The hopping Hall-mobility II - percolation theory in comparison with random walk theory

~

S o l l d S t a t e Communications, V o l . 4 3 , N o . 6 , pp.419-422, 1982. P r i n t e d in Great B r i t a i n .

0038-1098/82/300419-06503.00/0 Pergamon P r e s s Ltd.

THE HOPPINGHALL-MOBILITY I I - PERCOLATION THEORYIN COHPARISON WITH RANDOMWALKTHEORY

M. GrUnewald, H. MUller, D. WUrtz Fachbereich Physik, University of Marburg, F.R. Germany

(Received 22 March 1982 by M. Cardona) We present numerical calculations for the Hall mobility UH in spatial and energetical disordered hopping systems based on percolation theory and on the more sophisticated -andum v,alk theory. The density and temperature dependence of PH is st.udied. For a constant density of states ,H follows approximately a T'Z/~ behaviour in the low density and low temperature regime.

In a previous paperz (hereafter refered) to as (~TW) we have presented a calculation of the hopptno Hall mobility PH in the framework of oercolatton theory (PT) considering a spat i a l l y and eneroetically random site distribution. In that paper we remarked e x p l i c i t l y that within PT there is an uncertainty in the preexponential factors of the conductivity Oxx and the Hall conductivity Oxy. PT does not allow for an evaluation of prefactors unambiguously. To obtain the correct prefactors of the quantities Oxx, Oxy und "H use is made of the recently developeo~ Randum Walk Theory (RWT)2. In this paper we f i r s t discuss the results of (GMTW) in coe4~arison with recent calculations of Frtedmn and Pollak 3 (FP) and then present nu~rical calculations of the Hall mobility using the RWT. P{rcolatton theory starts from the equivalence of the OC-Hopptng conduction problem to a resistor network and assumes that nearly all of the current is carried by a c r i t i c a l path, determined essentially by the largest impedance on thts path. This concept i t certainly j u s t i fied i f the fluctuation of the i,~edances is sufficiently large. Then the conductivity Oxx of the ~ l e system is approximately given by °xx = ZcI " E1 " -

-

°xx=Zcl

With 1/4 = 2 (~ TO

The c r i t i c a l impedance Z c t s obtained by the "percolatio0-opttmization 4,s from the impedences Z ~ = ei131"tj building up the network. £iJ is the two stte-therml equilibrium jump rate between sites t and J, given by6

rij -

V2o e'Eahctj I .

Aij

(4)

)1/4 .

(5)

To obtain the absolute value of Oxx we have to know the characteristic length-E, which can roughly be taken as the separation of sites where critical paths intersect. But this quantity cannot be determined by percolation theory alone and there exist several different models in the literature (for a discussion see ref. 7) to calculate the prefactor oo containing the length L. All these models rest on the assumption that oo is a weak function of temperature T and so the conductivity shows the T-U~-behavlour. Here we are not further interested in the quantity L since we know that it cancels out in calculating the Hall effect within PT e. The Hall conductivity oxv results if higher order jump-processes are considered where one or more intermediate sites are involved s. So the Hall effect is due to at least three site hops at triads within the system of percolation paths. On the basis of this model the Hall mobtlit~ .H was calculated by SUttger and Bryksi n" yieldin~

(1)

-

~=e 2 1. exp{_(~)l/4} = ET" 1~ v2o = °o exp{_(~)l/4)T

: oxy .

Z "1

c

/

H ~rIp~

\

\riprpj+rpjrji+rjir ip/c (s)

(2)

"S Oxx~z ~

(3)

ArtpHj is the magnetic fteld dependent part of the three site transition rate with the intermediate site p given by Holstein 9

With

Atj-exp(- ~ (IEtI+IEjI

+ ]Ei-EjI))

,

where v2o ts a frequency prefactor, a"] the decay length of the localized states and B= /kT. ~t and.Et are the position,of e.ne~y msut~,d relative to the fermi levem) ot the site t , respectively. For a constant dens i t y of states N(E)= NF one obtains the famous Mutt law

~

H

Aripj " ~0 ~ Hz ½ (~pt X~pS)z • e-a(JJ[ipl +

419

l~jl + l~il)

