PHYSICS

I. Z. FISHER

ON THE MOBILITY OF ELECTRONS AND HOLES IN A LIQUID SEMICONDUCTOR

(Presented by Academician M. A. Leontovich, 12 VI 1957)

Let us consider the problem of a single “excess electron” or a single hole in an atomic semiconductor from the point of view of the tight-binding theory. Let \(\mathbf{R}_k\) \((k = 1, 2, \ldots, N)\) be the instantaneous values of the coordinates of the nuclei of the atoms of the system, numbered in some order. Let \(\Psi_k\) be the wave function of a nonstationary state of the entire system, constructed from one-atomic wave functions (and antisymmetrized with respect to the coordinates of all electrons), in which the \(k\)-th atom is a \((+)\)- or \((-)\)-ion, while the remaining \(N - 1\) atoms are normal. Then an arbitrary state of the system with one excess electron or one hole will be

\[ \Psi = \sum_{k=1}^{N} a_k(t)\Psi_k, \tag{1} \]

so that \(a_k(t) \equiv a(\mathbf{R}_k,t)\) may be regarded as the wave function of the excess electron or hole, while the quantity

\[ dW(\mathbf{R}_k;t) = |a_k(t)|^2 dV \tag{2} \]

as the probability of finding the excess electron or hole at the instant \(t\) in the neighborhood of the point \(\mathbf{r}=\mathbf{R}_k\).

Let at \(t=0\) we have \(a_1=1\), \(a_k=0\) for \(k \ge 2\), and choose \(\mathbf{R}_1=0\). If the medium is not ideally ordered and the defects of its structure (of thermal, chemical, or mechanical origin) are, on the average, distributed everywhere uniformly and isotropically, then for sufficiently large \(t\), not too small \(dV\), and as \(N \to \infty\), asymptotically one must have

\[ dW(\mathbf{R};t) \sim (4\pi Dt)^{-3/2} e^{-\mathbf{R}^2/4Dt} dV. \tag{3} \]

For example, if there exists a mean free path \(l\), then the distribution (3) is established at distances \(|\mathbf{R}| \gg l\). In this case, for the diffusion coefficient we have \(D \sim \frac{1}{3}lv\), where \(v\) is the mean velocity of the electron or hole.

In what follows we are interested in the diffusion of electrons or holes in a liquid semiconductor. In this case there is no mean free path, since the very concept presupposes that the wave functions of the electron or hole are, to a good approximation, plane waves and that \(l \gg \lambda\), where \(\lambda\) is the wavelength. For many semiconductors this inequality is not satisfied even at room temperatures, while near the melting temperature for all semiconductors one has \(l \lesssim \lambda\) or even \(l \ll \lambda\). Consequently, here there are neither plane waves nor a “mean free path.” Still more does the “mean free path” lose its meaning after melting of the semiconductor.

To determine the diffusion coefficient \(D\) in our problem, we shall proceed from the observation that a distribution of the type (3) can also be obtained in a certain auxiliary (and purely formal) problem of the wandering of a classical—

of the particle over a given set of points \(\mathbf R_k\) with a prescribed a priori probability of transition \(\mathbf R_k \to \mathbf R_i\). Various specifications of the probabilistic process describing the wandering are possible here, differing in the degree of aftereffect (the influence of already realized steps of the wandering on the probability of subsequent steps). Among this set of probabilistic processes there will be one such process that leads exactly to a value of \(D\) coinciding with the value of the diffusion coefficient in the quantum-mechanical problem of the behavior of an electron or a hole in a real liquid semiconductor. We shall call it the equivalent probabilistic process.

It seems natural to assume that for real liquids, except perhaps in the immediate vicinity of the melting–crystallization line, the equivalent probabilistic process is, to a sufficient accuracy, Markovian. Indeed, the question evidently reduces to how well the quasimomentum of an electron or hole is an integral of the motion. The smaller its uncertainty \(\Delta p\), the stronger the aftereffect in the probabilities of successive transitions \(\Psi_i \to \Psi_k\), and, for example, at \(\Delta p = 0\) we would have strict determinacy of the process. Conversely, the larger \(\Delta p\), the more likely it is for the electron or hole to deviate from the initial direction of motion. But the degree of conservation (or nonconservation) of quasimomentum as an integral of motion depends on the invariance of the Hamiltonian of the electronic subsystem with respect to the group of translations. In our case, in contrast to crystals, such invariance is completely absent for the liquid as a whole and is practically absent also “in the small.” Indeed, information known from X-ray studies of simple liquids concerning their short-range order indicates the absence in a liquid of repetition of the instantaneous structure under rectilinear translations even within several assumed “elementary cells.”

