TEMPERATURE DEPENDENCE OF THE HOPPING HALL MOBILITY IN SPATIALLY AND ENERGETICALLY DISORDERED SYSTEMS

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1 TEMPERATURE DEPENDENCE OF THE HOPPING HALL MOBILITY IN SPATIALLY AND ENERGETICALLY DISORDERED SYSTEMS M. Grünewald, H. Müller, P. Thomas, D. Würtz To cite this version: M. Grünewald, H. Müller, P. Thomas, D. Würtz. TEMPERATURE DEPENDENCE OF THE HOP- PING HALL MOBILITY IN SPATIALLY AND ENERGETICALLY DISORDERED SYSTEMS. Journal de Physique Colloques, 1981, 42 (C4), pp.c4-99-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1981 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE CoZZoque C4, suppz6ment au noio, Tome 42, octobre 1981 page C4-99 TEMPERATURE DEPENDENCE OF THF HOPPING HALL MOBILITY IN SPATIALLY AND ENERGETICALLY DISORDERED SYSTEMS M. Grunewald, H. Miller, P. Thomas and D. Wirtz E'achbereich Physik, Universitat Marburg, P. R. G. Abstract.- We present an averagi.ng procedure for the calculation of the Hall mobility for spatial and energetical disordered hopping systems in the framework of percolation theory. For a constant density of states function the temperature dependence of the Hall mobility follows a T-'j4- law a slope which is by a factor of about 8/3 smaller than that of the coriductivity. The comparison the self consistent Random Walk Theory of transport yields qualitatively the same results. Introduction.- We consider hopping transport in model systems where sites are randomly distributed in space and site energies are distributed accordi-ng to a given density of states function N(E). The elementary process governing the Hall Effect is assumed to be the three site interference process due to Holstein'. The model we consider for the calculation of the DC-Hall-mobility!.IH was introduced by Bottger and Bryksin (BB)~ and Friedman and Pollak (FP) 3. The problem of R-hopping conduction (no energy fluctuations), they tackle, was transformed irto the calculation of the average transverse current which develops in the equivalent random resistor network extrinsic currents. In the spirit of percolation theory the average has then been performed in different ways over the percolation path and the results for!.ih of BB and FP were quite differe~t. On the other hand, Movaghar et a1.4 derived an expression for the conductivity in disordered systems starting from a completely different point of view, namely the microscopic description of a random walk. Movaghar, Pohlmann and w"rtz5 extended this theory to yield the transverse conduct-ivity. They showed that the random walk expression for!.ih in the low density regime is quite close to that one found by BB. They have the same dependence on the site density and differ only by a constant prefactor of about 30. But one should remember that in percolation theory it is somewhat difficult to calculate prefactors. The different approaches and results are discussed in detail in ref. 5. The models so far considered deal the so called R-hopping problem where only spatial disorder is?resent. In this paper we follow the work of BB but we allow for fluctuations in the site cnergies as well. In the framework of percolation theory we present an averaging procedure to calculate the Hall mobility for a given density of states function. The basic Eormalism for calculating the Aall mobility is given, in more detail in ref. 6. As an example for systems including energy fluctuations we consider a constant density of states, yielding Mott's law for the conductivity In n/oo = - (T~/T)'I4. For uh we find In UH/UH, = - (T~/T)'I4 Tg : 1':. The magnitude of uh is rather small (2 10-~cm~/~s) if parameters characteristlc for hopping near the Fermi level in amorphous semiconductors are chosen. The conductivity ox,.- conduction is eq-alent to the calculation of the conductivity of a resistor network. The impedances of this network Zij are determined by the two site thermal eqnili.brium jumpe rate 1'.. between sites i and j ari j Z.. = e 6 r.. = e B V * ~ e. A,. (1) As shown by Miller and brah hams^, the problem of DC-hopping Article published online by EDP Sciences and available at

3 C4-100 JOURNAL DE PHYSIQUE where v20 is a frequency prefactor and a-' the decay length of the localized states. Percolation theory is based on the concept that nearly all of the current is carried by a critical path (or a system of critical paths), the conductivity of which is determined essentially by the largest impedance on this path (or on a section of it). This critical impedance will be written in terms of a critical exponent S/~T as The conductivity of the whole system is then approximately where the characteristic length L cannot be determined by percolation theory alone. It can roughly be taken as the separation of sites where critical paths intersect. The critical path is formed by impedances all smaller than Z,. The number n(ei,s) of such impedances linked tc a given site i energy Ei is obtained by integrating over the volume of the "cone capped cylinders" r ~(E.,E.) 3 1 (1~~ a 3 I = - ( s - ~ + I E. ~ + IE~-E.[)). (6) Percolation sets in if the average number of bonds per site is equal to percolation number vc = 2.7 l. <... >c means averaging over the path which is done by multiplying and integrating over E. ' 1 lo. For constant N(E) = N we obtain via eqs. (5), (7), (3), and (4) the famous ~ott law To 1/4 In ox, s S6 = (-1 T (9) ( a3 ) 1/4 (10) To kno The Hall conductivity.- oxy and Hall mobility UH result if higher order jump-processes are considered where one or more intermediate sites are involved. So the Hall effect is due to at least three-site hops at triads in the system of percolation paths. On the basis of this model (BB) obtain for the transverse conductivity oxy ( the magnetic field H being in z-direction) The brackets <...>, denote the average over all possible spatial and energetical paths. aidifj is the magnetic field dependent part of the three site transition rate

