Results of Experimental Studies of Electron Mobility in Semiconductors and Dielectrics in Strong Electric Fields
Abstract Generated abstract
This paper reviews experimental evidence on how electron mobility in semiconductors and dielectrics depends on electric field strength, with attention to deviations from Ohm’s law in strong fields. Drawing on studies of photoconductivity and high-field transport in mica, quartz, alkali-halide crystals, and especially n-type germanium, it compares reported regimes in which mobility is constant, decreases inversely with field, or decreases as the inverse square root of field. The discussion argues that current saturation in strong fields can be interpreted as evidence for reduced electron mobility, not only changes in carrier concentration. The paper concludes that two broad regions can be distinguished: an Ohmic region with approximately constant mobility and a high-field region where mobility generally decreases, although some theoretical exceptions are noted for ionic semiconductors.
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RESULTS OF EXPERIMENTAL STUDIES OF ELECTRON MOBILITY IN SEMICONDUCTORS AND DIELECTRICS IN STRONG ELECTRIC FIELDS
E. K. Zavadovskaya
(Presented by the scientific seminar on dielectrics of TPI)
In view of the fact that there are different points of view on the dependence of electron mobility on the field strength in crystals, experimental studies of electron mobility in semiconductors and dielectrics are of interest.
The available experimental data on electron mobility in dielectrics are not numerous. The experiments of V. I. Pruzhinina-Granovskaya [1] on the change in the resistance of mica in a magnetic field and those of P. E. Sarzhevsky [2] on the study of quartz in strong electric fields are known. According to these authors, electron mobility does not remain constant in high fields, and under certain conditions decreases with increasing field strength according to the laws \(u \sim \frac{1}{E}\) and \(u \sim \frac{1}{\sqrt{E}}\).
For dielectrics in strong fields there is a deviation from Ohm’s law. It is expressed in the constancy of the current when the field strength is varied.
N. P. Kalabukhov [3] observed a saturation current in the photoconductivity of X-rayed NaCl. In his experiments, a constant value of the photocurrent set in at a field strength of \(\sim 10^3\) V/cm and followed the region in which Ohm’s law is satisfied.
N. P. Kalabukhov explains the saturation current by the fact that the probability of capture of electrons by atoms depends on the velocity of electron motion. In the region of saturation current the electrons, apparently, reach that velocity at which the probability of their capture by atoms is greatest. In this region of field strengths the scattering of electrons increases, their number in the total flux in the direction of the field decreases, and the current density retains a constant value. N. P. Kalabukhov also observed a saturation current in the photoconductivity of colored KCl and KI crystals.
It should be added to the explanation of the saturation current in dielectrics given by N. P. Kalabukhov that, with increasing field strength, not only the concentration of electrons but also the mobility of electrons may change. And the change in electron mobility with field strength may prove to be more significant in the phenomenon of saturation current.
In elastic collisions an electron can transfer to traps or lattice sites a part of its kinetic energy equal to \(\frac{2m}{M} W_k\). With an increase in the velocity of electron motion, the amount of energy transferred increases. If the velocity of the electrons assumes a constant value
over some interval of field strength, then, consequently, the mobility of the electrons must vary inversely proportional to the field strength.
A. Hippel [4] observed saturation of the photocurrent in colored KCl crystals. The saturation current was established at a field strength of 600 kV/cm. The observation was carried out up to a field strength of 900 kV/cm. At 920 kV/cm breakdown of the specimen occurred. The current was measured at constant illumination of the specimen and, consequently, at an unchanged concentration of electrons in the specimen.
Thus, the phenomenon of photocurrent saturation in dielectrics is a fact indicating a change in the mobility of electrons with the field strength in dielectrics.
More detailed studies have been made of the mobility of electrons in semiconductors, in particular in germanium.
E. D. Ryder [5] measured the mobility of electrons in n-type germanium in the range of field strengths up to \(2 \cdot 10^4\) V/cm. In his experiments measures were taken to reduce thermal effects to a minimum. The voltage was applied to the specimen in the form of separate pulses of duration \(10^{-7}\) sec. The experiment showed that, for germanium at room temperature, Ohm’s law is valid up to fields of \(6 \cdot 10^2\) V/cm. With increasing field, constancy of the current density \(i\) is observed—i.e., \(E = \mathrm{const}\). Consequently, if the current density retains a constant value as the field strength increases, then either the mobility or the concentration of electrons in germanium must decrease with increasing field. However, in the region of field strengths where the current density remains constant, the electron velocity will also have a constant value (\(v = 10^7\) cm/sec). This circumstance indicates that the electron mobility decreases with field strength.
In the region of Ohm’s law the electron mobility retains a constant value. In stronger fields, where the current density is a constant quantity, the electron mobility decreases with increasing field strength according to the condition
\[ \mu \sim \frac{1}{E}. \]
There is a known experimental study of the mobility of electrons in n-type germanium carried out by E. M. Conwell and E. D. Ryder [6]. They made their measurements with pulsed application of voltage. In these experiments the concentration of electrons remained constant. E. M. Conwell and E. D. Ryder observed a change in the mobility of electrons in germanium with changing field strength.
According to their experimental data it follows that the mobility of electrons in weak fields is determined by scattering of electrons by impurities and is proportional to the cube of the electron velocity. Therefore, in the region of field where the current begins to increase more strongly than follows from Ohm’s law, at first an increase in electron mobility is observed. Then, when the principal factor determining the electron mobility becomes scattering of electrons by the lattice, the mobility decreases with increasing field according to the law
\[ u \sim \frac{1}{\sqrt{E}}. \]
This character of the change in electron mobility with field strength was experimentally observed in germanium by E. M. Conwell and E. D. Ryder.
