulov, and [JETP 84, 388 (1997)]. M. S. Soskin, Usp. Fiz. Nauk 129, 113 (1979) [Sov. 4. F. S. Chen, J. T. La Macchia, and D. V. Fraser, Appl. Phys. Usp. 22, 742 (1979)]. Phys. Lett. 13, 223 (1968). 11. P. A. Prudkovskiœ, O. V. Skugarevskiœ, and A. N. Penin, 5. K. N. Zabrodin and A. N. Penin, Kvantovaya Élektron. Zh. Éksp. Teor. Fiz. 112, 1490 (1997) [JETP 85, 812 (Moscow) 18, 622 (1991). (1997)]. 12. P. A. Prudkovskiœ and A. N. Penin, Pis’ma Zh. Éksp. 6. M. Cronin-Golomb, J. O. White, B. Fisher, and A. Yariv, Teor. Fiz. 70, 660 (1999) [JETP Lett. 70, 673 (1999)]. Opt. Lett. 7, 313 (1982). 7. A. Novicov, S. Odoulov, O. Oleinik, and B. Sturman, Ferroelectrics 66, 1 (1986). Translated by V. Sakun

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JETP Letters, Vol. 77, No. 7, 2003, pp. 358Ð361. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 426Ð429. Original Russian Text Copyright © 2003 by Arzhannikov, Astrelin, Burdakov, Ivanov, Koœdan, Mekler, Postupaev, Rovenskikh, Polosatkin, Sinitskiœ.

Direct Observation of Anomalously Low Longitudinal Electron Heat Conductivity in the Course of Collective Relaxation of a High-Current Relativistic Electron Beam in Plasma A. V. Arzhannikov, V. T. Astrelin, A. V. Burdakov*, I. A. Ivanov, V. S. Koœdan, K. I. Mekler, V. V. Postupaev, A. F. Rovenskikh, S. V. Polosatkin, and S. L. Sinitskiœ Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia * e-mail: [email protected] Received January 28, 2002; in ﬁnal form, February 18, 2003

The experimental results on a multiple-mirror trap GOL-3 with a short section of reduced magnetic ﬁeld (“magnetic pit”) are presented. The reduced speciﬁc energy release from a relativistic electron beam in the pit brings about a region with a temperature several times lower than in the surrounding plasma. The existence of the low- temperature region directly demonstrates that the longitudinal electron heat conductivity is suppressed in the collective electron-beam interaction with plasma. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 52.25.Fi; 52.40.Mj; 52.55.Jd

1. Introduction. It is generally believed in plasma experiments on the collective interaction of nanosecond physics that turbulence enhances transport coefficients electron beams with plasma (see, e.g., [5]); however, and, correspondingly, reduces plasma temperature and due to the short duration of the process (~50 ns), they energy confinement time. This is largely caused by the did not appreciably alter the dynamics of beam-heated fact that the overwhelming majority of experimental plasma. Various mechanisms of anomalous collision and theoretical studies on the physics of fusion plasma frequency in turbulent plasma (see, e.g., [6] and review mainly deal with devices of the tokamak type or other [7]) and in the electron-beam systems are known (see, toroidal systems, in which the plasma energy is lost e.g., review [8]). In our case, the following mechanism transverse to a magnetic field. As distinct from such can be responsible for this anomaly: during the collec- devices, high electron temperature in open magnetic tive relaxation, the electron beam excites intense reso- systems, as a rule, cannot be attained because of the nance (Langmuir) plasma oscillations. In the case of energy loss through the longitudinal electron heat-con- nonlinear relaxation of these oscillations, whose level ductivity channel. The reactor horizons of open systems can achieve the modulation instability threshold, an largely depend on whether these losses can be reduced intense ion sound can arise, together with density mod- or not. ulations associated with Langmuir collapse processes In the GOL-3 experiments with plasma heating by a (a detailed study of turbulent processes accompanying high-current microsecond relativistic electron beam the relativistic electron-beam relaxation in a magne- [1], an electron temperature of ~1 keV was achieved, tized plasma was performed on a device with different and, after modernization of this device [2], up to 2Ð parameters [9, 10]). 15 Ð3 3 keV at a density of ~10 cm . Both the absolute Recently, the magnetic system of the GOL-3 device value of temperature and its nonuniform distribution has been modernized. The main purpose of the modern- along the device length during ~5 µs could not be ization was to produce corrugated magnetic-field sec- explained by the assumption that the plasma energy is tions at the solenoid ends (with the aim of studying lost due to the classical electron heat conductivity. To plasma heating and confinement in a multiple-mirror explain these experimental results, it was assumed in magnetic system). In the course of these developments, [3] that an abnormally high electron-scattering fre- an experiment with a reduced magnetic-field section in quency caused by a strong turbulence occurring in the the center of a solenoid was also carried out. It is the course of electron-beam injection into plasma reduces purpose of this work to discuss some results of this the longitudinal electron heat conductivity by two to experiment. three orders of magnitude, as compared to the classical conductivity (this problem is considered in more detail 2. Statement of the experiment and diagnostics. in [4]). Such effects were probably observed in the In the experiments under discussion, the solenoid of the

DIRECT OBSERVATION 359

GOL-3 device was comprised of 110 coils with independent power supply and a total length of 12 m and arranged as three sections. A four-meter section with a uniform magnetic field of 4.8 T was in the center. The corrugated-field sections, each composed of 20 corru- gation periods with a maximum field of 5.2 T, a minimum field of 3.2 T, and a step of 22 cm, were adjacent at both its sides. The solenoid was ended by single magnetic mirrors with a field of 9 T. The initial hydrogen or deuterium plasma with an average density of ~1015 cmÐ3 and a temperature of 2Ð3 eV along the length was produced using a special linear discharge in a 10-cm-diameter metallic vacuum chamber placed inside the solenoid. Then a relativistic electron beam was injected into this plasma, with the following parameters: electron energy ~0.9 MeV, current ~25 kA, base duration ~8 µs, and energy content ~120 kJ. The beam diameter in the uniform-field section was about 6 cm. 15 (10

In the experiments, a special region with a reduced nT magnetic field was created in the central section of the solenoid, with the center at Z = 659 cm (hereafter, the longitudinal coordinate is measured from the middle of the input mirror) and a distance between neighboring coils 4 cm larger than in the remaining part of the solenoid. The diameter of the vacuum chamber was also increased to 150 mm in a region of length 42 cm. The coils adjacent to Z = 659 cm had an independent power supply and could be switched off in some experiments. The magnetic field at this point was 3.8 T for the Fig. 1. (a) Dependence of U on t. (1) Diode voltage and (2) bremsstrahlung radiation from the exit collector; (b) and switched-on adjacent coils and 1.3 T for the switched- (c) plasma pressure measured near the reduced-field section off coils. In the latter case, the magnetic field formed a (numbers near the curves correspond to the longitudinal local trap (“magnetic pit”) with a length of about coordinates of the coils): (b) B = 3.8 T in the central section 30 cm. with the center at 659 cm and (c) B = 1.3 T at the same site. Plasma heating and confinement were examined using several diagnostics (see, e.g. [2]). In this work, We first consider the case with the completely we will mainly focus on the results of diamagnetic mea- switched-on solenoid (Fig. 1b). After the beam injec- surements (about 25 coils were placed along the length tion, the plasma pressure increases until the injected of the device). Each of the “standard” coils was a loop beam power starts to decrease. As pointed out above, enveloping the plasma column (the walls of vacuum plasma electrons are heated in the course of beam injec- chamber can be considered as pit-conducting for times tion because of the collective effects that are mainly due discussed in this work). The coil placed in the mag- to the excitation of Langmuir oscillations. The excess netic-pit region had a different design and was shaped of beam instability increment over the effective elec- like a ring segment. It was placed at Z = 659 cm in a tron-collision frequency in plasma is the necessary con- diagnostic port near the plasma column boundary. This dition for the efficient beam relaxation. This condition fact should be taken into account in the discussion of signifies that, for a given plasma density, the beam cur- the experimental results (the absolute calibration accu- rent density must exceed a certain critical value. In the racy for this coil in a magnetic field of 1.3 T was low GOL-3 experiments, the maximal plasma density for because of the difficulties associated with taking into which electrons are efficiently heated is equal to about account the skin currents in the device elements in this 2 × 1015 cmÐ3 for the beam current density on the scale regime; it can be estimated at 30%). of 1 kA/cm2. 3. Results and discussion. The typical signal oscil- As in [1, 2], the heating in our experiments was lograms for the strong and weak field in the central highly nonuniform along the device length, especially magnetic pit are presented in the Fig. 1. For reference, at few starting meters; however, the signal shapes of the the signals from a voltage divider of the beam generator diamagnetic coils were similar to the one shown in and a detector of hard X-ray emission from the surface Fig. 1b for the Z = 701 cm coil. The signal amplitude of of an output beam collector are also shown in Fig. 1. this diamagnetic coil corresponds to the cross-sectional

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360 ARZHANNIKOV et al.

15 3 average value D = neTe + niTi ~ 10 keV/cm of the code (its first version was used in [4]) show that, due to plasma column (here, n and T are the concentration and the appearance of a longitudinal pressure gradient near temperature, respectively, and e and i refer, respec- the magnetic pit at the heating stage, plasma is “accu- tively, to electrons and ions). In the central section of mulated” in this region even during the beam pulse. the device, the pressure longitudinal gradient in plasma Due to the energy thermalization of these counterprop- is already low, and the signals of five diamagnetic coils agating plasma flows, the plasma pressure in the mag- placed in a uniform magnetic field from Z = 475 cm to netic-pit region may be higher (for a while) than in the Z = 844 cm coincide with each other to within 5% at the regions adjacent to the trap. plasma heating stage. The coil signal shape at Z = 659 cm in a field of 3.8 T is slightly different from the 4. Conclusions. A short section with a magnetic others (Fig. 1b). Plasma is heated slowly at this point, field of 1.3 T (i.e., 3.7 times lower than in the adjacent because the ratio of electron-beam density to the regions with a uniform magnetic field) has been formed plasma electron density (which, all things being the in a middle of the solenoid of the GOL-3 device. Due to same, determines the beam relaxation efficiency in a decrease in the local electron density of a relativistic plasma; see, e.g., [11]) is reduced due to the expansion beam in this section, beam-induced plasma heating of the magnetic-field tube in this region. After the ter- almost did not occur in the weak-field region. As a mination of heating, plasma is rapidly cooled, the tem- result, a short region with a temperature several times perature levels off along the length, and the reading of lower than in the surrounding plasma was formed dur- the Z = 659 cm coil becomes close to the readings of the ing the course of beam injection into the initially homo- neighboring coils. geneous plasma. The measured longitudinal plasma pressure gradient corresponded to an electron tempera- A different behavior of plasma pressure in the cen- ture gradient of ~1 keV/40 cm (for the above-men- tral section is observed if the field in the magnetic pit is tioned parameters of beam-induced plasma heating and weak after switching off the adjacent solenoid coils characteristic time of few microseconds, the ion tem- (Fig. 1c). On the whole, the character of plasma heating perature is much lower than the electron temperature in this case does not change over most of the solenoid, [1]). The existence of a low-temperature region is direct and the plasma pressure distribution along the device evidence of the suppression of longitudinal electron length is also unchanged. However, substantial changes heat conductivity at the stage of collective beamÐ in the pressure dynamics are detected by the coil at Z = plasma interaction and plasma heating. After heating, 659 cm. The magnetic field in the central section the heat conductivity regains its classical value (for decreases to ~1.3 T, and the beam-current density details, see [3, 4]), which is manifested by a rapid becomes 3.7 times lower than in the region of uniform increase in plasma pressure in the weak-field section field, so that the beam practically does not interact with and simultaneous plasma cooling in the remaining part plasma at this site. As a result, the plasma pressure, of the device. being determined by the electron temperature in the course of beam injection, is 5 to 10 times lower in the We are grateful to R.Yu. Akent’ev, V.G. Ivanenko, low-field section than the pressure detected by the coils V.V. Konyukhov, S.A. Kuznetsov, A.G. Makarov, at a distance of ~40 cm in the region of uniform field. It S.S. Perin, Yu.S. Sulyaev, V.D. Stepanov, and is important that, due to the suppression of the longitu- A.A. Shoshin for assistance in preparing and conduct- dinal electron heat conductivity (see Introduction), the ing experiments on the GOL-3 device and energy transfer to the region of magnetic-field mini- I.A. Kotel’nikov for helpful discussions. This work was mum from the neighboring regions of hot plasma is supported by the Ministry of Science of the Russian almost zero. A high electron-temperature gradient Federation and the Russian Foundation for Basic along the length (up to 2.5 keV/m) exists during the Research, project no. 00-02-17649. heating stage and characterizes the degree of suppression of electron heat conductivity over this time. After the termination of heating, the effect of anomalously REFERENCES low heat conductivity disappears, the effective collision frequency in plasma becomes classical, and the temper- 1. A. V. Burdakov, S. G. Voropaev, V. S. Koœdan, et al., Zh. Éksp. Teor. Fiz. 109, 2078 (1996) [JETP 82, 1120 ature along the plasma-column length starts to rapidly (1996)]. level off. At this instant of time, plasma pressure in the magnetic pit rises drastically (although it drops in the 2. A. V. Arzhannikov, V. T. Astrelin, A. V. Burdakov, et al., remaining regions of the device), and β in the magnetic Trans. Fusion Technol. 39 (1T), 17 (2001). pit reaches a value of 15Ð20%. 3. A. V. Burdakov and V. V. Postupaev, Preprint No. 92-9, Yet another two circumstances are noteworthy. IYaF SO RAN (Inst. of Nuclear Physics, Siberian Divi- First, the plasma pressure in this region of central sec- sion, Russian Academy of Sciences, Novosibirsk, 1992). tion slightly increases immediately also after heating in 4. V. T. Astrelin, A. V. Burdakov, and V. V. Postupaev, Fiz. a high magnetic field (Fig. 1b). Second, preliminary Plazmy (Moscow) 24, 450 (1998) [Plasma Phys. Rep. hydrodynamic calculations using the modified ISW 24, 414 (1998)].

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5. A. V. Arzhannikov, A. V. Burdakov, V. S. Koidan, and 9. V. S. Burmasov, L. N. Vyacheslavov, I. V. Kandaurov, L. N. Vyacheslavov, Phys. Scr. T 2/2 (1), 303 (1982). et al., Fiz. Plazmy (Moscow) 23, 142 (1997) [Plasma 6. M. V. Babykin, P. P. Gavrin, E. K. Zavoœskiœ, et al., Zh. Phys. Rep. 23, 126 (1997)]. Éksp. Teor. Fiz. 52, 643 (1967) [Sov. Phys. JETP 25, 421 10. V. S. Burmasov, V. F. Gurko, I. V. Kandaurov, et al., in (1967)]. Abstracts of 28th EPS Conference on Controlled Fusion 7. A. A. Galeev and R. Z. Sagdeev, in Basic Plasma Phys- and Plasma Physics (Madeira, Portugal, 2001), p. 521. ics, Ed. by A. A. Galeev and R. N. Sudan (Énergoatomiz- dat, Moscow, 1984; North-Holland, Amsterdam, 1984), 11. B. N. Breizman and D. D. Ryutov, Nucl. Fusion 14, 873 Vol. 2 (Suppl.). (1974). 8. R. N. Sudan, in Basic Plasma Physics, Ed. by A. A. Ga- leev and R. N. Sudan (Énergoatomizdat, Moscow, 1984; North-Holland, Amsterdam, 1984), Vol. 2 (Suppl.). Translated by V. Sakun

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JETP Letters, Vol. 77, No. 7, 2003, pp. 362Ð365. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 430Ð433. Original Russian Text Copyright © 2003 by Guseva, Gureev, Kolbasov, Korshunov, Martynenko, Petrov, Khripunov.

Subthreshold Sputtering at High Temperatures M. I. Guseva, V. M. Gureev, B. N. Kolbasov, S. N. Korshunov, Yu. V. Martynenko, V. B. Petrov, and B. I. Khripunov Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received February 25, 2003

Abstract—The sputtering of tungsten from a target at a temperature of 1470 K during irradiation by 5-eV deuterium ions in a steady-state dense plasma is discovered. The literature values of the threshold for the sputtering of tungsten by deuterium ions are 160Ð200 eV. The tungsten sputtering coefﬁcient measured by the loss of weight is found to be 1.5 × 10Ð4 atom/ion at a deuterium ion energy of 5 eV. Previously, such a sputtering coef- ﬁcient was usually observed at energies of 250 eV. The sputtering is accompanied by a change in the target surface relief, i.e., by the etching of the grain boundaries and the formation of a wavy structure on the tungsten surface. The subthreshold sputtering at a high temperature is explained by the possible sputtering of adsorbed tungsten atoms that are released from the traps around the interstitial atoms and come to the target surface from the space between the grains. The wavy structure on the surface results from the merging of adsorbed atoms into ordered clusters. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 52.55.Rk; 61.80.Jh; 81.15.Cd

INTRODUCTION sputtering of tungsten is high. In the concept of a gas The sputtering of materials by fast ions is a phenom- divertor that was developed in [4] in order to reduce the enon that has been studied fairly well and whose major divertor erosion, a tungsten divertor is exposed to a features seem to be well established. Thus, it is well dense plasma with a temperature of several electron- known that, at ion energies below a certain threshold volts. In the context of the above problems, we have carried out experiments aimed at modeling the opera- energy Etr, there is no sputtering [1]. Since the sputtering results from binary elastic collisions between the tion of a gas divertor for ITER by investigating the sput- ions and the atoms of the target, the existence of the tering of tungsten from a target at temperatures of 1200 threshold is a consequence of the condition that the to 1470 K during the irradiation by 5-eV deuterium ions maximum energy that the ion can transfer to the target in a steady-state dense plasma in the LENTA device. atom in a collision, The phenomena revealed in our experiments are also important for the problem of the passage of space ∆ λ λ ()2 Emax ==E , 4MiMa/ Mi + Ma , objects through the Earth’s atmosphere. Under atmo- is higher than the binding energy of the target atoms to spheric conditions, the relative kinetic energy of the gas the target surface (the sublimation energy U). For light atoms and molecules is several electronvolts, and the ions and target materials with high atomic numbers, the surface of the space object is heated to a high tempera- sputtering threshold is especially high. The theoretical ture. From a practical standpoint, the most interesting threshold energy for the sputtering of tungsten (U = phenomena are the etching of the grain boundaries and 8.7 eV) by deuterium ions is equal to 201 eV, while the possible accompanying changes in the mechanical experimental measurements yielded thresholds of properties of the grain material. 178 eV [2] and 160 eV [3]. In this work, we have detected the sputtering of tungsten from a target at a EXPERIMENT temperature of 1470 K during irradiation by 5-eV deuterium ions in a steady-state dense plasma. Our experiments on modeling a gas divertor in the The phenomenon observed is important not only LENTA device were carried out with tungsten samples from the standpoint of general physics but also from the in a deuterium plasma under beamÐplasma discharge point of view of practical applications in which low- conditions in a longitudinal magnetic field (see Fig. 1) energy ions interact with a material at a high tempera- [5]. The experiment is organized as follows. In the dis- ture. Thus, in the International Thermonuclear Experi- charge zone of the device, an electron beam generates a mental Reactor (ITER), it is proposed to utilize a tung- steady-state plasma flow (a beamÐplasma discharge) sten divertor capable of operating at high temperatures with an electron density of 1012Ð1013 cmÐ3 and electron [3]. The tungsten components of the divertor are temperature of 5Ð20 eV. The plasma flows along the expected to undergo erosion only during plasma disrup- magnetic field and enters the adjacent region—the tions [3], because the energy threshold for the physical interaction zone (or the “gas target”), which is fed with

