CHAPTER 3 Non-Ohmic Microwave Hopping Conductivity Yu.M. GAL PERIN, V.L. GUREVICH A.F. Ioffe Institute of the Academy of Sciences, Leningrad, USSR and D.A. PARSHIN Technical University, Leningrad, USSR cg Hopping Transport in Solids Edited by M. Pollak and B. Shklovskii Elsevier Science Publishers B.v., 1991 81 Contents 1. Introduction .......................................................................................................................................... . 2. Linear (Ohmic) microwave hopping conductivity .................................................................. .. 2.1. The resonant absorption ......................................................................................................... . 2.2. Population relaxation time .................................................................................................... .. 2.3. The non-resonant (relaxational) absorption ..................................................................... . 2.4. Magnetic field effects ................................................................................................................ .. 3. Non-linear resonant microwave hopping conductivity ......................................................... .. 3.1. Qualitative considerations ...................................................................................................... . 3.2. Basic equations ........................................................................................................................... . 3.3. Intensity dependence of absorption (small amplitudes) ............................................... . 3.4. The width of a burned hole ................................................................................................... .. 3.5. The region of high intensities (F ~ F,) ................................................................................ 3.6. Concluding remarks .................................................................................................................. . 4. Non-linear relaxational (non-resonant) absorption ............................................................... .. 4.1. Qualitative picture .................................................................................................................... .. 4.2. The general theory of non-linear absorption .................................................................. .. 4.3. Small amplitudes ....................................................................................................................... .. 4.4. Magnetic field effects ................................................................................................................ .. 5. Conclusions ........................................................................................................................................... . References .................................................................................................................................................... .. 82 83 84 84 86 89 92 94 96 102 103 105 107 109 110 110 114 116 119 121 122 1. Introduction Non-linear phenomena are of principal importance in the study of microwave hopping conductivity. On the one hand, the non-Ohmic behaviour of the conductivity can manifest itself at extremely low intensities of the microwave field, so that in some situations one should take special care to observe the linear regime. On the other hand, study of the non-linear regime may provide valuable information concerning the structure of the impurity band and the rates of the various relaxational processes. As was shown by Pollak and Geballe (1961), microwave hopping conductivity of a doped semiconductor is determined by transitions in the so-called close pairs. Each such pair consists of one impurity centre which is occupied by an electron and one impurity centre which is empty, the distance between them being much smaller than the average distance, f. Here f = (4nN /3) - 1/ 3, (1 ) N being the total impurity density. This so-called two-site approximation will be used throughout this chapter. It is valid provided OJ ~ 0"(0) where OJ is the frequency of the field variation while 0"(0) is the static hopping conductivity (see, e.g., Efros and Shklovskii 1985 and Long 1982). In non-linear absorption the electron occupancies in the pairs are shifted. This shift is equivalent to enhancement of the average energy of bound electrons or, in other words, to a rise in their effective temperature. Thus we are discussing here a typical hot-electron phenomenon. This chapter is organized as follows. First, we shall briefly remind readers about the mechanisms of linear hopping conductivity. We shall give the expressions for the real part of the hopping conductivity, O"(OJ), which is directly related to the absorption coefficient, rx rx == 4nO"/cfi, where c is the velocity of light and e is the dielectric susceptibility. In section 2 much attention will be payed to two aspects. The first is the magnetic field dependence of the absorption. It has been studied experimentally quite recently. The second is the relaxation of the pairs via their interaction with phonons. We wish to emphasize that we need to know this in detail to discuss practically all the non-linear phenomena, as well as some linear phenomena. However, there has been no systematic discussion of this interaction in the various types of semiconductors in a review so far. The two following sections are devoted to a discussion of the two contributions to the non-linear absorption, the resonant and relaxational parts. The influence of the magnetic field on both contributions is also discussed. 83 84 Yu.M. Galperin et al. 2. Linear (Ohmic) microwave hopping conductivity In this section we shall give a brief survey of linear (Ohmic) microwave hopping conductivity. There are several excellent review articles on this topic [see, e.g., Efros and Shklovskii (1985), Long (1982) and the literature quoted therein], so we shall restrict ourselves with some facts we shall need in the following sections. Two problems will be discussed here in some detail. The first is the interaction between close pairs of electron states and thermal phonons. The exact form of this interaction can be important for the description of both linear and nonlinear phenomena. However, we do not know of a complete enough analysis of this problem in the literature. The second problem concerns the influence of magnetic fields on the hopping conductivity. Several semiconductors have large enough values of the Bohr radius, and a sufficiently weak magnetic field can deform the electronic wavefunction. As a result, hopping conductivity becomes dependent on magnetic field. We shall see that investigation of such a dependence can provide an effective tool with which to study relaxation processes within the localized electron system. On the other hand, study of the magnetic field dependence is very useful in the course of investigations of acoustic attenuation. In particular, this helps to discriminate between electronic and non-electronic contributions to the absorption coefficient. 2.1. The resonant absorption The resonant absorption is due to the direct electron transitions between lower and higher levels of a close pair. In the impurity band of a doped semiconductor there is a broad distribution of both the differences of one-site energies ({J = ({J1 - ({J2 and the tunnel integrals A(r) (i.e., the off-diagonal matrix elements describing the tunnelling of an electron between the two sites). Such a situation seems to exist in a number of amorphous semiconductors as well. Thus, for a given w there are a number of pairs with the interlevel spacing (the energy) E=«({J2+A2)1/2, (A = Aoe- r /a ) close to liw. Here Ao is of the order of the electron coupling energy while a is the localization radius of the state. Henceforth such pairs will be called resonant ones. Of all the pairs only those that have only one electron contribute to the absorption. According to Shklovskii and Efros (1981) to calculate the number of such pairs one should take into account the Coulomb repulsion between the electrons. For two electrons to be localized at both sites of the pair with the intersite distance r an additional energy of the order of e2 /er is required. In our review article we, for simplicity, shall consider only the case where this energy is Non-Ohmic microwave hopping conductivity 85 much larger than both flO) and kB T: e2 fer ~ flO), kB T. (2) In this case the linear (Ohmic) microwave hopping conductivity is given by 4 2 (j(res)(O)) = n ag2 e O)r 3 tanh(~). (3) o 3 e w 2kB T Here g is the one-electron density of states while rw=aIOg(~) ( 4) is the minimum 'arm' of the pair with the interlevel spacing flO). (j~es)(O)) appears to be a smooth function of 0) although the contribution of any particular pair with the energy E is a sharp function of the difference E - flO), its width being determined by the electron-phonon relaxation. However, after the summation over all the resonant pairs all the specific information concerning the relaxational processes is integrated out. The factor tanh(flO)l2k BT) has a transparent physical meaning: it is the equilibrium population difference between the lower and the upper levels of the resonant pair. If the microwave amplitude ,.go is large enough this difference decreases and the absorption becomes smaller. As a result, experimental investigation of the non-linear behaviour may give valuable information concerning the non-equilibrium population and relaxation of the resonant pairs with a given energy. We shall discuss two types of such experiments: (i) To discover the intensity dependence of the microwave attenuation coefficient a. * (ii) The so-called hole burning: the frequency dependence of the absorption of a weak probe signal of some frequency 0)1 is studied in the presence of a relatively strong pumping signal of slightly different frequency 0). The pumping signal saturates the population of the resonant pairs which results both in an intensity dependence of its own absorption and the absorption of the probe signal of frequency 0)1' In fact, a 'hole' appears in the energy distribution of the resonant pairs. Both phenomena are due to a group of non-equilibrium pairs, their energies being close to the resonant one, flO). To analyze a non-equilibrium distribution function of the resonant pairs one should know the rate of the relaxation processes, l/r. This will be discussed in the next section. *Biittger and Bryksin (1975) were the first to discuss the intensity-dependent microwave hopping conductivity in semiconductors, though they did not explicitly