SOME COMPUTATIONAL PROBLEMS OF THE THEORY OF DISORDERED SEMICONDUCTORS

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SOME COMPUTATIONAL PROBLEMS OF THE THEORY OF DISORDERED SEMICONDUCTORS

V. Bonch-Bruevich

To cite this version:

V. Bonch-Bruevich. SOME COMPUTATIONAL PROBLEMS OF THE THEORY OF DISOR- DERED SEMICONDUCTORS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-153-C3-156.

�10.1051/jphyscol:1972322�. �jpa-00215056�

SOME COMPUTATIONAL PROBLEMS OF THE THEORY OF DISORDERED SEMICONDUCTORS

V. L. BONCH-BRUEVICH

Faculty of Physics, Moscow University, Moscow, B-234, U. S. S. R.

Some computational problems of the theory of the energy spectra and electron transport of disordered semiconductors are discussed.

1. Introduction. Theorem on the discrete energy levels in the forbidden band of a disordered semiconduc- tor. - One of the main specific features of disordered systems is the random nature of the static force field acting upon the charge carriers. This leads to two peculiarities of the theoretical treatment of such systems.

First, all the observable quantities are usually made up as a result of some averaging over the random field. Therefore the averaging procedure is to be includ- ed in the tlleory and the problem arises of finding the functional S[U] which determines the probability of a certain potential function U(x) to be realised

The answer to the problem depends necessarily upon some model-like assumptions. However it turns out that some theoretical results depend not upon the particular form of the functional 5[U] but rather upon some general regularity properties.

Second, the very statement of the problem of the energy spectrum suffers some changes in presence of a random field : the problem becomes a probabilis- tic one. This refers in particular to the famous problem of the discrete levels in a forbidden band of a disor- dered semiconductor (2) : one has to calculate the probability, Q,, of the states belonging to L2 to arise in a given random field. To this end it is convenient [5]

to consider first some particular configuration of the field and find a sufficient condition of the discrete energy levels to arise. The condition might be written down in a form of a certain inequality. The probability of the latter to be realised is just Q,.

The inequality in question would be trivial in case of a spherically symmetric field tending to zero at

(1)

In a macroscopically homogeneous system the random potential energy of the charge carrier, U(x), may be normalised so that < U >

0, the angular brackets symbolising the average over the random field [I]. We adopt such a normalisa- tion in what follows.

(2)

The notions of allowed and forbidden bands in a disor- dered system may be defined in a general way [2]. Note that the forbidden band

defined according to [23 is just the

mobility

gap

as introduced in [3], [4].

infinity. Then the discrete levels would correspond to the negative energy eigenvalues W and the lower bound of Q , would be easily obtained using a conven- tional variation principle. In case of actual interest three complications arise :

a) The coordinate of a localisation center is not specified physically in a macroscopically homogeneous system. Thus the words c( at infinity

need to be clarified.

b) The random field is not obliged to possess a spherical symmetry.

c) The random potential U(x) may have no definite limit at x tending towards any point in space.

The first complication is trivial and is solved tri- vially : since there is an obvious degeneracy with respect to the localisation center coordinates the values of the latter may be just fixed by agreement. We choose them as an origin ; thereby the meaning is given to an expression cr at infinity

The second and third complications make it neces- sary to reformulate the initial inequality somewhat.

This was done in reference [5] where the probability Q , was shown to be finite under rather broad assump- tions concerning the probability functional 5[U]

Note that the states in question remain localised not because the relevant wave functions do not overlap but in spite of this : the random levels are not obliged to resonate.

The complications b) and c) have the following consequences.

First, the discrete energy eigenvalues are referred to a certain cc random zero D. This is of no importance

(3)

Note however that the definition of a limit of an

infini- tely large

sample needs some care in the present problem.

There are in fact two limiting processes. The first of them means that the distances are considered wich are big compared to a localisation radius. The second is the thermodynamic one.

(4)

It is more convenient to impose the explicit assumptions not on the functional 3 itself but on the characteristic function

A(z) - ( ^exp ( ^{- iz} dkU(k) l ( k ) ) ^{) .}

Here U ( k ) is the Fourier transform of a random function U(x) while J(k) is some regular function.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972322

C3-154 V. L. BONCH-BRUEVICH

as far as one is interested in the electronic processes inside the sample. However the presence of a

random zero >> might show up in the phenomena of the type of an exoemission when there is some fixed energy level (like that of an electron at rest in a vacuum far enough from the sample).

Second, the cr coexistence >> becomes possible of a discrete and a continuous spectra : provided the sample is macroscopically big both types of electron states may correspond to the same energy eigenvalue in different parts of the sample. However this fact does not influence the zero temperature static conduc- tivity, a : according to the general theorem [2], [6] the charge carriers having such an energy do not contri- bute to a at T = 0. At T Z 0 they contribute to hopping only [7].

The estimate of the total concentration, v, of the bound states in a gap may be estimated as well [5].