But only such repetition could lead to the existence of quasimomentum of an electron or hole and to its conservation “in the small.” Therefore, in a real liquid the probabilities of successive quantum transitions \(\Psi_i \to \Psi_k\) are practically independent of one another, which leads to the Markovian or almost Markovian character of the equivalent probabilistic process introduced above.

In this case, for the diffusion coefficient of an electron or hole we obtain \((^1)\)

\[ D=\frac{1}{6}\langle \nu\rangle \langle R^2\rangle, \tag{4} \]

where \(\langle \nu\rangle\) is the mean number of transitions per unit time and \(\langle R^2\rangle\) is the mean square of the displacement length of the electron (hole) in one transition. For the mobility coefficient, from the Einstein relation, we obtain

\[ u=\frac{e}{kT}D=\frac{e}{6kT}\langle \nu\rangle \langle R^2\rangle . \tag{5} \]

The quantities entering (4) and (5) can be calculated in the following way. We shall first regard the positions of the atoms as fixed. Then the probability per unit time \(w_{ik}\) of the transition \(\Psi_i \to \Psi_k\) is proportional to the absolute value of the “transfer integral” \(L_{ik}\) from the theory of the polar model of an electronic conductor of Shubin—Vonsovsky \((^2)\). If there were only two atoms, then \(L_{ik}\) would coincide with the “exchange integral” of the theory of the \((\pm)\)-ion of a diatomic molecule, which determines the proportionality coefficient between \(L_{ik}\) and \(w_{ik}\) \((^3)\):

\[ w_{ik}=\frac{2}{\pi\hbar}\,|L_{ik}|. \tag{6} \]

Hence, for the total number of transitions per unit time from \(\Psi_i\) to all possible \(\Psi_k\), with subsequent averaging over all \(\Psi_i\), we obtain

\[ \langle \nu \rangle = \frac{1}{N}\sum_{i=1}^{N} \left( \sum_{\substack{k=1\\(k\ne i)}}^{N} w_{ik} \right) = \frac{2}{\pi \hbar N} \sum_{i=1}^{N}\sum_{\substack{k=1\\k\ne i}}^{N} |L_{ik}|. \tag{7} \]

Similarly, for \(\langle R^2\rangle\) we find

\[ \langle R^2\rangle = \langle |\mathbf R_i-\mathbf R_k|^2\rangle = \frac{ \sum_{i=1}^{N}\sum_{\substack{k=1\\(k\ne i)}}^{N} |\mathbf R_i-\mathbf R_k|^2 |L_{ik}| }{ \sum_{i=1}^{N}\sum_{\substack{k=1\\k\ne i}}^{N} |L_{ik}| }, \tag{8} \]

and, thus, for the mobility of an electron or a hole with fixed atoms, from (5), (7), and (8) we obtain

\[ u_1 = \frac{e}{3\pi \hbar kT} \frac{1}{N} \sum_{i=1}^{N}\sum_{\substack{k=1\\(k\ne i)}}^{N} |\mathbf R_i-\mathbf R_k|^2 |L_{ik}|. \tag{9} \]

To obtain the actual values of \(u\) and \(D\), it is necessary to take into account the thermal motion of the atoms of the liquid and the probability of their mutual arrangements. Both are achieved, evidently, by averaging the result (9) over the Gibbs distribution for the configurations of the atoms \(dW(\mathbf R_1,\mathbf R_2,\ldots,\ldots,\mathbf R_N)\). If \(L_{ik}=L(|\mathbf R_i-\mathbf R_k|)\) were a strictly binary quantity, this averaging could in fact easily be carried out with the aid of the radial distribution function of atoms \(g(r)\), which may be regarded as known from X-ray diffraction studies, since

\[ dW(\mathbf R_1,\mathbf R_2) = g(|\mathbf R_1-\mathbf R_2|) \frac{d^3\mathbf R_1\,d^3\mathbf R_2}{V^2}, \tag{10} \]

where \(V\) is the volume of the system. Then from (9) and (10) it would follow that

\[ u = \frac{4e}{3v\hbar kT} \int_{0}^{\infty} |L(r)|\,g(r)\,r^4 dr, \tag{11} \]

where \(v=V/N\) is the mean volume per atom in the liquid.