take into account the Coulomb repulsion of the electrons of one pair. 86 Yu.M. Galperin et al. 2.2. Population relaxation time Let us define the population relaxation time r by the balance equation: on ot n-no ( 5) r where n is the population of the upper level of a pair, no = [1 + exp(E/k8 T)] -1 is the equilibrium value of n. The time r is determined by transitions between the levels of the pair accompanied by emission or absorption of a phonon with energy E. The corresponding expression depends on the pair parameters, E and r, as well as on the temperature T. The dependence on r enters through the factor (A/E)2 [cf. with Bottger and Bryksin (1976)] r(~, r) = (~y rmi:(E) , (6) where rmin(E) is the minimal relaxation time for the pairs with a given value of the interlevel spacing, E. This time is equal to the relaxation time r of a pair with cp = 0, A = E which corresponds to the 'arm' r E = a 10g(Ao/E) of the pair. The origin of the factor (A/E)2 can be described as follows. In the absence of the interaction the Hamiltonian of a pair in the site representation can be written as .it~ = ~(CP (7) 2 A where (Ji are the Pauli matrices. Let us now consider the interaction of a pair with the strain produced by an acoustic wave. Under the influence of a strain the energy of each component of a pair acquires a term of the form (8) Here j = 1, 2 is the number of the component of a pair, uik(rj) is the strain tensor at the point rj while D<j) is the deformational potential tensor (DIP = D/l ik in the isotropic case). Thus the interaction Hamiltonian in the same representation has the form (9) After diagonalization of the Hamiltonian transformed to ~p _ .nint - l(D(1) 2 (1) _ ik Uik D(2) (! (2») E ik Uik (J3 *'0 the interaction Hamiltonian +~ E (J1 ) . '#:nt is (10) Thus we see that the off-diagonal matrix element is proportional to A/ E which Non-Ohmic microwave hopping conductivity 87 results in eq. (6). If the energy E = (<p2 + A2)1/2 is fixed the relaxation rate II, has its maximum at <p = 0, A = E. This is why we denote the corresponding value of, as 'min' To estimate 'min(E) we consider two cases. (a) The deformational potentials of the components of a pair are so different that their difference is of the order of the deformational potentials themselves, a few electron volts, IDI - D21 ~ D 1 , D2 ~ D. This situation can arise in amorphous semiconductors. In this case we have (see Galperin et al. 1983) 1 1 E3 ( E ) 'min(E) = 'o(T) (kB T)3 coth 2kB T . (11) One can easily visualize the physical meaning of this equation. The onephonon transition probability should be proportional to the phonon density of states which, in its turn, is proportional 1'0 w 2 • An additional factor w appears because of the frequency dependence of the interaction matrix element, so that one finally obtains a factor w 3 . Because of the energy conservation E = hw this factor turns into E3. On the other hand, cothCk~T )=coth(2::T )=2N w + 1, where N w is the Planck function. This factor appears because the relaxation can go both via phonon absorption (the probability is proportional to N w) and via phonon emission (the probability is proportional to N w + 1). Now (12) where Ec is some characteristic energy at which the uncertainty hi, in the energy of the upper level (at T = and A = E) reaches a value of the order of the level separation E (Gurevich and Parshin 1982a, b). This energy can be estimated as ° Ec c::: (ph 3 s5 )1/2ID. (13) Here p is the mass density of the semiconductor while s is the sound velocity. Assuming D ~ 1-2 eV, we find Ec to be 10-20 K. For E > Ec the relaxation of the pair under consideration can no longer be described by the one-phonon approximation, and the coupling to phonons becomes strong. Expression (11) has been derived under the assumption that the wave vector, qE = Elhs, of the phonon which is emitted or absorbed by a pair is far smaller than lla. With increasing E, this assumption breaks down, and eq. (11) acquires an extra factor [1 + (ElkB 7;YJ -4, where kB 7;, = 2hsla. At E> kB 7;, the phonon deformation field oscillates quickly over a distance a, the probability of transition decreasing with the increase of E. The above condition for applicability of the one-phonon approximation is valid for Ec < kB 7;,. In the opposite 88 Yu.M. Galperin et al. case, this approximation can be used over the entire energy interval. The value of 7;, depends on the particular semiconductor and can vary over a broad range, 10-100 K. (b) The deformational potentials of the two components of the pair differ only slightly or do not differ at all: D1 = D 2 • This situation can occur in doped crystalline semiconductors. To reach an understanding of this case we consider transitions caused by a phonon with an energy E and thus with a wave vector qE' If the parameter qEr where r = r 1 - r 2 is small, the displacements of the levels Duu(rd and Du ii (r 2 ) are different only due to a small phase difference between uii(r 1 ) and uii(r 2 ). Transitions become possible only because of this phase difference. Calculations show that this can be taken into account (in the isotropic case) by the additional factor* 1 - sin x/x, (x = E/kB T.), T. = 7;,a/2r = hs/r. (14) The final result is ( 15) There is yet another important case in which the piezoelectric interaction is predominant in the energy range of interest. This case can arise in essentially all known crystalline semiconductors which lack a center of inversion. The piezoelectric interaction leads to the following expression for the time r 1 = 1- -A2 - F (E)[ -1+ (E -- r rW) kB TE kB T. kB 7;, )2J-4 coth (-2kBE-T) (16) Here F(x) = 1 at X:P 1, and F(x)::::::: x 2 at x ~ 1; rW) is a relaxation time determined by the piezoelectric interaction. In order of magnitude, this time is given by l/rW) = 4nxe2 kB T/fN s, where X is the square of the electromechanical coupling constant averaged over all directions. ,.- Now let us summarize the results we shall need. The general expression for 1/r(E, r) can be expressed in the following form, _1_ _ En (~) A2(r) coth(E/2kBT) [1 + (E/kB 7;,)2J4 ' r(E, r) - fJn <Pn kB T. ( 17) where the exponent n as well as the factor fJn and the function <Pn(x) depend on the interaction mechanism and are given in the table 1. *Galperin et al. (1983) have erroneously written a different factor in their paper. 89 Non-Ohmic microwave hopping conductivity Table 1 Mechanism of interaction n fJn <Pn(xl x<%1 D j i'D2 Di+D~ (D j -D2l2 81tph 4 s s 2(Di +DD x~1 DA D2 D j =D 2 =D -1 PA tx 2 41tph 4 s5 fJ 3 (41te 351t - £ - yC 1 ph 2 ~ 4) + 3s~ ix 2 In this table f3 is the piezoelectric coefficient (calculations were made for cubic symmetry T or T d ), s((St) are the velocities of longitudinal (transverse) sound, and DA and PA denote the deformational and piezoelectric interactions, respectively. Other notations are as given above. In the isotropic case and in the absence of a magnetic field the tunnelling amplitude A depends on the modulus of r only. However, in the magnetic field H this quantity depends also on the angle between the directions of rand H (see section 2.4). 2.3. The non-resonant (relaxational) absorption The relaxational absorption results from a modulation caused by the alternating electric field in the interlevel spacing, E, of the pair. In the external alternating field C( t) the difference between the one-site energies, <P = <PI - <P2' acquires an increment eC(t) . r. As a result, we have for the interlevel spacing ( 18) This expression was derived in the adiabatic approximation. In general, the conditions for its applicability are (cf. Galperin 1983, Laikhtman 1984) hldE/dtl hw ~ E. ~ E, (19) (20) These conditions mean that the changes in the perturbation must be slow enough not to cause direct quantum transitions. The energy modulation given by eq. (18) shifts the populations nand 1 - n of the two levels, the change lagging in phase behind the energy variation. As a result, the energy of the alternating field is dissipated. The occupation numbers n are determined by the balance equation (5). In the 90 Yu.M. Ga/perin et al. linear approximation the power q absorbed by a pair is given by an expression of the Debye type: (21) Note that eq. (21) depends on the pair's parameters: the interlevel spacing E and the tunnel integral A(r). Thus the summation of the contributions of all the pairs can be expressed in the form of an integral over E and A weighted by a distribution function of these parameters. As was shown by Shklovskii and Efros (1981) if the inequality (2) holds one can assume the distribution of E to be constant in the region of interest. On the other hand, we consider the impurity centers to be distributed at random in space. Taking into account the exponential dependence of A on r, we obtain with logarithmic accuracy that the distribution function of A is proportional to A - 1. As is shown in section 2, ' can be expressed in the form (6). As a result, the total relaxational absorption is proportional to Here we have omitted the logarithmic factors that can be taken out of the integral. Let us begin with a discussion of the case T < 7;,. One can see that in this case the result depends on the relation between the frequency OJ and the maximum scattering rate at E ~ kB T which we shall denote by 1j'min( T). If OJ'min( T) ~ 1 the predominant pairs are those with, of the order of 'mine T) and with a characteristic 'arm' Since the only pairs taking part in absorption are those which have a single electron we find (see Shklovskii and Efros 1981) (22) A study of the temperature dependence of (T~el)(OJ) under these conditions can reveal which case (a or b) holds in a particular semiconducting material. This is also an important consideration for the interpretation of experiments on the acoustic properties of semiconductors. Under the condition OJ'min( T) q 1 the predominant pairs are those with Non-Ohmic microwave hopping conductivity E ~ kB Tand r ~ 91 l/w. The characteristic value of the pair's arm is now equal to rc = a 10g{Ao/kB T[wrmin(T)]1/2} (note that rc ~ r T ). As a result, one can find (Shklovskii and Efros 1981): (T~el)(w) ~(e4/£)g2awr~. (23) This expression is analogous to the equation for the sound and microwave absorption coefficient in glasses due to the interaction with the tunnelling twolevel system (JackIe 1972). As we have stated at the beginning of this section, eq. (21), as well as the equations based on it, is valid in the adiabatic approximation, eq. (20). As the energies of the pairs giving the main contribution to the absorption are of the order of kB T the resulting expressions, eqs. (22) and (23), are applicable, provided hw ~ kBT. At a first glance