To this end however an explicit form of the func- tional T is needed. In case of a Gaussian field one obtains :

where T is the Euler function, m is an electron effective mass, JI, = < ^{U 2} >.

2. Low temperature asymptotics of the hopping conductivity. - Existence of a multitude of discrete random levels leads to some specific features of elec- tron transport phenomena. In particular Mott 181 showed that under certain conditions the conven- tional exponential form of the temperature depen- dence of the hopping conductivity, ^oh, was replaced by a relation

Here To is a constant, T is the temperature in energy units. This result was rederived in references [9], [lo].

However it was based on the conventional exponential formula for the hopping probability. The latter assump- tion causes no doubts if a macroscoping fluctuation is needed for a jump ; it is valid as well in case of one-phonon transitions if the energy distances between the relevant levels exceed T. However the contrary situation is as well possible if the density of states, p, near the Fermi level F is big enough. In this case no exponential - like form of ah(T) is to be expected. The relevant calculation was performed in [ l l ] under the following conditions :

Here y(F) is the inverse localisation radius of an electron having the energy W = F (< 0 in the normali- sation adopted), W is the characteristic energy which determines the rate of decrease of the density of states on penetrating into the localised states region.

Here and in what follows the energy is referred to the renormalised band edge, so F < 0 in the case of interest. Conditions (2) might be fulfilled in substances possessing not too high a gap if the

amount of disorder >> is high enough.

Up to the relation between the applied and acting fields the hopping conductivity was found to be a linear function of T :

The problem of expressing the acting field in terms of the applied one (the latter being defined as just the voltage across the sample over its length) seems to be not quite trivial in case of the systems considered.

However no exponential-like temperature dependence is expected to arise from this factor under the condi- tions of interest.

Under the same conditions the temperature depen- dence of the form (3) is obtained as well for the Hall conductivity. However the constant factors are gene- rally different in the two cases.

It is worth noting that the result (3) is not based upon any model-like assumptions except those contain- ed in (2). On the other hand the numerical value of the coefficient as well as the very sign of the Hall coefficient does depend upon such assumptions.

Moreover the reliable calculation of this factor seems to necessitate the use of computers if only one does not want to employ rather far going approximations.

In this respect the question of what in particular is to be calculated seems to be rather acute in the theory of disordered systems. The point is that these are just the systems where the assertion of a generally indirect correlation between the density of states and the transport coefficients [2] seems to be of especial importance. For example the factor multi- plying T in eq. (3) contains not only the density of states, but as well the two-level correlation function

@(R' - R"; W', W"). This is the conditional proba- bility to find the energy eigenvalue W" attached to a localisation center near the point R" if there is a level W' with a localisation center R'. Analogous equation for the Hall conductivity contains the relevant three-level correlation function.

Calculation of such correlation functions seems to be one of the most urgent computational problems of the theory of disordered systems.

3. Density of states and the two-level correlation function

- To illustrate the computational pro- blems that arise we consider the calculation of some averaged quantities of interest. We limit ourselves by

( 5 )

This section is based on the results obtained by

A. G. Mironov and the author.

the

maximally disordered

systems (6) described by The trajectories e(t) are to satisfy the conditions an additive Hamiltonian and by the Gaussian form

of 5 [ U ] . The quantities are the density of states, <(t) = Y , t(0) = x . ^{(1 0)}

p ( ~ ) , and the two-level orr relation function Doing the averaging over the random field under the They are given by sign of the functional integral

one obtains :

p(Wf) p(Wf') @(R' - R" ; W', w") =

=4 < Im G,(Rf, R ' ; W')Im G,(R", RU;-W") > .

Here SZ is the fundamental volume, G,, is the retarded anticommutator Green's function defined and norma- lised according to [12] ; R ⁼ 2 m = l. Eq. (5) is valid if the localisation radii of electrons having the energies W' and W are small compared to I R' - R" I

and to the characteristic length which determines the decay rate of c? - I on increasing the distance

I R' - R"I.

Another quantity of interest is

This is known to be related to the combined density of states.

To see this it is convenient to use the standard bili- near expansion of G, in the eigenfunctions +,(x) of the Schroedinger equation in the field U(x). In case of interest the set of quantum numbers 1 contains just the energy W and the coordinates of the localisation center R ; $,(x) = $w(x - R). It is easily seen that

where

In particular at x = y f (W, 0) = 1 and (up to an irrelevant factor) the r. h. s. of (7) reduces to the combined density of states.

To calculate G, it is convenient to use the functional integral representation 1131 :

( 6 )

The system is called << maximally disordered

if the field

configurations possessing some microscopic symmetry form a set of measure zero.

where

while Y(x - y) is the correlation function of the random field :

It has to be considered as some prescribed function in the problem considered.

