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Experimental Confirmation of the Validity of the Einstein Relation for Charge Carriers in Semiconductors[^1]
In the course of intensive investigations carried out in recent years on the electrophysical properties of “monatomic” semiconductors—germanium and silicon—a number of propositions of semiconductor theory have been experimentally confirmed. We have in mind experiments on the introduction into Ge and Si crystals of known small amounts of donor and acceptor impurities[^2], the results of studies of the optical properties of these crystals in the infrared region[^3], and also direct evidence for the comparatively long-lived existence of nonequilibrium (“foreign”) charge carriers, which is of great importance in the theory of rectification and amplification of alternating currents by semiconductor triodes[^4].
In the present note experiments will be described that made it possible to verify the validity of the equation known as the “Einstein relation.” Since the “Einstein relation” is a direct consequence of the Boltzmann distribution of charged particles over energy, the result obtained should be regarded as evidence for the existence of such a distribution both for nonequilibrium electrons in the conduction band and for nonequilibrium holes in the filled energy band of the crystal.
The “Einstein relation” connects the mobility \(\mu\) and the diffusion constant \(D\) of charged particles:
\[ \frac{D}{\mu} = \frac{kT}{e_0}, \tag{1} \]
where \(k\) is Boltzmann’s constant, \(T\) is the absolute temperature, and \(e_0\) is the charge of the particle.
Before the publication of Einstein’s work[^5], equation (1) had been given by Nernst[^6] and Townsend[^7]. Accordingly, it would be more correct to call it the “Nernst–Townsend–Einstein equation.”
Let us assume that the particles under consideration are in equilibrium in a field \(E\), varying only along the \(X\) axis, so that
\[ E = - \frac{\partial \varphi}{\partial x}. \tag{2} \]
According to Boltzmann’s law, the number of particles \(n(x)\) per unit volume at a certain value of \(x\) is equal to
\[ n(x)=\mathrm{const}\cdot e^{-\frac{e_0\varphi}{kT}} . \tag{3} \]
In addition, \(n(x)\) can be determined from the condition of absence of current
\[ \mu n e_0 E-e_0D\frac{dn}{dx}=0, \tag{4} \]
which, after integration, gives
\[ n=\mathrm{const}\, e^{-\frac{\mu\varphi}{D}} . \tag{5} \]
Comparing (5) with (3), we obtain
\[ \frac{D}{\mu}=\frac{kT}{e_0}, \]
i.e., equation (1).
The experiments in which the validity of (1) was checked were carried out as follows: to a homogeneous germanium crystal, having the form of a long bar, a constant electric
Figure (schematic; visible labels translated): vertical axis \(n(x)\), \(n(t)\); horizontal axis \(x\), \(t\); pulse widths \(\Delta t_1\), \(\Delta t_2\); intervals \(t_1\), \(t_2\). Blocks and labels: “generator of rectangular pulses”; contacts \(A\) and \(K\); field \(E\); “Germanium”; “amplifier”; “C.O.”.
field was applied (see figure). Two metallic points \(A\) and \(K\) were pressed against the surface of the crystal, serving respectively: one for creating an excess concentration of “foreign” charge carriers, and the other for detecting this excess concentration*).
A short-time \((\Delta t_0)\) current pulse through the rectifying contact \(A\) created in the crystal, at first, a sharply bounded region of excess concentration of nonequilibrium carriers, displaced (in the case of holes in germanium with intrinsic electron conductivity) in the direction of the field \(E\).
*) For a detailed description of the pulse technique of similar experiments, see \({}^{8}\). The experiments were carried out for the purpose of direct measurement of the mobility of charge carriers.
During the time \(t\) corresponding to the displacement of the pulse along the crystal, as a result of irregular diffusion motion, the width of the pulse of carriers increases. The decrease in amplitude cannot serve as a quantitative characteristic, owing to the inevitable decrease in the number of carriers due to recombination.
By measuring the width of the pulse recorded on the oscilloscope screen at two different positions of the contact \(K\), one can determine the ratio of the diffusion constant to the mobility as
\[ \frac{D}{\mu}=\frac{U(\Delta t_1^2-\Delta t_2^2)}{11.08\,t_1(t_1-t_2)}, \tag{6} \]
where \(\Delta t_1\) is the pulse width corresponding to half the amplitude at the transit time \(t_1\); \(\Delta t_2\) is the pulse width corresponding to the time \(t_2\); \(U\) is the potential difference between the points \(A\) and \(K\) for the transit time \(t_1\).
Since the determined ratio \(\frac{D}{\mu}\) depends on the difference of the squares of the measured quantities, individual measurements do not give sufficiently reliable results.
In view of the fact that the data of the numerous measurements carried out by the authors obey the law of normal distribution, the method of least squares was used to determine the most probable value.
The mean value of \(\frac{D}{\mu}\), obtained for electrons in \(p\)-germanium with an accuracy exceeding 1%, coincided with the result obtained for holes in \(n\)-germanium.
The following results were obtained*):
\[ \frac{D}{\mu}=0.0268\pm0.0013\ \text{eV},\quad T=303\pm1^\circ\ \text{abs},\quad \frac{kT}{e_0}=0.0262\pm0.0001\ \text{eV}. \]
This may be regarded as proof of the validity of equation (1). Experiments confirming Einstein’s relation were carried out earlier for ions and colloidal particles. The new experimental confirmation, pertaining to nonequilibrium electrons and holes in crystals, appears sufficiently important and interesting.
V. V.
CITED LITERATURE
- A. Giordano et al. (article signed by 61 authors), Phys. Rev. 88, 1368 (1952).
- K. Lark-Horovitz, Phys. Rev. 82, 763 (1951).
- H. Briggs, JOSA 42, 686 (1952).
- V. S. Vavilov, UFN 40, 120 (1949); UFN 46, 96 (1952).
- A. Einstein, Ann. d. Phys. 17, 549 (1905). N. Mott and R. Gurney, Electronic Processes in Ionic Crystals, 1950, p. 79.
- W. Nernst, Zeits. Physik Chem. 9, 613 (1884).
- J. Townsend, Trans. Roy. Soc. A 193, 129 (1900).
- J. Haynes and W. Westphal, Phys. Rev. 85, 680 (1952).
*) The accuracy of determining the quantity \(\frac{kT}{e}\), equal to \(\pm 0.0001\ \text{eV}\), is given with a large margin, based on the possible scatter of the temperature values of the experiment.