SUBTHRESHOLD SPUTTERING AT HIGH TEMPERATURES 363 neutral gas at a pressure of up to 10 mtorr. As the plasma interacts with the gas, its parameters change considerably. In the interaction process, the energy of the initial plasma flow is dissipated; in particular, the electron temperature decreases substantially (to 0.5Ð 1.5 eV), as is the case in the divertor of a tokamak operating with gas injection (the phenomenon of “plasma detachment”) [4]. As a result, the energy content of the plasma that comes to the surface of the cooled collector plate (Fig. 1) is reduced. Tungsten samples (W + 0.04% Mo) are placed at and near the collector plate surface and are oriented along and across the magnetic field. The temperature of the irradiated surface of the samples is determined by the energy of the incident plasma flow. Fig. 1. Schematic of the experiment on modeling a gas The sample temperature, which is controlled by ther- divertor by irradiating tungsten samples in the LENTA mocouples and an optical pyrometer, changes from device: (1) mass spectrometer, (2) Langmuir probes, (3) cathode, (4) anode, (5) optical monochromator, (6) 1200 to 1470 eV. In our experiments, the energy of the Mach region, (7) movable calorimeter, (8) samples, and (9) ions bombarding the tungsten surface corresponds to collector plate. the potential jump across the wall sheath and is about 5 eV (in which case the electron temperature in the deuterium plasma is about Te ~ 1.5 eV). The plasma flow density is determined from the density of the ion satu- ration current measured by Langmuir probes near the target surface and also from direct measurements of the ion current to the samples. The ion flow density is maintained at a level of (1Ð5) × 1021 mÐ2 sÐ1, and the irradiation dose is about 5 × 1025Ð1026 mÐ2, which corresponds to the dose that is expected to be accumulated by the divertor plate surface in ITER during one cycle of normal operation. Before irradiation, the tungsten samples were polished both mechanically and electrolytically. Also, the samples were weighted before and after irradiation. The sputtering coefficient, which was defined as the average number of target atoms sputtered by one ion, was determined from the loss of weight and the accu- 25KV X1000 5231 JEOLS 10.0 U mulated dose. The microstructure of the tungsten surfaces was investigated using a scanning electron microscope. The Fig. 2. Microstructure of an electrolytically polished tung- surface of a polished sample is shown in Fig. 2. sten surface. The tungsten sputtering coefficient measured by determining the loss of weight at a deuterium ion energy of 5 eV and a tungsten temperature of 1470 K In our experiments, tungsten samples were also irra- was found to be (1Ð1.5) × 10Ð4 atom/ion. It should be diated in a plasma at lower temperatures, all other con- noted that these values of the tungsten sputtering coef- ditions being the same. At temperatures below 1250 K, + no tungsten sputtering was detected. The microstruc- ficient are characteristic of 250-eV D ions. This new ture of the tungsten surface irradiated at temperatures result is quite unexpected. of 1210Ð1250 K is shown in Fig. 4, in which the etching As tungsten is sputtered, its surface structure of tungsten is not observed. Consequently, the effect of changes. Figure 3 presents the microstructure of the etching of a tungsten surface by low-energy deuterium tungsten surface after irradiation at a temperature of ions also has a threshold in terms of the temperature. 1470 K by a flow of 5-eV D+ ions in a steady-state deuterium plasma. The topography of the irradiated surface shows that it has been deeply eroded. The plasma THEORETICAL MODEL etches the surfaces of the grains, making the grains and The most natural assumption is that, at low (sub- their orientation apparent. The tungsten surface threshold) ion energies, the only atoms that can be sput- becomes wavy; the waves originate and propagate from tered are weakly bonded atoms that are adsorbed on the the grain boundaries. surface and whose binding energy to the surface is

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364 GUSEVA et al.

25KV X1000 6093 JEOLS 10.0 U 25KV X1000 3065 JEOLS 10.0 U

Fig. 3. Microstructure of a tungsten surface after irradiation Fig. 4. Microstructure of a tungsten surface after irradiation at a temperature of 1470 K in a steady-state plasma by a at temperatures of 1210Ð1250 K in a steady-state plasma by flow of deuterium ions with the energy E = 5 eV and the cur- a flow of deuterium ions with the energy E = 5 eV and the rent density J = (1Ð5) × 1021 mÐ2 sÐ1, the irradiation dose current density J = (1Ð5) × 1021 mÐ2 sÐ1, the irradiation dose being 1026 mÐ2. being 1026 mÐ2. about ϕ ≈ 0.4U. The coupling between the adsorbed In thermodynamic equilibrium, the density of adsorbed ions that merge into clusters, as well as their binding to atoms at the surface is very low: the surface, is stronger. Therefore, the sputtering of clusters can be neglected. na= Ð2 exp[]Ð()U Ð ϕ /kT ≈ 10Ð3 cmÐ2. At the same time, the energy transferred from a deu- However, at high temperatures, adsorbed atoms can terium ion with the energy E = 5 eV to a tungsten ion come from the boundaries of the grains; i.e., interstitial can be significantly lower than the binding energy of an atoms can move from the grain boundaries to the sur- adsorbed tungsten atom to the surface, face and can become adsorbed atoms. This is evidenced ∆E ==λE ≈ 0.215 eV Ӷ ϕ 3.4 eV. by the etching of the grain boundaries. This indicates that, at such energies, the elastic sputter- The boundaries of the grains, which are usually far ing of the adsorbed atoms is also impossible. from being perfect crystals, are composed of chaoti- cally arranged atoms, which, however, are trapped at However, the weak coupling of an adsorbed atom to the boundary or, in other words, are in bound states. As the atoms of the metal also indicates that its energy lev- the temperature increases, these atoms are detrapped els are narrower than the zone in the metal. This makes and start to migrate along the boundary of the grain. possible the sputtering of adsorbed atoms by slow ions. The grain boundaries are etched as a result of escape of The mechanism for this sputtering is as follows. The the interstitial atoms from the intergrain space either at field of an incident ion excites an electron of an a high temperature or under ion irradiation. adsorbed atom from the ground state to a repulsing term. Physically, an incident ion attracts an electron and The number of interstitial atoms that escape from a causes it to move from a position between an adsorbed unit area of the grain boundary to the surface per unit time as a result of thermal excitation is equal to q = atom and a metal (in this position, the electron binds the τ adsorbed atom to the metal) to a position above the Cd/ , where C is the density of interstitial atoms in the adsorbed atom. This possible mechanism, which is an space between the grains (this density is assumed to be analogue of the mechanism for the desorption of impu- constant over the entire intergrain space because of the rity atoms from metal surfaces, acts as if the binding of high mobility of the interstitial atoms), d is the width of τ νÐ1 an adsorbed atom to the surface were eliminated by the the intergrain space, = exp(ÐEa/T) is the time dur- electron excitation. In terms of the cross section σ for ing which an interstitial atom escapes from the inter- this process (estimates show that σ ≈ 10Ð15 cm2) and the grain space to the surface as a result of thermal excitation, ν is the oscillation frequency of the atoms, and density n of adsorbed atoms at the surface, the sputter- ≈ ing coefficient is equal to Ea 1 eV is the activation energy (this energy is set equal to the Peierls potential [6] for the motion of an Y = σn. atom along a dislocation. In this case, the number Q of

JETP LETTERS Vol. 77 No. 7 2003 SUBTHRESHOLD SPUTTERING AT HIGH TEMPERATURES 365 adsorbed atoms that come to a unit area of the surface affect the density of interstitial atoms in the space per unit time is equal to between the grains and, consequently, the value of the sputtering coefficient. Qq==/rC()/τ ()d/r , where r is the size of the grains. The adsorbed atoms are either sputtered with prob- CONCLUSIONS σ α 2 ability j s or merge with probability Dn to form the (i) We have discovered the phenomenon of the sput- grain structures on the surface. Here, the cross section tering of tungsten from a target at a temperature of σ 1470 K during the irradiation by 5-eV deuterium ions s for the sputtering of adsorbed atoms by an incident ion is estimated to be (1/3Ð1) × 10Ð15 cm2; D ≈ in a steady-state dense plasma. The known threshold 10−6 cm2/s is the coefficient of diffusion of adsorbed for tungsten sputtering by deuterium ions is 160 eV. For α π 5-eV deuterium ions, the tungsten sputtering coefficient atoms along the surface [7, 8]; and = 4 R/a, where R × Ð4 is the distance within which an adsorbed atom is is equal to 1.5 10 atom/ion. trapped at the surface by another adsorbed atom and a (ii) The sputtering is accompanied by the following α 2 ӷ σ two processes: the etching of the grain boundaries and is the atomic size. Since Dn j sn, we have C = (Q/αD)1/2. Estimates show that the density C can be as the formation of a wavy structure on the tungsten sur- high as about ~1012 cmÐ2. In this case, the sputtering face. The waves originate and propagate from the grain σ ≈ × Ð3 boundaries. coefficient is equal to Y = sC (1/3Ð1) 10 , which satisfactorily explains the experimentally measured (iii) The subthreshold sputtering at a high tempera- values of this coefficient. ture is explained by the possible sputtering of adsorbed tungsten atoms that are released from the traps around The formation of wavy structures near the grain the interstitial atoms and come to the target surface boundaries can be attributed to the merging of individ- from the space between the grains. The wavy structure ual adsorbed atoms into extended clusters, which are on the surface results from the merging of adsorbed stretched along the direction of diffusion of the atoms into ordered clusters. adsorbed atoms from the boundary of the grain. This mechanism for the formation of clusters is analogous to that for the formation of a porous lattice during the REFERENCES vacancy expansion, the only difference being that, in 1. Sputtering by Particle Bombardment, Ed. by R. Behrisch our case, the two-dimensional geometry and the (Springer, Berlin, 1981; Mir, Moscow, 1984). directed diffusive flow of the adsorbed atoms lead to the formation of clusters stretched along the flow direction. 2. J. W. Davis, V. R. Barabash, V. R. Makhankov, et al., The wavelength of the wavy structures is on the order J. Nucl. Mater. 258Ð263, 308 (1998). of the diffusion length of the adsorbed atoms over a 3. M. I. Guseva, A. L. Suvorov, S. N. Korshunov, and N. E. Lazarev, J. Nucl. Mater. 266Ð269, 222 (1999). period of time t1 during which they are trapped in the λ ≈ 1/2 4. Nucl. Fusion 39, 2391 (1999). wave; i.e., we have (Dt1) . Setting t1 = ν−1 ν ≈ 13 Ð1 5. N. V. Antonov, B. I. Khripunov, V. B. Petrov, et al., exp(Et /kT), where 10 s is the oscillation fre- ≈ J. Nucl. Mater. 220Ð222, 943 (1995). quency of the atom and Et 0.4U = 3.5 eV is its binding energy at the wave crest, we obtain λ ≈ 1 µm. 6. T. Suzuki, H. Yosinaga, and S. Takeucti, Dislocation Dynamics and Plasticity (Syokabo, Tokyo, 1986; Mir, It should be emphasized that, in the model proposed Moscow, 1989), Chap. 4. here, the effects of subthreshold sputtering and a 7. A. N. Orlov and Yu. V. Trushin, Energy of Point Defects change in the surface structure are possible only for in Metals (Énergoatomizdat, Moscow, 1983). polycrystals. Monocrystals do not possess a grain struc- 8. V. P. Zhdanov, Elementary Physicochemical Processes ture and do not have sufficiently intense sources of on Surface (Nauka, Novosibirsk, 1988). adsorbed atoms. According to the model, the sputtering coefficient increases with decreasing grain size, Y ~ rÐ1/2. The way the polycrystalline material is obtained can Translated by G. Shepekina

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JETP Letters, Vol. 77, No. 7, 2003, pp. 366Ð370. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 434Ð438. Original Russian Text Copyright © 2003 by Kiselev, Kikoin, Molenkamp.

Dynamically Induced Kondo Effect in Double Quantum Dots M. N. Kiselev1, *, K. A. Kikoin2, and L. W. Molenkamp3 1 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, D-97074 Wärzburg, Germany *e-mail: [email protected] 2 Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 3 Physikalisches Institut (EP-3), Universität Wärzburg, D-97074 Wärzburg, Germany Received February 18, 2003

A new mechanism of resonance Kondo tunneling via a composite quantum dot (QD) is proposed. It is shown that, owing to the hidden dynamic spin symmetry, the Kondo effect can be induced by a ﬁnite voltage eV applied to the contacts at an even number N of electrons in a QD with zero spin in the ground state. As an example, a double QD is considered in a parallel geometry with N = 2, which possesses the SO(4) type symmetry characteristic of a singletÐtriplet pair. In this system, the Kondo peak of conductance appears at an eV value compensating for the exchange splitting. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 72.10.Fk; 72.15.Qm; 73.63.Kv

The Kondo effect, originally observed in the form of involved in the algebra of the corresponding dynamic an anomalously strong resonance scattering of elec- group. As a result, the effective Hamiltonian describing trons by magnetic impurities in metals, has proved to be the Kondo tunneling acquires a more complicated form a universal mechanism of interaction between an elec- than that of the sd-exchange Hamiltonian describing tron gas and localized quantum objects possessing the Kondo effect in metals: all the above vectors con- internal degrees of freedom [1]. In particular, a mag- tribute to the cotunneling with spin reversal via the QD. netic impurity within the barrier between two metal A theory of the dynamic symmetry of composite (dou- contacts, as well as a quantum dot (QD) with uncom- ble and triple) QDs has been recently developed in [8], pensated spin occurring in a tunneling junction where it is also demonstrated how an external magnetic between metal electrodes, can account for anomalously field or the gate electric field can influence the dynamic high tunneling transparency of the barrier for electrons symmetry. from the contacts [2, 3]. Shortly after the experimental Below, we consider a new class of phenomena discovery of such Kondo resonances in the tunneling related to the dynamic symmetry of QDs. It will be via planar QDs [4], it was established that the spectrum demonstrated that violation of the thermodynamic of phenomena related to the effective magnetic equilibrium between contacts may induce resonance exchange in QDs is by no means restricted to simple Kondo tunneling not observed in the equilibrium sys- passage from the problem of Kondo scattering to that of tem. The nonequilibrium Kondo effect in QDs at a Kondo transfer. In particular, it was found that the res- finite voltage applied between the source and sink has onance Kondo tunneling via QDs with an even number been extensively studied (see, e.g., [9]). In most cases, of electrons and zero total spin is possible under the however, researchers were interested in the influence of action of an external magnetic field [5] or the electric nonequilibrium conditions on the Kondo effect existing field of a gate [6]. in the equilibrium state. In such a situation, relaxation The large variety of manifestations of the Kondo of the system related to the finite lifetime of excited effect in QDs is related to the fact that these nanodi- states probably hinders attaining a strong coupling mensional objects are essentially a kind of artificial regime (see, e.g., the discussion in [10, 11]). atom possessing complicated spectra. The tunneling of We are interested in a different situation, whereby electrons from outside via QDs breaks their spin sym- no channel of nonequilibrium induced spin relaxation metry and induces transitions to low-lying excited exists in the ground state (e.g., for S = 0). In this case, states. The transitions at energies comparable with the the spin degrees of freedom are excited only in QDs Kondo temperature (TK) are involved into resonance possessing a dynamic symmetry. Such a symmetry is interactions and modify the pattern of the Kondo scal- inherent, for example, in a double quantum dot (DQD) ing as compared to that typical of the canonical Kondo structure experimentally realized and studied recently effect in metals. Thus, within the limits of the Kondo [12]. In the simplest nontrivial case, the DQD contains energy scale, it is necessary to take into account the two electrons occupying energy levels according to the dynamic symmetry of a given QD [7]. This symmetry HeitlerÐLondon scheme (Fig. 1). The system occurs is determined both by the spin and by other vectors under the conditions of a strong Coulomb blockade Q

DYNAMICALLY INDUCED KONDO EFFECT IN DOUBLE QUANTUM DOTS 367 suppressing the tunneling v between potential wells. (a)(A) The spectrum of spin states represents a singletÐtriplet Source (SÐT) pair with zero spin in the ground state (because the effective exchange between the two valleys, I ≈ F2 Q2 v2/Q, has an antiferromagnetic character). As demon- 2 strated previously [6, 8], an isolated DQD represents a C C1 quantum spin rotator with the SO(4) symmetry, in contrast to the SU(2) symmetry of a QD with odd occupa- F1 Q1 tion usually considered in the theory of the Kondo type nm tunneling. The SO(4) group is generated by the spin 400 vector S and the vector P describing the SÐT transition Drain matrix. The Hamiltonian of an isolated QD can be written as 0.5 1.0 1.5 2.0 µm |〉〈| |〉η 〈|η Hd = ES S S + ∑ET T T , (1) η (b) δ η ± where ET = ES + ; = , 0 are the projections of the spin total S = 1; and δ = I is the exchange coupling. Since the direct tunneling W of electrons from contacts to the DQD is suppressed by the Coulomb blockade Q, the charge transfer is possible only by means of second- order processes (cotunneling). The effective Hamilto- nian describing these processes is as follows [6, 8]: () Hint = J∑ SP+ sαα', αα ' (2) † τ † ˆ sαα' ==∑ckασˆck'α'σ', nαα' ∑ckασ1ck'α'σ, kk' kk' where the subscript α = L, R denotes electrons in the Fig. 1. (a) A parallel double quantum dot configuration left- and right-hand contacts, respectively; τˆ are the (Hofmann et al. [12]) and (b) an energy band diagram illus- trating tunneling and cotunneling processes contributing to ˆ ≈ 2 ⑀ the differential conductance in this system. Pauli matrices; 1 is the unit matrix; and J W /( F Ð ES/2)is the effective constant of exchange between the DQD and contacts (we neglect a difference between tunneling parameters in the S and T states of the QD). where j, k, l are the coordinate indices and ejkl is the The S and P vectors defined above are written in matrix LeviÐCivita tensor. form as follows: As can be seen, both S and P vectors are involved in  0100 1000 the tunneling with spin reversal. However, since the + 0010 z 0000 threshold energy for excitation of the spin degrees of S ==2, S , freedom is δ, the spin scattering under equilibrium con- 0000 00Ð0 1 δ ditions is effective only provided that TK > . It will be 0000 0000 shown below that this threshold can also be surmounted Ӷ δ in the opposite limit, TK , at a finite source–sink  ≈ δ 0001 0000 voltage eV compensating for the SÐT splitting + 0000 z 0001 energy. In the weak coupling regime, T > TK, we use the P ==2, P Ð, 0000 0000 thermodynamic perturbation theory and assume that electrons in the contacts obey the Fermi statistics with 00Ð0 1 0100 µ µ the chemical potentials R and R + eV in the right- and where a singlet state corresponds to the last row. The left-hand contacts, respectively. We can also assume that weak tunneling currents do not violate thermody- corresponding algebra (o4) is described by the commu- tations relations namic quasi-equilibrium (the validity of this approach is discussed below). []S , S ==ie S , []P , P ie S , j k jkl l j k jkl l In order to construct the perturbation theory, let us [], P j Sk = iejklPl, perform fermionization of the generators of SO(4)