(7) .Aipj

PERCOLATION

420

THEORY IN COMPARISON WITH RANDOM WALK THEORY

Vol• 43, No. 6

n

with

Aipj =~{e~IEilAji~ip+eBlEPIAipApj+e61EjlApj~ji } (8) where H. is the magnetic field chosen in z-direction, ~3o is a frequency prefactor and the angular brackets <.-.>c denote the average over all possible spatlal and energetical paths in the system. The configurational average of eq. (6) was performed numerically by GMTW~n the sense of PT, where the so called "cone capped cylinder" provide the cut-off of the three-site integrals. The result, is shown in fig. I and 1og(pH/po) vs. (To/T)zl~ gives nearly a straight line. TBis clearly shows that the main contribution to p is due to equilateral triangles with sides o~ length c and site energies c. Thesemean critical quantities can be found using the percolation averaging

~1~ 5

TW

xm

where t = (To/T)z/h and f(n,x)= 1- e'X.m~O~ . Here the TO of FP is by a factor of 1.15 greater than ours given in eq. (5). The FP-result for PH is also shown in fig..1 and eq. (10) is surprisingly close to a T-Z/4-behaviour in the plotted temperature range. These results are based on the configuratlonal averaged expression (6) for PH where the correlation length L - in contrast to the conductivlty - cancels out. BUttger and Bryksin could only achieve this by approximately decoupling certain prefactors in the calculatlon of oxy and oxx. So there is s t i l l some uncertainty in the correct magnitude of PH. To overcome this uncertainty we calculated the Hall mobility in the framework of the more sophisticated RandomWalk Theory. The RandomWalk Theor~ differs from the method based on the resistor network calculations slnce i t is a microscopic approach and starts from an evaluatlon of the conductlwtles Oxx and Oxy using the master equatlon. The microscoplc point of view has the great advantage, that i t is valid for all densities and temperatures. Such a theory for the Hall effect has been developed by Movaghar, Pohlmann and WUrtzI° (MPW). The Hall effect is at least a three slte phenomenon. So MPWformulated a three site selfconsistent approximation in analogy to the two-site selfcensistent theory for the conductivity Oxx2,zz. The final results of their theo~ (in the DC-llmit) can be summarized by the set of equatlonsZO,z2,z3 °xx = e2B~ I d~lj~jdEidEjN(Ei)N(Ej)rij°l(Ej) r i j + Ol(Ej)

-8

is

2'0

(11)

o = e2B f d~id~j dEodEidEJ N(Eo)N(Ei)N(Ej) xy 1 H

3's

(~I x~J )z Ariojn1(Ej ) °1(E°)

[rji+rjo+Ol(Ej)][roj+ro1+o1(Eo)]- rjoroj (12)

Fig. 1 Hall mobility log(uH/Uo) vs (To/T)I/~ in the framework of PT. The dots are our numerical values• The upper curve is a s t r a i g h t l i n e f i t . The lower broken l i n e represents the approximation of FP.

procedure (GMTW) and we get PH ~ exp{-(~ )1/4}

(9)

in agreement with the numrical result, rBT~-is the TO appearing in the conductivity axx duced by a factor of (8/3) W= 50. Recent analytic calculations by FP based on the same model lead to a power law for UH3: FP 2.6 x 1013 11,~) 1.92 x 102 ~H = %{ t13/4 [4. f( --t-l-/iI~

ti/2

•

+t-n7 T

4 x 105 }

~l(El)

ap

t3/4 (10)

J

d~i j dEj N(Ej) r i j Ol(Ej) r l j + o1(Ej)

(13)

where ap = e-1 is a connectivity factor ~ Now a l l one has to do is to calculate ol(E ) s e l f consistently from eq. (13) and then to i n s e r t into eqs. (11) and (12)• F i r s t l e t us consider the l i m i t of high densities and high temperatures which can be done a n a l y t i c a l l y in the case of an i n f i n i t e bandwidth. In this l i m i t 01 dominates in the denominator of eqs. (11) and (12) and we can d i r e c t l y solve f o r axx and Oxy leading to

u~cA :

3.12 x 103 f(13,½) - 1.74 x 104

•

ol(E ) satisfies the integral equation

55= -NFkT Uo T (~-~J-)

(14)

=v_~ e with uo v20 ~ " This r e s u l t f o r PH is the d i f f u s i o n equation Hall m o b i l i t y and essent i a l l y the "random phase" r e s u l t TM. I t is e a s i l y obtained also by using straightforward

Vol. 43, No. 6

PERCOLATION THEORY IN COMPARISON WITH RANDON WALK THEORY

~.H

perturbation theory, and is therefoFe called "quasicrystalline approximation" (QCA) in analogy to the theory of liquid metals. On the other hand in the case of low dens i t i e s and low temperatures we calculated uH numerically from eqs. (11-13). Going over to dimensionless variables one can show that the selfconsistency equation (13) and the resulting dimensionless Hall mobility PH/Po from (11) and (12) are only functions of (~3/NFkT) in this regime. In f i g . 2 we therefore plot log(~H/po) vs. (~3/NFkT)I/~. For low ~3/NFkT the Hall mobility approaches the QCA value and for h~gh ~3/NFkT we nearly obtain a T-I/h-behaviour. As compared to the percolation results of GMTW we find similar shapes of the two curves but the difference in the absolute value. For representative values of Nc = lOl9cm3/eV, T = 300 K, ~-i = 10 M and po " 1.5 cm2/Vs we estimate from

loj °xY ~

1 0

-!' OCA

-2-

_e -3-

\\ \\. RWT

-2

\ "6"

\

\ %

-7.