this is the only case of interest because one can see from eqs. (3), (22) and (23) that already at hw:::::: kB T that the resonant contribution exceeds the relaxational one. Nevertheless, because of the extremely small critical fields for saturation of the resonant absorption we shall also examine the conductivity in the quantum frequency range, hw ~ kB T. Indeed we shall see below that the critical intensities for the non-linear resonant absorption and for the non-resonant case can differ by several orders of magnitude. Consequently, even in the quantum frequency range a region of intensities may exist where absorption is determined by the linear non-resonant contribution. At hw ~ kB T one should use the quantum-mechanical theory to calculate the non-resonant contribution to (To(w). It can be shown that the main contribution is due to pairs with E < hw and r:::::: r W' For such pairs we have wr(hw, r w) ~ 1 and the conductivity can be calculated in the second approximation of the perturbation theory in the same way as in the calculation of the acoustic and electromagnetic absorption by two-level systems in glasses (Gurevich and Parshin 1982a, b). The results at kB 7;, ~ hw:::::: kB Tare rather cumbersome but at kB 7;, ~ hw ~ kB Tthey can be described in order of magnitude by eq. (22) where kB Tshould be replaced by hw while r T is replaced by r W" A special remark may be given for the case r~i~( hw) ~ w (corresponding to piezoelectric interactions for hw ~ kB T,')' Here (T~el)( w) ~ w as for Wrmin( T) ~ 1. In the quantum case, however, (T~el)(w) differs from eq. (23) by a factor of [wr min ( hw)] - 1, which does not depend on w. This factor is small if perturbation theory can be used to describe the electron-phonon interaction. Let us now turn to a discussion of the case T ~ 7;,. 92 Yu.M. Galperin et al. As the pairs with E> kB T" have very large relaxation times they do not contribute to the absorption. The contribution is dominated by the pairs with E < kB T". If WTmin(k BT,,) ~ 1 we have for the characteristic arm of these pairs and (24) As 1/Tmin(kB T,,) ~ T for T ~ T" the absorption is in this case independent of frequency and temperature. For WTmin(k BT,,) ~ 1 the absorption is dominated by the pairs with the arms of the order of rc = a 10g{Aolk BT,,[WTmin(k BT,,)]1/2}. For these pairs wT(kB T", rJ ~ 1, so that rc > rT' As a result, we obtain (}~el)(w) ~ (e4/8)ag2wr~(T"IT). This means that the absorption is proportional to temperature. (25) W and decreases with 2.4. Magnetic field effects An external magnetic field H leads to a deformation of the impurity electron wavefunction, the deformation being different in the directions parallel to Hand perpendicular to it. Consequently the overlap integral A becomes dependent on the angle between the 'arm' of a pair rand H. To estimate the role of the magnetic field let us recall briefly the procedure of the calculation of () 0 (w) (or the absorption coefficient a). The total power absorbed, Q, is the sum of the contributions of all the actual pairs which is dependent on the product of the quantity le80 • rl2 and a function of rp and A. This function is determined by the absorption mechanism and should be calculated taking into account the above-mentioned effect of Coulomb correlation (see Efros and Shklovskii 1985):* it is proportional to the quantity (see Galperin and Priev 1986) S(n, A) =- R 3( 0 logoR ) A ' (26) where n = rlr, while R(n, A) is determined by the equation A(Rn) = A. (27) *The calculation of the microwave hopping conductivity in a magnetic field was first done by Klimkovich et al. (1983). In this paper the Coulomb correlation, which is important for obtaining the correct dependences on T, wand H, has not been taken into account. Non-Ohmic microwave hopping conductivity 93 It is clear that absorption is proportional to the integral J= f dn 2 -nsS(n,AJ. (28) 4n Here ns = (n· iff)/Iff where Iff is the AC electric field while Ae is a characteristic value of A. Ae is dependent on the absorption mechanism and equal to A(r r T or re): the quantities r r T and re are determined above. In the absence of a magnetic field, 0)' 0)' (29) while, in the presence of a magnetic field, J depends on the magnitude and direction of H. We shall restrict ourselves with the two limiting cases of weak and strong magnetic fields with respect to the value Ho=ch/ea 2. (30) At H = Ho the magnetic length 2 = (ch/eH) 1/2 is equal to the localization radius, a. Thus the ratio Ho/H = (2/a)2 characterizes the degree of deformation of an electronic wavefunction created by the magnetic field. To calculate the quantity given by eq. (28) one should solve eq. (27). At H ~ H 0 and H ~ H 0 it is possible to use the corresponding asymptotic expressions for the overlap integral A(r) (see Shklovskii and Efros 1984). As a result, we obtain (Galperin and Priev 1986) at H ~ H 0: JII(H) - J(O) = _ ~(~)4 fE2 '" H2 J(O) 10 2 e , J.l(H) - J(O) J(O) and at = _ ~(~)4 f!!2: '" H2 2 2 (31) (32) e H~Ho JII(H) J(O) = 3(aH)2(~)2(_1 ) '" H- 4 / 3 , a a fEe J.l(H) = 6(~)4(_1 )10g[(~)2 fE ] '" H- 2 10g H. J(O) a ~ 2 e (33) (34) Here the indices II (1.) refer to the cases iff I H and iff 1. H, respectively, a = h/(2mE H) 1/2, m is the electronic effective mass and EH is the binding energy of a donor in a magnetic field. If one takes into account that EH '" H 1 / 3 is a good approximation at H ~ Ho (see Shklovskii and Efros 1984) then JII(H) '" H- 4 / 3 . The experimental dependence of acoustic absorption [which is proportional to O"(w)] on H- 4 / 3 in strongly compensated samples ofn-InSb is shown in fig. 1 (Galperin et al. 1986). According to estimates the relaxational absorption 94 Yu.M. GaJperin et al. 1_....J~_-'1_....L~_8 ...., _fL.f--.l(f~.... ~4_ _11 - t/,s 3 X {O, i<Oe -1 Fig. 1. Magnetic field dependence of sound absorption in InSb (Galperin et al. 1986): No - N A = 1.3 X 1013 cm- 3 , N o = 1.2 x 10 14 cm- 3 Frequency, w/2n (in MHz): 1, 150; 2, 205; 3,400; 4, 448; 5, 570; 6, 750. dominates in this experimental situation, the condition oYr min ( T) ~ 1 being met. As we have seen in this case the absorption coefficient should be roughly proportional to wand should be almost independent of T. The ratio of the absorption coefficient to w is shown in fig. 2a. One can see that this quantity is universal for all the frequencies investigated. The temperature dependence of absorption is shown in fig. 2b. 3. Non-linear resonant microwave hopping conductivity To begin with, let us recall that the factor tanh(hw/2k B T) in eq. (3) for O"ges)(w) has a quite simple physical meaning. It is the difference between the populations of the upper and lower levels of the pairs with interlevel spacings E = hw. We shall call these pairs resonant ones. If the amplitude go of the electric field is high enough then this difference decreases and this decrease is believed to be the source of the non-linear behaviour. At the first sight, it is quite simple to calculate the non-linear resonant conductivity. One should analyze the equation for the density matrices of the resonant pairs taking into account their interaction with thermal phonons. The corresponding equation is analogous to the Bloch equation for a spin interacting with an alternating magnetic field. However, according to our estimates direct relaxation processes due to emission and absorption of thermal phonons cannot explain both the order of magnitude of the threshold of the non-linear resonant phenomena and their 95 Non-Ohmic microwave hopping conductivity (;(#)-6(0) w 10 <!o .30 -1 ~o ~ _ __ o r -______~~~O----------~~~,O--------------~_.-p -2 6 A A ___ T.K A -6 o 0 02 -I G(II-ooJ-G(/1:0), ciS/em Fig. 2. (a) Magnetic field dependence of G/w (G is the sound absorption coefficient) in InSb (Galperin et al. 1986). The points correspond to various frequencies of ultrasound from 100 up to 800 MHz. (b) Temperature dependence of sound absorption coefficient, G (Galperin et al. 1986). Frequency, w/2n (in MHz): 1, 157; 2, 750. frequency, temperature and intensity dependences. The origin of such a situation is the phenomenon of spectral diffusion which acts side by side with the direct relaxation processes and makes the whole physical picture rather complicated. The phenomenon of spectral diffusion was discussed by Klauder and Anderson (1962) in the theory of magnetic resonance. An analogous idea was used by Joffrin and Levelut (1975) and by Hunklinger and Arnold (1976) in connection with low-temperature resonant acoustic absorption in glasses. 96 Yu.M. Galperin et al. The physical picture of this phenomenon can be described as follows. Let us consider a resonant pair, i.e., a pair with interlevel spacing E close to hw. This resonant pair interacts with neighbouring thermal pairs, i.e., the pairs with the energy E:E: kB T. The interaction and consequently the interlevel spacing E of the resonant pair depends on the occupation numbers of the neighbouring thermal pairs. Due to interaction with the phonons the thermal pairs make transitions at random moments. As a result, the interlevel spacing E appears to be a random function of time. This is how the transitions of thermal pairs smear the resonance and enhance the effective number of pairs that take part in the resonant absorption. The role of spectral diffusion for non-linear hopping conductivity was first outlined by the authors (Galperin et al. 1983), while the quantitative theory of this phenomenon was developed by them in Galperin et al. (1988a, b). The following survey will be organized as follows. In section 3.1 we shall give qualitative estimates. The basic equations will be discussed in section 3.2, while the non-linear resonant contribution to o-(w) and the phenomenon of hole burning will be considered in sections 3.3 and 3.4, respectively. 3.1. Qualitative considerations To begin with, let us consider the characteristic parameters of the problem. The interaction of an alternating field with a resonant pair is determined by the offdiagonal matrix element !hF where the frequency F is related to the amplitude $0 of the field in the following way: hF = e$o' rA(r)/E. (35) Note that F is the Rabi frequency for the resonant pair. It characterizes the frequency of the coherent oscillations of the level occupancies under the action of a resonant field (see, e.g., Landau and Lifshitz 1977). In the absence of interactions among the pairs the absorption coefficient a depends on the relation between the Rabi frequency F and the characteristic value y of the intrinsic damping of the resonant pairs. The latter is given by y = 1/r(hw, r w)' (36) The contribution of a pair to a is proportional to the product of the population difference with the characteristic width of its absorption band. At F ~ y, the population difference is independent of F and is equal to its equilibrium value tanh(hw/2k BT). The absorption band in this case has the Lorentzian shape and the characteristic width y. The main contribution to the absorption is described by the linear theory - eq. (3): in the next approximation a correction of the order of (Ffy)2 appears.