Placing the origin at the point x, conditions (10) reduce now to

t(t) = <(0) = 0 . ^(10')

Further on, putting w = W' - W " , (1 1') where (')

c, ⁼ eap { ^- 5: ^dr, 1' d72 ~ [ E ( T I ) - e(%)l -

0

Now the trajectories 5, e', are to satisfy the conditions

Eq. (11) and (11') seem to be convenient both to study some general relations and to develope some approximate schemes differing from the conventional ones. One of such methods [13] consists in approximat- ing the functions e ( ~ ) , e'(~') by trigonometric polino- mials with subsequent calculation of the multiple integrals by, say, a Monte-Carlo method. Note however

(7)

Such a device was used by various anthorslsee ; for example [14]-[17].

(8)

Eq. (12') is written in the one band approximation.

Having in mind the interband transitions w should be replaced by w - A and the factors m/mc and m/mv should be introduced in the first and second terms of A2 respectively, A, mc and mv being the renormalised gap and the conduction and valence bands effective masses.

11

C3-156 V. L. BONCH-BRUEVICH that the mathematical problem of approximating the

functional integral by a multiple integral of finite order might be not quite trivial.

Another method which allows as well an analytical treatment of the problem might be called

the strong random field approximation

It is justified provided

where (in ordinary units)

a-

being the correlation radius which determines the decay rate of the function (13) on increasing the dis- tance, I x - y [ ('). The inequality (14') holds in parti- cular near the renormalised edge - that is just where various previous approximations were inconsistent.

The functional integrals (11) and (11') may now be calculated approximately by a kind of the steepest descent method (cf. [18]). For simplicity we adopt yet another condition

where (in ordinary units)

Note however that, as contrasted to (14'), inequa- lity (14") is not needed in principle ; it is used just to simplify the calculation.

Using (14'), (14") the density of states may be cal- culated explicitly. At negative energies (corresponding to the tail region) we obtain (in ordinary units) :

(9)

To be specific we use and exponential correlation function in what follows :

Y ( X - Y )

V I ~ X P ( - - a l x - Y I } .

Thus there is an energy region (W 5 El, E2) below the renormalised band edge where the density of states is almost constant and not small. The usual rapid decay of the density of states in the gap begins near I WI - - E l .

In particular in case of a heavily doped semiconduc- tor we have

Here n is an effective impurity concentration

a labelling various imputiry types, present in concen- trations n, and having the charge Za I e I per impurity atom,

is the dielectric constant of a sample, r, is the screening radius.

Then

where a, = &A2 / me2, WB = e2 / 2 c a ~ are the Bohr radius and Bohr energy in a crystal. In case of a complete degeneracy eq. (17) reduce to

Note however that the result (16) is quite independent of the usual (say, semiclassical) approximations made previously in the theory of degenerate semiconductors.

This result is valid in case of a Gaussian random field of an arbitrary origin provided the inequali- ties (1 57, (1 5") are satisfied.

[I] BONCH-BRUEVICH (V. L.), Jouun. Non. Cryst. Sol., 1970, 4, 410.

121 BONCH-BRUEVICH (V. L.), Theory of Cond. Matter.

JAEA, Vienna (1969), p. 989 ; Fyzika i Technika Polupuovodnikov, 1968,2, 363.

[3] MOTT (N. F.), Festkorpe~probleme, 1969,9, 22.

[4] COHEN (M. H.), FRITSCHE (H.), OVSHINSKY (S. R.), Phys. Rev. Lett., 1969, 22, 1065.

[5] BONCH-BRUEVICH (V. L.), JETP, September, 1971.

[6] BONCH-BRUEVICH (V. L.), Phys. Lett., 1965, 18, 260.

[7] BONCH-BRUEVICH (V. L.), JETP, 1970, 59, 985 ; Mat. Res. Bull., 1970, 5 , 555.

[8] MOTT (N. F.), Phil. Mag., 1969,19,835.

[9] AMBEGAOKAR (V.), HALPERIN (B. J.), LANGER (J. S.), Preprint no 13-71. University of Helsinki, 1970 (10).

('0)

The author is obliged to Prof. V. Ambegaokar for the

reprint.

References

[lo] BRENIG (W.), WOLFLE (P.), DOHLER (G.), Phys. Lett., 1971, 35 A, 77.

[11] BONCH-BRUEVICH (V. L.), KEIPER (R.), in press.

[I21 BONCH-BRUEVICH (V. L.), TYABLIKOV (S. V.), The Green Function Method in Statistical Mechanics.

North-Holland Publ. Comp. Amsterdam, 1962.

[13] FEYNMANN (R. P.), HIBBS (A. R.), Quantum Mechanics and Path integrals. McGraw Hill Book Company, New York, 1965.

[14] EDWARDS (S. F.), GULYAEV (JU. V.), PYOC. Phys. Soc., 1964, 83, 495.

[IS] CHAPLIK (A. V.), JETP, 1967,53, 1371.

[I61 JONES (R.), LUKES (T.), Proc. Roy. Soc., 1969, A 309, 457.

[17] JONES (R.), J. Phys. C. Solid St. Phys., 1970, 3, 190.

[IS] SURIS (R. A.), Fyzika Tverdogo Tela, 1962, 4, 1154.