- Email: [email protected]
Hopping Hall conductivity — A numerical study
Physica B 167 (1990) 1-6 North-Holland
HOPPING HALL C O N D U C T I V I T Y - A NUMERICAL STUDY
R. NI~METH 1 Universitiit zu K6ln, lnstitut fiir Theoretische Physik, Ziilpicher Str. 77, D-5000 K6ln 41, FRG Received 15 January 1990 Revised manuscript received 26 April 1990
We investigate numerically the temperature dependence of the Hall conductivity in two and three dimensions with different electronic densities of states. For relatively high temperatures a simple three-site optimization procedure fits the results rather well. There are deviations from the exponential dependence at lower temperatures which can be explained with the increased role of fluctuations. We demonstrate also the nonapplicabitity of the percolation model concerning the fluctuations of physical quantities and comment the present experimental situation as well.
1. Introduction
The Hall coefficient, which is a useful measure for charge carrier concentration in materials with conduction electrons, may provide further insight into the transport of isolators as well, although it is not coupled directly to the charge carrier concentration any more. The simple Drude model, which yields the classical relationship between the Hall coefficient and carrier concentration, is no longer valid in the insulator phase, where the conduction takes place through phonon-assisted hopping between localized states. Thus it is not even immediately obvious, whether there is a Hall effect in the hopping regime at all. The route of charge carriers moving in extended states can be easily influenced by an external magnetic field, but for carriers hopping between localized states it is not trivial how a magnetic field can change the direction of motion. Holstein [1] was the first to calculate a nonvanishing Hall coefficient in the impurity conduction regime. His results were generalized later and used to create an equivalent resistor network
1 Present address: Institut fiir Festk6rperforschung der KFA J/ilich; Postfach 1913, D-5170 J/ilich 1, FRG.
with site or site-and-energy disorder [2-5]. However, the experimental verification of the theoretical predictions about e.g. the temperature dependence has been absent so far. The first qualitative measurements on the temperature dependence were published very recently [6, 7] and we have not had any opportunity so far to decide between the different analytical approaches [2-
5]. Motivated by the experimental decision possibility we present a numerical study on the temperature dependence of the Hall mobility associated with hopping conduction. The calculation is performed in the framework of the percolation theory considering spatially and energetically random site distribution with different densities of states. At higher temperatures we find satisfactory agreement with the three-site optimization procedure of ref. [8], but we do not see a simple exponential behaviour generally. This feature will be explained with the increased role of the fluctuations at low temperatures. In section 2, we calculate the different probability densities of the percolation network and derive the temperature dependence of the diagonal conductivity. In section 3, we present the results of the numerical integration and compare them to former ones. Section 4 involves a discussion and a comparison with experiments.
0921-4526/90/$03.50 (~ 1990- Elsevier Science Publishers B.V. (North-Holland)
R. NOmeth / Hopping Hall conductivity 2. T h e p e r c o l a t i o n n e t w o r k
The problem of the DC hopping conduction can be transformed into the calculation of the conductivity of an equivalent resistor network [2, 9]. The impedances Zq (between site i and j) forming this network are: Zij = e2/31Vij .
Here/3 = 1/kBT and Fq is the two site jump rate
[lO1: Fq = const
e - 2°~rilaij
,
Aij = e-~(IEi-EjI+IEiI+IEII)/2 ,
where ~ denotes the inverse localization length (we choose /x--0 for the chemical potential). The conductivity of the whole system will be determined by the critical, i.e. just percolating network and its impedance can be expressed with a critical exponent ~ in the following way:
where ( . . . ) means the average with n(E, ~) as distribution function:
( f ) = ~ dEn(E, ~)N(E)f(E) f dEn(E, ¢)N(E) If f depends on several energies (which is the case by e.g. the Hall mobility) then we have to use one n(E, ~) function for each of these variables. The concrete values of u are known from the percolation theory (see e.g. ref. [2] and references therein): u ( 2 D ) = 4 . 5 and v ( 3 D ) = 2.7. In this model the n(E, ~) appears as the randomness parameter and plays a similar role as the bond concentration in the simple percolation problem. Now we calculate the n(E, ~) functions in the following four cases: (i) 3D and N(E) = N o (constant DOS in three dimensions);
(ii) Z~ x = const/3
4~ N o 3 2 (/3/2~)3(sc-[E])3(~ + [El).
n(E, ~ ) -
3D and N(E)= p3EZ;
e t3e .
The critical network is built up from impedances smaller than or equal to Z c. The number of such impedances n(E i, ~) linked to a given site i is obtained by integrating over a cylinder-like volume [2, 3]:
n(Ei, ¢) = f dEj drqN(Ej)4crr,2j
n(E, ~ ) - 4rr P3 (/3/2a)3(~:- [El)3 3 60 × (2~ 3 +
(iii)
n(E, se) (iv)
2 where O is the Heaviside step function and N(E) is the density of states (DOS). (In this article we use the bond percolation model, but there are some other possible choices, e.g. the original volume model of ref. [10], but the results should be qualitatively independent of the concrete choice.) Percolation sets in if the average number of bonds per site is equal to the percolation threshold v:
v = (n(E i, ~)) ,
2 D and
9~IEI 2 +
N(E) =
9]E13)
.
