PHYSICS

G. AKRAMOV

DEGENERATE PINCH IN InSb

(Presented by Academician M. A. Leontovich, 23 X 1964)

In many experiments (see, for example, (¹, ²)) it has been found that in an electron–hole plasma of InSb conditions are realized under which the phenomenon of current breakdown exists (the so-called pinch effect). In (²) the radii of pinch channels were determined for 8 different InSb samples; moreover, the dependence was shown of the ratio of the pinch radius \(r\) to the current \(I\) in the plasma on the supplied power \(W\) inside the plasma. The values of \(r\) calculated from the experimental data vary within the range \((1.6 \div 4)\cdot 10^{-3}\) cm. In (²) the particle density reaches \(10^{19}\ \text{cm}^{-3}\), and the plasma temperature is of the order of \(10^{3}\ ^\circ\mathrm{K}\). Under these conditions there is strong degeneracy of the electron–hole plasma of InSb, and one should use the formulas of the quantum statistics of a degenerate Fermi gas, rather than the classical formulas used by the author of (²). In the present work, under very simplified assumptions (the model of a free electron–hole gas), a formula is obtained expressing the relation between the radius and the current in the pinch.

Fig. 1. Theoretical curve of the dependence of \(r\) on \(I\) according to formula (5) (for \(\alpha\) of order unity) on a double logarithmic scale. Points are experimental data from (²). The numbers at the points are the numbers of the pinch channels (see (²)).

Applying the plasma equilibrium equation

\[ \nabla p = \frac{1}{c}[\mathbf{j}\mathbf{H}] \tag{1} \]

and the equation

\[ j = \frac{c}{4\pi}\operatorname{rot}\mathbf{H} \tag{2} \]

to a cylindrical pinch, we obtain

\[ 2\pi c^{2}r^{2}(P_e + P_{\text{д}}) = I^{2}, \tag{3} \]

where \(P_e\) and \(P_{\text{д}}\) are the electron and hole pressures averaged over the cross section of the plasma cord; \(I\) is the current in the cord. The pressures of electrons \(P_e\) and holes \(P_{\text{д}}\) are determined by the formulas (see, for example, (³))

\[ P_{e,\text{д}} = {}^{1}/_{5}(3\pi^{2})^{2/3}\frac{\hbar^{2}}{m^{*}_{e,\text{д}}}\, n_{e,\text{д}}^{5/3}. \tag{4} \]

Eliminating \(n\) by the formula \(n = I / e\pi r^2 v_D\), where \(v_D\) is the electron drift velocity, we obtain the desired dependence between the radius and the current in the pinch

\[ r = aI^{-1/4}, \tag{5} \]

\[ a = (0.4)^{3/4}(3\pi)^{1/2}(c\hbar)^{3/2}(ev_D)^{-5/4}m_e^{*-3/4}. \tag{6} \]

For InSb, \(m_e^* = 0.013m\), \(m_h^* = 0.18m\); \(m\) is the mass of a free electron.

To compare the result obtained with the experimental data \({}^{(2)}\), it is necessary to determine the magnitude of the electron drift velocity \(v_D\). For this purpose we use the results of Ref. \({}^{(4)}\), where it is shown that in semiconductors the current reaches saturation when the electron drift velocity \(v_D\) reaches the speed of sound \(v_S\), i.e., the value \((2 \div 5)\cdot 10^5\) cm/sec. In order for our formulas (5) and (6) to agree with experiment \({}^{(2)}\), it is necessary to put in (6) \(v_D \simeq 10v_S\); in this case \(a\) turns out to be of order unity (see Fig. 1).

Moscow State University
named after M. V. Lomonosov

Received
25 VII 1964

CITED LITERATURE

\({}^{1}\) V. D. Osipov, A. N. Khvoshchev, ZhETF, 43, 1179 (1962).
\({}^{2}\) V. Ancker-Johnson, J. E. Drummond, Phys. Rev., 131, 1961 (1963).
\({}^{3}\) L. Landau, E. Lifshitz, Statistical Physics, Moscow—Leningrad, 1951, p. 185.
\({}^{4}\) R. W. Shith, Phys. Rev. Letters, 9, 87 (1962).