* *Throughout this chapter we consider the resonant pairs as if all their dipole moments were oriented parallel to the AC field. Actually they are distributed over various directions. Although we do not take into account this distribution explicitly this does not change our qualitative conclusions. Non-Ohmic microwave hopping conductivity 97 If F?- Y then the population difference decreases as IlF2 (i.e., it is proportional to the reciprocal intensity of the wave) while the characteristic spectral width increases proportional to F. As a result, the absorption coefficient appears to be proportional to IIF. Thus the critical amplitude Fe which corresponds to the onset of non-linear absorption is equal to y. Correspondingly, the width of a burned hole is of the order of y at F ~ Fe and of the order of F at F?- Fe. The spectral diffusion changes this picture drastically. To analyze its effects, let us estimate the characteristic deviation of a resonant pair's energy, E, due to jumps in a neighbouring thermal pair, the distance between the pairs being equal to R. The variation of the interlevel spacing is equal to the interaction energy of two electric dipoles with electric moments er OJ and erT separated by a distance R: e 2 r,,,rTIsR 3 == AIR3. The characteristic value of the interaction energy is of the order of AliP where R is the average distance between the thermal pairs of importance. By this we mean those thermal pairs that can contribute to the spectral diffusion. These are the pairs whose transition rates, r, are larger than y. All other pairs can be considered as 'static' and may be disregarded. On the other hand, we know that for E ~ kB T" the rate of relaxation is enhanced with energy E. Therefore, for temperatures T < T" those pairs have the highest relaxation rate r 0 whose energies E (as well as the tunnel parameters A) are of the order of kB T (pairs with even larger energies are not excited and make no transitions). Thus at T < T" the density of the thermal pairs, IIR 3 , goes as the first power of the temperature: IlR3 = PTk B T, (37) where P T = 4ng2 (e 2 IsrT)ar} is in fact the density of states for the thermal pairs. It depends weakly (logarithmically) on temperature. If T> T" the transitions of the pairs with E ?- kB T" are so rare that these pairs cannot contribute to the spectral diffusion. In this case to determine the concentration of the thermal pairs of importance one should replace T in eq. (37) by T". The interaction parameter AI R3 is an important characteristic of spectral diffusion. It characterizes the width of an interval of the resonant pair's frequency deviation, E - hw, due to interaction with the thermal pairs. Let us introduce a quantity 'd having units of time which is determined by the relation AIR3 = hl'd, At T < T" hl'd = t}Tk B T, (38) 98 Yu.M. Ga/perin et al. where (39) For a lightly doped crystalline semiconductor with intermediate degree of compensation one can obtain instead of eq. (39) (40) where N is the impurity concentration. This expression can be obtained from eq. (39) by the use of eq. (111) for the density of electronic states, goff (see Efros and Shklovskii 1985, Galperin et al. 1983). To get an idea about the order of magnitude of '1T one should take into account that within the framework of the two-site approximation the quantities rwand rT should be much smaller than the average distance between impurities, N - 1/3. Thus one can see that if the two-site approximation is valid then '1T ~ 1. (41) There exists another mechanism of interpair interaction which is due to elastic strains induced by thermal pairs. According to our estimates (Galperin et al. 1983) the contribution of this mechanism does not exceed the order-ofmagnitude estimates given by (39) and (40). The quantity l/T d is much larger than y and F in many interesting experimental situations. If this is so the dynamical broadening F of the resonant pair's spectral band is not important and the spectral diffusion plays the principal role. As we shall see below an important role in the theory of spectral diffusion is played by a relation between the quantity l/Td and the characteristic transition rate roof the thermal pairs of importance. At T < 7;" r 0 = l/Tmin( kB T) while at T> 7;, r0 = l/T min ( kB 7;,). As the transition rate depends on the interlevel spacing the quantities y and r 0 are different. The explicit expressions for r 0 can be extracted from table 1. The physical meaning of the dimensionless parameter r 0 Td can be described as follows. At small times t ~ l/r 0 a resonant pair walks away from the resonance according to a linear law (see fig. 3):* IE(t) - E(O)I ~ hr ot/Td. ( 42) *The linear time dependence (rather than the diffusional square root one) results from the strong correlation between the value of the jump and its probability: the probability of small jumps is much higher than of large ones. 99 Non-Ohmic microwave hopping conductivity E (t) - E(O) t Fig. 3. Time dependence of interlevel spacing of a resonant pair due to jumps in the adjacent thermal pairs. To explain the origin of this law (Klauder and Anderson 1962) let us consider a box of spherical form (see fig. 4) with linear dimensions of the order of R t ~ R, the resonant pair being in its centre. There are about (Rtf R)3 thermal pairs in this box with characteristic transition rates r o. If the condition (R t / R)3 rot ~ 1 is fulfilled at least one thermal pair in the box suffers a transition during the time t with the probability of the order of unity. In fact this condition determines the value of R t . The corresponding shift of the resonant pair's interlevel spacing E is A/Rf ~ hrot/rd' Equation (42) follows immediately from this estimate. To analyze a coherent phenomenon such as resonant absorption one should take into account the phase relations. In other words, one should consider the phase ¢ of the off-diagonal part of the resonant pair's density matrix. The phase coherence is destroyed by the spectral diffusion which in this way determines the .. . . '... .....• ';.. 0 . . ....-- ... . . . ..... • o o • o •• .. ·. · . . e.· .':.. ·..-a... .... . . . . . ··........ •• • o 0 0" •• .. ," '. '0 o • 0 o •• Fig. 4. A resonant pair (in the centre) and its thermal environment. Yu.M. Galperin et al. 100 width of the resonance for a particular pair. The random phase variation, A1>, during the time interval t due to the spectral diffusion is given by A1> ~ T ot 2 /Td' The time of the phase destruction, T¢, is determined by the condition A1> is given by ~ 1 and ( 43) This expression is valid if the condition T¢ ~ 1/To is met, i.e., if (44) because the law (42) is valid only for t~ l/T o. Let us now introduce the concept of the resonance region. This is the energy interval where the value of the phase is preserved with sufficient accuracy. As the time of phase relaxation is T¢, inserting this time in eq. (42) we conclude that the width of the resonant region is h/T¢. If T OTd ~ 1 the phase 1> essentially cannot change during the time t < l/T o. Thus the characteristic phase-breaking time T¢ is much longer than 1/T o. On the other hand, at large t ~ 1/T 0 the characteristic value of the energy deviation IE(t) - E(O)I is (on average) independent of the time because the order of magnitude of this quantity cannot exceed h/Td' In other words, in this case the energy deviation IE(t) - E(O)I experiences a random walk over an interval of width h/T d . Thus the phase-breaking time T¢ should be determined by the width of the spectral interval and be of the order of Td , T¢ ~ Td ~ 1/ To. Obviously the width of the resonant region is in this case of the order of h/Td' i.e., it is virtually the whole region of the spectral diffusion. Let us now discuss the temperature dependence of the dimensionless parameter T oTd' It follows from the considerations given above that the dimensionless parameter T OTd increases with temperature. There is a characteristic temperature 'T.i determined by the equation ToTd = 1. Its value depends on the parameters of the semiconductor. Consequently, the case of high temperatures, T~ 'T.i, corresponds to the inequality ToTd ~ 1 while the case of low temperatures corresponds to the condition ToTd ~ l. Now we can estimate the critical amplitude Fe for the saturation of resonant absorption as well as the width of a burned hole. Let us concentrate on the case where the spectral diffusion is important. At T ~ Td and l/T d ~ y all the resonant pairs in the interval of the width h/Td are in a non-equilibrium state. The characteristic rate of the population increase is F2Td while the relaxation rate due to phonons is y. Equating these two rates one obtains for a stationary case the estimate ( 45) Non-Ohmic microwave hopping conductivity 101 The width of a burned hole is evidently equal to l/rd which appears to be much larger than Fe. At T ~ 7d one can discriminate between two limiting cases where the spectral diffusion is important: (46) and (47) In the first case the linear regime occurs if the Rabi oscillations are virtually non-existent, i.e., their period I/F is much larger than the phase-breaking time r",. In the second case, F ~ l/r"" during the passage of the resonance region there are a number of Rabi oscillations, the non-linearity is strong and the level occupancies are almost equalized. This means that the critical amplitude for the onset of non-linear absorption is l/r",: Fe ~ l/r", ~(rO/rd)1/2. (48) The corresponding estimate for the width of a burned hole can be obtained as follows. At F ~ Fe a resonant pair becomes excited with a probability of the order of unity during its passage through the resonant region of the width h/r", ~ h(r O/rd) 1/2 . Then it leaves the resonant region but remains excited during the time t ~ l/y ~ 1/r 0 which is much larger than r",. Inserting t ~ I/y into eq. (42) we arrive at the estimate r O/yrd for the burned hole width. In the second case, eq. (47), one can obtain the estimate of interest with the help of the following qualitative picture (Laikhtman 1986). As in the previous case the region of random variation of the resonant pair's energy, h/rd' is much larger than the width of the resonance, h( r O/rd) 1/2. Thus the resonant pair makes many excursions from the resonant region due to random changes of energy E. Each time its average population increases by a small amount, F2r~ = F2rd/rO ~ 1. The total number of such excursions during a lifetime IIY is r oIY ~ 1, and the total change of the resonant pair's population is (F2rJro)(ro/Y). Equating this quantity to unity we obtain an estimate for Fe: (49) which coincides with that for high temperatures, T~ 7d. The burned hole's width is of the order of l/rd in this case. As we shall see below the exact expressions for Fe differ from the order-ofmagnitude estimates (45), (48) and (49) by large logarithmic factors which depend on the temperature T, the frequency ill and the parameters of the semiconductor. At high intensities, F ~ Fe as will be shown in section 3.5, the total power 102 i Yu.M. Galperin et al. i h et al. 3 absorbed by resonant pairs in 1 cm is independent of F. Thus the absorption coefficient IX varies as I/F2, i.e., is inversely proportional to the intensity. Note that in the absence of the spectral diffusion IX ~ 1/F, i.e., is inversely proportional to the square root of intensity. This is the main difference between the behaviour of the absorption coefficient in the presence and in the absence of the spectral diffusion. Sometimes the phenomenon of spectral diffusion is analyzed on the basis of Bloch equations for the density matrix by the introduction of a time of phase relaxation, '2' We wish to emphasize that such an approach is in principle incapable of giving the dependence IX ~ I/F2 for high intensities and therefore can only be of limited value. 