No;
No = ~ -~- ( / 3 / 2 a ) 2 ( s ~ - [El)2(2s~ + I E I ) .
2D and N(E)= p2E;
102 n(E, ~ ) = w-6( / 3 / 2 ~ ) 2 ( ~ - IEI)2(~2 + 2IEI2) "
Having these functions we can evaluate the /3~: dependence of the diagonal conductivity. For P2 and P3 we choose: 102 =
2K 2 4 we
and
P3-
3K 3 6 , "lYe
because we want to compare our results to the ones of ref. [11a, b], where the influence of the Coulomb interaction on the DOS and on the
R. NOmeth I Hopping Hall conductivity
diagonal conductivity was treated. We want to emphasize, however, that our treatment is not the exact one, what the Coulomb interaction concerns. One should take care about the electron correlations at each hop as was done in ref. [llb]. Ours is more a mean field like approximation for the interacting electrons, but it may give some insight into the behaviour of the original system. Besides, we can simply imagine models with power law DOS, which was first treated in ref. [12]. On the other hand it is still a problem, whether we are allowed to use the one electron hopping picture in the interacting case or we should take into consideration the many particle transitions as well [13]. At T = 0 the predictions of the theory of ref. [11] are in excellent agreement with computer simulations (see ref. [lla, b] and references therein) and we shall use the above DOS in the following. After these comments let us treat the temperature dependence of the diagonal conductivity in the above listed cases. Performing the indicated averages we arrive at the following results: (i)
/3~ = 2 . 0 3 ( f l a 3 / N o ) 1/4 ,
(ii)
fl~ = 2.82~/-fl e2a
As a next point we want to treat the hopping in magnetic field (directed perpendicular to the electric one). To get a nonvanishing off-diagonal conductivity one has to take into account hopping processes of higher order, i.e. when the electronic transitions involve more than two sites. This problem was first investigated by Holstein [1], who showed, that in weak magentic fields the three-site processes dominate. It means e.g. that we have to calculate the three-site overlap integrals by the tunnelling term. Utilizing that the influence of the magnetic field can be described by a phase factor in the wave function, one can calculate explicitly the three-site transition probabilities, W,.pj [2, 10]: Wip j = WoAipjAip j e-'~(rip+rm+rii )
where V¢0 is a constant, A~pj is given by: Aip j = l ( AjiAip e ~rE,t + ApjAji e~lEJI -[- aipApj et31EPI) ,
and Aip j results from the Lorentz force: A,pj = ½Hl(rpi , r j)l •
K
(iii)
/3~
=
3 ( [ 3 o t 2 / N o ) 1/3 ,
(iv) /3~?=3.97
e
.
The case (i), which is the most investigated one, is the same as in ref. [2]. The numerical factor of ref. [11] in case (ii) is 1.68, which shows that this mean field like approximation is a rather good estimation. We could choose, of course, such numerical factors in the expression of P2 and P3 (instead of 2/w and 3/~r) which reproduce the results of ref. [11], but these factors do not appear in the Hall mobility and so we do not bother about them. These temperature dependences have been calculated to show the effectiveness of this approximation and to check the correctness of the n ( E , ~) functions.
3
However, not all the three-site clusters contribute, but only the configurations where no short cuts are present [2]. Applying Kirchoff's rules one can deduce the off-diagonal conductivity of the sample:
We evaluate the Hall mobility (/~H) instead of Orxy Orxy ~LH -- Ho.xx '
since we do not want to treat the explicit field dependence. During the averaging process the following integration has to be performed:
4
R, N~meth / Hopping Hall conductivity
<.n= f dEif dE;f dE,f × X
f
f d4
)n(e;)n(Ej).(Ep) f(Ei,
El, Ep,
ri, r/, rp)
x[fdEifdEjfdEpfdr x
We use two different averaging processes as possible explanation of the numerical data: (a) we take the values of the simple Mort optimization for the diagonal conductivity (these values were used in ref. [3] to explain the numerical data found), and (b) we use a three-site optimization [8]. In both cases we find a simple lntz H = -e/3~ dependence where e is a numerical parameter. Its values are the following in the different cases
fdr3jfdr3p
N(Ei)N(Ej)N(Ep)n(Ei)n(E,)n
X
Moreover one has inequalities
to take
care
about
the
O
< 2~ ( ~ - (Ig'l + lEml + lgm - E'[) for each pair of indices. By the analytic evaluation of some of the integrals we can simplify the above expression (e.g. one can drop the denominator if we are interested only in the /3~ dependence), but we have to carry out numerical integration, and so we do not go into the details of the structure of the above integrals. Since we have a high dimensional integral we chose the Monte Carlo integration method to evaluate it. The results and interpretations can be found in the next section.