JETP LETTERS Vol. 77 No. 7 2003 368 KISELEV et al. ⑀ ⑀ Replacing kL by kL Ð eV at the vertex (Fig. 2b), we obtain Γ()2b ω ∼ 2ν (){}ω,,()δ LR () J ln D/max eV Ð T . ε where D ~ F is the truncation parameter determining the effective width of continuum between contacts, ν is the density of states on the Fermi level, and f(ε) is the Fermi function. As can be seen, a bias compensating for the exchange splitting |eV Ð δ| Ӷ max[eV, δ] gives rise to a logarithmic singularity (typical of the Kondo effect) independent of eV. In the second vertex correc- Fig. 2. Perturbation theory diagrams determining (a) J, (b, tion (Fig. 2c), the compensation is absent and the cor- d) the main corrections to J, and (c, e) corrections contain- δ ӷ ω ing a lower power of the logarithm. Solid curves refer to responding contribution for eV ~ T, can be esti- electron between contacts; dashed curves refer to the QD mated as states. Γ()2c ω ∼ 2ν ()()δ Ӷ Γ()2b ω LR () J ln D/ eV + LR (). group—a generalizing procedure originally suggested Analogous estimates for the diagrams in Figs. 2d and for SU(2) group [13, 14]: 2e give () + ()† † Ð ()† † Γ 2d ()ω ∼ J3ν2 ln2 ()D/max{}ω,,()eV Ð δ T , S ==2 f 0 f Ð1 + f 1 f 0 , S 2 f Ð1 f 0 + f 0 f 1 , LR () + ()† † Ð ()† † Γ 2e ()ω ∼ J3ν2 ln()D/max{}ω,,()eV Ð δ T (6) P ==2 f 1 f s Ð f s f Ð1 , P 2 f s f 1 Ð f Ð1 f s , LR † † z ()† † (3) × ln()D/max{}ω, eV, T . Sz ==f 1 f 1 Ð f Ð1 f Ð1, P Ð f 0 f s + f s f 0 . Only the first of these contributions survives in the main † Here, f ±1 are the operators of creation for fermions logarithmic approximation. Thus, a logarithmic singu- + larity in the tunneling amplitude is actually restored by with the spin projections 1 and Ð1; f 0 and fs are the applying an electric field to a DQD with zero spin in the operators of creation for zero-spin fermions in the trip- ground state, whereby a sequence of divergent tiling let and singlet states, respectively. Representation (3) diagrams degenerates into the sequence of ladder dia- automatically takes into account the local kinematic grams. † constraint ∑Λ f Λ f Λ = 1. A diagram technique of the The perturbation theory diagrams at T > TK can be perturbation theory is provided by the temperature summed using the renorm group method, which is Green’s functions for electrons between contacts, applicable under both equilibrium and nonequilibrium τ − 〈〉, τ † , conditions [15]. The set of renorm group equations for GL, R(k, ) = T τcLR, σ()k cLR, σ()k 0 , and for fermi- ΛΛ' the tunneling vertices Jαα obtained upon reduction of Ᏻ τ 〈〉τ † ' ons in the DQD, Λ( ) = ÐT τ f Λ()f Λ()0 . The Fou- the high-energy part of the spectrum is as follows: rier transform in imaginary time yields T ST dJ 2 dJ 0 ⑀ ()⑀ ⑀ µ Ð1 ------LL ==Ðν()JT , ------LL ÐνJST JT , Gkα()n = i n Ð kα + LR, , dDln LL dDln LL LL 0 Ð1 Ᏻ ω ()ω η ,, (4) T ST η()m ==i m Ð ET , Ð101, dJ dJ ------LR ==ÐνJT JT , ------LR ÐνJST JT , (7) Ᏻ0 ⑀ ()⑀ Ð1 dDln LL LR dDln LL LR s()n = i n Ð ES , ⑀ π ω π S where n = 2 T(n + 1/2), m = 2 T(m + 1/3) [13, 14]. dJLR 1 ST TS 1 ST TS ------= ---ν J , J , + ---J , J , . Figure 2 shows the main diagrams of the perturba- dDln 2 LL + LR Ð 2 LL z LR z tion theory. The first diagrams (Figs. 2b and 2c) determining the Kondo tiling in a standard theory give the Solving these equation with the boundary conditions ΛΛ' following expressions for the renormalized exchange Jαα'()D = J , we obtain vertices: TT J ST J ⑀ Jαα, ' ==------ν (), Jαα, ' ------ν (), Γ()2b ω ∼ 2 1 Ð f ()kL Ð eV 1 Ð JDln /T 1 Ð JDln /T LR () J ∑ω------⑀ µ δ-, (8) Ð kL + L Ð () k SS 3ν 2 ln D/T (5) J LR = J Ð --- J ------ν (). ⑀ 4 1 Ð JDln /T Γ()2c ω ∼ 2 f ()kL Ð eV LR () J ∑ω------⑀ µ δ. The structure of the renorm group equations (Fig. 3) Ð kL ++L k shows that the Kondo singularity arising in the T-chan-

JETP LETTERS Vol. 77 No. 7 2003 DYNAMICALLY INDUCED KONDO EFFECT IN DOUBLE QUANTUM DOTS 369 nel influences conductance in the S-channel. This influence is related to the presence of the operator P in the tunneling Hamiltonian, which breaks the spin symmetry of the DQD. Thus, the Kondo effect in this system exists only due to the dynamic symmetry inherent in the DQD. ST 2 The differential conductance G(eV, T)/G0 ~ J LR 2 πប [15], where G0 = e / , is a function of the universal parameters T/TK and eV/TK, ∼ Ð2 ()[]()δ , G/G0 ln max eV Ð T /T K , (9) with a maximum at eV Ð δ = 0 (see Fig. 4). Thus, in con- Fig. 3. Irreducible diagrams determining renorm group trast to the usual situation [9], whereby the Kondo peak equations. Cross-hatched squares and circles represent ver- (representing a zero bias anomaly) exhibits evolution or tices of the TÐT and SÐT transitions, respectively; other splitting at finite eV values, the Kondo peak in our case notations as in Fig. 2. δ appears at a threshold bias of eV0 = . Owing to this threshold character, the peak is asymmetric (cf., e.g., [11]). The asymmetry, as well as the broadening of the Kondo resonance are related to nonequilibrium character of the tunneling process. In contrast to the usual situation [11, 16], when relaxation takes place in the ground state of the QD, the spin triplet in our case appears only as a virtual state (Fig. 3) and, hence, the nonequilibrium effects are not as destructive. Both relaxation and asymmetry are determined by the imaginary part of the self-energy of the Green’s function Ᏻ ω T( ). Figure 5 shows diagrams describing the self- energy in the same order as the renormalization of vertices. The diagrams of Figs 5a and 5b determine the ប τ ω main contribution to the imaginary part / d. For ~ eV, this contribution amounts to ~(eV)(J/D)2 and con- kK tains no logarithmic corrections. Such corrections Fig. 4. The Kondo peak of the differential conductance as a appear in the third order, but still outside the limits of function of two universal parameters eV/TK and T/TK (inset) δ ប τ the main logarithmic approximation, and are estimated and a curve for /TK = 10 and / TK = 0.1. as eV(J/D)3ln(D/eV). As a result, we obtain as an estimate ប τ ∼ ()ν ST 2[]()S ()() / d eV J0 1 + OJ0 ln D/ eV .

Comparing this damping to TK and taking into account that (under the resonance conditions) eV ~ δ ~ J, we arrive at the following condition for the existence of the T anomalous Kondo peak at a finite bias: δδ()2 Ӷ Ӷ δ T /D T K . (10) Here, the right-hand inequality resembles the Doniach criterion for the stability of a Kondo singlet with Fig. 5. The diagrams determining the main contribution to respect to antiferromagnetic correlations (see, e.g., [1]). ប τ / TK (aÐd) (see text). The diagrams (eÐf) describe thresh- The conditions (10) are satisfied in a broad range of old processes leading to the Kondo peak asymmetry. parameters since δ/D Ӷ 1. Another contribution, related to the reoccupation of levels as a result of the tunneling of nonequilibrium electrons, leads to asymmetry of the resonance line. second-order processes, possess a threshold character ប τ These processes are described by the diagrams in and give small contributions to / d. As can be seen Figs. 5e and 5f, in which at least one of the virtual sates from Fig. 4, the asymmetry is small even at a significant is triplet. Such transitions, as well as the corresponding damping.

JETP LETTERS Vol. 77 No. 7 2003 370 KISELEV et al.

Thus, we have described a situation in which the 540 (1998); F. Simmel, R. H. Blick, J. P. Kotthaus, et al., Kondo effect exists only under nonequilibrium condi- Phys. Rev. Lett. 83, 804 (1999). tions and is induced by an external voltage applied to 5. M. Pustilnik, Y. Avishai, and K. Kikoin, Phys. Rev. Lett. electrodes in tunneling contact with a composite QD. In 84, 1756 (2000); M. Eto and Yu. Nazarov, Phys. Rev. this case, the Kondo-type tunneling is induced by Lett. 85, 1306 (2000). dynamic processes of the excitation of low-lying spin 6. K. Kikoin and Y. Avishai, Phys. Rev. Lett. 86, 2090 states of the QD, the ground state of which is a spin sin- (2001). glet. The simplest example of such a system is offered 7. I. A. Malkin and V. I. Man’ko, Dynamic Symmetries and by a double QD with even occupation under the condi- Coherent States of Quantum Systems (Nauka, Moscow, tions of strong Coulomb blockade. The spin symmetry 1979). of such a QD is essentially that of a quantum spin rota- 8. K. Kikoin and Y. Avishai, Phys. Rev. B 65, 115 329 tor. Since a singlet ground state in quantum mechanics (2002); T. Kuzmenko, K. Kikoin, and Y. Avishai, Phys. is always accompanied by triplet excitations, this situa- Rev. Lett. B 89, 156602 (2002). tion is not very unusual and can probably be manifested 9. Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 as a peak in the differential conductance, observed at a (1992); Phys. Rev. B 49, 11040 (1994); T.-K. Ng, Phys. Rev. Lett. 76, 487 (1996); M. H. Hettler, J. Kroha, and nonzero bias in a Coulomb window with an even num- S. Hershfield, Phys. Rev. B 58, 5649 (1998). ber of electrons. Such a peak should be distinguished from a maximum corresponding to the cotunneling via 10. O. Parcollet and C. Hooley, Phys. Rev. B 66, 085315 (2002); P. Coleman and W. Mao, cond-mat/0203001; excited electron states [17]. cond-mat/0205004. This study was supported in part (M.N.K.) by the 11. A. Rosch, J. Paaske, J. Kroha, and P. Wölfle, Phys. Rev. European Commission (LF Project) via the Weizman Lett. 90, 076804 (2003). Institute (Israel) (contract #HPRI-CT-1999-00069) and 12. L. W. Molenkamp, K. Flensberg, and M. Kemerink, by the Deutsche Forschungsgemeinschaft (SFB-410). Phys. Rev. Lett. 75, 4282 (1995); F. Hofmann, T. Hein- The authors are grateful to P. Wölfle for fruitful discus- zel, D. A. Wharam, et al., Phys. Rev. B 51, 13872 sions. (1995). 13. V. N. Popov and S. A. Fedotov, Zh. Éksp. Teor. Fiz. 94 (3), 183 (1988) [Sov. Phys. JETP 67, 535 (1988)]. REFERENCES 14. M. N. Kiselev and R. Oppermann, Phys. Rev. Lett. 85, 1. D. Cox and A. Zawadowski, Adv. Phys. 47, 599 (1998); 5631 (2000). L. Kouwenhoven and L. Glazman, Phys. World 253, 33 15. P. W. Anderson, J. Phys. C 3, 2435 (1970); A. Kaminski, (2001). Yu. V. Nazarov, and L. I. Glazman, Phys. Rev. Lett. 83, 384 (1999). 2. J. Appelbaum, Phys. Rev. Lett. 17, 91 (1966). 16. A. Rosch, J. Kroha, and P. Wölfle, Phys. Rev. Lett. 87, 3. L. I. Glazman and M. E. Raikh, Pis’ma Zh. Éksp. Teor. 156802 (2001). Fiz. 47, 378 (1988) [JETP Lett. 47, 452 (1988)]; 17. S. De Franceschi, S. Sasaki, J. M. Elzerman, et al., Phys. T.-K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988). Rev. Lett. 86, 878 (2001). 4. D. Goldhaber-Gordon, J. Göres, M. A. Kastner, et al., Phys. Rev. Lett. 81, 5225 (1998); S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, Translated by P. Pozdeev

JETP LETTERS Vol. 77 No. 7 2003

JETP Letters, Vol. 77, No. 7, 2003, pp. 371Ð375. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 439Ð444. Original Russian Text Copyright © 2003 by Dubonos, Kuznetsov, Zhilyaev, Nikulov, Firsov.

Observation of the External-ac-Current-Induced dc Voltage Proportional to the Steady Current in Superconducting Loops S. V. Dubonos, V. I. Kuznetsov, I. N. Zhilyaev, A. V. Nikulov, and A. A. Firsov Institute of Microelectronic Technology and Ultra-High-Purity Materials, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia Received February 20, 2003

A dc voltage induced by an external ac current was observed in a system of asymmetric aluminum loops at temperatures corresponding to 0.95Ð0.98 of the superconducting transition temperature. The voltage magnitude and sign change periodically in a magnetic ﬁeld with a period corresponding to the magnetic ﬂux quantum through the loop. The amplitude of these oscillations depends nonmonotonically on the amplitude of ac current and is almost independent of its frequency in the range from 100 Hz to 1 MHz. The observed phenomenon is interpreted as the result of displacing the loop into a dynamic resistive state by the external current, where the loop is “switched” back and forth between the closed superconducting state with a nonzero steady current and the nonclosed state with a nonzero resistance along the loop circle. It is shown that voltages are summed up in a system of loops connected in series. For systems with one, three, and twenty loops, the voltage reaches 10, 40, and 300 µV, respectively. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 74.40.+k

It is known that, due to the momentum circulation 〈ρ/s〉 = dlρ/sl differing from the average resis- ls ∫ls s 〈ρ 〉 ρ ( v ) v Φ πប tance Rl/l = /s l = dl /sl along the entire circle. If °∫dlp = °∫dlm + 2eA = ml°∫d +2e = n2 (1) °∫l l l l the analogy with the ordinary current applies to the steady current, then a constant potential difference pro- in superconducting loops, the current I = sj = s2en v Φ Φ p p s s portional to Ip( / 0) will be observed in the segment of is stable. Its magnitude and sign show periodic depen- inhomogeneous superconducting loop with I ≠ 0 and Φ p dence on the magnetic flux through the loop, with a R > 0. period corresponding to the magnetic-flux quantum l Φ πប Φ Φ ∝ Φ Φ 0 = /e, because the quantized velocity circulation The voltage oscillations V( / 0) Ip( / 0) of this type were observed in our work [4] for the segments of 2πបΦ v an asymmetric aluminum loop. We observed them in a °∫dl s = ------n Ð ------Φ (2) m 0 narrow temperature range of (0.988Ð0.994)Tc, where Tc l corresponds to the midpoint of the resistive transition; of superconducting pairs takes values corresponding to their amplitude increased with lowering temperature in Φ Φ this interval. Such a dependence testifies that V(Φ/Φ ) energy minimum; i.e., the value of (n Ð / 0) varies, 0 with varying Φ, within the interval from Ð1/2 to 1/2 [1]. is induced by external noises. The purpose of this work In the stationary state with a time-independent density is to elucidate how noises of various frequency and Φ Φ of superconducting pairs, the steady current can flow intensity act on V( / 0). With this aim, the effect of the ∆ π only in a closed superconducting state, i.e., if there is ac current Iac = Isin(2 ft) varying in frequency f and ∆ Φ Φ phase coherence along the entire loop circle and the amplitude I on V( / 0) was studied at lower temper- resistance Rl along the circle is zero. atures of (0.95Ð0.98)Tc where the low-intensity random Φ Φ However, according to the LittleÐParks experiment noise shows a weak dependence on V( / 0) because of ≠ the higher critical current. [2] and theory [3], steady current Ip 0 is observed not only in the superconducting state with Rl = 0 but also Systems composed of 3 and 20 (Fig. 1) asymmetric aluminum loops with the superconducting transition tem- for Rl > 0. It is known that, if a current I = ∫ dlE/Rl ≈ ≈ Ω ᮀ °l perature Tc 1.3 K, resistance of a square of 0.5 / at ≈ induced by the Faraday electromotive force dlE = 4.2 K, and the resistance ratio R(300 K)/R(4.2 K) 2 °∫l were used for the study. We used a loop system, rather −dΦ/dt flows along the loop circle, then the potential than a single loop, as in [4], with the aim to verify 〈ρ 〉 〈ρ 〉 difference V = ( /s ls Ð /s l)lsI should be nonzero in whether the voltages are summed in the loops con- the segment ls with the average resistance Rls/ls = nected in series. All loops had a diameter of 4 µm and

0021-3640/03/7707-0371 $24.00 © 2003 MAIK “Nauka/Interperiodica” 372 DUBONOS et al. perpendicular to the sample plane was produced by a superconducting magnet. Each structure had ten contacts, two of which (e.g., IÐI in Fig. 1) were used to deliver ac current Iac with a frequency from f = 100 Hz to f = 1 MHz and the amplitude ∆I up to 50 µA, and the others (e.g., V1ÐV8 in Fig. 1) served as potential contacts. Measurements showed that, for any frequency in the interval from f = 100 Hz to f = 1 MHz, Iac induced Φ Φ oscillations of V( / 0) if the amplitude of ac current ∆ exceeded some critical value Icr (Fig. 2). Similar to the critical superconducting current Ic, this critical value decreased with increasing temperature and magnetic field. This is reflected in Fig. 2 by the fact that the oscil- Φ Φ ∆ µ lations of V( / 0) appear for I = 3 A only at high Φ Φ / 0 values. Similar to the LittleÐParks oscillations Φ Φ (see, e.g., [5]), the oscillations of V( / 0) are damped Φ Φ at large / 0 (Fig. 2), which is explained by the field- induced suppression of superconductivity and, hence, Fig. 1. Photograph of a system of series 20 aluminum asym- Ip in finite-thickness strips. metric loops with the current I and potential Vi contacts. Φ Φ ∆ ∆ The oscillations of V( / 0) observed at I > Icr are Φ Φ similar to the Ip( / 0) oscillations [1]. Both curves Φ Φ Φ Φ pass through zero at = n 0 and = (n + 0.5) 0. This is explained by the fact that the quantum value n of pair momentum circulation in a closed superconducting state acquires, with an overwhelming probability, the only integer value corresponding to the energy mini- Φ Φ mum, except for the vicinity of = (n + 0.5) 0, where, V)

µ simultaneously, the energies of two states with oppo- (

V sitely directed velocities are minimal. For this reason, 3 µA Φ Φ ∝ Φ Φ ∝ 〈 〉 Φ Φ V( / 0) Ip( / 0) ( n Ð / 0), where the mean 〈 〉 ≈ Φ Φ 〈 〉 Φ Φ ≈ n n for / 0 far from (n + 0.5), ( n Ð / 0) 0 for Φ Φ ≈ | | / 0 (n + 0.5), and the voltage maxima V max occur Φ Φ ≈ Φ Φ ≈ at / 0 (n + 0.25) and / 0 (n + 0.75). | | V max depends nonmonotonically on the ac current amplitude ∆I both in a single loop (Fig. 2) and in a loop Fig. 2. Dc voltage oscillations in a magnetic field, as mea- system and shows no appreciable dependence on the Iac sured for one of the loops at T = 1.280 K ≈ 0.97T and the frequency over the whole range studied from f = 100 Hz c | | external current Iac with frequency f = 2.03 kHz and ampli- to f = 1 MHz. After the amplitude V max rapidly reaches ∆ µ ∆ tude I = 3, 3.5, 4, 5, 7, 12, and 25 A. All curves except for its maximum at a current amplitude close to Icr, it ∆I = 3.5 µA are vertically shifted. decreases monotonically upon further increase in ∆I ∆ | | ∆ (Fig. 2). At large I, the V max( I) dependence is close | | ∝ ∆ to V max 1/ I. Measurements showed that, both the a thickness of 40 nm. They were asymmetric; their | | ∆ µ position of the V max( I) curve maximum and the criti- halves lw and ln had different widths ww = 0.4 m and ∆ ∆ µ cal value Icr shift to lower I values with increasing wn = 0.2 m (Fig. 1) and, respectively, the cross-sec- ≈ µ 2 ≈ µ 2 temperature. tions sw 0.016 m and sn 0.008 m . The micro- structures were fabricated on a Si substrate using elec- In loops with similar asymmetry and size, the volt- Φ Φ tron-beam lithography based on a JEOL-840A electron age V( / 0) should oscillate synchronously in a mag- microscope. Exposure was performed at a voltage of netic field. For this reason, there is a fundamental pos- 25 kV and a current of 30 pA. The electron beam was sibility of summing this voltage in a system of series controlled by a PC using the PROXY program package. loops. Our studied showed that this possibility can easily be realized. We managed to observe oscillations | | µ Measurements were made in a standard cryostat with amplitude V max up to 10 V (Fig. 2) in a single with He4 as a coolant, whose evacuation allowed the loop, up to 40 µV in a system of three loops, and up to temperature to be lowered to 1.2 K. A magnetic field 300 µV in a system of 20 loops (Fig. 3).