4 thc intermediate site p, given by Holstein1. For small electron phonon coupling where v30 is a frequency prefactor and AiEj is the projection in the Z-plane of the vector area spanned by the triad (ipj). Inserting eqs. (12), (4). and (1) (2) into eq. (11) and dcviding by axxeh we obtain the Hall mobility u~ The configurational average <...>c is performed as the restrictions 6 1 I?,,,>, and r,l a (5-- ( IE,~ + 1 ~ + ~ IEm-Enl)). 1 2 The temperature dependence of ph may be deduced approximately by the following argument. Thc main contribution to uh is due to equilateral triangles sides of length <R>, and site energies <E>, related to the critical exponent EB by The mean critical quantities <R>c and <E>c are obtained from the averaging procedure as prescribed by eq. (0) but taking care that always eq. (15) is fulfilled. We obtain for 3 To 1/4 a<r>, = (-) T (16) which is idcntical to the average hopping distance obtained from the simple "Mottoptimization"". Thus the Hall. mobility uh from eq. (13) results in 2. e-b<.ea<r>c+r<e>c = -a<r>c (17) uh follows a law where T: is the To appearing in the conductivity eq. (9) reduced by a factor of This result is confirmed by a direct numerical calculation of eq. (14) using Monte Carlo inte ration techniques. Fig. 1 shows the numerical rcsults for log p~ vs. (T,/T) '1' for a constant density of states N(E) =No. The slope in this plot is indeed 3; 3/8 and the val.ue of LI is very small. For the followinq parameters u30/:2(~ ' 0.1 (FP), a-i 10 8, N= 101e/cm3eV, we ab- tain for the prefactor po ==- 7 2 cm2/vs. Setting T = 300K we compute via eq. (16) (To/T) " 16 and witx2eig?ui the Hall mobility uh to We remark in agreement: (BB) that there is an uncertainty in the preexponential factors of the conductivity ox, and Hall conductivity BXy because in percolation theory there exists no unambiguous scheme for calculatir~g prefactors. If one is interested in the correct prefactors one should use the more sophisticated

5 JOURNAL DE PHYSIQTJE random walk theory4 r 5. The Random Walk Theory formulated for the Hall effect in the R-hopping problems can also easily be extended to the case of variable range hopping. The averaging procedure over all spatial and energetical coordinates of the system is somewhat more difficult from the numerical point of view. First numerical results show in the low temper-, ature and low density regime the same qualitative behaviour (~-l/'-law) as obtained by the percolation theory described in this paper. Fig. 1 log(yh/po) vs. (T~/T) The results of the numerical calculation are indicated by dots. O We have presented an averaging procedure for the Hall mobility for spatial and energetical disordered hopping systems in the framework of percolation theory. The treatment applied in this paper is based on a model of the Hall effect where three site processes dominate and where the transition rates are describable by eq. (1), e.g. due to s-like wavefunctions. For a constant density of states model the absolute value of the Hall mobility p~ is quite sensitive to the value of To/T. The temperature dependence of y~ follows a T-'l4- law a slope which is approximately by a factor 8/3 smaller than that of the conductivity. 1. HOLSTEIN T., Phys.Rev. 124 (1961) BOTTGER H., BRYKSIN V., phys.stat.so1. (b) 90 (1977) FRIEDMAN L., POLLAK M., Phil.Mag. B 38 ( MOVAGHAR B., POHLMANN B., SAUER G.W., phys.stat. sol. (b) 97 (1980) 533; MOVAGHAR B., SCHIRMACHER W., J. of Phys. C 14 (1981) MOVAGHAR B., POHLMANN B., WURTZ D., J. of Phys. C (in press) 6. GRUNEWALD M., MULLER H., THOMAS P., W~RTZ D., Solid State Comm. (in press) 7. MILLER A., ABRAHAMS E., Phys.Rev. 120 (1960) AMGEBAOKAR V., HALPERIN B.I., LANGER J.S., PhyS-Rev. B 4 (1971) POLLAK M., J. Non-Cryst. Solids 11 (1972) 1 and in "The Metal Non-Metal Transition in Disordered Systems", ed. by L. Friedman and P. Tunstall, Proc. 19th Scottish universities Summer School, p OVERHOF H., phys.stat.so1. (b) 67 (1975) PIKE G.E., SEAGER C.H., Phys-Rev. B 10 (1974) MOTT N.F., DAVIS E.A., Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford, =Edition 1979, p. 34