(i)
Ea= 3 ,
Eb=(-32)3/4--1=0.354
,
(ii)
ea= ¼, eb=(3)'/2-1=0.225,
(iii)
e,,=.{,
eb = ( 3 ) 2 / 3 - 1 = 0 . 3 1 0 ,
(iv)
E.=¼,
eb=(3)1/2--1=0.225.
At smaller values of/3~: (i.e. at higher temperatures) we can fit the curves with a straight line (the dashed lines in fig. 1). We find the following slopes (fig. 1):
(i)
0.36-+0.01,
(iii)
0.31+0.01,
(ii)
0.19_+0.01,
(iv)
0.20_+0.01.
4
~ l l l l l l l l l l l l l l l l l ] l l l l
I II
I
I~T
E o
3. Results and discussion In this section we present the numerical data and compare them to former works. The simplest approximation, which we can compare to our results (see the figures), is the following: we substitute some average values for the actual ones of E and r in the integral taking into consideration that the equation
fl~ = 2 a r o + f i E o has to be satisfied (E 0 and r 0 denote the average values). Then we arrive at the equation /.£H ~
e-~r0
-E ~:
\0.
4 ~-'+_ -6
EL
-In
"o.
o "+--~ -~ ~ +
-0 -io
0
12 A
A
-14 -16 ~II]IIII[III]JL i0 15 20
~11111l~1
R5
(To /
30
35
[
40
I
45
T) x
Fig. 1. The logarithm of the Hall conductivity (in arbitrary units) vs. (To/T) x where (i) x = 0 . 2 5 , ©, (ii) 0.5, +; (iii) 0.33, E3, and (iv) 0.5, A. The dotted lines are the fits of the simple exponential dependence.
R. N~meth / Hopping Hall conductivity
(The errors were estimated from the MC integration.) These values are in fairly good agreement with the above approximate values and confirm the three-site optimization procedure [8]. However, there are some other common characteristics of the above figures which can be found in refs. [4] and [5] as well: one sees a larger Hall mobility at low temperatures as could be expected from the continuation of the simple exponential behaviour. We think that it can be explained, at least qualitatively, if we take into account the direction fluctuations of the electron hops. During the diagonal conduction (H = 0) the electrons hop forward, backward and in all the possible directions with an effective drift parallel to the electric field. At each hop the electron has to find an end-site within the energy interval 8E~-kBT. As long as 8E is rather large, the electron has the opportunity to choose among many possible end-places. In this regime the optimization procedure works good since the electron will choose the more probable tracks. As we decrease the temperature the number of the possible end-sites decreases as well and the electron has less chance to compensate a possible hop in one of the off directions. Such an "offdiagonal" hop will be, of course, compensated on longer distances and it does not affect the diagonal conductivity. However, if we switch on a magnetic field, which prefers one of the off directions, then these fluctuations will be "activated" and we can see a larger off diagonal conductivity. This trend is demonstrated by all of the curves in fig. 1. To show explicitly the importance of the fluctuation we calculated the average of the nth power o f / z n for n = 1, 2 and 3 in three dimension with constant DOS (fig. 2). For high temperatures the curves can be fitted with a straight line (which corresponds to a simple exponential function) with the following slopes: n=l,
0.36-+0.01,
n=2,
0.60_+0.02,
n=3,
0.75___0.03.
8
ii1~
5
i,~
I
,,i,,flllJb,l,ll~llll,
I
_\
4
_~
\
2
k\ _
\
o~ A El \ "ID
V
--
~\
(~
~I2L /tl zx
-8
o
-I0
o
o o
o
-12 -14
o 13
II LI I l l l l l l
i0
15
t llll
I Illll~ll
20
25
30
r ll~lilll
35
40
I
45
( T o / T ) 1/4 Fig. 2. The different moments of the Hall conductivity vs. (To~T) '1~ in case (i): n = l ©, n = 2 ; [] and n = 3 ; A.
The errors result, as before, from the MC integration. There are two important points, which should be emphasized at this stage: (a) ((tz 2 ) (/Zn) z ~ 0 , i.e. there is no self-averaging within this model, and (b) the slopes cannot be fitted with a straight line as a function of n. Point (a) might be a bit surprising at first sight, since all the physical (measurable) quantities should have a definite value in the bulk limit. The inequality ( ( / Z 2 ) - - ( / X H ) 2 ) ~ 0 has, of course, no serious consequences regarding the real samples; it is a peculiarity of the applied model. It means that the conjecture of ref. [10] about the applicability of the percolation network model works rather good for the averages, but has no relevance concerning the fluctuations. Since the large exponents and low temperatures play a similar role, it follows immediately, that the conjecture does not work at low temperatures. Point (b) also strengthens the importance of the fluctuations. The anomalous scaling of the exponents reminds us of the multifractal behaviour of the underlying critical network [14]. The multifractality means, that the different parts of the system, i.e. the fluctuations of the available spatial regimes are important. This is the first explicit appearance of this effect, although it was argued in ref. [15] that the percola-
6
R. N~meth / Hopping Hall conductivity
tion picture m a y be w r o n g if the electron does not have the o p p o r t u n i t y to choose a m o n g different sites.