JETP LETTERS Vol. 77 No. 7 2003 OBSERVATION OF THE EXTERNAL-AC-CURRENT-INDUCED DC VOLTAGE 373

As with every rectification, the observed “rectification” of the external ac current Iac is possible only if the currentÐvoltage characteristic (CVC) is asymmetric about the external current (Fig. 4). The observed periodic change in the sign and magnitude of this asymme- V) try in a magnetic field occurs in a steady current, µ (

Φ Φ ∝ Φ Φ V because V( / 0) Ip( / 0). This change can be explained by the fact that the steady current flowing along the loop circle increases the overall current in one of its halves (cf. Fig. 4, bottom right), as a result of which the criticality arises at lower Iac values. In a closed superconducting state, the distribution of Iac = Iw + In over loop halves lw and ln is determined by the quantization of velocity circulation (2): l v Ð l v = Fig. 3. Magnetic-field induced oscillations of the dc voltage n n w w ≈ l I /s 2en Ð l I /s 2en = l j /2en Ð l j /2en = in a system of 20 loops at T = 1.245 K 0.97Tc for the exter- n n n sn w w w sw n n sn w w sw ∆ µ πប 〈 〉 Φ Φ nal current Iac with f = 1.2 kHz and I = 3.2 A and in a sys- (2 /m)( n Ð / 0). Since the lengths lw and ln of the ≈ tem of three loops at T = 1.264 K 0.96Tc for the external halves and the corresponding superconducting pair ∆ µ current Iac with f = 555 kHz and I = 4.5 A. The second densities nsn and nsw are the same, the current densities µ 〈 〉 curve is shifted vertically by 370 V. in both halves are the same for Ip = 0 (i.e., for ( n Ð Φ Φ / 0) = 0 and equal to jn = jw = jac = Iac/(sn + sw)), so (a) that they simultaneously reach the critical value at Iac/(sn + sw) = jc. The steady current alters the current densities in the loop halves (Fig. 4). Depending on the c+ directions of Ip and Iac, the density increases either in one or another half. In Fig. 4, the density increases in the narrow half, when Ip flows counter-clockwise while I flows from left to right, or vice versa. The current V) ac µ (

densities in the loop halves change differently, by Ip/sn V and Ip/sw. For this reason, the CVC is asymmetric about the external current I at Φ ≠ nΦ and Φ ≠ (n + 0.5)Φ , ac 0 0 Ð while the asymmetry value and its sign change with c changing Ip. For an external current directed along Ip in the narrow half, the critical value Iac c+ /(sn + sw) = jc Ð Ip/sn is achieved at lower Iac values than for Iac directed (b) (c) along Ip in the broadened half, where the criticality is achieved at Iac cÐ /(sn + sw) = jc Ð Ip/sw (Fig. 4). The difference Iac cÐ Ð Iac c+ = (sn + sw)(jc Ð Ip/sw) Ð (sn + ≈ sw)(jc Ð Ip/sn) = (sw/sn Ð sn/sw)Ip 1.5Ip. According to the ≈ µ ≈ experimental data, Iac c+ 0.45 A and Iac cÐ µ ≈ 1.0 A, as is shown in the CVC in Fig. 4 for T 0.99Tc, ≈ µ ≈ Ip = (Iac cÐ Ð Iac c+ )/1.5 0.37 A, and (sn + sw)jc µ | | ≈ ≈ µ 1.6 A. At Iac (sn + sw)jc 1.6 A, the CVC shows a | | singularity, and, at larger Iac values, it coincides with its normal-state shape (Fig. 4). ≈ Φ ≠ Fig. 4. (a) CVC of a system of two loops at T 0.99Tc, The effect of a persistent current on the transition to Φ Φ ≠ Φ n 0 and (n + 0.5) 0; (b) variation of the current density the resistive state is illustrated by three upper oscillo- ≠ in the loop halves for Ip 0; and figure (c) clarifies the non- grams in Fig. 5, which were recorded at temperature monotonic dependence of the oscillation amplitude ≈ ∆ µ Φ Φ ∆ T 0.96Tc and external current amplitude I = 9 A. At V( / 0) on I. Φ = 0, the resistive state is not observed. This signifies ≈ that, at a temperature of 0.96Tc, the critical current den- a single I direction, which is shown in Fig. 5. This sig- ∆ ≈ µ 2 ≈ × 8 2 ac sity jc exceeds I/sw 9 A/0.016 m 5.6 10 A/m , ∆ ∆ nifies that I(sn + sw) + Ip/sn > jc and I(sn + sw) + Ip/sw < because the loop-connecting strip cross section is equal j . From the first inequality and the j > ∆I/s inequality to s (Fig. 1). For Φ = 0.75Φ corresponding to the c c w w 0 ∆ 2 maximum of Ip, the resistive state is observed only for obtained above, it follows that Ip > Isn /sw(sn + sw) =

JETP LETTERS Vol. 77 No. 7 2003 374 DUBONOS et al. ∆ tudes, I(sn + sw) + Ip/sw > jc, the resistive state is T observed for both Iac directions (Fig. 5). The resistive state induced by a persistent current cannot be stationary, because, e.g., the current in this case should be damped upon the transition of, e.g., ln to the resistive state with Rn > 0, which signifies that the current through ln decreases to jn < jc. For this reason, both superconducting stationary states with different connectivities are unstable at jc Ð Ip/sn < Iac/(sn + sw) < jc, so that the loop should switch between them with the eigenfrequency ω, which is determined by the relaxation time to the equilibrium superconducting state [6]. A half of this dynamical resistive state, in which Iac and Ip are directed oppositely and Iac/(sn + sw) Ð Ip/sw < Iac/(sn + sw) < jc, is in the superconducting state, whereas its other half, where they coincide, is switched between the states with R = 0 and R > 0. Upon switching, the pair momentum ∆p changes in the superconductivity half (e.g., lw), and, as a result, the external current Iac = Iw + In is redistributed between the loop halves. This change proceeds under the action of Fig. 5. Four pairs of voltage oscillograms measured in a sys- electric field dp/dt = mdv /dt + 2edA/dt = 2eV /l = tem of three loops, together with the currents induced this s w w voltage. 2eRnIn/lw with Rn > 0, while the initial state is restored due to the quantization of momentum circulation (1) after returning ln to the superconducting state [6]. This fact makes it clear why the pair momentum does not change, on the average, despite the fact that the time- dc 〈 〉 ≠ averaged voltage Vdc = Vw t 0. Since its one-direc- tional change is caused by an electric field, one has V) µ ( ∆ ω V dc = p lw/2e. (3) ∆ The change p in momentum is limited, because In can change only from Iacsn/(sn + sw) + Ip to zero, while Iw cannot exceed the critical value sw jc. According to the ∆ initial restriction, Iw < Iac/3 + Ip, and, according to the ∆ second, Iw < 3Ip. As the steady current decreases to zero, the pair-momentum circulation changes by πប 〈 〉 Φ Φ ∆ πប 〈 〉 Φ Φ 2 ( n Ð / 0). Hence, p < 3(2 /l)( n Ð / 0). From this inequality and Eq. (3) it follows that, for the Fig. 6. The V(Φ/Φ ) oscillation amplitude as a function of µ 0 voltage Vdc = 70 V, which was observed by us for n Ð the amplitude ∆I of the external current with f = 2.03 kHz, Φ Φ ω ∆ ≈ / 0 = 0.25, the switching frequency = 2eVdc/ plw > as measured for one of the loops at T = 1.280 K 0.97Tc. ∆Θ ∆Θ Θ ∆τ 4/3(e/πប)(l/l )V = 8/3(483.6 MHz/µV)V should The curve shows the Vdc( n Ð w)/ = Vdc depen- w dc dc µ ≈ µ ≈ dence calculated for Ic = 4.5 A, Ip 0.5 A, and Vdc exceed 90 GHz. 25 µV. Changes in the momentum ∆p = ∫ dpt()d /dt = tR > 0

∫ > dt2eVm/lw and current In can be lower than their ∆ ≈ µ ≈ Φ Φ tR 0 I/6 1.5 A for T 0.96Tc and = 0.75 0. In a higher magnetic ﬁeld with Φ = 4.25Φ corresponding limiting values, if the residence time tR > 0 in the resis- 0 tive state is shorter than the current relaxation time, to the opposite direction of Ip, the steady resistive state which is determined by the kinetic (lwm/2e)dvs/dt = is observed for the oppositely directed Iac, while the 2 (lwm/s4e ns)dIw/dt = LkindIw/dt and geometric lwdA/dt = resistive state in the other direction is unstable (Fig. 5), ≈ LdIw/dt inductances and resistance Rn. Since Lkin/L evidencing the suppression of j by a magnetic ﬁeld. c λ2 /s for a thin superconductor [6], one has in the weak The amplitude ∆I = 9 µA corresponds to the maximum L | | ∆ ≈ ≈ µ 2 Ӷ λ2 of the V max( I) curve for T 0.96Tc. At larger ampli- screening case with sw 0.016 m L (T) =

JETP LETTERS Vol. 77 No. 7 2003 OBSERVATION OF THE EXTERNAL-AC-CURRENT-INDUCED DC VOLTAGE 375 ∆Θ ∆Θ λ2 (0)/(1 Ð T/T ) ≈ 0.0025 µm2/(1 Ð T/T ) ≈ 0.1 µm2 that explained by a decrease in n and w (Fig. 4). The L c c dependence V (∆Θ Ð ∆Θ )/Θ = V ∆τ(∆I), where ≈ 2 ≈ dc n w dc the kinetic inductance Lkin m0.5l/4e nswsw ∆τ ∆ ∆Θ ∆Θ Θ ( I) = ( n Ð w)/ is calculated with parameters µ λ2 ≈ × Ð11 ≈ µ ≈ µ | | ∆ 00.5l L (T)/sw 4 10 H exceeds the geometric Ic 4.5 A and Ip 0.5 A and describes the V max( I) ≈ Ω ≈ ≈ ≈ µ inductance. The normal-state resistance Rn, n 15 of dependence (Fig. 6) in the Vdc, n Vdc, w Vdc 25 V the narrow half corresponds to the relaxation time approximation rather well. ≈ × Ð11 Φ Φ Lkin/Rn 0.3 10 s. The measured values Vdc = In summary, the oscillations of V( / 0) voltage 〈 〉 〈 〉 〈 〉 Vw t = RnIn t < Rn, n In t indicate that the time-averaged observed in [4] could be induced by a broad-spectrum current, e.g., through the narrow half 〈I 〉 > V /R noise. Since the critical current and the critical value n t dc n, n ∆ exceeds I s /(s + s ) and is close to I s /(s + s ) + I Icr decrease to zero as Tc is approached, these oscilla- ac n n w ac n n w p ≈ in the superconducting state. For example, for the CVC tions can be induced at T Tc by a low-intensity noise, ≈ µ ≈ shown in Fig. 4, Vdc 7.4 V per one loop at Iac and a system of asymmetric superconducting loops can µ 〈 〉 ≈ µ 0.7 A, which corresponds to In t > Vdc/Rn, n 0.49 A, be used for their detection. ≈ µ ≈ Iacsn/(sn + sw) 0.23 A, and Iacsn/(sn + sw) + Ip We are grateful to V.A. Tulin for discussion of the 0.6 µA. results. This work was supported by the Presidium of Inasmuch as the loop is in the superconducting state the Russian Academy of Sciences within the frame- with R = 0 at |I |/(s + s ) < j Ð I /s and undergoes work of the fundamental program “Low-Dimensional l ac n w c p n Quantum Structures.” transition to the usual resistive state with Ip = 0 at | | Iac /(sn + sw) > jc, only the dynamic resistive state with 〈 〉 ≠ 〈 〉 Ip t 0 and Rl t > 0 can contribute to the dc voltage REFERENCES V(Φ/Φ ) ∝ I . For this reason, the time-averaged volt- 0 p 1. M. Tinkham, Introduction to Superconductivity Φ Φ Θ (McGraw-Hill, New York, 1975; Atomizdat, Moscow, age V( / 0) = ∫Θ dtVdc/ can be estimated at V = ∆Θ ∆Θ Θ ∆Θ ∆Θ 1980). (Vdc, n n Ð Vdc, w w)/ , where n and w are the Θ 2. W. A. Little and R. D. Parks, Phys. Rev. Lett. 9, 9 (1962). portions of the period = 1/f during which jc Ð Ip/sn < |∆ π | 3. I. O. Kulik, Zh. Éksp. Teor. Fiz. 58, 2171 (1970) [Sov. Isin(2 ft) /(sn + sw) < jc and jc Ð Ip/sw < Phys. JETP 31, 1172 (1970)]. |∆ π | Isin(2 ft) /(sn + sw) < jc (Fig. 4), and Vdc, n and Vdc, w 4. S. V. Dubonos, V. I. Kuznetsov, A. V. Nikulov, and are the average voltages induced during these times. V. A. Tulin, in Abstracts of All-Russian Scientiﬁc and Φ Φ The oscillation amplitude V( / 0) is maximal at jc Ð Technical Conference on Micro- and Nano-Electronics ∆ Ip/sn < I/(sn + sw) < jc Ð Ip/sw, where the voltage is 2001 (2001), Vol. 2, P2-25; S. V. Dubonos, V. I. Kuz- induced only for one of the ac current directions netsov, and A. V. Nikulov, in Proceedings of 10th Inter- ∆Θ Θ ∆ national Symposium on Nanostructures: Physics and (Fig. 5) and V = Vdc, n n/ . For I/(sn + sw) > jc + Ip/sw, ∆Θ ∆Θ Θ Technology (Physicotechnical Inst., St. Petersburg, the time-average voltage V = (Vdc, n n Ð Vdc, w w)/ 2002), p. 350. decreases sharply because of the appearance of the ∆Θ Θ 5. H. Vloeberghs, V. V. Moshchalkov, C. van Haesendonch, resistive state with ÐVdc, w w/ for the oppositely et al., Phys. Rev. Lett. 69, 1268 (1992). directed Iac (Fig. 5). However, the oscillation amplitude Φ Φ ∆Θ ∆Θ 6. A. V. Nikulov, Phys. Rev. B 64, 012505 (2001). for V( / 0) is still nonzero, because n and w are different in magnitude (Fig. 4). The further decrease of this amplitude with increasing ∆I (Figs. 2, 6) can be Translated by V. Sakun

JETP LETTERS Vol. 77 No. 7 2003

JETP Letters, Vol. 77, No. 7, 2003, pp. 376Ð380. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 445Ð449. Original Russian Text Copyright © 2003 by Yakimov, Dvurechenskiœ, Nikiforov, Bloshkin.

Phononless Hopping Conduction in Two-Dimensional Layers of Quantum Dots A. I. Yakimov*, A. V. Dvurechenskiœ*, A. I. Nikiforov*, and A. A. Bloshkin** * Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected] ** Novosibirsk State University, Novosibirsk, 630090 Russia Received February 20, 2003

Regularities are studied in charge transport due to the hopping conduction of holes along two-dimensional layers of Ge quantum dots in Si. It is shown that the temperature dependence of the conductivity obeys the EfrosÐ Shklovskii law. It is found that the effective localization radius of charge carriers in quantum dots varies nonmonotonically upon filling quantum dots with holes, which is explained by the successive filling of electron shells. The preexponential factor of the hopping conductivity ceases to depend on temperature at low temperatures (T < 10 K) and oscillates as the degree of filling quantum dots with holes varies, assuming values divisible by the conductance quantum e2/h. The results obtained indicate that a transition from phonon-assisted hopping conduction to phononless charge transfer occurs as the temperature decreases. The Coulomb interaction of localized charge carriers has a dominant role in these phononless processes. © 2003 MAIK “Nauka/Interperi- odica”. PACS numbers: 73.63.Kv; 72.20.Ec

INTRODUCTION [4]. It was predicted theoretically [5, 6] and then demonstrated in experiments with GaAs/AlxGa1 Ð xAs [7] If the resistance of a two-dimensional disordered and Ge/Si heterostructures with Ge quantum dots system is larger than a value determined by the resis- 2 (QDs) [8] with an artificial screen introduced in the sys- tance quantum h/e , where h is Planck’s constant and e tem that the occurrence of long-range Coulomb interac- is the elementary charge unit, then the system occurs on tion leads to the EfrosÐShklovskii law the dielectric side of the metalÐinsulator transition and []()1/2 its conductance G tends to zero as the temperature T GT()= G0 exp Ð T 0/T , (2) decreases [1, 2]. In the regime of the strong localization 2 κξ κ of charge carriers, when the localization radius ξ is where T0 = 6.2e /kB [6], where is the relative much smaller than the distance between localized dielectric constant. Within the mechanism of phonon- states, charge transfer is carried out through tunneling assisted hopping conduction, the preexponential factor γ m γ hops of electrons from one center to another, and the G0 takes the form G0 = T , where is a temperature- hopping length at low temperatures increases with independent parameter reflecting the characteristic fre- decreasing temperature [3]. The occurrence of disorder quency of “attempts” of electron hops. Theoretical cal- of certain origin in a system is the reason for the spread culations [6, 9] and experimental studies [7, 10, 11] of energy levels corresponding to various localized indicate that the exponent m has a value of about m ~ Ð1. states. Therefore, in a transition between localization In 1994, Aleiner, Polyakov, and Shklovskii hypoth- centers, an electron is forced to absorb or emit phonons. esized that under certain conditions electronÐelectron Under conditions of this conventional phonon-assisted interaction rather than phonons can stimulate electron mechanism of hopping transport, the temperature transitions between localized states [12]. In this case, dependence of the conductance takes the form the preexponential factor G0 does not depend on tem- []()x perature. The question of the effect of electronÐelectron GT()= G0 exp Ð T 0/T , (1) interaction on the conductivity of two-dimensional systems became especially pressing after the repeated dis- where the parameter T0 is determined by the properties covery of the metalÐinsulator transition [1]. For sys- of the material and the exponent x < 1 is determined by tems with small disorder in which the fluctuation the energy dependence of the density of states at the amplitude of the Coulomb potential is larger than the Fermi level g(Ef). If the electronÐelectron interaction in characteristic disorder energy because of random the system is insignificant and g(Ef) = const, then x = charge exchange between groups of closely spaced 1/3 (the Mott law for a two-dimensional system) and localized states, the phononless mechanism of hopping ξ2 T0 = 13.8/kBg(Ef) , where kB is the Boltzmann constant conduction was proposed by Kozub, Baranovskii, and