3.2. Basic equations The resonant hopping conductivity is determined by the density matrix of a resonant pair eiwt _if ). ~. The rane ral diffusio !rmined by I 1~I/rois ! IVro. iresonance rrved with jtime in eq. (50) Ipt change In the resonant approximation, the equations for its elements have the form: I '4> is mud I-n an at= -y(n-no)-FRef ( 51) and af + 1.(W -at E(t)) f h - + -2yf = -F2 (2n - 1) . (52) ~istic valu lof the til ~/'d' In otl f1 random i should b< '4>:::: rbfof'd' the orde I Here E(t) = E + h Llw(t), Llw(t) = I Jl~l(t), (53) Idependen (54) psideratiol ih tempera I where hJ I is the interaction energy between the resonant and the Ith thermal pairs, and ~l(t) is the random telegraph process (see fig. 5). The latter can acquire the values + 1 and - 1 at random times, the average rate of such jumps being r l • We assume the different functions ~l(t) to be uncorrelated ones. This assumption allows us to perform independent averaging over these functions. One can see that the random function Llw(t) is not Gaussian. The real part of the conductivity 0"( w) is determined by the imaginary part of the pair susceptibility. 1m X(w), which is connected with the off-diagonal elements of the density matrix by the relation 2 1m X(w) = FV I . Re <j)¢. (55) ~tion rO'd Isequently, 'ity r Oed ~ I rO'd ~ 1. ~de Fe for !hole. Let lairs in the fristic rate pnons is y late Non-Ohmic microwave hopping conductivity - - 103 r--- r-- t -I '--- '- Fig. 5. The random telegraph process ((tl. Here the angular brackets <... >~ mean the average over realizations of all the telegraph processes ~lt): one should sum over all the resonant pairs in the volume V. We shall assume that the spatial configurations of the thermal pairs and their transition rates Tl are uncorrelated with the parameters of the resonant pair. In this case, instead of eq. (55), we have 1m x(w) = - 2P _0) F fco dE Re «f)~>c. (56) 0 Here the quantity PO) = g2(e 2/er w)4nar~ (57) <... plays the role of the effective density of states for resonant pairs, while >c means the configurational average. To be more rigorous, one should average over the tunnelling amplitudes A(r) of the resonant pairs (because F and y depend on this quantity) as well as average over all directions of the dipole moments of the pairs. However, it can be shown that this procedure produces no important changes and we shall omit it for brevity. 3.3. Intensity dependence of absorption (small amplitudes) To calculate the critical amplitude F characterizing the onset of non-linear behaviour let us iterate the set of eqs. (51) and (52) in powers of F. The first nonvanishing iteration gives 1m x(w) = nhP w( 1 - ;; + ... )tanh (2::T ) = 1m Xo(W)( 1 - ;; + ... ) (58) 104 Yu.M. Galperin et al. where (see Galperin et al. 1988a, b) 1 F2 = foo dT fro dT' exp[ -yeT + T')]<K(T, T')c (59) cOO and K(T, T') = (cos(I2r+r' dt LlW(t)B(t)))/ (60) where 1, (0 ~ t ~ T) B(t) = 0, { -1, (T<t~T+T') (61) (T + T' < t ~ 2T + T'). It is the function K( T, T') that contains all the information concerning the spectral diffusion. It can be calculated exactly for a telegraph process with the help of the theory of stochastic differential equations (see Kl'yatzkin 1980). We have K(T, T') = n kl(T, T'), (62) I r where kl is the contribution of the lth thermal pair: k(T, T') = e- 2rr ( cosh (IT) + ~Sinh(IT) + ~: e- 2r(r+r') sinh2(IT). (63) Here I = (r 2 - J 2)1/2 and we have omitted the indices 1for the quantities r l and JI • The expression (63) is valid both for r > J and r < J. The configurational average c is the average over the distances R between thermal pairs and the resonant pair as well as over the rates r of the thermal pair transitions. To calculate the first average we assume that the pairs are randomly and uniformly distributed in space. The distribution of r [i.e., the distribution of A(r)] has the form <... ) (1jr)(1- r;ro)-1/2. We shall omit the factor (1 - r; r 0) -1/2 for brevity. One can show that the results for the limiting cases remain unchanged after this. Making use of the Holtsmark method for the configurational average [see, e.g., the review by Chandrasekhar (1943)] we obtain <K(T, T')c = exp[ -SeT, T')jT d ], (64) where SeT, T') = 'XldJffodr f 0 p 0 r [1 - k(T, T')], (65) Non-Ohmic microwave hopping conductivity 105 where k(r, r') is determined by eq. (63). Combining eqs. (64) and (59) we obtain the result for Fc. The final expressions for this (as well as for the width of a burned hole Av) are given in table 2. These expressions differ from the estimates (45), (48) and (49) made by large logarithmic factors. The existence of such factors has a transparent physical meaning: the main contribution to the spectral diffusion belongs neither to the thermal pairs that are nearest neighbours nor to some other thermal pairs situated at some characteristic distance R. Rather, there is a relatively large number of pairs situated at some relatively large interval of R-variation that is responsible for the phenomenon. This brings about special mathematical difficulties for the solution of this problem for three-dimensional (3D) case. One can see from table 2 that the spectral diffusion is unimportant in the cases 1 and 3 while in the other cases it plays the principal role. 3.4. The width of a burned hole To analyze the phenomenon of hole burning one should calculate the change AQ of a small probe signal's attenuation in the presence of an intense wave of frequency w. In the cases where spectral diffusion is important the main contribution to AQ is due to the deviation of populations of resonant pairs, An(t): AQ = -1CW I V-I L h2 FiAn(t) O[hWI - (66) E(t)], WI being the frequency of a probe signal. Table 2 Condition N 'I ~ l/T d T~ F; ~v '12 max ('I, F) (Lorentzian) (1) T;, ny -log 2Td 'lTd 2 'I ~ l/Td 3 'I ~ (FoITd) 1/2 '12 Fo ~ 'I ~ (FO/rd) 1/2 2nFo (Fo) --log -2- 4 T ~ T;, Td (1) n -log 2Td 'lTd (Lorentzian) max(y, F) (Lorentzian) 'I Td FO/YT d (non- Lorentzian) 5 y ~ Fo ~ (FO/rd)1/2 ny (Fo) -log 2Td r -log - ) nCo 2Td Y (Lorentzian) Yu.M. Galperin et al. 106 Iterating the eqs. (51) and (52) in powers of F we obtain (Galperin et al. 1988a, b) LlQ = - B LO') dr e -yt/2 cos where v = OJ - OJ 1 (Ivl 2 ~ VT foX> dr' e -yt' <L( r, r') )e, (67) OJ), B = tnOJn Fi F2 P tanh( nOJ/2kB T) OJ t (68) t L(r,r') = leXP(irLlOJ(t)-if dt' LlOJ(t')))' \ t-t-t' ~ ' (69) The function L( r, r') and its configurational average can be calculated along the same lines as in the previous section. As a result, we obtain <L(r, r')e = exp[ - V(r, r')/r d ] (70) r d~ Jor (71) V( r, r') = Fo co Jo J dr [1- cp( r, r')], r and cp(r, r') = e -Ft cos Jr( cosh Jr + fSinh Jr) + e -nt+ 2t')fsin Jr sinh Jr. (72) Equations (67)-(72) permit us in principle to calculate the form of a burned hole provided the parameters of the semiconductor and the characteristics of the pumping signal are known. On the other hand, these equations can be investigated analytically for the same limiting cases that are discussed in section 3.1 (the same is, of course, true for Fe investigated in the previous section). The analysis (see Galperin et al. 1988a, b) gives results for the cases 2 and 5 (see table 2); the form of the hole is Lorentzian while its width is given in table 2. In case 4 the form of the burned hole appears to be much more complicated and is given by 2 A fOOd exp( -nror /2r d ) (73) ilQ = - B r cos VT • o y+(nrOr/r d ) The profile of this function is given in fig. 6. Its asymptotic form at large values of Ivl goes as 1/V2, i.e., it is the same as for the Lorentzian function. One can prove that this is a general feature of the 3D case. Finally, let us compare the hole's width Llv with the critical intensity Fe. One can see from table 2 that in all the cases where spectral diffusion is important the following order-of-magnitude estimate Fe~(yLlv)1/2 is valid. (74) 107 Non-Ohmic microwave hopping conductivity 1(Z) -3 -2 -f 3 exp [ -(~) 2. 1 - /(z) = -~ 1f (") dx 10 1 1 1f z2 +1 cosxz 2] 10 1+x 2-/(z)=---- -I Fig. 6. Shape of a burned hole in the quantum case: 1, at ro ~ y ~ (rO/rd)1/2(y-l(2nro/rd)1/2 2, the Lorentzian with the same asymptotics and area under the curve. 3.5. The region of high intensities (F ~ = 10); Fe) Now we shall discuss briefly the qualitative picture of absorption at high intensities (F ~ FJ. The average power absorbed by resonant pairs can be expressed as Q= - w vLhF<Re f \, (75) which can be transformed with the help of eq. (51) to the form Q= hwyPw <... LX! dE «n - no»t. (76) Here )t is the notation for time averaging. This expression is in fact a consequence of energy conservation. Therefore the quantity Q is determined by the characteristic width of the region of integration over E and by the value of «n - nO»t in this region. Let us restrict ourselves to the cases where the spectral diffusion is important. (In other cases one can use the Bloch equations with the corresponding longitudinal and transverse relaxation times, '1 and, 2; '2 = 2, 1 = 2/y). Thus for high temperatures (T~ 7d) we are interested in the case y ~ l/'d' In this case, as we have seen, the characteristic region of integration over E (i.e., the 108 Yu.M. Galperin et al. width of a burned hole) is of the order of h/c d while the quantity <en - no) \ is equal to t - no for F?