~. l

i

l

|

2

4

6

8

is

We have presented numerical calculations of the Hall effect in spatial and energetical disordered Hopping systems in the framework of RWT and compared i t to PT. The Hall mobility was evaluated in the RWT for a l l densities and temperatures within the three site model. The high density/temperature l i m i t is exact and for low densities/temperatures the result is quite close to PT. The dependence of uH on (~3/NFkT) in this region is similar to results of GMTW and FP. The absolute value of UH is of course quite sensitive to the value of (a3/NFkT). The RWT is considered to be superlor to PT, since i t g~ves the correct prefactors. I t w i l l be

2

0

>

ioroj + r o j r j i + r j i r i o c

and carrying out the configuratlonal average in the sense of PT. I t is interestlng to note that the convergence of the three site integrals is ensured by the existence of cl in the denominator of the RWT eq. (11), while in PT the integrals extend e x p l i c i t e l y over a f i n i t e four dimensional energy - space domain.

f i g . 2 PH = 10-Wcm~/Vs for the percolation result whereas the RWT gives a Hall mobility greater by about one order of magnitude. This discrepancy in the absolute values also shows up in the R-hopping caseI0. I t is worth noting that the PT l i m i t can be obtained by putting o1= 0 in the denominator of eq. (12).

°4-

421

~

,

i

!

"% l

,

w

,

10 12 14 16 18 20 22 24 ! G3 ~V~

tNF--7 .;

Fig. 2 Hal 1 mobi 1i ty 1og (PH/Po) vs (a3/NFkT) 1/~ The f u l l curve is the numerical r e s u l t in the framework of Rkrr. The dashed l i n e shows the high density and high temperature QCA result. The lower broken curves are due to PT of OMTW and FP. The arrow indicates the value of (a3/NFkT)I/W for T =~300 K, NF= 1019/cm3eV and ~'I=10 )~.

422

PERCOLATION THEORY IN COMPARISON WITH RANDOM~.LK TI~OR¥

Vol. 43, No. 6

Acknowledgement This work was supported in part (D.W.) by the Deutsche Forschungsgemeinschaft. We thank B. Movaghar and P. Thomas for many helpful discussions.

interesting to compare now the theoretical predictions with computer simulations for the Hall mobility as i t was done previously in the case of R-Hoppingl ° .

References 1. N. 6rUnewald, H. NUller, P. Thorns and D. WUrtz, Solid State C o l . 38, 1011 (1981) 2. 8. Novaghar and W. Schtrmcher, J. of Phys. C 14, 859 (1981) 3. L. Friedman and M. Pollak, Phil.Hag. B 44, 487 (1981) In t h e i r result eq. (84) ~ correct the sign of the third tern which is a misprint of eq. (73) 4. M. Pollak, J. Non-Cryst. Solids 11, 1 (1972) 5. H. Overhof, phys.stat.sol. (b) 67, 709 (1975) 6. V. Ambegaokar, B.I. Halperin, and J.S. Langet, Phys.Rev. B 4, 2612 (1971) 7. K. Hayden, Thesis 1979, University of Warwick

8. H. BUttger and V. Bryksin, phys.stat.sol. (b) 80, 569 (1977) 9. T. Holstein, Phys.Rev. 124, 1329 (1961) 10. B. Novaghar, B. Pohlmnn and D. WUrtz, J. of Phys. C 14, 5127 (1981) 11. B. Havaghar, B. Pohlmnn and G.W. Sauer, phys.stat.sol. (b) 97, 533 (1980) 12. B. Movaghar, Proc. of the 9th Int. Conf. on Amorphous and Liquid Semiconductors, 1981 Grenoble, p. 73, Journal de Physique, Suppl. no 10, Ed. 8.K. Chakraverty and D. Kaplan 13. D. t{drtz, Thesis 1981, Universit~it Marburg 14. L. Friedman, J. of Non-Cryst. Solids 6, 329 (1971)