PHONONLESS HOPPING CONDUCTION IN TWO-DIMENSIONAL LAYERS 377

Shlimak [13]. Because of random acts of electron hops FORMATION OF GE/SI HETEROSTRUCTURES between states, the Coulomb potential in the surround- WITH QDs ing system fluctuates in time. In turn, the interaction of Samples were grown by molecular-beam epitaxy on this potential with carriers bound on the centers and Si(001) substrates with a resistivity of 1000 Ω cm participating in hopping charge transfer changes their doped with boron up to a concentration of ~1013 cmÐ3. energy and, hence, the probability of tunneling transi- The growth temperature of Si layers was, respectively, tions. At certain instants of time a resonance situation 800 and 500°C before and after the deposition of the Ge arises when two spatially close levels coincide in layer. A Ge layer eight monolayers thick (≈10 Å) was energy and the charge carrier passes from one center to introduced into the middle of the 90-nm Si layer grown another. In this case, phonons are not necessary for on the substrate at a temperature of 300°C. The Ge nan- charge transport, and Coulomb interaction is the factor oclusters formed in this case had the shape of pyramids. assisting hopping charge transfer. The authors of [13] The average sizes of the pyramid base in the growth showed that, under conditions when this phononless plane were 10 nm, the height was ~1 nm, and the layer × 11 Ð2 mechanism is manifested, the temperature dependence density of nanoclusters was 4 10 cm [19]. The controlled filling of Ge islands with holes was carried of conductivity must be described by an equation simi- out by introducing a Si layer δ-doped with boron in the lar to Eq. (2) with a temperature-independent preexpo- 2 samples at a distance of 5 nm below the QD layer. nential factor having a value of order e /h. Because the ionization energy of boron impurities in silicon comprises only 45 meV, and the energies of the Recently, experimental works appeared carried out first ten hole levels in germanium pyramids of this size with Si MIS transistors [14], Si/SiGe quantum wells counted from the Si valence band top are 200Ð320 meV [15], GaAs/AlGaAs heterostructures [2, 16], and Ge/Si [20], holes at temperatures below room temperature quantum dots [8], in which two-dimensional hopping leave the impurities and fill levels in QDs. The concen- conduction of the type given by Eq. (2) was observed tration of boron in various samples varied from 2 × but with a temperature-independent preexponential fac- 1011 cmÐ2 to 2.4 × 1012 cmÐ2, which allowed the average 2 2 tor equal to either e /h [8, 14, 16] or 2e /h [2, 15]. These number of holes Nh per one Ge QD to be varied from results were contradictory to the model of phonon- 0.5 to 6 at a step of 0.5. Ohmic contacts were formed by assisted hopping conduction in two-dimensional disor- sputtering Al islands on the sample surface followed by ° dered systems and pointed to the necessity of searching heating the structure at a temperature of 400 C in a for new mechanisms of hopping charge transport. The nitrogen atmosphere. The current was measured at dif- drawback of the works cited above was the fact that ferent T at voltages no higher than 20 mV, which corresponded to the ohmic region of currentÐvoltage charac- G0 = const was a priori assumed in the analysis of teristics throughout the entire temperature range stud- experimental data, whereas it was necessary to show ied in this work. that actually m = 0. Because of this instance, the problem of experimentally revealing phononless hopping conduction has not yet been solved. EXPERIMENTAL RESULT Experimental temperature dependences of the con- In our opinion, Ge/Si heterostructures containing 2 layers of self-assembled Ge quantum dots that are ductance in e /h units are given in Fig. 1 in Arrhenius formed during the heteroepitaxy of elastically strained coordinates for samples with different numbers of holes Շ 2 systems [17] are best suitable for this purpose as the Nh. In all samples at low temperatures G(T) e /h, object of investigations, because (1) it is possible to which is characteristic of the dielectric side of the metalÐinsulator transition. In order to obtain detailed obtain dense arrays with a QD layer density of up to 12 Ð2 information on the functional dependence G(T), we 10 cm [18], in which the hopping transport of holes used the differential method for an analysis of the between QDs is the dominating mechanism of current dimensionless conductance activation energy [21] flow at low temperatures [8]; (2) because of the speci- ∂ ∂ x w(T) = lnG(T)/ lnT = m + x(T0/T) . In this approach, ficity of heteroepitaxy, Ge nanoclusters lie exactly in Ӷ x if m x(T0/T) , then logw(T) = A Ð xlogT, and A = one plane (growth plane); that is, there is no disorder xlogT0 + logx. Constructing logw(T) as a function of factor associated with disordering in the structure logT, one can find the exponent x from the slope of the growth direction; (3) as distinct from the majority of straight line. The parameter A can be found by the inter- impurities in semiconductors, assemblies of QDs can section point of the straight line with the ordinate axis, serve as a system of multicharged localization centers, which gives the characteristic temperature T0 = in which the role of Coulomb potentials is especially (10A/x)1/x. Typical plots of logw(T) versus log T for important. The goal of this work was to determine the some samples are given in Fig. 2. At T Շ 10 K, a linear mechanism and regularities of charge transfer in layers relationship is observed between logw(T) and logT, Ӷ x of Ge/Si quantum dots. pointing to the fact that m x(T0/T) at these tempera-

JETP LETTERS Vol. 77 No. 7 2003 378 YAKIMOV et al.

Fig. 1. Temperature dependences of the conductance in Fig. 2. Temperature dependences of the logarithmic deriva- samples differing in the average number of holes per one Ge tive w(T) = ∂lnG(T)/∂lnT for samples differing in the aver- QD. age number of holes per one Ge QD. tures. From the slope angle of the approximating simulation of the hole energy spectrum and hole wave straight lines (solid lines in Fig. 2), we found that the functions in pyramidal Ge nanoclusters in Si was per- exponent x takes approximately the same value x = formed by Dvurechenskii, Nenashev, and Yakimov 0.51 ± 0.05 for all the 12 samples in accordance with [20]. The hole ground state has an s-type symmetry and Eq. (2). is doubly degenerate by the spin direction. The first Ӎ excited state possesses a p-type symmetry with the Because it was already stated that x 0.5, the non- degree of degeneracy equal to 4. One pair is associated linear regression method can be used for further deter- with the spin degree of freedom, and another one, with mining the exponent m in the region of low tempera- the equivalence of crystallographic directions along tures. With this aim, the experimental dependences G(T) at T < 10 K were approximated by the equation which the diagonals of the Ge pyramid base are ori- γ m 1/2 γ ented [20]. It is evident in Fig. 4a that, at the very begin- G(T) = T exp[Ð(T0/T) ], and the parameters , m, and ning of filling each of the shells (Nh = 0.5 for the s shell T0 were varied until the coincidence of the calculated ± and Nh = 3Ð4 for the p shell), m = Ð(0.75 0.05); that and experimental curves. The results of this procedure is, conduction is phonon-assisted; however, at the end are presented in Fig. 3. Symbols correspond to experi- of filling, a considerable contribution from phononless mental data, and solid lines depict calculated curves. It was found that m lies in the region Ð(0.11 ± 0.09) processes arises even at high temperatures (m 0). (Fig. 4a). The closeness of m to zero allows a conclu- The most probable hypothesis that can explain this sion to be made that actually the preexponential factor behavior is that the intensity of charge exchange G is virtually temperature-independent at low temper- between localized electron states is small when the 0 shell is completely empty or fully filled (in the first atures. case, there is no mobile charge, and in the second, there A similar approach to determining the temperature is no vacant places with close energy levels). Therefore, dependence G0(T) was applied to the region of high the mechanism of hopping transport stimulated by temperatures (10 < T < 40 K). The high-temperature Coulomb interaction [13] is suppressed. The depen- exponent m in samples differing in the numbers of dence of the effective localization radius ξ shown in holes Nh is given in Fig. 4a. In order to understand the Fig. 4b gives evidence in favor of this interpretation. The value of ξ was found from the relationship T = observed dependence m(Nh), it is necessary to consider 0 2 κξ κ the structure of electron shells in Ge QDs. A numerical 6.2e /kB with = 12 for Si. In the situation of hop-

JETP LETTERS Vol. 77 No. 7 2003 PHONONLESS HOPPING CONDUCTION IN TWO-DIMENSIONAL LAYERS 379

High T

Fig. 3. Temperature dependences of the conductance constructed on the logGÐTÐ1/2 coordinates. Symbols correspond to experimental points, solid lines are approxima- tions of the experimental data using the equation G(T) = γ m 1/2 γ T exp[Ð(T0/T) ]. T0, , and m are variable parameters. Fig. 4. Dependence of (a) the exponent m characterizing the ξ temperature dependences of the preexponential factor G0 in ping conduction over the impurity band, is as a rule the region of high (10 < T < 40 K) and low (T < 10 K) tem- nothing more nor less than the Bohr radius of the peratures, (b) the effective localization radius ξ, and (c) the charge carrier at an impurity. In the case of QDs, when preexponential factor G0 at low temperatures (T < 10 K) on in each of them there are several bound states, electrons the average number of holes in QDs Nh. can pass, for example, from the ground state of one QD to the excited state of another QD. In this case, the effective localization radius reflects the spatial range of cesses in two-dimensional disordered systems in the overlap between the wave functions in the initial and strong localization regime are necessary. final states. It is evident in Fig. 4b that the lowest value In conclusion, let us make another comment. The ξ of is attained in a sample with Nh = 3. This means that number of phonons with a particular energy decreases the filling of the s shell is actually completed at these with decreasing temperature. As a consequence, two values of Nh, and holes in hopping are forced to pass alternatives arise. First, an electron is forced to hop into the p shell with a completely different electron onto a spatially more distant state with a minimum dif- configuration. ference in energy. Second, it can make a hop onto a closer state by changing the energy of the state through Let us turn back to the analysis of the preexponential the Coulomb interaction with centers with fluctuating factor in the region of low temperatures. The depen- charges. Apparently, the probability of the second pro- dence G0(Nh) is presented in Fig. 4c. It turned out that cess at low T is higher. For this reason, a change in hop- the prefactor actually has a value of order e2/h in agree- ping conduction mechanisms takes place on cooling the ment with the theory of electronÐelectron interaction; system. however, it is not constant but varies upon varying the The authors are grateful to S.D. Baranovskiœ for use- degree Nh of filling the QD with holes, assuming values ful discussions. divisible by e2/h. Currently, there is no preconceived This work was supported by the Russian Foundation explanation of the oscillating behavior of G0, and fur- for Basic Research, project no. 03-02-16526, and the ther theoretical and experimental investigations into the program “Universities of Russia,” project effects of Coulomb interaction in charge-transfer pro- no. UR.01.01.019.

JETP LETTERS Vol. 77 No. 7 2003 380 YAKIMOV et al.

REFERENCES 12. I. L. Aleiner, D. G. Polyakov, and B. I. Shklovskii, in Proceedings of the 22nd International Conference on the 1. S. V. Kravchenko, W. E. Mason, G. E. Bowker, et al., Physics of Semiconductors, Vancouver, 1994, Ed. by Phys. Rev. B 51, 7038 (1995). D. S. Lockwood (World Sci., Singapore, 1994), p. 787. 2. M. Y. Simmons, A. R. Hamilton, M. Pepper, et al., Phys. 13. V. I. Kozub, S. D. Baranovskii, and I. S. Shlimak, Solid Rev. Lett. 80, 1292 (1998). State Commun. 113, 587 (2000). 3. N. Mott, J. Non-Cryst. Solids 1, 1 (1968). 14. W. Mason, S. V. Kravchenko, G. E. Bowker, and J. Fur- 4. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of neaux, Phys. Rev. B 52, 7857 (1995). Doped Semiconductors (Nauka, Moscow, 1979; Springer, 15. J. Lam, M. D’Iorio, D. Brown, and H. Lafontaine, Phys. New York, 1984). Rev. B 56, R12741 (1997). 5. A. L. Efros and B. I. Shklovskii, J. Phys. C 8, L49 16. S. I. Khondaker, I. S. Shlimak, J. T. Nicholls, et al., Phys. (1975). Rev. B 59, 4580 (1999). 6. D. N. Tsigankov and A. L. Efros, Phys. Rev. Lett. 88, 17. O. P. Pchelyakov, Yu. B. Bolkhovityanov, A. V. Dvurech- 176602 (2002). enskiœ, et al., Fiz. Tekh. Poluprovodn. (St. Petersburg) 7. F. W. Van Keuls, X. L. Hu, H. W. Jiang, and A. J. Damm, 34, 1281 (2000) [Semiconductors 34, 1229 (2000)]. Phys. Rev. B 56, 1161 (1997). 18. A. I. Yakimov, A. V. Dvurechenskii, A. I. Nikiforov, et al., Phys. Rev. B 67, 125318 (2003). 8. A. I. Yakimov, A. V. Dvurechenskii, V. V. Kirienko, et al., Phys. Rev. B 61, 10868 (2000). 19. N. P. Stepina, R. Beyer, A. I. Yakimov, et al., Phys. Low- Dimens. Semicond. Struct. 11/12, 262 (2001). 9. E. I. Levin, V. L. Nguen, B. I. Shklovskiœ, and A. Éfros, Zh. Éksp. Teor. Fiz. 92, 1499 (1987) [Sov. Phys. JETP 20. A. V. Dvurechenskii, A. V. Nenashev, and A. I. Yakimov, 65, 842 (1987)]. Nanotechnology 13, 75 (2002). 21. A. G. Zabrodskiœ and K. N. Zinov’eva, Zh. Éksp. Teor. 10. G. Ebert, K. von Klitzing, C. Probst, et al., Solid State Fiz. 86, 727 (1984) [Sov. Phys. JETP 59, 425 (1984)]. Commun. 45, 625 (1983). 11. A. Briggs, Y. Guldner, J. P. Vieren, et al., Phys. Rev. B 27, 6549 (1983). Translated by A. Bagatur’yants

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JETP Letters, Vol. 77, No. 7, 2003, pp. 381Ð384. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 450Ð454. Original Russian Text Copyright © 2003 by Val’kov, Dzebisashvili.

Modification of the Superconducting Order Parameter D(k) by Long-Range Interactions V. V. Val’kov1, 2, 3, * and D. M. Dzebisashvili1, 3 1 Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia *e-mail: [email protected] 2 Krasnoyarsk State Technical University, Krasnoyarsk, 660074 Russia 3 Krasnoyarsk State University, Krasnoyarsk, 660075 Russia Received December 9, 2002; in ﬁnal form, February 21, 2003

It is demonstrated that the inclusion of long-range intersite interactions qualitatively modiﬁes the dependence of a superconducting gap on quasimomentum for both s- and d-symmetry types. In particular, the order parameter of a superconducting phase with the d symmetry type depends on two amplitudes and has the form x2 Ð y2 ∆ ∆ ∆ (k) = 1(coskx Ð cosky) + 2(cos2kx Ð cos2ky). In this case, the theoretical dependence of the critical temperature on the degree of doping agrees with the experimental dependence. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 74.20.-z

1. It is known that the (tÐJ) model [1] properly demonstrated in many works dealing with the quasipar- describes, on a qualitative level, the magnetic pairing ticle energy spectrum (see, e.g., [9Ð13]). In these cases, mechanism in high-temperature superconductors (see, the theoretical positions agreed satisfactorily with the e.g., review [2]). If this model is developed on the basis APRES data. In particular, it was pointed out that the of the Hubbard model in the strong-correlation regime, inclusion of frustrated bonds (J2 > 0) is important for the effective Hamiltonian Heff includes so-called three- the description of the evolution of spectral dependence center terms [3, 4]. In [5], it was shown that the three- in the presence of doping [12]. Since, as was mentioned center terms H(3)make a weak contribution to the dis- above, the exchange interaction parameters play the persion curves of energy spectrum. This result is quite role of coupling constants in the magnetic mechanism of superconducting pairing, one can expect that J and natural, because the corrections H(3) to the hopping 2 parameters contain the additional smallness. A different J3 may inﬂuence both the functional form of the order situation appears in the analysis of a superconducting parameter and the conditions for the appearance of phase. In the case of the magnetic pairing mechanism, superconducting state. the exchange interaction plays the role of coupling con- Below, the effective Hamiltonian based on the stant. The energy parameters in the three-center terms strong-correlation Hubbard model (extended (tÐJ) are of the same order of magnitude. For this reason, the model with three-center interactions) is used to demon- contribution H(3) to the self-consistent equation for the strate that the exchange interactions between the spins superconducting gap becomes appreciable. The inﬂu- of non-nearest neighbors have an appreciable effect ence of three-center terms on the formation of super- both on the quasimomentum dependence of the order conductivity was studied in [6, 7]. It was shown in [7] parameter and on the form of the equation for the super- that the inclusion of three-center terms results in the conducting gap and critical temperature Tc. renormalization of the coupling constant. This substan- 2. The Hubbard Hamiltonian tially reduces the region of a superconducting phase with the d 2 2 symmetry type of order parameter [8]. ()εµ+ x Ð y H = ∑ Ð a fσa fσ σ Beyond the nearest-neighbor approximation, the f (1) effective Hamiltonian includes the exchange interac- + tion between the spins of quasiparticles separated by a + ∑t fma fσamσ + Un∑ ˆ f ↑nˆ f ↓ distance larger than the lattice parameter. The important fmσ fσ role of quasiparticle hopping between the sites from the is chosen as the initial model, and it is assumed that distant coordination spheres and the exchange interac- three hopping parameters are nonzero: t , δ = Ðt , tions between the non-nearest-neighboring spins was ff+ 1 1

382 VAL’KOV, DZEBISASHVILI t , δ = Ðt , and t , δ = Ðt , where δ are the radius representation, then, as was shown by Zaœtsev et al. in ff+ 2 2 ff+ 3 3 i 〈〉0σ σ2 〈〉σ2 σ2 vectors of the sites from the ith coordination sphere. [17], the anomalous means X f Xm and X f Xm It is well known that, in the regime of strong electron also appear, which are caused by the transitions from ӷ | | correlations (U tfm ) and for concentrations n < 1, one the lower to the upper Hubbard subband and the transi- can pass, in the Hubbard operator representation, to the tions inside the upper subband that make contribution effective Hamiltonian of the form at finite U. The formal absence of these anomalous means in our approach does not mean that we ignore ()εµ σσ σ0 0σ H = ∑ Ð X f + ∑t fmX f Xm these processes. The matter is that, when passing to fσ fmσ Heff, all calculations are carried out in the new representation, for which the indicated processes are taken into t t σσ σσ σσ σσ + ∑ ------fm mf ()X X Ð X X account by different operator structures. This statement U f m f m (2) can be clarified as follows. fmσ 〈〉02 〈〉0σ 02 We first consider the X f and X f Xm means t t σ0 σσ 0σ σ0 σσ 0σ + ∑ ------fm mg ()X X X Ð X X X U f m g f m g that are associated with the transitions from the lower fmgσ to upper Hubbard subband (in this case, it is tacitly ()fg≠ assumed that the representation is induced by the orig- with a quadratic accuracy in |t |/U [7, 2, 8]. The nota- inal Hamiltonian). The transition to Heff implies the uni- fm + tion is standard, and its meaning can be found in the tary transformation (S = ÐS) cited works and in review [2]. Note, nevertheless, that H H ==exp()S HSexp()Ð ; H|〉Ψ E |〉Ψ , the last term in the Hamiltonian depends on three sites eff n n n and describes the correlated hopping. H |〉Φ ==E |〉Φ , |〉Φ exp()ΨS |〉, By using the diagram technique for the Hubbard eff n n n n n operators [14, 15] or the method of irreducible Green’s after which the Hamiltonian, the basis functions, and all functions in the atomic representation with anomalous operators, whose averaging gives the physical quanti- means [16], one arrives at the self-consistency equation ties of interest, change their form. For example, the for the superconducting order parameter (SOP) ∆ with 02 0σ σ2 q operators X and X X are transformed as allowance made for the three-center terms [8]: f f m 02  X02 X˜ ∆ 1 n()n tktq f f k = ----∑2tq ++--- Jkq+ + JkqÐ 41Ð ------N  2 2 U q ()()0↓ 0↑ 0↑ 0↓ {}()3 (3) = Ð ∑ t fg/U X f Xg Ð X f Xg + Otfg/U , 2  ∆ g tq J0 q Eq Ð n---- Ð ----- ------tanh------. U 2  2Eq 2T 0σ σ2 ˜ 0σ ˜ σ2 ()ησ X f Xm X f Xm = Ð∑ tmg/U () In this equation, tk and Jk are the Fourier transforms of g 2 0σ σσ 0σ 0σ σσ 0σ 3 the parameters tfm and 2t fm /U. The energy of Bogoli- × (){}() X f Xm Xg Ð X f Xm Xg + Otfm/U . ubov quasiparticles is denoted by Ek = The transformation laws clearly demonstrate that, in ()ε˜ µ 2 ∆ 2 k Ð + k , and the renormalized electronic the new representation, the above-mentioned anoma- spectrum is lous means are not ignored within the linear accuracy in t (tfm/U), and their contribution is determined by the ε˜ ()n k 1 k = 1 Ð n/2 1 Ð ------tk Ð ---- means of operators which are responsible only for the 2U N transitions between the states without pairs. In essence, this is a particular case that follows from the general  × n []()()tq statement made in [18]. As for the anomalous means ∑tq ++---JkqÐ 2 Ð n tk + 1 Ð n tq ---- nqσ, 2 U associated with the transitions inside the upper Hub- q bard band, one can readily verify that the corresponding ()ϑ ϑ ()()Ð1 contribution is nonzero only in the quadratic (and not nqσ = 1 Ð q /2 + q expEq/T + 1 , (4) linear) approximation in (tfm/U). For this reason, these ϑ ξ ξ ε µ q ==q/Eq, q ˜ q Ð . means make no contribution to our theory. Note that, when deriving Eq. (3) from H , only As known, Eq. (3) has solutions differing in the type eff of ∆ symmetry. We will consider the influence of long- 〈〉0↓ 0↑ k X f Xm anomalous means appear. If, however, one range hopping separately on the type of SOP symmetry starts from the Hamiltonian (1) written in the atomic and on the k dependence for a given symmetry.