> Fe. As a result, we have the following estimate for Q: Q : : :; hwyPw(1/rd) tanh(hw/2k B T) :::::; h2P wwF; tanh(hw/2k B T), (77) where Fe is determined by eq. (45). To obtain the absorption coefficient rt. one should divide Q by the intensity of the electromagnetic wave proportional to F2. Consequently, at F?> Fe we have (78) Thus at F?> Fe the absorption coefficient is inversely proportional to the intensity of the wave. We shall see that such a dependence is valid in all the cases where spectral diffusion is important, while in the cases where spectral diffusion can be neglected rt. ~ rt.o(FclF), i.e., the absorption is inversely proportional to the square root of intensity. At low temperatures (T q 1d) in case 4 (see table 2) the width of the region of integration over E is of the order of h/rd because this is the width of the spectral diffusion region. One can estimate the characteristic value of <en - no)t from the following considerations. We have seen that a resonant pair changes its population in the resonant region of the width h/rq, ~ her o/r d ) 1/2 and passes this region during the time rq, :::::; (rd/ r 0) 1/2. It remains in the excited state during the time ljy ?> rq,. Then it becomes de-excited via emission of a phonon. During its further passage through the region of spectral diffusion it has the equilibrium population no. The resonant pair reaches the resonant region again after a time l/r0 ?> l/y. Thus the relative part of the time spent in the excited state (with n = t) is of the order of r o/y q 1. Consequently, we obtain <en - no) >t = Ci - noHr oM· Taking this into account and making use of eq. (48) for Fe we obtain the estimate (78). In the case 5 (table 2) a resonant pair makes a lot of excursions into the resonant region during its lifetime ljy. Each time its population increases by a small quantity F2r~ = F2rJ r o. Thus to equalize the occupancies of both levels many such excursions are needed, their number being of the order of 1/F2r~, so the characteristic time for this process is of the order of 1/roF2r~. On the other hand, we have l/r oF2r~ = 1/rdF2:::::; (l/y)(F;/F2), where Fe is determined by eq. (49). Thus at F?> Fe the lifetime l/y is much larger than the pumping time 1/roF2r~ and a resonant pair spends the main part of its lifetime in the excited state. As a result, <en - no)\:::::; t - no, while the characteristic width of the region of integration over E is again h/r d • Taking this fact into account we obtain the estimate (78) again. Non-Ohmic microwave hopping conductivity 109 Thus in all cases the estimate given by eq. (78) is valid. However, for different cases one should use in it different expressions for the critical amplitude F c' 3.6. Concluding remarks Finally, we want to discuss briefly the experimental situation. We have seen that several parameters enter the theory and a lot of different limiting cases can be met. To understand what situation is the case in a given material at a given frequency wand temperature T one should compare the three parameters, r 0' Y and l/Td' As was mentioned above the first two parameters can be expressed in terms of the relaxation time T( E, r) [see eq. (17) and table 1] as follows 1 Y= ----,--T(liw, r w) , (79) The difference between the expression for r 0 at T < 7;, and at T> 7;, is due to the fact that pairs with E> kB 7;, are weakly coupled to the phonons. The rate of their transitions is small enough and these pairs are of no importance for the spectral diffusion. The quantity Td is determined by eq. (38) at T < 7;,; at T> 7;, one should replace T and r T in eqs. (38)-(40) by 7;, and ra , respectively. All the above mentioned considerations are based on the suggestion that the two levels of a pair are well defined. This is so if the inequality (80) is met. For a deformational interaction at D1 -:f. D2 and kB 7;, > Ec = /311/2 (see table 1) this condition breaks down in the temperature region EclkB to 7;,. The theory developed breaks down also if lir 0 > kB Tor y > w. A corresponding limitation can be easily obtained with the help of eq. (79) and table 1. Our previous considerations were based on the assumption that the phonon system of a semiconductor is in equilibrium. In some situations a group of phonons with frequencies close to w can become strongly non-equilibrium. In such a situation the theory should be generalized (Gurevich and Rzaev 1987, Parshin and Rzaev 1987a, b) and some new interesting phenomena appear. The most interesting is, probably, the spectral diffusion of phonons which is due to the spectral diffusion of pairs. We have seen that the values of the critical intensity and the burned hole width depend critically on the frequency w, the temperature T and on the semiconductor's parameters. These values can fall within a very wide interval. For example, at T = 1 K and w/2n = 1 GHz the critical amplitude e1c corresponding to the value of F c can vary between 10- 7 and 10- 2 V cm - 1 depending on the sample's parameters. One may also note that there are a number of non-linear resonant phenomena in hopping conductivity not considered in this review. Perhaps, most interesting Yu.M. Galperin et al. 110 among these are the various echo phenomena, both electrical and acoustical. However, a detailed discussion of these deserves a special review. In our opinion, a systematic experimental investigation of the non-linear resonant phenomena in the hopping conductivity of semiconductors is called for. 4. Non-linear relaxational (non-resonant) absorption Let us assume the electric field $( t) to be so large that it produces the energy modulation in a pair to be larger than kB T: (81) Then the relaxational absorption depends on the amplitude 1&'0: specifically, it decreases as this amplitude increases. In the non-linear regime the distance rein this inequality can, in general, depend logarithmically on the wave amplitude: we shall analyze this dependence below. This section will be organized as follows. At first we shall restrict ourselves to the case of high fields. A qualitative discussion as well as the quantitative theory will be given. Then we shall discuss the non-linear corrections to o-o(w) if the condition d ~ kB T is met. These corrections can be of interest because there are situations where the condition (81) cannot be met. For example, it is typical for doped semiconductors with shallow donors (or acceptors). An intense electromagnetic wave can ionize the impurity states and the free carriers can dominate in absorption. Thus there is a mechanism of non-linear behaviour which can compete with the relaxational one. One should note that the process of ionization occurs if the field amplitude exceeds some threshold value. This is why one can study weak non-linearities of relaxational absorption which are not masked by the ionization of impurities if the intensity is lower than the abovementioned threshold. 4.1. Qualitative picture To begin with, let us consider a qualitative picture in the case of the deformational interaction for Dl i= D 2 • We shall discuss the general situation below. Furthermore, analyzing the non-linear relaxational absorption we shall restrict discussion to the case d ~ kB I;, (the opposite case seems to be extremely difficult to achieve in experiments). To analyze the physics involved let us recall that the interlevel spacing E in the presence of an alternating field $(t) has the form of eq. (18) and becomes time dependent. This dependence for d > kB T is depicted in fig. 7. The power q absorbed by a pair is . q = <nE) w == 2n f27t / W 0 dE dt n(t) dr' (82) 111 Non-Ohmic microwave hopping conductivity E(t) I I -~ T I t to 2% 41 Fig. 7. Time dependence of the energy of a thermal pair under the action of a strong alternating field, d'!> kB T. where the occupation numbers n(t) are determined by eq. (5). We wish to emphasize that it is the quantity E(t) that enters the expressions for no(E) and !(E, r). Let us analyze the important limiting cases. Under condition (81) the level separation E is of the order of kB T only during the short time intervals (see fig. 7) M ~ kB Tldw ~ (83) 1/w. It is during these time intervals that thermal phonons can excite the pair (if it is initially in the lower energy state). The characteristic relaxation time with respect to these processes is !min( T). During the rest part of the period the pairs have energy E > kB T and can only relax by emitting phonons with energies E(t) and cannot absorb them. The characteristic relaxation time of the pairs with respect to these processes is !(d, r T ) ~ !min(T). We can thus distinguish three characteristic limiting cases: M ~ !min(T), (84) ,1.t ~ !min(T); w!(d, r T ) ~ I, (85) and (86) Let us begin with the region of relatively low frequencies (84). The inequality (84) is equivalent to the condition W!min( T) ~ kB Tid ~ 1. 112 Yu.M. Galperin et al. Because of the inequality (84) the pairs with ,(kBT, r) ~ Llt relax immediately after crossing the kB Tlayer. The energy of the emitted (and absorbed) phonons, averaged over the period, is of the order of kB T. The conductivity is dominated by pairs with 'arms' of length re determined from the condition ,(kB T, rJ = M: a ( A5 ). rc = -log 2 w'min(T)dkBT The characteristic density of such pairs with (87) qJ = IqJl - qJ21 ~ dis g2( e 2Isrc)r;ad [we are assuming e 2 lere ~ d; pairs with a large initial splitting qJ > d have during the whole period an energy gap exceeding kB T and do not contribute to O"(w)]. On the average over a period, such a pair transfers an energy of the order of kB T to the phonon subsystem. Calculating the power absorbed by such pairs per unit volume of a semiconductor, and dividing it by t~5, we find the conductivity to be (88) where the numerical factor is found from a rigorous theory (see section 4.2). If 'min( T) ~ Llt, the picture is considerably more complicated. In this case an unexcited pair which has traversed the kB T layer has a small probability, Lltl'min( T), to be excited via absorption of a thermal phonon. However, if it has been excited it emits a phonon of much larger energy because the level separation increases with time t. Just how large this energy is depends on how rapidly the relaxation time ,(E, r) falls off with increasing E. In the case at hand, i.e., the case Dl =P D2 with kBT~E<kBT", we have [see eq.