4. Conclusion and discussion We p e r f o r m e d numerical calculations about the t e m p e r a t u r e d e p e n d e n c e of the Hall mobility in four different cases. A t relative high t e m p e r a tures we f o u n d a linear d e p e n d e n c e of ln/x H as a function of/3~ (exponent of the diagonal conductivity). T h e results confirm the three-site optimization p r o c e d u r e of ref. [8] and agree with the results of ref. [3], where the two works overlap. To explain the w e a k e r d e p e n d e n c e at lower temperatures we speculate that it m a y be due to the depressed fluctuation of the available site energies. Besides that we d e m o n s t r a t e d , calculating the averages of higher powers, that the percolation m o d e l is no m o r e valid concerning the fluctuations of the different quantities. K o o n and C a s t n e r [6] f o u n d in their experim e n t for the slope of ln/x H (see fig. 1) 0.37 + 0.02, which is in a g r e e m e n t with o u r numerical results and with b o t h of the m e n t i o n e d approximations. Their o t h e r finding, n a m e l y that the slope decreases a p p r o a c h i n g the m e t a l - i n s u l a t o r transition, can also be explained on the basis of the above data, if one assumes that the C o u l o m b interaction plays also s o m e role at lower concentrations and its influence will be depressed if the c o n c e n t r a t i o n increases. R o y et al. [7] f o u n d also a Mott-like b e h a v i o u r in their experiment, although the slope was bigger than the values predicted above or the ones of ref. [6]. T h e y m e a s u r e d the diagonal c o n d u c tivity as well and f o u n d a positive m a g n e t o conductivity. Using the expression for /xH one can easily see that the positive m a g n e t o c o n d u c tivity increases the slope. H o w e v e r , at present there is no t h e o r y which can m a k e a prediction about the t e m p e r a t u r e and field d e p e n d e n c e involving both effects of the magnetic field. We have treated only the D C Hall-effect so far. It is possible to give a universal description
for all frequencies e.g. within the so called 'ext e n d e d pair a p p r o x i m a t i o n ' [16]. In this m e t h o d one takes into a c c o u n t three sites explicitly and treats the rest of the system in a mean-field like m a n n e r . A t the m o m e n t there is no direct generalization of the percolation description to finite frequencies. U n f o r t u n a t e l y , there are no o t h e r experiments which could be qualitatively c o m p a r e d to the results above. We hope, h o w e v e r , that this p a p e r motivates further effort in this field and that our predictions will be c h e c k e d experimentally in the near future. I thank B. Miihlschlegel for useful discussions.
References [1] T. Holstein, Phys. Rev. 124 (1961) 1329. [2] H. B6ttger and V. Bryksin, Hopping conduction in solids (VHC, 1985). [3] M. Gr/inewald, H. M/iller, P. Thomas and D. W/irtz, Solid State Commun. 38 (1981) 1011. [4] M. Griinewald, H. M/iller and D. Wfirtz, Solid State Commun. 43 (1982) 419. [5] L. Friedman and M. Pollak, Phil. Mag. B 44 (1981) 487. [6] D.W. Koon and T.G. Castner, Solid State Commun 64 (1987) 11. [7] A. Roy, M. Levy, X.M. Guo, M.P. Sarachik, R. Ledesma and L.L. Isaacs, Phys. Rev. B 39 (1989) 10185. [8] R. N6meth and B. Mfihlschlegel, Solid State Commun. 66 (1988) 999. [9] A. Miller and E. Abrahams, Phys. Rev. 120 (1960) 745. [10] V. Ambegaokar, B.I. Halperin and J.S. Langer, Phys. Rev. B 4 (1971) 2612. [11] A.L. Efros and B.I. Shklovskii (a) in: Electron-electron interaction in disordered systems, A.L. Efros and M. Pollak, eds. (North-Holland, Amsterdam, 1985). (b) Electronic properties of doped semiconductors (Springer, Berlin, 1984). [12] M. Pollak, J. Non-Cryst. Solids 11 (1972) 1. [13] M. Pollak and M. Ortuno, in: Electron-electron interaction in disordered systems, A.L. Efros and M. Pollak, eds. (North-Holland, Amsterdam, 1985). [14] L. De Arcangelis, S. Redner and A. Coniglio, Phys. Rev. B 31 (1985) 4725. [15] A.S. Skal and B.I. Shklovskii, Sov. Phys. Semicond 8 1975) 1029. [16] P.N. Butcher, J.A. Mclnnes and S. Summerfield, Phil. Mag. 48 (1983) 551.
