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. We assume the relaxation
processes in the phonon system to be effective enough that one can consider
phonons as equilibrium ones. On the other hand, at low temperatures the
phenomenon of a phonon bottleneck may be, in principle, important for the
non-linear absorption we are discussing. The role of non-equilibrium phonons
has been discussed in detail for TLS in glasses (see Gurevich and Rzayev 1987,
Parshin and Rzayev 1988) and in semiconductors (Parshin and Rzaev 1987).
So to make possible a direct comparison with the results discussed in the present
review one should be careful to choose the experimental conditions to eliminate
the influence of the non-equilibrium phonons.
In conclusion we want to repeat that many of the results and conclusions of
this review also apply to the absorption of sound in semiconductors in the
hopping conductivity regime.
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