JETP LETTERS Vol. 77 No. 7 2003 MODIFICATION OF THE SUPERCONDUCTING ORDER PARAMETER ∆(k) 383

(a) s Symmetry; due to the presence of three non- ≠ zero t1, t2, t3 0, the solution to Eq. (3) for this symme- ∆ try type is given by k, which is expressed as ∆ ∆ ∆ ∆ ∆ k = 0 +++1S1()k 2S2()k 3S3().k (5) Hereafter, the following invariants are used for brevity:

() 3 S1()k = coskxak+ cos ya /2, 10 ()() S2()k = cos kxa cos kya , () S3()k = cos 2kxa + cos 2kya /2. The order parameter in the form of Eq. (5) is the solution to the integral Eq. (3), if the unknown coefficients ∆ i satisfy the set of four equations 3 3 3 Fig. 1. Effect of hopping to the third coordination sphere on n n α ∆ ∆ the concentration dependence of Tc for different 3 = t3/t1. 0 = ∑ ∑ 2GljT j + --- J0Gl Ð ---- ∑ GlijT iT j l | | 2 U Parameters: t2/t1 = Ð0.2 and t1 /U = 0.2. l = 0 j = 1 ij, = 1 (),, T j ==Ð4t j, j 123, ∆ ∆ if the amplitudes 1 and 2 are the solutions to the fol- 3 3 (6) lowing two equations: ∆ () ∆ m = 4∑ nJmGml + 1 Ð n/2 T m ∑ GliT i/U l, j = 2 2 2  t t l = 0 i = 1 ∆ ===λ ∑ Φ ∆ ; λ n ---1- , λ n ---3- , i i ij j 1 U 2 U 2 ,, j = 1 Jm ==2tm/U, m 123, (9) 1 E where Φ = ----∑ϕ ()q ϕ ()q tanh ------q . nm N n m 2T Gi ==Gi00, Gij Gij0, q ≠ ∆ It follows that, for t3 0, 1is always nonzero. The 1 Ψ Gijl = ----∑Si()q S j()q Sl()q q, condition for the compatibility of this set of equations N leads to the equation q () ()1 Ð λ Φ ()λ1 Ð λ Φ = λ Φ2 , (10) Ψ tanh Eq/2T 1 11 2 22 1 2 12 q ==------, S0()k 1. 2Eq which determines, in particular, the critical tempera- From the set of Eqs. (6) it follows that, in the limit of ture. One can see that the well-known equation for the infinitely strong repulsion, for which the superconduct- critical temperature in the (tÐJ*) model is obtained only ing pairing is governed only by the Zaœtsev kinematic for t3 = 0. mechanism, the order parameter includes only the con- 3. Because of the lack of volume, we only present tributions linear in t; the results of numerical analysis for the influence of ∆ ∆ (b) dxy symmetry; k = 1sin(kxa)sin(kya) This sym- distant hopping on the characteristics of superconduct- ∆ metry type is absent for the SOP in the nearest-neighbor ing state for k of the d 2 2 -type symmetry. Figure 1 ≠ x Ð y approximation and appears only if t2 0. The corre- ∆ shows the concentration dependence of the transition sponding SOP amplitude 1 is found from the transcen- temperature to the d 2 2 phase with inclusion of the dental equation x Ð y parameter t3. One can see that the electron hopping ()2 2 Ψ 14= nJ2/N ∑ sin qx sin qy q ; (7) from the third coordination sphere exerts a substantial q effect on the position of the maximum of the Tc(n) curve. It is notable that the experimentally observed sit- (c) d 2 2 symmetry. Due to the distant hopping (t2, x Ð y uation with a Tc maximum at n ~ 0.8 can easily be real- ≠ ∆ ∆ t3 0), the well-known SOP k = 0(cos(kxa) Ð ized. Curves 1 and 2 in Fig. 2 demonstrate the effect of cos(kya)) is impossible, because it does not satisfy the modified Eq. (10) with long-range interactions on the integral Eq. (3). The solution to this equation can be critical temperature. Curve 1 is constructed using the represented in a two-parameter form solution to the complete Eq. (10), and curve 2 is obtained on the assumption that λ = 0. ∆ ∆ ϕ ∆ ϕ 2 k = 1 1()k + 2 2(),k (8) In the superconducting phase (T < Tc), the ampli- ϕ ()() () , ∆ ∆ l()k ==cos lkxa Ð cos lkya , l 12, tudes 1 and 2 are nonzero and change synchronously

JETP LETTERS Vol. 77 No. 7 2003 384 VAL’KOV, DZEBISASHVILI

done by Zaœtsev et al. in [17]. However, if one is interested in the leading approximation with respect to (tfm/U), the variant presented in this work is simpler. It is this fact that allows the inﬂuence of the long-range hopping on the possible symmetry type of order parameter and on the modiﬁcation of the quasimomentum

3 dependence to be analyzed for any symmetry type. 10 This work was supported by the Russian Foundation for Basic Research + KKFN “Eniseœ” (project no. 02- 02-97705) and the program “Quantum Macrophysics” of the Presidium of the Russian Academy of Sciences. One of us (D.M.D.) is grateful to the Foundation for the Assistance of the Native Science and Siberian Division of the Russian Academy of Sciences (Lavrent’ev Com- Fig. 2. Concentration dependence Tc(n): t2/t1 = 0.4, t3/t1 = petition of Youth Projects). | | 0.3, and t1 /U = 0.2. For the values for curves 1 and 2, see the text. REFERENCES 1. P. W. Anderson, Science 235, 1196 (1987). 2. Yu. A. Izyumov, Usp. Fiz. Nauk 167, 465 (1997) [Phys. Usp. 40, 445 (1997)]. 3. L. P. Bulaevskiœ, É. L. Nagaev, and D. L. Khomskiœ, Zh. Éksp. Teor. Fiz. 54, 1562 (1968) [Sov. Phys. JETP 27, 836 (1968)].

3 4. K. A. Chao, J. Spalek, and A. M. Oles, J. Phys. C 10,

10 L271 (1977). 5. V. Yu. Yushankhay, V. S. Oudovehko, and R. Hayn, Phys. Rev. B 55, 15562 (1997). 6. J. E. Hirsch, Phys. Lett. A 136, 163 (1989). 7. V. Yu. Yushankhay, G. M. Vujicic, and R. B. Zakula, Phys. Lett. A 151, 254 (1990). 8. V. V. Val’kov, T. A. Val’kova, D. M. Dzebisashvili, and S. G. Ovchinnikov, Pis’ma Zh. Éksp. Teor. Fiz. 75, 450 Fig. 3. Temperature dependence of the amplitudes ∆ and |∆ | 1 (2002) [JETP Lett. 75, 378 (2002)]. 2 for n = 0.84. Parameters are as in Fig. 2. 9. A. Nazarenko, K. J. E. Vos, S. Haas, et al., Phys. Rev. B 51, 8676 (1995). with temperature. An example of the behavior of this 10. O. P. Sushkov, G. A. Sawatzky, R. Eder, and H. Eskes, type is demonstrated in Fig. 3. One can see that, over Phys. Rev. B 56, 11769 (1997). the entire temperature range where the superconducting 11. A. F. Barabanov, O. V. Urazaev, A. A. Kovalev, and |∆ | ∆ solution occurs, the temperature of 2 ( 2 < 0) is the L. A. Maksimov, Pis’ma Zh. Éksp. Teor. Fiz. 68, 386 ∆ same as for 1. (1998) [JETP Lett. 68, 412 (1998)]. 12. R. Hayn, A. F. Barabanov, J. Schulenburg, and J. Richter, Note in conclusion that the effect of long-range hop- Phys. Rev. B 53, 11714 (1996). ping both on the occurrence of superconducting state and on the momentum dependence of the order param- 13. T. Tohyama and S. Maekawa, Supercond. Sci. Technol. eter has been demonstrated in this work by the example 13, R17 (2000). of effective Hamiltonian obtained from the Hubbard 14. R. O. Zaœtsev, Zh. Éksp. Teor. Fiz. 70, 1100 (1976) [Sov. model in the strong electron-correlation regime. In this Phys. JETP 43, 574 (1976)]. case, there is a correspondence between t and 15. R. O. Zaœtsev and V. A. Ivanov, Pis’ma Zh. Éksp. Teor. 3 Fiz. 46 (S1), 140 (1987) [JETP Lett. 46, S116 (1987)]. exchange parameter J3. Nevertheless, a situation is often considered where the hopping parameters and the 16. N. M. Plakida, V. Yu. Yushankhay, and I. V. Stasyuk, exchange constants are thought to be independent. In Physica C (Amsterdam) 162Ð164, 787 (1989). this case, a situation can in principle be realized where 17. R. O. Zaœtsev, V. A. Ivanov, and Yu. V. Mikhaœlova, Fiz. the magnitude of long-range exchange interactions will Met. Metalloved. 68, 1108 (1989). not be related to the hopping amplitudes. 18. N. N. Bogolyubov, Selected Papers (Naukova Dumka, Kiev, 1970), Vol. 2, p. 423. It is worth noting that the more general results could be obtained using the original Hamiltonian (1), if the set of four equations is written in the form as it was Translated by V. Sakun

JETP LETTERS Vol. 77 No. 7 2003

JETP Letters, Vol. 77, No. 7, 2003, pp. 385Ð388. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 455Ð458. Original Russian Text Copyright © 2003 by Nasyrov, Novikov, Gritsenko, Yoon, Kim.

Multiphonon Ionization of Deep Centers in Amorphous Silicon Nitride: Experiment and Numerical Simulations K. A. Nasyrov*, Yu. N. Novikov**, V. A. Gritsenko**, S. Y. Yoon***, and C. W. Kim*** * Institute of Automatics and Electrometry, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia ** Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected] *** Samsung Advanced Institute of Technology, Suwon, 440-600 Korea Received March 3, 2003

The conductivity of amorphous silicon nitride has been studied experimentally in a wide range of electric ﬁelds and temperatures. The experimental results are in a quantitative agreement with the theory of multiphonon ionization of deep centers for the bipolar model of conductivity. The best agreement between experiment and the calculation has been obtained for the same parameters of deep electron and hole centers. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 77.22.-d; 77.84.Bw

The majority of amorphous dielectrics, such as injection from the metal; see Fig. 1a. On the other hand, Si3N4, Ce3N4, BN, Ta2O5, HfO2, Y3O3, and TiO2, con- in this work experimental results are compared with the tain a high density of deep centers (traps). The conduc- more general bipolar model, in which the injection of tivity of such dielectrics in strong electric fields (106Ð electrons from silicon, the injection of holes from the 107 V/cm) is limited by the ionization of deep centers. metal, and the recombination of free electrons with Silicon nitride is the dielectric that is most comprehen- localized holes and free holes with localized electrons sively studied from the viewpoint of the mechanism of are taken into account; see Fig. 1b. The goal of this charge transfer. work is to experimentally study the mechanism of charge transfer in Si N over a wide range of tempera- Amorphous silicon nitride (Si3N4) is characterized 3 4 by a high density (>1019 cmÐ3) of electron and hole tures and fields and to quantitatively compare the traps with a giant confinement time of electrons and experiment with the calculation based on the bipolar holes in a localized state (>10 years at 300 K) [1]. Cur- rently, it is commonly accepted that the ionization of deep centers in Si3N4 is confined to the Frenkel effect [2Ð5]. However, it was demonstrated in [4Ð8] that the interpretation of Si3N4 conductivity within the Frenkel model gives an anomalously small value of the frequency factor ν ≈ 106Ð109 sÐ1. The frequency factor was estimated in the original paper by Frenkel at a level of ν ≈ 1015 sÐ1 [9]. Moreover, the formal agreement with the modified Frenkel model that takes into account tunnel ionization can be obtained only at an anomalously large value of the effective tunneling mass m* = 4me [8]. At the same time, experiment gives a value of the effective tunneling mass in silicon nitride close to m* = 0.4me [10]. As distinct from dielectrics, the ionization of deep centers in semiconductors is interpreted within the theory of multiphonon ionization [11]. In [8], experimental results on the conductivity of metal nitrideÐoxideÐsemiconductor (MNOS) structures were quantitatively compared with the theory of multiphonon ionization for the unipolar model of con- Fig. 1. Energy diagram for (a) unipolar and (b) bipolar models of conductivity for an MNOS structure at a positive volt- ductivity, which takes into account only electron injec- age on an aluminum electrode. Symbols (Ð) and (+) mark tion from silicon and does not take into account hole respectively electrons and holes captured by a trap.

386 NASYROV et al.

Fig. 2. Experimental (dots) and calculated (solid lines) cur- Fig. 3. Experimental (dots) and calculated (solid lines) tem- rentÐvoltage characteristics for the model of the mul- perature dependences of the current for the model of the tiphonon ionization of traps in Si3N4 at a positive voltage on multiphonon ionization of traps for a positive voltage on an an aluminum electrode. The calculation was carried out for aluminum electrode. The calculation was carried out for the e h σe σh σ the same parameters of electron and hole traps σ = σ = same parameters of electron and hole traps = = r = σ × Ð13 2 e h e × Ð13 2 e h e h r = 5 10 cm , Wt = Wt = Wt = 1.7 eV, Wopt = 5 10 cm , Wt = Wt = Wt = 1.7 eV, Wopt = Wopt = h e h e h e Wopt = Wopt = 2.7 eV, Wph = Wph = Wph = 0.045 eV, Wopt = 2.7 eV, Wph = Wph = Wph = 0.045 eV, and Nt = e h × 19 Ð3 h × 19 Ð3 and Nt = Nt = Nt = 7 10 cm . Nt = Nt = 7 10 cm . conductivity model and the multiphonon mechanism of equation and the Poisson equation taking into account trap ionization in silicon nitride. the nonuniform electric ﬁeld distribution in silicon Samples of MNOS structures were manufactured on nitride a p-type silicon substrate with a resistivity of 20 Ω cm ∂ , ∂ , nx() t 1 jxt()σeν , ()e , grown using the Czochralski method. A thin, tunnel------∂ = ------∂ - Ð nx() t Nt Ð nt()xt transparent oxide layer 18-Å thick was grown at a tem- t e x (1) ° , , σ ν , , perature of 760 C. Silicon nitride 670-Å thick was + nt()xtPx() t Ð r nx() tpt(),xt obtained at a temperature of 700°C by the deposition method in a low-pressure reactor. The SiH Cl /NH ∂n ()xt, e e 2 2 3 ------t - = σ νnx(), t()N Ð n ()xt, ratio was 0.1. Aluminum electrodes with an area of 5 × ∂t t t (2) Ð3 2 10 cm were sputtered through a mask. , , σ ν , , Ð nt()xtPx() t Ð r nx() tnt(),xt CurrentÐvoltage characteristics (Fig. 2) were mea- ∂ , sured experimentally at different temperatures, and ∂ px(), t 1 j p()xt h h ------= ------Ð σ ν px(), t()N Ð p ()xt, temperature dependences of the current (Fig. 3) were ∂t e ∂x t t measured for different voltages. The experimental (3) , , σ ν , , dependences were obtained in a cryostat cooled by liq- + pt()xtPx() t Ð r px() tnt(),xt uid nitrogen with controlled temperature in the range ∂ , 77Ð450 K. The currentÐvoltage characteristics were pt()xt h h ------= σ ν px(), t()N Ð p ()xt, measured with voltage varied at a rate of 0.02 V/s. The ∂t t t (4) heating rate of samples was ~5 K/min. All the measure- , , σ ν (), , ments were carried out for positive voltages on the gate. Ð pt()xtPx() t Ð r pxtnt(),xt At this voltage polarity, both the injection of electrons ∂ (), , from the silicon substrate and the injection of holes F nt()xtÐ pt()xt ------∂ = Ðe------εε -. (5) from aluminum take place. x 0 A one-dimensional bipolar model of Si N conduc- 3 4 Here, n and nt are the concentrations of free and trapped tivity was used for comparison with experiment. Previ- electrons, p and pt are respectively the concentrations of ously, this model was used in [12]. Charge transfer was e h described with the use of the ShockleyÐReadÐHall free and trapped holes, Nt and Nt are the concentra-

JETP LETTERS Vol. 77 No. 7 2003 MULTIPHONON IONIZATION OF DEEP CENTERS 387 tions of electron and hole traps, F(x, t) is the local electric ﬁeld, e is the electron charge, σe, h is the capture σ cross section of a trap, r is the recombination cross section between free and trapped carriers of opposite sign, v is the drift velocity, and ε = 7.5 is the low-frequency permittivity of Si3N4. The following values were used in this work for the capture and recombina- σe σh σ × Ð13 2 tion cross sections = = r = 5 10 cm [6, 7, 13]. The drift velocity of electrons is related to the current density by the equation j = env. The multiphonon ionization model [11, 14] was used for the ionization probability P of a trap in Si3N4

eF  4 2m 3/2 P = ------expÐ------ប Wopt 22m∗W 3 eF opt (6) ∗ m Wph ()Wph +4------Wopt Wopt Ð Wt coth------ , ប2 F2 2T Fig. 4. Experimental dependence j/F versus 1/F for the tem- where Wopt and Wt are the optical and thermal ionization perature T = 77 K constructed on a semilogarithmic scale. energies of a trap, Wph is the phonon energy, m* is the effective mass of a carrier. The same effective masses The lowest field Fmin necessary for such tunneling is equal to 0.5m0 (where m0 is the free electron mass) were chosen in modeling for electrons and holes. The elec- determined from the relationship tron and hole injection from the silicon substrate and 2ω 2m∗W aluminum electrode, respectively, was calculated based ph Fmin = ------, (8) on the FowlerÐNordheim mechanism. e We carried out two series of experiments, in one of where ω is the frequency of the trap “nucleus,” which ω ≈ ប which the currentÐvoltage characteristic was recorded can be estimated from the relationship Wph/ . The × 6 at a fixed sample temperature, and the temperature estimation gives Fmin = 5 10 V/cm. dependence of the current through the MNOS was The experimental currentÐvoltage characteristic recorded in the second series at a constant applied volt- constructed on the ln( j/F)Ð1/F coordinates for the tem- age. The currentÐvoltage characteristics (Fig. 2) were perature T = 77 K and electric fields higher than 5 × measured at temperatures T = 77, 300, and 410 K in the 6 voltage range 30Ð55 V, which corresponds to average 10 V/cm is shown in Fig. 4. The average field F = V/d × 6 (V is voltage, and d is the nitride thickness) is plotted in electric fields of ~(4–8) 10 V/cm in silicon nitride. It Fig. 4. On these coordinates, the rectification of the is evident in the figure that the current increases expo- experimental curve is observed. The optical energy of nentially with increasing voltage on the aluminum elec- the trap W = 2.5 eV was estimated by the slope of the trode. opt curve for m* = 0.5. The temperature dependences of the current were It was found that the best agreement with the exper- measured in the temperature range 77Ð410 K for volt- iment is obtained for the same parameters of electron ages of 44, 37, and 30 V and were plotted on the Arrhe- e h e h nius coordinates ln(j) Ð TÐ1 (Fig. 3). The figure demon- and hole traps: Wt = Wt = Wt = 1.7 eV, Wopt = Wopt = strates that the current weakly depends on the tempera- W = 2.7 eV, We = Wh = W = 0.045 eV, and N e = ture at T < 200 K. The weak temperature dependence of opt ph ph ph t h × 19 Ð3 the current points to the tunneling mechanism of trap Nt = Nt = 7 10 cm . The currentÐvoltage charac- ionization. The same behavior of the measured currents teristics calculated for temperatures of 77, 300, and with temperature was observed previously in [2, 3]. 410 K are shown in Fig. 2 (solid lines), and temperature The direct tunneling of carriers through the triangu- dependences of the current for voltages U (44, 37, and lar barrier without the participation of phonons is the 30 V) on the aluminum electrode are shown in Fig. 3 main mechanism of trap ionization at low temperatures. (solid lines). The largest discrepancy between the cal- The ionization probability in the presence of an electric culated and experimental data was observed for volt- field is given by the equation [15] ages less than 35 V. This discrepancy can be explained by slow current relaxation in silicon nitride, whose ∗ eF 4 2m 3/2 nature remains unclarified [16]. The value Wopt = 2.5 eV P = ------exp Ð------W . (7) ∗ 3 បeF opt estimated by the slope of the currentÐvoltage character- 22m Wopt istic (Fig. 4) turned out to be smaller than the value

JETP LETTERS Vol. 77 No. 7 2003 388 NASYROV et al.