(l7) and table 1J ,(E, r T ) = 'min(T)(kBTIE). The characteristic time t in which the pair emits a phonon is given by t* = ,[E(t*), rT]. (89) As an estimate we may assume that the energy E(t) is proportional to t, E(t) ~ dwt, in the relaxation region. We then find that (90) The limiting cases (85) and (86) correspond to different relations between t* and 11w. At high frequencies [eq. (86)] the inequality t* ~ 11w holds. In this case the excited pair does not have time to emit a phonon during a period and instead emits it after several periods, on the average after a time interval ,(d, rT) ~ 1/w. The average energy of emitted phonons being of the order of d, the power 113 Non-Ohmic microwave hopping conductivity absorbed by one pair is q ~ wd dtlr(kB T, r). The main contribution to 0'( w) is due to the pairs with r( kB T, r) with r ~ r T • As a result, after summing over all the pairs, we find 2e3 2 kBT a () w = 076 . g - arT e rmin(T)iffO ~ O'o(w)kBT eiffOrT ~ rmin( T), i.e., (91) where 0'0 is given by eq. (22) while the numerical factor can be obtained from the exact expression (see the next section). In the case of intermediate frequencies (85), kB Tid ~ wrmin(T) ~ dlkB T, the inequality t* ~ 1/w is fulfilled. Therefore, a pair excited in the kB T layer should most probably emit a phonon during a time t* much shorter than the period. The characteristic energy of the emitted phonons being of the order of dw t* in this case, the power absorbed by one pair is q= M dwt* 1 r(kBT,r)w- ~ wk B T (M )1/2 r(kBT,r) As in the previous case the pairs with r ~ r T [and r(kB T, r) ~ rmin(T)] give the main contribution. As a result, we have e 5/2 1 O'(w) = 1 99g2_ar3/2w1/2(k T)3 /2 . e T B iff6/2 r~~( T) . (92) The frequency dependence of the non-linear conductivity O'(w) for the case considered above is depicted schematically in fig. 8. The broken line represents O'o(w). G' (w) ..,.- - ;' I "'-UJ. -@~ -.:¥ / 00 I-L-_-r---"--..i. T;"u,(T) .&I d f t'min(Tj I t;,in (T) d ee T Fig. 8. Frequency dependence of the hopping conductivity 0"(0)) (schematic). The broken line represents 0"0(0)). 114 Yu.M. Galperin et al. Let us estimate the characteristic value, iffe2 ~ kB Tler e , of the field amplitude which is required to observe non-linear behaviour of relax ationa I absorption. With T = 0.3 K and r T = 2 x 10- 6 cm we have iffe2 ~ 10 V cm -1. The strong non-linearity of the absorption can be observed if this field is below the threshold for impurity breakdown. Note that iffe2 decreases proportionally to T as the temperature is lowered. iffo 4.2. The general theory of non-linear absorption Behaviour of the non-linear conductivity depends strongly on the form of the functional dependence of the relaxation time r on the interlevel separation E. As we have seen above (section 2.2 and table 1) in several important cases this functional dependence can be described as r(E, rT ) = rmin(T)(kB TIE)" tanh(El2k BT). (93) We have v = 1 for the deformational interaction with D1 :f. D2 at E ~ kB 7;, and with D1 = D2 at kB T,. ~ E ~ kB 7;, as well as for the piezoelectric interaction at E ~ kB T,.. For D1 = D2 at E ~ kB T,. we have v = 3 while for the piezoelectric interaction at kB T,. ~ E ~ kB 7;" V = - 1. If v > 0 we can write the estimate (90) in the following way: t* = [r m in(T)(At)"]1/(1 +v). In the same way we obtain a( ())) '" g 2 e(2v+3)/(v+1) '" ())1/(V+1)(k T)(2v+1)/(V+1) ar( v + 2)/( v + 1) ---cc:--c~.,-::-B.,-,--,,-~_ _ T iff~V+ 1)/(v+ 1)r:;(!~+ 1)( T) I: instead of eq. (92). In particular, for v = 3 we have '" a ( ()) ) 9 4 2e / 5/4 ())1/4(kB T) 7/4 ",g -arT e fP7/4 3/4(T) rmm 00 To obtain the numerical factors given above one can use the following expression for the power absorbed by one pair (Galperin 1983, Laikhtman 1984): () II q = 8nk BT x 2"/,,, 0 E(t)E(t - t') dt dt cosh 2 [E(t - t')/2kB T] exp , [ (I l-exp - 2"f'" o dtff -r( tff) )J-1 (It' 0 dt ret - ff ) tff) (94) This expression follows from eqs. (5), (19), (20) and (82): r depends on the time t through the functional dependence E(t). The integrals in (94) can be simplified in various limiting cases making it possible to evaluate the numerical factors in the expressions for a( ())). Non-Ohmic microwave hopping conductivity 115 Now we are left with the case of piezoelectric interaction, v = -1. This situation is very similar to the case of metallic glasses where the functional dependence T(E) is the same: it has been considered in detail by the authors (Galperin et al. 1984a, b). To discuss this case let us introduce the ratio AE = ~tE~ = T( E, r) 2 A (r) dkB TWT min ( T) coth (~) 2kB T (95) where tE = E/dw is the characteristic time of the interlevel spacing variation. The ratio AE is some characteristic of the probability of relaxation of a pair during time tEo The probability is of the order of AE at AE ~ 1 and of the order of unity at AE;;::; 1. As a function of time AE has a maximum Amax = A/AI where Al = dWT min ( T) at the moments to satisfies the equation E(t o) = A. The minimum value of AE' Amin = (A/A2)Z, where A z = [dk BTWTmin(T)Jl/2, is reached at E c:::: kB T. Another important property of )oE is that it is independent of E for E > kB T. One can discriminate between two limiting cases. The simplest is the case of high frequencies (96) in which kB T ~ A2 ~ AI' Because the main contribution is given by the pairs with A < kB T we have Amax ~ 1. As a result, the probability of excitation during an excursion to the thermal region is small ( :::::; Amax); it is maximal for pairs with A c:::: kB T. Consequently, these pairs give the main contribution. Such a pair, if excited, takes an energy of the order of kB T from the phonon system and remains excited during many (:::::; Tmin ( T)/ Llt) periods. The probability of its deexcitation being independent of the energy E, the average energy which returns to the phonon system in the course of de-excitation is of the order of d ~ kB T. Therefore, on the average, the pair experiences one act of excitation during the time (2n/W)(T min (T)/M) and one act of de-excitation. As a result, we obtain an order-of-magnitude estimate which coincides with eq. (91). The case of low frequencies, (97) is much more complicated. In this situation the inequalities A I ~ A z ~ kB T hold. These conditions mean that a pair with A c:::: kB T can be excited and de-excited many times during its excursion to the thermal layer UE ~ 1). Outside this layer such a pair is immediately de-excited by phonons P'E ~ 1) and is in the ground state for E> kB T. As a result, the population of such a pair is close to the equilibrium one and these pairs are inefficient for the absorption. The main contribution in such a situation is due to the pairs with A ~ kB T. As the rate of their relaxation is small enough, these pairs can have non-equilibrium populations. Yu.M. Galperin et al. 116 The absorption is dominated by the pairs with Amin ~ 1, i.e., with A::::: A 2 • We shall call them intermediate pairs: the condition r(kB T, r) = Ilt is satisfied for them. The intermediate pairs are responsible for an interesting physical phenomenon that may be called the protraction of relaxation. An intermediate pair can remain in the excited state with a probability of the order of unity after its escape from the thermal layer. Because )'E is independent of Eat E> kB T this pair can be de-excited by the phonons and returns an energy E ::::: kB T with a probability of the order of unity. At the same time it can protract its relaxation until the values E ::::: d of the interlevel spacing are reached. Correspondingly, there are two contributions to the power absorbed: the first is of the order of dOJkB T while the second is of the order of dOJd. One may anticipate that it is possible to neglect the first because we discuss the case d ~ kB T. Indeed, the order of magnitude of absorption is determined by the second item, the total absorption being of the order of the linear case. However, the first item, though small, has a more pronounced amplitude dependence. As a result (Galperin et al. 1984b) (98) where a l and a 2 are numerical factors of the order of unity while O'o(OJ) is given by eq. (23). One can show that the contributions of 'slow' pairs (with A ~ A 2 ) and of 'fast' ones (with A ~ A 2 ) are sufficiently small. 4.3. Small amplitudes At a first glance the region of small amplitudes, d~ kBT, (99) is not too interesting and informative. Indeed, we have seen that in the linear regime pairs with E ~ kB T play the main role. Consequently, one can suppose that due to a small non-linearity some corrections to the linear hopping conductivity of the order of (d/kB T)2 may appear. As will be shown below this conclusion is in general incorrect. The reason for this can be explained as follows. Along with the pairs having E ~ kB T there are pairs with a small interlevel spacing, E ~ kB T. Their relative number being small, these do not contribute significantly to the linear conductivity 0' o( OJ). At the same time these pairs may dominate the non-linear behaviour because the influence of the external AC field on them is far stronger than on the thermal ones. As a result, in some limiting cases practically all the non-linear behaviour can be dominated by the pairs with E ~ kB T. We want to point out that the experimental study of such a Non-Ohmic microwave hopping conductivity 117 phenomenon gives an opportunity to investigate the pairs with E ~ kB T. To determine when these pairs are effective let us estimate the contribution of pairs with E < E* to the linear absorption where E* is much smaller than kB T but otherwise arbitrary. As was pointed out in section 2.3 the linear non-resonant contribution to a( w), Oa 0, due to the pairs in the region of energies E, E + dE is proportional to 2 r 1 oao oc 1 +w(wr)2 cosh 2 (E/2k BT) dE. We can replace the factor cosh 2 (E/2k BT) by unity because we are interested in the pairs with E ~ kB T. In what follows the energy dependence of r is very important. This dependence can be described in some interesting limiting cases by eqs. (6) and (93). Here we shall consider the simplest situation with Dl =f. D 2 • At small energies, E ~ kB T, kB 7;" for this case r is independent of E: l/r = (l/ro)(A/k BT)2, where ro is given by eq. (12). Summing over all the pairs with E < E* we obtain the following estimate: oa0 ~ -lIE' dE II Too lIE' dE tan - 2 dp w r pJT=P 1 + (wr) 2 oc - T 0 1( E 2 ) 2 . wro(kBT) (100) Here we have transformed the integration variable r to p = [A(r)/E]2 and have omitted the logarithmic factors. Strictly speaking, non-linear corrections to the linear contribution oa o( w) [eq. (100)] can be evaluated by expanding of the general equation (94) in powers of (d/E) (Galperin and Priev 1988). At the same time it is clear that an order-ofmagnitude estimate should be ant = a(w) - ao(w) oc ~ IE' dE( ~) \an -1 CkB ~:wrJ. (101) Indeed, the second factor in the integrand is the linear contribution of