PHENOMENOLOGICAL CONSIDERATION OF TRANSIENT PROCESSES IN A METAL—SEMICONDUCTOR (DIELECTRIC)—METAL SYSTEM

Yu. N. SUKHUSHIN, S. A. ZAIDMAN

(Presented by the scientific-technical seminar of the Department of Radio Engineering)

Transient processes in nonconducting materials in an electric field are of great importance in the study of a wide range of electrophysical phenomena, such as electrical conductivity, electroluminescence, the electret state, electrical breakdown, the excitation of various nonelectrical processes in an electric field, and many others. The solution of this problem is complicated by the great diversity of phenomena occurring in an electric field in a substance, which is especially clearly expressed in the electrophysics of high-resistance materials. Known approaches, based, as a rule, on the simultaneous solution of the differential equations of Poisson, continuity, and kinetic equations, for example [1, 2, 3], lead to cumbersome expressions or are not solved at all in general form, which makes it difficult to understand the physics of the phenomena. In this work a method is proposed for considering transient processes on the basis of a phenomenological approach, in which, on the basis of the known physical regularities of the behavior of ordinary charges in a solid, the transient process is expressed as the distribution of the field and resistance between the electrodes, as well as the current through the substance. The solution is intended to be carried out by the method of successive approximations, involving in the consideration an ever broader range of phenomena.

In the present, first, approximation the following premises are taken as the basis:

1. The potential distribution is determined exclusively by contact phenomena.

2. Injection and extraction of nonequilibrium carriers, as well as their recombination, are absent.

3. The potential distribution before the moment of voltage application in the region of the space charge is taken to be linear.

$$ \Delta \tilde{\varphi}(x) = \Delta \varphi_k \left(1 - \frac{x}{l_d}\right). \tag{1} $$

1. The motion of the space-charge region is determined only by the drift of carriers.

2. Active current is considered.

Let us consider, for definiteness, the case of an electronic semiconductor whose work function is greater than the work function of the metal at the con-

contact. The concentration of electrons in the space-charge region is expressed by the known relation

\[ n=n_0 \exp[-\tilde{\Delta\varphi}(x)], \]

or, taking (1) into account,

\[ n=n_0 \exp\left[-\tilde{\Delta\varphi}_{\mathrm{k}}\left(1-\frac{x}{l_0}\right)\right]. \tag{2} \]

The resistance of such a contact layer will be

\[ R_{\mathrm{k}}=\int_{0}^{l_0}\frac{dx}{q\mu n(x)\cdot S} =\frac{l_0}{q\mu n_0\tilde{\Delta\varphi}\cdot S} \left(e^{\tilde{\Delta\varphi}}-1\right), \tag{3} \]

and the resistance of the region free of ordinary charges is

\[ R_0=\frac{l-2l_0}{q\mu n_0\cdot S}. \tag{4} \]

The resistance of the entire sample will then be

\[ R=R_0+2R_{\mathrm{k}} \]

or

\[ R=\frac{2l_0\left(e^{\tilde{\Delta\varphi}}-1\right)+\tilde{\Delta f}(l-l_0)} {q\mu n_0\cdot S\cdot \tilde{\Delta f}}. \tag{5} \]

The condition of absence of injection, extraction, and recombination of carriers is equivalent to the condition that the resistance of the sample remain constant; therefore the current–voltage characteristic for our approximation will have the form

\[ I_{\mathrm{p}}(t)= \frac{2l_0\left(e^{\tilde{\Delta\varphi}}-1\right)+\tilde{\Delta\varphi}(l-l_0)} {q\mu n_0\cdot S\cdot \Delta\tilde f}\cdot V_{\mathrm{in}}(t). \tag{6-1} \]

Thus, condition (2) is equivalent to the condition that Ohm’s law be preserved in the transient process.

The external field is distributed in proportion to the resistances of the individual regions, that is,

\[ V_i=\frac{R_i}{R}V_{\mathrm{in}}, \]

or, for the regions under consideration,

\[ V_{\mathrm{k}}(t)= \frac{l_0\cdot V_{\mathrm{in}}(t)} {2l_0+(l-2l_0)\dfrac{\tilde{\Delta f}}{e^{\tilde{\Delta\varphi}}-1}} \pm \Delta\varphi_{\mathrm{k}}, \tag{7} \]

\[ V_0(t)= \frac{(l-2l_0)V_{\mathrm{in}}(t)} {2l_0\left(e^{\tilde{\Delta\varphi}}-1\right)+\tilde{\Delta\varphi}(l-2l_0)}. \]

The sign “+” is for the voltage drop at the cathode, and “−” at the anode. It is seen that the potential distribution is described by the law of variation of the external voltage.

To express the distribution of the field strength inside the sample over the regions under consideration, we use condition (4); we shall consider

assume that the space-charge region moves through the specimen according to the law [4].

\[ l_k(t)=l_0\pm \mu E_k(t)\cdot t, \tag{8} \]

or

\[ l_k(t)=l_0\pm \mu \frac{V_k(t)}{l_k(t)}\cdot t, \]

whence

\[ l_k(t)=\frac{l_0}{2}+\sqrt{\frac{l_0^2}{4}\pm \mu V_k(t)\cdot t}. \tag{9} \]

Here the sign “+” refers to the cathode region, and the sign “−” to the anode region.