Wopt = 2.7 eV obtained from an accurate simulation of REFERENCES the experiment (Figs. 2, 3). This small discrepancy can 1. V. A. Gritsenko, Atomic and Electronic Structure of be explained by the fact that the ionization probability Amorphous Insulators in Silicon MIS Devices (Nauka, of a trap was estimated with the use of the average value Novosibirsk, 1993). of the electric field (F = V/d) in the first case and with 2. S. M. Sze, J. Appl. Phys. 18, 2951 (1967). the use of the local (depending on the coordinate) elec- 3. V. A. Gritsenko and A. V. Rzhanov, Zh. Tekh. Fiz. 46, tric field obtained as an accurate solution of the Poisson 2155 (1976) [Sov. Phys. Tech. Phys. 21, 1263 (1976)]. equation taking into account the nonuniformity of the 4. R. A. Williams and M. M. E. Beguwala, IEEE Trans. trapped charge in the bulk of silicon nitride in the sec- Electron Devices 25, 1019 (1978). ond case. 5. S. Manzini, J. Appl. Phys. 62, 3278 (1987). 6. V. A. Gritsenko, E. E. Meerson, I. V. Travkov, et al., Mik- Thus, the experiment on charge transfer in silicon roélektronika 16 (1), 42 (1987). nitride carried out over a wide range of electric fields 7. H. Bachhofer, H. Reisinger, E. Bertagnolli, et al., and temperatures is quantitatively described by the the- J. Appl. Phys. 89, 2791 (2001). ory of the multiphonon ionization of traps. An agree- 8. K. A. Nasyrov, V. A. Gritsenko, M. K. Kirn, et al., IEEE ment with the experiment was obtained for the bipolar Electron Device Lett. 23, 336 (2002). model of conductivity with the same parameters (con- 9. Ya. I. Frenkel’, Zh. Éksp. Teor. Fiz. 8, 1292 (1938). centration, capture cross section, and optical and ther- 10. V. A. Gritsenko, E. E. Meerson, and Yu. N. Morokov, mal ionization energies) of electron and hole traps in Phys. Rev. B 57, R2081 (1998). silicon nitride. 11. V. N. Abakumov, V. I. Perel’, and I. N. Yassievich, Non- radiative Recombination in Semiconductors (S.-Peterb. The large difference found between the thermal and Inst. Yad. Fiz. Ross. Akad. Nauk, St. Petersburg, 1997). optical ionization energies within the theory of mul- 12. G. V. Gadiyak, M. S. Obrekht, and S. P. Sinitsa, Mik- tiphonon ionization is evidently due to the occurrence roélektronika 14, 512 (1985). of a strong polaron effect in Si N . Previously, the 13. F. L. Hampton and J. R. Cricchi, Appl. Phys. Lett. 35, 3 4 802 (1979). polaron model of electron and hole traps in Si N was 3 4 14. S. S. Makram-Ebeid and M. Lannoo, Phys. Rev. B 25, discussed in [1, 17, 18]. According to this model, elec- 6406 (1982). trons and holes in Si3N4 are captured by a minimal sili- 15. S. D. Ganichev, I. N. Yassievich, and V. Prettl, Fiz. con cluster, namely, a SiÐSi bond. The polaron model Tverd. Tela (St. Petersburg) 39, 1905 (1997) [Phys. Solid suggests that a SiÐSi bond or a silicon cluster composed State 39, 1703 (1997)]. of several silicon atoms is a deep center for electrons, a 16. V. A. Gritsenko, E. E. Meerson, and S. P. Sinitsa, Phys. deep center for holes, and a recombination center. A Status Solidi A 48, 31 (1978). quantum-chemical simulation of a SiÐSi bond in Si N 17. V. A. Gritsenko and P. A. Pundur, Fiz. Tverd. Tela (Len- 3 4 ingrad) 28, 3239 (1986) [Sov. Phys. Solid State 28, 1829 carried out in [19] qualitatively confirms this hypothe- (1986)]. sis. 18. P. A. Pundur, J. G. Shvalgin, and V. A. Gritsenko, Phys. This work was supported by the National Program Status Solidi A 94, K701 (1986). of the Korean Ministry of Science and Technology on 19. V. A. Gritsenko, H. Wong, J. B. Xu, et al., J. Appl. Phys. Terabit-Scale Nanoelectronics and by the Siberian 86, 3234 (1999). Branch of the Russian Academy of Sciences, integration project no. 116. Translated by A. Bagatur’yants

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JETP Letters, Vol. 77, No. 7, 2003, pp. 389Ð392. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 459Ð463. Original Russian Text Copyright © 2003 by Shamirzaev, Gilinsky, Bakarov, Toropov, Ténné, Zhuravlev, Borczyskowski, Zahn.

Millisecond Photoluminescence Kinetics in a System of Direct-Bandgap InAs Quantum Dots in an AlAs Matrix T. S. Shamirzaev*, A. M. Gilinsky*, A. K. Bakarov*, A. I. Toropov*, D. A. Ténné*, K. S. Zhuravlev*, C. von Borczyskowski**, and D. R. T. Zahn** * Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected] ** Institut für Physik, TU Chemnitz, D-09107 Chemnitz, Germany Received March 4, 2003

Anomalously long millisecond kinetics of photoluminescence (PL) is observed at low temperatures (4.2Ð50 K) in direct-bandgap InAs quantum dots formed in an AlAs matrix. An increase in temperature leads to a decrease in the duration of PL decay down to several nanoseconds at 300 K, whereas the integral PL intensity remains constant up to 210 K. In order to explain the experimental results, a model is proposed that takes into account the singletÐtriplet splitting of exciton levels in small quantum dots. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 78.55.Cr; 78.67.Hc

The relaxation and recombination of excitons in taxy on semi-insulating GaAs substrates with (100) ori- self-assembling InAs quantum dots have been actively entation using a Riber-32P setup. Samples consisted of studied in recent years, because this system is promis- five layers of InAs quantum dots separated with AlAs ing for creating high-performance light-emitting layers 8-nm thick. The amount of InAs deposited in the devices [1]. Most attention has been concentrated on growth of each layer with quantum dots was equivalent quantum dots formed in a GaAs matrix [2, 3]. It has to two monolayers. The layers with quantum dots were been stated that the radiative recombination time of grown at a temperature of 480°C. A detailed description excitons in this system comprises several nanoseconds of the growth process is given in [5]. The sizes of quan- [2]. At the same time, the recombination dynamics of tum dots were estimated by electronic images of cross excitons in a system of InAs quantum dots formed in a sections of samples and varied from 60 to 240 Å in the wide-bandgap AlAs matrix has remained poorly under- plane perpendicular to the growth direction and from stood. Only one work devoted to studying the transient 20 to 60 Å in the growth direction. As distinct from photoluminescence (PL) of InAs quantum dots in an InAs quantum dots in a GaAs matrix, InAs quantum AlAs matrix has been published to date [4], whose dots in an AlAs matrix form into a system with the dot 11 Ð2 authors note that photoluminescence in this system at density NQDs higher than 10 cm , and a decrease in liquid-helium temperature demonstrates microsecond growth temperature leads to an increase in density and decay times. a decrease in dot sizes [5Ð8]. Because the growth tem- This work is devoted to studying the temperature perature of our samples was lower than that in [7], dependence of the transient and steady-state PL of InAs whose authors report obtaining samples with NQDs = quantum dots formed in an AlAs matrix. We have 7 × 1011 cmÐ2, we believe that the density of quantum shown for the first time that the radiative recombination dots in the system under study is at least not lower. of excitons at low temperatures (4.2Ð50 K) is character- Transient and steady-state PL was measured. The ized in this system by an anomalously long, millisec- excitation of PL was carried out by Ar+, Ti : Sapphire, ond decay time. With increasing temperature, the dura- and semiconductor laser radiation, and the exciting tion of PL decay decreases by five orders of magnitude, photon energy was both larger and smaller than the reaching several nanoseconds at 300 K. The experi- ()AlAs mental results have been explained within a model tak- bandgap of the AlAs matrix Eg . Steady-state PL ing into account the singletÐtriplet splitting of exciton was excited by Ar+ laser radiation (hν = 2.54 eV > levels in small quantum dots arranged in a system of AlAs ν closely packed quantum dots with local coupling Eg ) or by a semiconductor laser with h = 1.82 eV < between dots. AlAs Eg . The excitation of transient PL was carried out by Structures with self-assembled InAs quantum dots rectangular pulses of the semiconductor laser (hν = in an AlAs matrix were grown by molecular-beam epi- 1.82 eV) or 200-fs pulses of a Ti : Sapphire laser (hν =

390 SHAMIRZAEV et al.

Fig. 2. Temperature dependence of the maximum position of the line caused by recombination in InAs/AlAs quantum Fig. 1. PL spectrum of InAs/AlAs quantum dots at 4.2 K. dots. The solid line is a guide to the eye.

AlAs 3.10 eV > Eg ) operating in the mode locking regime. energy of both larger and smaller than the bandgap The signal of steady-state and transient PL in the micro- width of the AlAs matrix; (2) the kinetics is of nonex- and millisecond time range was recorded by a cooled ponential character throughout the entire temperature photomultiplier with an S20 type photocathode operat- range and is described by a power law I(t, T) ~ (1/t)γ(T); ing in the photon counting regime. The measurement of and (3) the duration of PL decay down to an intensity of PL kinetics in the submicrosecond time range was car- 2.5 × 10Ð3 of the initial magnitude is independent of ried out by a Hamamatsu C4334 streak camera (time temperature in the range 4.2Ð50 K and equals at least resolution, 15 ps) mounted on a CROMEX 250IS spec- 2 ms, as illustrated in Fig. 4. With increasing tempera- trometer. ture, the duration of decay decreases by ﬁve orders of It is seen in Fig. 1 that a line positioned at an energy magnitude and becomes equal to 25 ns at 300 K. of 1.684 eV and due to radiative recombination in quan- It is evident in Fig. 4 that the integrated intensity of tum dots dominates in the steady-state PL spectrum of PL remains constant in the temperature range 4.2Ð210 K, the structure with InAs/AlAs quantum dots measured at which indicates that the sharp change in decay time liquid-helium temperature with an excitation energy of with increasing temperature is not associated with the 2.54 eV and an excitation density of 2 W cmÐ2. An inclusion of additional channels of nonradiative recom- increase in excitation intensity leads to an increase in bination. A further increase in temperature up to 300 K the half-width and to a short-wavelength shift of the leads to a decrease in integral intensity by a factor of line. Similar behavior of the line generated by recombi- 15 due to the thermal activation of charge carriers into nation in InAs/AlAs quantum dots was observed previ- the wetting layer and/or the AlAs matrix [6, 9]. ously in [6]. This behavior is due to the redistribution of The authors of [4] observed microsecond decay of charge carriers among quantum dots of different size in the low-temperature (6 K) PL of InAs/AlAs quantum the system of closely packed, coupled quantum dots. dots upon excitation by light with a photon energy The temperature dependence of the position of the exceeding the bandgap width of the AlAs matrix. In the PL line maximum shown in Fig. 2 demonstrates a non- model of recombination proposed in [4], it was sug- monotonic character. The maximum position remains gested that the long decay of PL is explained by the spa- constant in the temperature range 4.2Ð50 K; as the tem- tial separation of charge carriers of different sign perature increases from 50 to 150 K, the line shifts between neighboring quantum dots, which occurs toward the short-wavelength side; and with a further because of electron scattering by the states of the X val- increase in temperature, the line exhibits a long-wave- ley of the AlAs matrix in the process of electron energy length shift. relaxation. The photoluminescence kinetics of InAs/AlAs The set of experimental data obtained in this work quantum dots at various temperatures is shown in cannot be explained within the model proposed in [4]. Fig. 3. We note the following features of PL kinetics: Actually, upon exciting PL by light with a photon (1) the shape of a kinetic curve and the duration of PL energy lower than the bandgap width of the matrix, the decay are independent of the exciting light photon nonequilibrium charge carriers are excited inside a energy and are similar for excitation with a photon quantum dot and cannot reach the X valley of AlAs;

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MILLISECOND PHOTOLUMINESCENCE KINETICS IN A SYSTEM 391 hence, the mechanism proposed in [6] is not realized. Moreover, this model does not contain temperature- dependent parameters and cannot explain the decrease by five orders of magnitude in the PL decay time with increasing temperature observed in our study. In order to explain the millisecond PL kinetics observed in this work we took into account the fine structure of exciton levels strongly localized in quantum dots [10]. An exciton as a system of two paired spins can exist either in an optically active singlet state τ with a short radiative lifetime C or in an optically inactive triplet state, which is characterized by a long life- τ ӷ τ time T C. Because of exchange interaction, the singlet and triplet exciton levels are split. The splitting ∆ between the ground-state triplet and higher lying singlet exciton levels in three-dimensional semiconductor materials and large quantum dots is smaller than kT Fig. 3. PL kinetics of InAs/AlAs quantum dots at various even at liquid-helium temperature. In this case, the temperatures (8, 150, and 300 K, curves 1Ð3, respectively). kinetics of exciton PL is exponential and is character- Curves 2 and 3 were measured upon exciting PL by light τ ized by the lifetime C. However, for excitons strongly with a phonon energy (3.10 eV) larger than the bandgap of localized in small quantum dots embedded in a wide- the AlAs matrix. Curve 1 consists of three parts marked in the figure with arrows. In section (a) measurements were bandgap matrix, as takes place for InAs quantum dots made by a streak camera, and PL was excited by light with in an AlAs matrix, the splitting ∆ strongly increases a photon energy (3.10 eV) larger than the bandgap width of [11, 12]. In such a dot, in the absence of competing the AlAs matrix. In sections (b) and (c), PL was measured channels of charge carrier escape from the dot, the radi- using a system of time-correlated photon counting, and PL ∆ was excited by light with a photon energy (1.82 eV) smaller ative lifetime of an exciton at low temperatures kT < than the bandgap width of the AlAs matrix. The dashed is determined by the exciton lifetime in the optically lines are guides to the eye. τ inactive triplet state T. In a system of isolated quantum dots with a spread of sizes, the dispersion of ∆ values leads to a spectral dependence of the PL decay time [11]. A completely different pattern is observed in a system of locally coupled InAs quantum dots in an AlAs matrix. Because the distance between quantum dots is small, the wave func- (s) tions of charge carriers in neighboring quantum dots strongly overlap; moreover, the potential barrier between dots can decrease locally [4, 7]. This leads to the migration of excitons from small dots to large ones. In this case, large dots with a small value of ∆ and, hence, a short lifetime provide a channel for exciton recombination. At the same time, small dots with a large value of ∆ and a long lifetime serve as a reservoir from which excitons find their way into large dots. In such a system of coupled quantum dots, the duration of PL decay in large dots is determined by the time during which excitons find their way from small dots into large dots rather than the radiative recombination time of Fig. 4. Temperature dependences of (1) the duration of PL decay down to an intensity comprising 2.5 × 10Ð3 of the ini- excitons in a dot. It is evident that this time coincides in tial value and (2) the PL integrated intensity of InAs/AlAs the order of magnitude with the lifetime of excitons in quantum dots. Solid lines are guides to the eye. small dots. Because the lifetime values in dots of different size are randomly distributed over a wide range τ τ from C to T, the kinetic curve becomes nonexponential an increase in the probability of exciton transfer in shape. The power law of decay observed in this work between coupled dots and, simultaneously, to a is typical for decay processes characterized by a large decrease in the lifetime of excitons in small dots due to set of characteristic lifetimes and was observed previ- thermal activation from the triplet state to the singlet ously for the recombination of excitons localized at state. At high temperatures, when the condition kT > ∆ composition fluctuations in indirect-bandgap AlGaAs is fulfilled in small dots, the PL decay time decreases τ solid solutions [13]. An increase in temperature leads to down to the value of C. The decrease in PL decay time

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392 SHAMIRZAEV et al. in the samples under study starts at temperatures This work was partially supported by grants of exceeding 50 K; hence, the magnitude of singletÐtriplet Volkswagen Foundation, project no. I/76 837; the Rus- splitting must comprise at least several millielectron sian Foundation for Basic Research, project no. 01-02- volts. Using the theory developed in [14] for crystals of 16969; and the Ministry of Industry, Science, and Tech- Td symmetry without regard for the long-range part of nology of the Russian Federation. exchange interaction, we estimated the splitting ∆ in small dots. Because InAs dots have a shape differing from a spherical one considered in [14], in our calcula- REFERENCES tions we used the radius of a sphere with a volume 1. Y. Arakawa and H. Sakaki, Appl. Phys. Lett. 40, 939 equal to the volume of a real dot as the dot radius R. The (1982). obtained splitting ∆ = 4.6 meV agrees well with our 2. R. Heitz, A. Kalburge, Q. Xie, et al., Phys. Rev. B 57, experimental data. 9050 (1998). The model of locally coupled quantum dots also 3. P. D. Buckle, P. Dawson, S. A. Hall, et al., J. Appl. Phys. provides an explanation for the temperature depen- 86, 2555 (1999). dence of the PL line position. An increase in tempera- 4. P. Dawson, Z. Ma, K. Pierz, and E. O. Gobel, Appl. Phys. ture results in an increase in the population of the opti- Lett. 81, 2349 (2002). cally active singlet state and, hence, in an increase in 5. D. A. Tenne, A. K. Bakarov, A. I. Toropov, and the probability of exciton recombination in small dots. D. R. T. Zahn, Physica E (Amsterdam) 13, 199 (2002). This process leads to a redistribution of excitons among 6. U. H. Lee, D. Lee, H. G. Lee, et al., Appl. Phys. Lett. 74, dots of different size. The short-wavelength shift of the 1597 (1999). PL line observed when the temperature increases in the 7. Z. Ma, K. Pierz, and P. Hinze, Appl. Phys. Lett. 79, 2564 range 50Ð150 K is thus due to an increase in the fraction (2001). of excitons recombining in the dots of smaller size. The 8. P. Ballet, J. B. Smathers, H. Yang, et al., J. Appl. Phys. long-wavelength shift of the line observed when the 90, 481 (2001). temperature increases above 150 K is due to a decrease 9. L. Zang, T. F. Boggess, D. G. Deppe, et al., Appl. Phys. in the bandgap of InAs and AlAs [6]. Lett. 76, 1222 (2000). Thus, the transient and steady-state PL of self- 10. H. Fu, Lin-Wang Wang, and A. Zunger, Phys. Rev. B 59, assembled InAs quantum dots in an AlAs matrix has 5568 (1999). been studied in this work. Anomalously long millisec- 11. P. D. J. Calcott, K. J. Nash, L. T. Canham, et al., J. Phys.: ond nonexponential PL decay has been observed at low Condens. Matter 5, L91 (1993). temperatures. The duration of decay signiﬁcantly 12. Al. L. Efros, M. Rosen, M. Kuno, et al., Phys. Rev. B 54, decreases as the temperature increases above 50 K and 4843 (1996). reaches a value of several nanoseconds at 300 K. The 13. M. V. Klein, M. D. Sturge, and E. Cohen, Phys. Rev. B experimental results have been explained within the 25, 4331 (1982). model of locally coupled quantum dots. The long decay 14. S. V. Gupalov and E. L. Ivchenko, Fiz. Tverd. Tela at low temperatures is due to the singletÐtriplet splitting (St. Petersburg) 42, 1976 (2000) [Phys. Solid State 42, of exciton levels in small quantum dots. 2030 (2000)]. The authors are grateful to Dr. A.V. Efanov for fruitful discussion of the results. Translated by A. Bagatur’yants

JETP LETTERS Vol. 77 No. 7 2003

JETP Letters, Vol. 77, No. 7, 2003, pp. 393Ð396. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 7, 2003, pp. 464Ð467. Original English Text Copyright © 2003 by Lesovik, Chtchelkatchev.