HOPPING HALL C O N D U C T I V I T Y - A NUMERICAL STUDY
R. NI~METH 1 Universitiit zu K6ln, lnstitut fiir Theoretische Physik, Ziilpicher Str. 77, D-5000 K6ln 41, FRG Received 15 January 1990 Revised manuscript received 26 April 1990
We investigate numerically the temperature dependence of the Hall conductivity in two and three dimensions with different electronic densities of states. For relatively high temperatures a simple three-site optimization procedure fits the results rather well. There are deviations from the exponential dependence at lower temperatures which can be explained with the increased role of fluctuations. We demonstrate also the nonapplicabitity of the percolation model concerning the fluctuations of physical quantities and comment the present experimental situation as well.
1. Introduction
The Hall coefficient, which is a useful measure for charge carrier concentration in materials with conduction electrons, may provide further insight into the transport of isolators as well, although it is not coupled directly to the charge carrier concentration any more. The simple Drude model, which yields the classical relationship between the Hall coefficient and carrier concentration, is no longer valid in the insulator phase, where the conduction takes place through phonon-assisted hopping between localized states. Thus it is not even immediately obvious, whether there is a Hall effect in the hopping regime at all. The route of charge carriers moving in extended states can be easily influenced by an external magnetic field, but for carriers hopping between localized states it is not trivial how a magnetic field can change the direction of motion. Holstein [1] was the first to calculate a nonvanishing Hall coefficient in the impurity conduction regime. His results were generalized later and used to create an equivalent resistor network
1 Present address: Institut fiir Festk6rperforschung der KFA J/ilich; Postfach 1913, D-5170 J/ilich 1, FRG.
with site or site-and-energy disorder [2-5]. However, the experimental verification of the theoretical predictions about e.g. the temperature dependence has been absent so far. The first qualitative measurements on the temperature dependence were published very recently [6, 7] and we have not had any opportunity so far to decide between the different analytical approaches [2-
5]. Motivated by the experimental decision possibility we present a numerical study on the temperature dependence of the Hall mobility associated with hopping conduction. The calculation is performed in the framework of the percolation theory considering spatially and energetically random site distribution with different densities of states. At higher temperatures we find satisfactory agreement with the three-site optimization procedure of ref. [8], but we do not see a simple exponential behaviour generally. This feature will be explained with the increased role of the fluctuations at low temperatures. In section 2, we calculate the different probability densities of the percolation network and derive the temperature dependence of the diagonal conductivity. In section 3, we present the results of the numerical integration and compare them to former ones. Section 4 involves a discussion and a comparison with experiments.
0921-4526/90/$03.50 (~ 1990- Elsevier Science Publishers B.V. (North-Holland)
R. NOmeth / Hopping Hall conductivity 2. T h e p e r c o l a t i o n n e t w o r k
The problem of the DC hopping conduction can be transformed into the calculation of the conductivity of an equivalent resistor network [2, 9]. The impedances Zq (between site i and j) forming this network are: Zij = e2/31Vij .
Here/3 = 1/kBT and Fq is the two site jump rate
[lO1: Fq = const
e - 2°~rilaij
,
Aij = e-~(IEi-EjI+IEiI+IEII)/2 ,
where ~ denotes the inverse localization length (we choose /x--0 for the chemical potential). The conductivity of the whole system will be determined by the critical, i.e. just percolating network and its impedance can be expressed with a critical exponent ~ in the following way:
where ( . . . ) means the average with n(E, ~) as distribution function:
( f ) = ~ dEn(E, ~)N(E)f(E) f dEn(E, ¢)N(E) If f depends on several energies (which is the case by e.g. the Hall mobility) then we have to use one n(E, ~) function for each of these variables. The concrete values of u are known from the percolation theory (see e.g. ref. [2] and references therein): u ( 2 D ) = 4 . 5 and v ( 3 D ) = 2.7. In this model the n(E, ~) appears as the randomness parameter and plays a similar role as the bond concentration in the simple percolation problem. Now we calculate the n(E, ~) functions in the following four cases: (i) 3D and N(E) = N o (constant DOS in three dimensions);
(ii) Z~ x = const/3
4~ N o 3 2 (/3/2~)3(sc-[E])3(~ + [El).
n(E, ~ ) -
3D and N(E)= p3EZ;
e t3e .
The critical network is built up from impedances smaller than or equal to Z c. The number of such impedances n(E i, ~) linked to a given site i is obtained by integrating over a cylinder-like volume [2, 3]:
n(Ei, ¢) = f dEj drqN(Ej)4crr,2j
n(E, ~ ) - 4rr P3 (/3/2a)3(~:- [El)3 3 60 × (2~ 3 +
(iii)
n(E, se) (iv)
2 where O is the Heaviside step function and N(E) is the density of states (DOS). (In this article we use the bond percolation model, but there are some other possible choices, e.g. the original volume model of ref. [10], but the results should be qualitatively independent of the concrete choice.) Percolation sets in if the average number of bonds per site is equal to the percolation threshold v:
v = (n(E i, ~)) ,
2 D and
9~IEI 2 +
N(E) =
9]E13)
.