The simultaneous solution of (7) and (9), and the condition \(E_i=\dfrac{V_i}{l_i}\pm E_{k0}\), gives the desired distribution of errors

\[ E_{k\mathrm{p}}= \frac{V_{\mathrm{in}}(t)} {\left[ 2l_0+(l-2l_0)\dfrac{\widetilde{\Delta\varphi}}{e^{\widetilde{\Delta\xi}}-1} \right]\Delta_k(t)} \pm \frac{\widetilde{\Delta\varphi_k}}{l_0\Delta_k(t)}, \tag{10-1} \]

where

\[ \Delta_k(t)=1+ \sqrt{ 1\pm \left[ \frac{V_{\mathrm{in}}(t)} {2l_0+(l-2l_0)\dfrac{\widetilde{\Delta f}}{e^{\widetilde{\Delta\xi}}-1}} \pm \Delta\varphi \right]\cdot \mu t }. \]

Analogous calculations for a hole semiconductor give the following expressions for the voltage and the current–voltage characteristic:

\[ I_p(t)= \frac{2l_0(1-\widetilde{\Delta\xi})+\widetilde{\Delta\varphi}(l-2l_0)} {q\mu n_0\cdot S\cdot \widetilde{\Delta f}} \cdot V_{\mathrm{in}}(t) \]

\[ \simeq \frac{l-2l_0}{q\mu n_0\cdot S}\cdot V_{\mathrm{in}}(t), \tag{6-2} \]

\[ E_{k\mathrm{p}}(t)= \frac{V_{\mathrm{in}}(t)} {\left[ 2l_0+(l-2l_0)\cdot\dfrac{\widetilde{\Delta\varphi}}{1-e^{-\widetilde{\Delta\varphi}}} \right]\Delta_{k\mathrm{p}}(t)} \pm \frac{\widetilde{\Delta\varphi_k}}{l_0\cdot \Delta_{k\mathrm{p}}(t)}, \tag{10-2} \]

where

\[ \Delta_{k\mathrm{p}}(t)=1+ \sqrt{ 1+ \left[ \frac{V_{\mathrm{in}}(t)} {2l_0+(l-2l_0)\dfrac{\widetilde{\Delta\varphi}}{1-e^{-\widetilde{\Delta\varphi}}}} \pm \widetilde{\Delta\varphi_k} \right]\mu\cdot t }. \]

In the case when \(\varphi_p>\varphi_\mu\), expressions (6-2) and (10-1) are valid for a hole semiconductor, and (6-1)—(10-1) for an electron semiconductor; it should only be taken into account that in the terms before \(\Delta\varphi_k\) the sign “+” will refer to the anode, and “−” to the cathode, while the sign before the square bracket in the expression \(\Delta(t)\) has the anode value.

The expressions obtained for the current–voltage characteristics correctly describe the rectification effect in a metal—semiconductor (dielectric)—metal system. The range of their applicability is limited by the processes of depletion of the layer at the anode; however, in this case condition (2) is evidently violated. The expressions obtained were used by one of the authors for a qualitative explanation of the development of prebreakdown conditions in certain high-resistance semiconductors [5]; from the point of view set forth here, it proved possible to explain a wide range of experimental data (the inclusion of the material and the polarity of the electrodes, the relative magnitude of the electric strength, and the influence of temperature). A detailed analysis of the relations obtained is the aim of further work.

Notation:

\[ \Delta \widetilde{\varphi}=\frac{\Delta \varphi(x)}{\varkappa T} \]
— distribution of the potential of the contact charge,

\[ l_0 \]
— length of the space-charge region,

\[ l \]
— length of the specimen,

\[ \Delta \widetilde{\varphi}_{\mathrm{k}}=\frac{\Delta \varphi_{\mathrm{k}}}{\varkappa T} \]
— contact potential difference,

\[ n \]
— electron concentration,

\[ n_0 \]
— electron concentration in the bulk,

\[ q \]
— electron charge,

\[ \mu \]
— mobility,

\[ S \]
— contact area,

\[ R_{\mathrm{k}} \]
— contact resistance,

\[ R_0 \]
— bulk resistance,

\[ I_{\mathrm{p}}(t) \]
— electron current,

\[ V_{\mathrm{vn}}(t) \]
— external voltage,

\[ V_{\mathrm{k}}(t) \]
— voltage drop at the contact,

\[ V_0(t) \]
— voltage drop in the bulk of the specimen,

\[ l_{\mathrm{k}}(t) \]
— length of the contact region,

\[ t \]
— time,

\[ T \]
— absolute temperature,

\[ \Delta(t) \]
— response function of the space-charge region to an external action.

References

1. A. I. Gubanov, Theory of the Rectifying Action of Semiconductors. GITTL, Moscow, 1956.

2. S. A. Eremin, O. K. Mokeev, Yu. R. Nosov. Charge-Storage Semiconductor Diodes and Their Application. “Sovetskoe radio,” 1966.

3. E. I. Androvich, FTP, I, issue 11, 1620 (1967).

4. S. M. Ryvkin, The Photoelectric Effect in Semiconductors. Fizmatgiz, Moscow, 1963.

5. Yu. I. Sukhushin, Yu. A. Zakharov. Physics of Combustion and Explosion (in press).

NOTED MISPRINTS