Quantum and Classical Binomial Distributions for the Charge Transmitted through Coherent Conductor¦ G. B. Lesovik1 and N. M. Chtchelkatchev1, 2 1 Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, 117940 Russia e-mail: [email protected] 2 Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow region, 142092 Russia e-mail: [email protected] Received February 27, 2003

We discuss controversial results for the statistics of charge transport through coherent conductors. Two distribution functions for the charge transmitted was obtained previously, one actually coincides with classical binomial distribution, the other is different, and we call it here quantum binomial distribution. We show that high- order charge correlators, determined by the either distribution functions, all can be measured in different setups. The high-order current correlators, starting with the third order, reveal (missed in previous studies) special oscillating frequency dependence on the scale of the inverted time ﬂight from the obstacle to the measuring point. Depending on setup, the oscillating terms give substantially different contributions. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 05.60.Gg; 72.10.Bg

In last years, new direction has appeared in quantum distribution function [2], the following definition was transport investigations—description of the statistics of adopted: a charge transmitted through a quantum conductor. The t0 distribution functions are usually investigated for a  χλ 〈〉{}λ ˆ λ ˆ charge Q transmitted during a large interval of time ()==exp i Q()t0 exp i ∫dtI()t (1) t0  through a certain cross-section of a conductor, and, 0 what is essential, all observables are calculated from (here, 〈…〉 denotes the ensemble averaging). This defi- the first principles. Despite of the appreciable amount nition is the most direct generalization of the classical of articles (see the review [1]) and obtained results, definition; it differs only in the replacement of the some questions, in particular concerning the measure- charge and current observables by the appropriate oper- ment theory, still remain unclear. One of the problems ators. Performing calculations, we considered large 〈〉〈〉ˆ n is that (in contrast to classical) the question arises in the time intervals at which the correlators Q()t0 = quantum case of how to define the observable which 〈〉〈〉t0 … t0 ˆ …ˆ ∫ dt1 ∫ dtn I()t1 I()tn are approximately equal to should be calculated. Technically, this uncertainty is 0 0 n n connected with the noncommutativity of the current 〈〉〈〉Qˆ ()t = t 〈〉〈〉Î , (2) operators at different times. As it appears, it is possible 0 0 0 n to present several definitions for the distribution func- where 〈〉〈〉Î0 0 is the irreducible current correlator of tion (DF) and characteristic function (CF) which (a) the nth order at zero frequency limit (the irreducible coincide in the classical case, (b) satisfy some general correlators satisfy the equation 〈〉exp{}iλQˆ ()t = n principles (in particular, correlators 〈〉Q prove to be ∞ n n t0 exp{}∑ ()iλ 〈〉〈〉Qˆ ()t /n! ). real), although lead to different answers in quantum n = 1 case. It is possible to understand which definition is The method of calculation used in [2] can be gener- “correct” only after analysis of a definite setup for (at alized to the case of finite temperatures, and (for normal least gedanken) measurement. single-channel conductors) we find  ⑀ In this paper, we consider several variants of mea- χλ d {()() ()= expt0g∫------ln 1 Ð nL 1 Ð nR + nLnR surements and the corresponding CF definitions. We  2πប will also comment the results obtained earlier. In the (3) first paper devoted to the microscopic description of the  ()χλ χ λ } ---+ nL 1 Ð nR ⑀()+ nR()1 Ð nL ⑀()Ð , ¦ This article was submitted by the authors in English. 

394 LESOVIK, CHTCHELKATCHEV where χ⑀()λ = cos{}λeT()⑀ + i T()⑀ sin{}λeT()⑀ , Since it was not quite clear how one can measure quan- g = 2 is the factor taking into account spin degeneracy, tum binomial DF and correlators of the type given in T is the transparency, and nL, R are the filling factors in Eq. (4), it was suggested in [4] that a spin located near the left and right reservoirs, respectively. This distribu- a wire can be used as the counter of electrons passed ≠ through it. Thus, it appears that the definition for CF tion (at kBT = 0, eV 0) will be called quantum binomial distribution. In the multichannel case, χ(λ) is the in this case differs from (1) by the presence of time Π χ λ χ λ ordering: product n n( ) of the characteristic functions n( ) t t corresponding to the transmission eigenvalues Tn. The 0 0 distribution function (3) formally describes fractional χλ()= 〈〉T˜ expiλ/2 Î()x dtTexp iλ/2 Î()x dt (6) charge transport. In the indirect measurements of a ∫ ∫ charge in the solid-state systems, the fractional charge 0 0 may, in principle, appear, for example, in the shot noise in this (and only in this) formula, the symbol T means under the conditions of fractional quantum Hall effect the usual time ordering, and T˜ means the ordering in [3]. In our case, the size “of a charge quantum” 2e T the opposite direction. is determined by the eigenvalue of the current operator, It was found in [4, 5] (see also references in [1]) for provided that it acts on the subspace of one-particle the third-order correlator of the transmitted charge that ± 〈〉± ˆ ± excitations with the given energy e T = I()t0 , 2eVt |±〉 〈〉〈〉Q3 = e3T()1 Ð T ()12Ð T ------0. (7) where the normalized one-particle excitations sat- t0 h isfy the condition 〈〉+ Î()Ðt = 0; see also [2]. If we 0 As we see, the third-order correlators (4) and (7) are take into account the logarithmic, in time t0, corrections essentially different. It would seem that there is nothing to the irreducible correlators, then the exact charge unexpected in such a distinction, because definitions quantization, which follows from the discrete distribu- (1) and (6) are different. However, for example, the tion function, will be replaced, apparently, by small third-order charge correlator, according to both defini- modulations in the continuous distribution function. If tions (1) and (6), actually contains the current correlator we restrict ourselves to the corrections (occurring from at small frequencies; but such a current correlator can be the vacuum fluctuations) to the pair correlator, then the calculated with the help of the first definition (1) cor- distribution function becomes rectly in the zero-frequency limit (5), and it can be checked independently using the same technique as was 1/2 ()0 2π used in [6] for the calculation of the pair correlator [7]. PQ()= ∑P ()ne T ------ប {}ω It appears that the dispersion of the third-order current G ln t0 n correlator at small frequencies (along with the differ- 2 × exp{}Ð()QneTÐ /2Gបln{}t ω , ence in the definitions) leads to different answers for 0 the third-order correlators. Really, at frequencies ω Ӷ eV/ប and x > 0, we have where P(0)(ne T ) is the discrete distribution function 1, 2, 3 〈〉〈〉πδ() ω ω ω without logarithmic corrections, G is the conductance, Iω ()x1 Iω ()x2 Iω ()x3 = 2 1 ++2 3 ω 1 2 3 and is a characteristic frequency scale of the conduc- ω v ω v ω v × ()i 1 x1/ F ++i 2 x2/ F i 3 x3/ F tance dispersion. T 1 Ð T e (8) ω 3 Using Eq. (3), we find for the third-order correlator: Ð2i 2 x2/v F 2e × []12Ð T Ð e eV------. ⑀ h 〈〉〈〉ˆ 3 d 3 2() Qt0 = Ðt0g∫------e T nL Ð nR 2πប For x1 = x2 = x3 = x, 2 × []{}[]() () 〈〉〈〉Iω ()x Iω ()x Iω ()x = 2πδ() ω ++ω ω 3 nL 1 Ð nR + nR()1 Ð nL Ð 1 Ð 2TnL Ð nR . 1 2 3 1 2 3

ω 3 (9) At small voltage (and for energy-independent trans- Ð2i 2 x/v F 2e × T()1 Ð T []12Ð T Ð e eV------. parency T), the correlator is proportional to V3, and at h large V we get [2] Formally, assuming that such a frequency dependence

3 3 2 2eVt is correct for all frequencies, we ﬁnd for the correlators 〈〉〈〉Q = Ð2e T ()1 Ð T ------0. (4) in the time representation: t0 h 3 〈〉〈〉, , , 2e () According to Eq. (2), the third-order current correlator It()1 x It()2 x It()3 x = eV------T 1 Ð T at zero frequency is h × ([]δ()δ() (10) Sym 1Ð 2T ti Ð t j t j Ð tk 〈〉〈〉3 3 2()2eV I0 = Ð2e T 1 Ð T ------. (5) h Ð δ()δ˜ti Ð ˜t j ())˜t j Ð ˜tk ,

JETP LETTERS Vol. 77 No. 7 2003 QUANTUM AND CLASSICAL BINOMIAL DISTRIBUTIONS 395 where the symbol Sym means symmetrization in the ated at the distance d from the wire and L from the scat- ≠ ≠ ˜ ≡ ˜ ˜ terer is larger than the decay time τ of the correlators. indices i j k; t2 t2 + 2x/vF, t1 = t1, t3 = t3 (if a real 0 frequency dependence is taken into account instead of If these requirements are violated, the answer will be the δ functions, there should stand functions which different. In the calculations described in [4], although τ 1 it was formally assumed that the distance L is zero, the decay with characteristic times t ~ 0, but, for simplic- δ limit was considered where L actually exceeded the ity, we will describe the case with functions). Substi- wave-packet size (which was also set equal zero). tuting this expression into the expression for the third- For the case where the spin detector is located near order charge correlator which follows from Eq. (6), we Ӷ v τ Ӷ v τ get the scatterer x F 0 and close to the wire d F 0, 〈〉〈〉3 2 the answer for Qt is proportional to ÐT (1 Ð T) and T T t2 T t2 T 0 3 3 it coincides with the answer (4) obtained from the quan- 〈〉〈〉Q = --- dt dt dt + dt dt dt T 4 ∫ 1∫ 2∫ 3 ∫ 2∫ 1∫ 3 tum distribution. Indeed, using the general expression 0 0 0 0 0 0 Ӷ v τ (11) for the correlator (8) at x1, 2, 3 F 0 we get T t1 t2 T t3 t2 〈〉〈〉Iω ()x Iω ()x Iω ()x 〈〉〈〉 1 1 2 2 3 3 + ∫dt1∫dt2∫dt3 + ∫dt3∫dt2∫dt1 It()1 It()2 It()3 , 3 (13) 2 2e 0 0 0 0 0 0 Ӎ Ð2πδ() ω ++ω ω 2T ()1 Ð T eV------. 1 2 3 h and, integrating over times, we find for finite x the answer (7) proportional to T(1 Ð T)(1 Ð 2T), which is Using this expression at ω Ӷ eV/ប and the definition typical of the classical binomial distribution (the same (11), we obtain the expression for 〈〉〈〉Q3 proportional phenomenon takes place for other irreducible high- t0 order correlators). to ÐT2(1 Ð T), as well as in the calculation with the use This occurs because the terms in the Eq. (10) con- of CF (1). δ ˜ The measurement with spin basically can be imple- taining functions depending on ti do not make contri- mented in practice with the help of muons, which can bution to the answer, because they are nonzero only be trapped near the conductor, and then the measure- when, simultaneously, t3 > t2 and t1 > t2 and the integra- ment of the direction of their decay would give the tion volume in Eq. (11) does not cover this sector. Con- angle of their spin rotation in a magnetic field. As an 〈〉〈〉3 additional example of the measuring procedure that, sider, e.g., the contribution to the correlator QT from the term in Eq. (10) proportional to basically, can be implemented practically, we have analyzed the measurement of the irreducible charge corre- δ ˜ ˜ δ()δ˜ ˜ ()δ()lators with the help of an ammeter represented by a ()t1 Ð t3 t3 Ð t2 = t1 Ð t3 t3 Ð2t2 Ð x/v F . (12) semiclassical system (for example, an oscillatory cir- ≈ ≈ From Eq. (12) it follows that the region where t1 t3 cuit) weakly interacting with the current in a quantum t2 + 2x/vF in Eq. (11) should give the leading contribu- conductor. The ammeter state is characterized by the tion to the integrals. But from the requirement x > 0 it magnitude φ. The interaction of the ammeter with a follows that t3 > t2 and t1 > t2. The region defined by quantum conductor is described by the interaction λφˆ λ these inequalities does not overlap with the integration Hamiltonian Hi = I()t , where is the interaction 〈〉〈〉3 volume in Eq. (11); thus, contribution (12) to QT is constant, Î()t is the current operator in a quantum con- zero. Similarly, it is possible to show that all terms pro- ductor, and Î()t = ∫Î()tx, fx()dx (we do not take into portional to δ()˜t Ð ˜t δ()˜t Ð ˜t in Eq. (10) do not make i j j k account the retardation effects); thus, the area of inte- 〈〉〈〉3 contribution to QT at x > 0. gration is determined by some kernel f(x). Correlators φ are expressed through the current correlators in a One can say that, when the incident wave packet quantum conductor as follows: first completely passes through the detector and, with a time delay, a part of the wave packet reflected from the λ n barrier goes back through the detector, the specific 〈〉〈〉()φ n  τ …τκ()τ ()t = Ð--- ∫d 1 d n t Ð 1 quantum interference disappears and the answer (7) is 2 true. But if the distance to the detector is small, the inci- c (14) × ()…κτ ()τ ()τ dent wave packet interferes with the reflected wave in sgn t Ð 1 t Ð n sgn t Ð n the measurement region, leading eventually to the × 〈〉〈〉T I()τ …I()τ , answer (4). So, Eq. (7) is true when the electron-flight c 1 n time from the scatterer to the spin of the detector situ- where the integration is performed along the usual Keldysh contour, and κ(τ) is the ammeter susceptibility. 1Characteristic time scale of the correlator decay can be estimated In the specific case where the ammeter represents an τ ប as 0 ~ /eV. We will consider the time dependence of the correlators in detail in a separate article. oscillator, the equation of motion for φ is úúφ + γφú +

JETP LETTERS Vol. 77 No. 7 2003 396 LESOVIK, CHTCHELKATCHEV

Ω2φ = λI(t)/M; the susceptibility κ(τ) = We are grateful to M. Reznikov and, especially, to ˜ ˜ ˜ 2 2 D. Ivanov for fruitful discussions. D. Ivanov paid our Θ(t)exp(−γt/2)sin(Ωt )/MΩ, where Ω = Ω Ð γ /4 . attention to the special coordinate and frequency γ ≈ Ω γ Ӷ τ The case 2 , 1/ 0 is the most interesting to us. dependence of the third-order correlators which proved Then, to be important for reviewing various conditions of measurement. We are also grateful to M. Feigelman for dω ,, 〈〉〈〉φ3 ≈ 〈〉λ〈〉3 3 123κω κω κω reading manuscript and useful remarks. ()0 I0 ∫------()1 ()2 ()3 ()2π 3 (15) This work was supported by the Russian Science 〈〉λ〈〉3 3 Support Foundation, the Russian Foundation for Basic ×δ() ω ω ω 2 I0 1 ++2 3 = ------3 4 -, Research, the Russian ministry of science (the project 27 M Ω “Physics of quantum computations”), and SNF (Swit- zerland). 〈〉λ〈〉n n n n! I 〈〉〈〉φ ()0 ≈ ------0 -, (16) nn + 1 MnΩn + 1 REFERENCES 〈〉〈〉n where I0 is the irreducible current correlator of 1. L. S. Levitov, cond-mat/0210284. order n, deﬁned by the quantum binomial distribution [in Eq. (16), we neglected the contribution from the 2. L. S. Levitov and G. B. Lesovik, Pis’ma Zh. Éksp. Teor. own thermal ammeter noise]. Fiz. 55, 534 (1992) [JETP Lett. 55, 555 (1992)]. Thus, by measuring irreducible correlators of coor- 3. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 72, 724 dinate φ of the ammeter, it is possible to measure the (1994); L. Saminadayar, D. C. Glattli, Y. Jin, et al., Phys. irreducible high-order current correlators in the zero- Rev. Lett. 79, 2526 (1997); R. de-Picciotto, M. Rezni- kov, M. Heiblum, et al., Nature 389, 162 (1997). frequency limit (in particular, 〈〉〈〉I3 ∝ ÐT2(1 Ð T)). 0 4. L. S. Levitov and G. B. Lesovik, cond-mat/9401004; Such measurements are possible also for less restrictive L. S. Levitov, H. W. Lee, and G. B. Lesovik, J. Math. requirements on the ammeter frequencies if the kernel Phys. 37, 4845 (1996). f(x) deﬁnes the area of an integration so that x Ӷ v τ . F 0 5. L. S. Levitov and G. B. Lesovik, Pis’ma Zh. Éksp. Teor. In the opposite limit, it is natural to expect “classical” Fiz. 58, 225 (1993) [JETP Lett. 58, 230 (1993)]. 〈〉〈〉3 ∝ answers for the correlators (in particular, I0 6. G. B. Lesovik, Pis’ma Zh. Éksp. Teor. Fiz. 49, 513 T(1 Ð T)(1 Ð 2T)). (1989) [JETP Lett. 49, 592 (1989)].

JETP LETTERS Vol. 77 No. 7 2003