pairs with a given value of E while the first one is the relative non-linear correction in the lowest non-vanishing order. One can see that at wro ~ 1 this integral is determined by its upper limit E*. This means that the main contribution to ant is given by pairs with E ::::; kB T. The correct estimate for ani in this case can be obtained by equating E* to kB T. Thus, (102) The case wro ~ 1 is far more interesting. As it follows from eq. (101) that in 118 Yu.M. Ga/perin et al. this case a characteristic energy, E1 ~ kB T( wto)1/2 ~ kB T, (103) appears, one obtains the following estimate for (Tn! d2 (Tut d2 (104) ~ ~ kBTE1 ~ (kBT)2(WtO)1/2' Thus at Wto ~ 1 the ratio dlE 1 rather than the much smaller quantity dlkB T plays the role of the dimensionless parameter of non-linearity. If d > E 1 , or (105) one cannot use the expansion of eq. (94) in powers of diE and consequently the order of magnitude estimate (101) becomes invalid. In this case the main contribution to (Tn! is due to the pairs with E ~ d. The contribution of these pairs to (Tut is of the order of their contribution to (To. Thus the ratio (Tnll(To is determined by the ratio of the numbers of pairs with E ~ d and E ~ kB T, respectively. As a result (106) We see that in this case the non-linear corrections are more pronounced. The estimates for other mechanisms of relaxation can be obtained from eq. (101) in a similar way by replacing to by the corresponding quantity dependent on E. To obtain numerical factors in the formulae one should use the exact eq. (94) for the power absorbed by a pair. To obtain a simplified expression one should extract from this the linear contribution. As the remaining non-linear contribution is determined by E~kBT at wtmin(T)~l, one can replace cosh- 2 [E (t - t')/2k B T] by 1. Then at small values of d the integrals can be evaluated with the help of a direct though rather cumbersome procedure (Galperin and Priev 1988). The resulting expressions for (J"t(w) are given in table 3. Here Aw,T = (g2e 6a6ff'~,Twle), ff'T = log(Aolk BT), ff'w = log[Aolk BT(wt o) 1/2]. We wish to emphasize that in the interval 1 ~ Wto ~ (dlkB T)2 the situation is rather unusual, i.e., the non-linear correction to the conductivity is positive. As we know, at even higher amplitudes (T(w) should decrease with the intensity. Therefore in this case one should expect a non-monotonic intensity dependence of the absorption. It is interesting to mention that the case considered here bears a rather strong resemblance to the situation in dielectric glasses. As is well known, their 119 Non-Ohmic microwave hopping conductivity Table 3 Interval of parameters -0.6Ay S5 roTo(kBT) S2 4.5xlO- 3 A 0 W (roTo) 1/2(k BT)2 iSoi -015A • 2 W kB Tea!l'w microwave (or acoustic) absorption is due to interaction with two-level systems (TLS) (see Anderson et al. 1972, Phillips 1972). The TLS also relax via thermal phonons and the dependence r(E) is the same as for the case discussed in this section. It may be worthwhile to mention that a problem formally very similar to that considered above, i.e., non-linear acoustic attenuation in glasses in the weak non-linearity regime was considered by Levelut and Schon (1986). However, they obtained a result drastically different from ours. The difference between our approaches originates in the different equations for the relaxation time r. In our notation, they use the equation 1 1 (A)2(E(t))!I (E(t)) kB T coth 2kBT ' ~ = rmin(T) E ( 107) where fl = 1 corresponds to metallic glasses while fl = 3 corresponds to the dielectric glasses. The second factor in this equation contains a 'bare' value of E which is independent of t while all the other factors contain E(t). We believe that E(t) should enter in every place in this equation and that there is no room (neither from pure mathematical considerations nor from physical analysis) for the 'bare' value E. This can also be shown by analysis of the density matrix equation of a TLS in the adiabatic approximation. This is why we cannot agree with the results obtained by Levelut and Schon (1986). We believe that the regime of weak non-linearity can be rather informative because the dependences of the non-linear contribution on wand T, as well as the critical amplitude are determined by the mechanism of pair relaxation. Consequently, we have an independent way to study these mechanisms. 4.4. Magnetic field effects To estimate the influence of the magnetic field on the non-linear relaxational absorption let us keep in mind that the only quantity which is affected by His the overlap integral A(r). This quantity becomes strongly dependent on the 120 Yu.M. Ga/perin et al. angle between H and the pair's arm r. On the other hand, the interaction energy e(80 'r) depends on the angle between rand 8 0 , Thus a dependence of O'(w) on the angle between Hand 8 0 appears. The procedure of calculation of the linear contribution to O'o(w) in the presence of a magnetic field has been outlined in section 2.4. The whole effect of the magnetic field can be expressed through the integral J [eq. (28)]. In general the intensity dependence of the absorption by the pairs is rather complicated. However, in the limiting cases of the low and high amplitudes it is a simple power function: (l08) In the regime of high amplitudes f3 = t for the intermediate frequencies [see relation (85)], while f3 = 1 in the other cases. In the regime of low amplitudes f3 = 3 at W!o ~ (d/kB T)2 and f3 = 4 at W!o ~ (d/kB T)2 (here we consider the amplitude dependence of QUI = Q - Qo). Consequently we have the integral J«(J)= dn f-n(JS(n 4n $ A) (109) 'c instead of eq. (28) where Ac is determined after eq. (28) (in the linear case Qo ex d 2 ; f3 = 2). The integrals (109) can be evaluated directly at A ~ a and the results are given in the table 4. We wish to emphasize that the external magnetic field changes the criterion for non-linear behaviour. The effective parameter for non-linearity d/kB T should be multiplied by aH / a for 8 0 II H and by A/a for tf!o ~ H (the last factor is the characteristic cosine of the angle between r and H). Thus the magnetic field leads to a decrease of the absorption and to a decrease of the relative non-linear contribution as well. The condition for strong non-linearity in the presence of a Table 4 f3 1 2: 3 4 JII(H)jJ(O) J""(H)jJ(O) A)S/2 log (aH) log H ex-- ex a ( A H S /4 Non-Ohmic microwave hopping conductivity 121 magnetic field takes the form kBT~ d(aH/a), (~o II H) { dUe/a), (~o.1 H). (110) In our previous discussion we have not considered an extra dependence of r on the magnetic field which exists along with the dependence through the parameter A(r). The extra dependence is produced by a shrinkage of the impurity state wavefunctions in the magnetic field. Indeed, the factor [1 + (E/kB T,.)Zr4 in eq. (17) is due to the oscillations of the phonon field over the localization length a, for E/kB T" is the ratio between a and the wavelength of a phonon with frequency OJ = E/h. The shrinkage of the wavefunctions should diminish the role of such oscillations and, consequently, the relaxation time. The latter becomes dependent on H for A ~ a and E> kB T: = 2sh/a H . An essential change in the final results occurs at T ~ T;. = hs/kBA ~ T" where the deformation field of the thermal phonons oscillates over the transverse dimension A of the electronic wavefunction. To obtain an order-of-magnitude estimate for this case one should use all the expressions for T ~ T" and replace T" by (T:T;Y/z. For other cases the extra dependence r(H) is not very important. Expenments with magnetic fields may present an extremely effective possibility of studying the various mechanisms of relaxation of close pairs as well as the structure of the impurity band in semiconductors. 5. Conclusions Thus the intensity dependence of microwave absorption in semiconductors has two stages. This is a general property of entities which contain two-level systems with a broad distribution of relaxation times. Typical for semiconductors is a very low threshold for the onset of non-linearity of the resonant absorption which is due to high values of the dipole moments of the close pairs. On the other hand, the non-linearity of the relaxational absorption should occur at much higher intensities and sometimes can even be unobservable because of the impurity breakdown. In other words the critical amplitudes ~cl and ~cz for the onset of both non-linearities in semiconductors may differ at the same temperature and frequency by several orders ofmagnitude.* We should also like to emphasize that the lower is the temperature the more pronounced are the non-linear phenomena discussed. In deriving the basic results we have been working primarily from the model *So far we do not know of any experiments in semiconductors where such a two-stage behaviour has been observed. However, behaviour of this type was observed in metallic glasses (amorphous metals) by Hikata et al. (1982). Interpretation of these experimental data along lines identical to those discussed above is given by Galperin et al. (1984a, b). 122 Yu.M. Galperin et al. considered by Shklovskii and Efros ( 1981). It was assumed there that the energy spread of the levels is of non-Coulombic origin and is much greater than the Coulomb interaction of the carriers over distances of the order of r, the average distance between the carriers. A model of this type can probably correctly describe the situation in some amorphous materials, but it is generally unjustified for describing the hopping conductivity in doped semiconductors, where the level spread results from the interaction of electrons with charged centers. However, it turns out that in doped crystalline semiconductors the frequency, temperature and amplitude dependences in which we are interested are the same as predicted by the model used above. In this case to obtain an order-ofmagnitude estimates for <1(w) one should replace the density of states g in the expressions given above by an effective density of states, geff, which depends on the concentrations of donors (N D ) and acceptors (N A ). Here we shall give the order-of-magnitude estimates for geff' for details see Galperin et al. (1983) and Efros and Shklovskii (1985). In the case of a weakly doped, weakly compensated semiconductor with NA ~ND' ( 111) This expression provides a correct order-of-magnitude estimate even for the case of intermediate compensation for N A c:::: N D as well. The case of strong compensation is more complicated. Nevertheless, one can use the estimate (111) to within logarithmic accuracy. We should like to stress one more important point. 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