No;
No = ~ -~- ( / 3 / 2 a ) 2 ( s ~ - [El)2(2s~ + I E I ) .
2D and N(E)= p2E;
102 n(E, ~ ) = w-6( / 3 / 2 ~ ) 2 ( ~ - IEI)2(~2 + 2IEI2) "
Having these functions we can evaluate the /3~: dependence of the diagonal conductivity. For P2 and P3 we choose: 102 =
2K 2 4 we
and
P3-
3K 3 6 , "lYe
because we want to compare our results to the ones of ref. [11a, b], where the influence of the Coulomb interaction on the DOS and on the
R. NOmeth I Hopping Hall conductivity
diagonal conductivity was treated. We want to emphasize, however, that our treatment is not the exact one, what the Coulomb interaction concerns. One should take care about the electron correlations at each hop as was done in ref. [llb]. Ours is more a mean field like approximation for the interacting electrons, but it may give some insight into the behaviour of the original system. Besides, we can simply imagine models with power law DOS, which was first treated in ref. [12]. On the other hand it is still a problem, whether we are allowed to use the one electron hopping picture in the interacting case or we should take into consideration the many particle transitions as well [13]. At T = 0 the predictions of the theory of ref. [11] are in excellent agreement with computer simulations (see ref. [lla, b] and references therein) and we shall use the above DOS in the following. After these comments let us treat the temperature dependence of the diagonal conductivity in the above listed cases. Performing the indicated averages we arrive at the following results: (i)
/3~ = 2 . 0 3 ( f l a 3 / N o ) 1/4 ,
(ii)
fl~ = 2.82~/-fl e2a
As a next point we want to treat the hopping in magnetic field (directed perpendicular to the electric one). To get a nonvanishing off-diagonal conductivity one has to take into account hopping processes of higher order, i.e. when the electronic transitions involve more than two sites. This problem was first investigated by Holstein [1], who showed, that in weak magentic fields the three-site processes dominate. It means e.g. that we have to calculate the three-site overlap integrals by the tunnelling term. Utilizing that the influence of the magnetic field can be described by a phase factor in the wave function, one can calculate explicitly the three-site transition probabilities, W,.pj [2, 10]: Wip j = WoAipjAip j e-'~(rip+rm+rii )
where V¢0 is a constant, A~pj is given by: Aip j = l ( AjiAip e ~rE,t + ApjAji e~lEJI -[- aipApj et31EPI) ,
and Aip j results from the Lorentz force: A,pj = ½Hl(rpi , r j)l •
K
(iii)
/3~
=
3 ( [ 3 o t 2 / N o ) 1/3 ,
(iv) /3~?=3.97
e
.
The case (i), which is the most investigated one, is the same as in ref. [2]. The numerical factor of ref. [11] in case (ii) is 1.68, which shows that this mean field like approximation is a rather good estimation. We could choose, of course, such numerical factors in the expression of P2 and P3 (instead of 2/w and 3/~r) which reproduce the results of ref. [11], but these factors do not appear in the Hall mobility and so we do not bother about them. These temperature dependences have been calculated to show the effectiveness of this approximation and to check the correctness of the n ( E , ~) functions.
3
However, not all the three-site clusters contribute, but only the configurations where no short cuts are present [2]. Applying Kirchoff's rules one can deduce the off-diagonal conductivity of the sample:
We evaluate the Hall mobility (/~H) instead of Orxy Orxy ~LH -- Ho.xx '
since we do not want to treat the explicit field dependence. During the averaging process the following integration has to be performed:
4
R, N~meth / Hopping Hall conductivity
<.n= f dEif dE;f dE,f × X
f
f d4
)n(e;)n(Ej).(Ep) f(Ei,
El, Ep,
ri, r/, rp)
x[fdEifdEjfdEpfdr x
We use two different averaging processes as possible explanation of the numerical data: (a) we take the values of the simple Mort optimization for the diagonal conductivity (these values were used in ref. [3] to explain the numerical data found), and (b) we use a three-site optimization [8]. In both cases we find a simple lntz H = -e/3~ dependence where e is a numerical parameter. Its values are the following in the different cases
fdr3jfdr3p
N(Ei)N(Ej)N(Ep)n(Ei)n(E,)n
X
Moreover one has inequalities
to take
care
about
the
O
< 2~ ( ~ - (Ig'l + lEml + lgm - E'[) for each pair of indices. By the analytic evaluation of some of the integrals we can simplify the above expression (e.g. one can drop the denominator if we are interested only in the /3~ dependence), but we have to carry out numerical integration, and so we do not go into the details of the structure of the above integrals. Since we have a high dimensional integral we chose the Monte Carlo integration method to evaluate it. The results and interpretations can be found in the next section.