However, in reality \(L_{ik}\) has a more complicated structure of the form

\[ L_{ik} = L^{(0)}(\mathbf R_i,\mathbf R_k) + \sum_{\substack{j=1\\(j\ne i,k)}}^{N} L^{(1)}(\mathbf R_i,\mathbf R_k;\mathbf R_j). \tag{12} \]

For example, for a hole in a model of monovalent atoms in an \(s\)-state, one obtains\({}^{(2)}\)

\[ \begin{aligned} L_{ik} &= \int \varphi_i(\mathbf r)\,G(\mathbf r-\mathbf R_k)\,\varphi_k(\mathbf r)\,d^3\mathbf r + \\ &= \sum_{j\ne i,k} \int \varphi_i(\mathbf r) \left\{ G(\mathbf r-\mathbf R_j) + e^2\int \frac{\varphi_j^2(\mathbf r')\,d^3\mathbf r'}{|\mathbf r-\mathbf r'|} \right\} \varphi_k(\mathbf r)\,d^3\mathbf r + \\ &\quad + \sum_{j\ne i,k} e^2 \iint \frac{\varphi_i(\mathbf r)\varphi_j(\mathbf r)\varphi_j(\mathbf r')\varphi_k(\mathbf r')} {|\mathbf r-\mathbf r'|} \,d^3\mathbf r\,d^3\mathbf r', \tag{13} \end{aligned} \]

where \(\varphi_i(\mathbf r)=\varphi(\mathbf r-\mathbf R_i)\) is the electronic wave function in the \(i\)-th atom; \(G(\mathbf r-\mathbf R_i)\) is the potential of the \(i\)-th atomic core (see also \((4)\)). Therefore (11) remains valid only in the case where \(L_{ik}\) is understood as the result of averaging (12) over all possible values of \(\mathbf R_j\) (with \(\mathbf R_i\) and \(\mathbf R_k\) fixed):

\[ L_{ik}=L^{(0)}(\mathbf R_i,\mathbf R_k)+(N-2)\,\overline{L^{(1)}(\mathbf R_i,\mathbf R_k;\mathbf R_j)}^{(\mathbf R_j)} . \tag{14} \]

It is difficult to carry out such an averaging exactly because of the lack of knowledge of the ternary function of the atomic distribution. However, this can be done approximately (but quite reliably) if one uses the “superposition approximation” from the theory of liquids \({}^{5}\), which gives

\[ \overline{L_{ik}^{(1)}}=\frac{1}{V}\int L^{(1)}(\mathbf R_i,\mathbf R_k;\mathbf R_j)\, g(|\mathbf R_j-\mathbf R_i|)\,g(|\mathbf R_j-\mathbf R_k|)\,d^3\mathbf R_j . \tag{15} \]

Then from (11), (12), (14), and (15) we finally obtain

\[ u=\frac{4e}{3\hbar vkT}\int_0^\infty g(r)\left|L^{(0)}(r)+\frac{1}{v}\overline{L^{(1)}}(r)\right|r^4\,dr . \tag{16} \]

Here it should be borne in mind that \(g(r)\) itself depends parametrically on \(T\) and \(v\), so that the temperature and density dependence of the mobility of an electron or a hole in an atomic liquid semiconductor is more complicated than is determined by the factor in front of the integral in (16).

A preliminary numerical estimate of the results of the theory with the aid of hydrogen \(\varphi(r)\)-functions in the model of a completely disordered liquid with \(g(r)\equiv 1\) leads, for values of \(v\) typical of real liquids, to a value of the diffusion coefficient \(D\sim 1\ \mathrm{cm}^2/\mathrm{sec}\), which seems to us quite satisfactory and encouraging.

Belorussian State University
named after V. I. Lenin

Received
10 VI 1957

References

\({}^{1}\) S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, IL, 1947.
\({}^{2}\) S. Shubin, S. Vonsovskii, Sow. Phys., 7, 292 (1935).
\({}^{3}\) D. I. Blokhintsev, Foundations of Quantum Mechanics, 1949.
\({}^{4}\) S. V. Vonsovskii, Uspekhi fiz. nauk, 48, 289 (1952).
\({}^{5}\) N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, 1946.