(i)
Ea= 3 ,
Eb=(-32)3/4--1=0.354
,
(ii)
ea= ¼, eb=(3)'/2-1=0.225,
(iii)
e,,=.{,
eb = ( 3 ) 2 / 3 - 1 = 0 . 3 1 0 ,
(iv)
E.=¼,
eb=(3)1/2--1=0.225.
At smaller values of/3~: (i.e. at higher temperatures) we can fit the curves with a straight line (the dashed lines in fig. 1). We find the following slopes (fig. 1):
(i)
0.36-+0.01,
(iii)
0.31+0.01,
(ii)
0.19_+0.01,
(iv)
0.20_+0.01.
4
~ l l l l l l l l l l l l l l l l l ] l l l l
I II
I
I~T
E o
3. Results and discussion In this section we present the numerical data and compare them to former works. The simplest approximation, which we can compare to our results (see the figures), is the following: we substitute some average values for the actual ones of E and r in the integral taking into consideration that the equation
fl~ = 2 a r o + f i E o has to be satisfied (E 0 and r 0 denote the average values). Then we arrive at the equation /.£H ~
e-~r0
-E ~:
\0.
4 ~-'+_ -6
EL
-In
"o.
o "+--~ -~ ~ +
-0 -io
0
12 A
A
-14 -16 ~II]IIII[III]JL i0 15 20
~11111l~1
R5
(To /
30
35
[
40
I
45
T) x
Fig. 1. The logarithm of the Hall conductivity (in arbitrary units) vs. (To/T) x where (i) x = 0 . 2 5 , ©, (ii) 0.5, +; (iii) 0.33, E3, and (iv) 0.5, A. The dotted lines are the fits of the simple exponential dependence.
R. N~meth / Hopping Hall conductivity
(The errors were estimated from the MC integration.) These values are in fairly good agreement with the above approximate values and confirm the three-site optimization procedure [8]. However, there are some other common characteristics of the above figures which can be found in refs. [4] and [5] as well: one sees a larger Hall mobility at low temperatures as could be expected from the continuation of the simple exponential behaviour. We think that it can be explained, at least qualitatively, if we take into account the direction fluctuations of the electron hops. During the diagonal conduction (H = 0) the electrons hop forward, backward and in all the possible directions with an effective drift parallel to the electric field. At each hop the electron has to find an end-site within the energy interval 8E~-kBT. As long as 8E is rather large, the electron has the opportunity to choose among many possible end-places. In this regime the optimization procedure works good since the electron will choose the more probable tracks. As we decrease the temperature the number of the possible end-sites decreases as well and the electron has less chance to compensate a possible hop in one of the off directions. Such an "offdiagonal" hop will be, of course, compensated on longer distances and it does not affect the diagonal conductivity. However, if we switch on a magnetic field, which prefers one of the off directions, then these fluctuations will be "activated" and we can see a larger off diagonal conductivity. This trend is demonstrated by all of the curves in fig. 1. To show explicitly the importance of the fluctuation we calculated the average of the nth power o f / z n for n = 1, 2 and 3 in three dimension with constant DOS (fig. 2). For high temperatures the curves can be fitted with a straight line (which corresponds to a simple exponential function) with the following slopes: n=l,
0.36-+0.01,
n=2,
0.60_+0.02,
n=3,
0.75___0.03.
8
ii1~
5
i,~
I
,,i,,flllJb,l,ll~llll,
I
_\
4
_~
\
2
k\ _
\
o~ A El \ "ID
V
--
~\
(~
~I2L /tl zx
-8
o
-I0
o
o o
o
-12 -14
o 13
II LI I l l l l l l
i0
15
t llll
I Illll~ll
20
25
30
r ll~lilll
35
40
I
45
( T o / T ) 1/4 Fig. 2. The different moments of the Hall conductivity vs. (To~T) '1~ in case (i): n = l ©, n = 2 ; [] and n = 3 ; A.
The errors result, as before, from the MC integration. There are two important points, which should be emphasized at this stage: (a) ((tz 2 ) (/Zn) z ~ 0 , i.e. there is no self-averaging within this model, and (b) the slopes cannot be fitted with a straight line as a function of n. Point (a) might be a bit surprising at first sight, since all the physical (measurable) quantities should have a definite value in the bulk limit. The inequality ( ( / Z 2 ) - - ( / X H ) 2 ) ~ 0 has, of course, no serious consequences regarding the real samples; it is a peculiarity of the applied model. It means that the conjecture of ref. [10] about the applicability of the percolation network model works rather good for the averages, but has no relevance concerning the fluctuations. Since the large exponents and low temperatures play a similar role, it follows immediately, that the conjecture does not work at low temperatures. Point (b) also strengthens the importance of the fluctuations. The anomalous scaling of the exponents reminds us of the multifractal behaviour of the underlying critical network [14]. The multifractality means, that the different parts of the system, i.e. the fluctuations of the available spatial regimes are important. This is the first explicit appearance of this effect, although it was argued in ref. [15] that the percola-
6
R. N~meth / Hopping Hall conductivity
tion picture m a y be w r o n g if the electron does not have the o p p o r t u n i t y to choose a m o n g different sites.
4. Conclusion and discussion We p e r f o r m e d numerical calculations about the t e m p e r a t u r e d e p e n d e n c e of the Hall mobility in four different cases. A t relative high t e m p e r a tures we f o u n d a linear d e p e n d e n c e of ln/x H as a function of/3~ (exponent of the diagonal conductivity). T h e results confirm the three-site optimization p r o c e d u r e of ref. [8] and agree with the results of ref. [3], where the two works overlap. To explain the w e a k e r d e p e n d e n c e at lower temperatures we speculate that it m a y be due to the depressed fluctuation of the available site energies. Besides that we d e m o n s t r a t e d , calculating the averages of higher powers, that the percolation m o d e l is no m o r e valid concerning the fluctuations of the different quantities. K o o n and C a s t n e r [6] f o u n d in their experim e n t for the slope of ln/x H (see fig. 1) 0.37 + 0.02, which is in a g r e e m e n t with o u r numerical results and with b o t h of the m e n t i o n e d approximations. Their o t h e r finding, n a m e l y that the slope decreases a p p r o a c h i n g the m e t a l - i n s u l a t o r transition, can also be explained on the basis of the above data, if one assumes that the C o u l o m b interaction plays also s o m e role at lower concentrations and its influence will be depressed if the c o n c e n t r a t i o n increases. R o y et al. [7] f o u n d also a Mott-like b e h a v i o u r in their experiment, although the slope was bigger than the values predicted above or the ones of ref. [6]. T h e y m e a s u r e d the diagonal c o n d u c tivity as well and f o u n d a positive m a g n e t o conductivity. Using the expression for /xH one can easily see that the positive m a g n e t o c o n d u c tivity increases the slope. H o w e v e r , at present there is no t h e o r y which can m a k e a prediction about the t e m p e r a t u r e and field d e p e n d e n c e involving both effects of the magnetic field. We have treated only the D C Hall-effect so far. It is possible to give a universal description
for all frequencies e.g. within the so called 'ext e n d e d pair a p p r o x i m a t i o n ' [16]. In this m e t h o d one takes into a c c o u n t three sites explicitly and treats the rest of the system in a mean-field like m a n n e r . A t the m o m e n t there is no direct generalization of the percolation description to finite frequencies. U n f o r t u n a t e l y , there are no o t h e r experiments which could be qualitatively c o m p a r e d to the results above. We hope, h o w e v e r , that this p a p e r motivates further effort in this field and that our predictions will be c h e c k e d experimentally in the near future. I thank B. Miihlschlegel for useful discussions.
References [1] T. Holstein, Phys. Rev. 124 (1961) 1329. [2] H. B6ttger and V. Bryksin, Hopping conduction in solids (VHC, 1985). [3] M. Gr/inewald, H. M/iller, P. Thomas and D. W/irtz, Solid State Commun. 38 (1981) 1011. [4] M. Griinewald, H. M/iller and D. Wfirtz, Solid State Commun. 43 (1982) 419. [5] L. Friedman and M. Pollak, Phil. Mag. B 44 (1981) 487. [6] D.W. Koon and T.G. Castner, Solid State Commun 64 (1987) 11. [7] A. Roy, M. Levy, X.M. Guo, M.P. Sarachik, R. Ledesma and L.L. Isaacs, Phys. Rev. B 39 (1989) 10185. [8] R. N6meth and B. Mfihlschlegel, Solid State Commun. 66 (1988) 999. [9] A. Miller and E. Abrahams, Phys. Rev. 120 (1960) 745. [10] V. Ambegaokar, B.I. Halperin and J.S. Langer, Phys. Rev. B 4 (1971) 2612. [11] A.L. Efros and B.I. Shklovskii (a) in: Electron-electron interaction in disordered systems, A.L. Efros and M. Pollak, eds. (North-Holland, Amsterdam, 1985). (b) Electronic properties of doped semiconductors (Springer, Berlin, 1984). [12] M. Pollak, J. Non-Cryst. Solids 11 (1972) 1. [13] M. Pollak and M. Ortuno, in: Electron-electron interaction in disordered systems, A.L. Efros and M. Pollak, eds. (North-Holland, Amsterdam, 1985). [14] L. De Arcangelis, S. Redner and A. Coniglio, Phys. Rev. B 31 (1985) 4725. [15] A.S. Skal and B.I. Shklovskii, Sov. Phys. Semicond 8 1975) 1029. [16] P.N. Butcher, J.A. Mclnnes and S. Summerfield, Phil. Mag. 48 (1983) 551.