E. D. Ryder and W. Shockley [7] observed a deviation from Ohm’s law in germanium at a field strength \(E \sim 6 \cdot 10^2\) V/cm. At this value of the field strength and above, the current remained constant. The concentration of electrons in their experiments remained unchanged up to a field strength \(E \sim 2 \cdot 10^4\) V/cm. In the region of Ohm’s law the electron mobility remained
constant. In stronger fields, where the current has a constant value, the electron mobility changes according to the condition \(u \sim \dfrac{1}{E}\).
E. D. Ryder and V. Shockley give a theoretical explanation of the results obtained. They show that the field strength at which a deviation from Ohm’s law is observed is determined by collisions of electrons with lattice vibrations. If the energy losses of the electrons are due to elastic vibrations, then the limiting field strength corresponds approximately to
\[ \frac{V_s}{u_0}=150\ \mathrm{V/cm}, \]
where \(V_s\) is the velocity of longitudinal sound waves, and \(u_0\) is the mobility in small fields. If energy losses due to optical vibrations of the lattice sites predominate, then the limiting field strength will be higher. The drift velocity at the limiting field strength is determined by the condition \(\left(\dfrac{h\nu}{m}\right)^{1/2}\) and takes a constant value, numerically equal to \(\sim 10^7\ \mathrm{cm/sec}\). This value of the velocity is well confirmed experimentally for germanium.
Thus, detailed investigations of germanium have shown that, for semiconductors, an experimentally observed decrease in electron mobility with increasing field occurs, which may proceed according to the law \(u \sim \dfrac{1}{E}\) or \(u \sim \dfrac{1}{\sqrt{E}}\).
The experimental and theoretical data presented on electron mobility in semiconductors and dielectrics make it possible to establish two regions in the dependence of electron mobility on field strength. In the first region, corresponding to Ohm’s law, the electron mobility remains constant as the field strength increases. In the second region, the electron mobility decreases as the field strength increases. The decrease in electron mobility occurs either according to the law \(u \sim \dfrac{1}{E}\) or \(u \sim \dfrac{1}{\sqrt{E}}\). For example, for the semiconductor germanium the limiting field strength is \(E = 6 \cdot 10^2\ \mathrm{V/cm}\). For mica, the electron mobility remains constant up to \(3 \cdot 10^5\ \mathrm{V/cm}\), and from \(E = 3 \cdot 10^5\ \mathrm{V/cm}\) to \(10^6\ \mathrm{V/cm}\) a decrease in electron mobility is observed.
The results of theoretical [8] and experimental studies of electron mobility in semiconductors and dielectrics can be briefly presented in the form of a table. Table 1 gives the dependence of electron mobility on field strength, obtained by various authors for some semiconductors and dielectrics.
The results of theoretical and experimental studies of semiconductors generally lead to a decrease in electron mobility with field strength. The exception is semiconductors with an ionic structure, according to the data of Davydov and Shmushkevich, which show that the electron mobility in such crystals should increase with field strength.
For dielectrics, we know of no theoretical works studying electron mobility. Experimental works on the study of electron mobility in dielectrics are few in number. They show that the electron mobility in dielectrics decreases with field strength—
Table 1
Dependence of electron mobility in semiconductors and dielectrics on field strength
| Authors | Materials | Region of investigation | Character of the dependence of electron mobility |
|---|---|---|---|
| 1. B. I. Davydov, E. M. Shmushkevich (1940) | Semiconductors with an atomic lattice | Weak fields Strong fields |
Mobility is constant Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
| 2. B. I. Davydov and E. M. Shmushkevich (1940) | Semiconductors with an ionic lattice | Weak fields Strong fields |
Mobility is constant Mobility increases with the field |
| 3. N. L. Pisarenko (1938) | Semiconductors with an atomic lattice | Strong fields | Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
| 4. L. Landau and A. Kompaneets (1935) | Semiconductors | Strong fields | Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
| 5. E. M. Conwell and E. D. Ryder | Semiconductor germanium | Beyond the limits of Ohm’s law | Mobility increases, then decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
| 6. W. Shockley (1951) | Semiconductor | Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
|
| 7. E. D. Ryder (1951) | Semiconductor germanium | a) up to $6 \cdot 10^2$ V/cm in the region of Ohm’s law b) from $6 \cdot 10^2$ V/cm beyond the limits of Ohm’s law. |
Mobility is constant Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ |
| 8. E. M. Conwell (1952) | Semiconductor germanium | In high fields the mobility changes. | |
| 9. A. F. Ioffe (1939) | Semiconductor Cu$_2$O | Up to $5 \cdot 10^4$ V/cm | Mobility is constant |
| 10. V. I. Pruzhanina-Granovskaya (1940) | Dielectric mica | From $1 \cdot 10^5$ V/cm—$3 \cdot 10^5$ V/cm From $3 \cdot 10^5$ V/cm—$5 \cdot 10^5$ V/cm |
Mobility decreases Mobility increases |
| 11. P. E. Sarzhevsky (1950) | Dielectric quartz | Up to $3 \cdot 10^5$ V/cm | Mobility decreases |
| 12. A. A. Vorob’ev, E. K. Zavadovskaya (1951) | Crystals of alkali-halide salts | In strong fields | Mobility decreases $u \sim \dfrac{1}{\sqrt{E}}$ $u \sim \dfrac{1}{E}$ |
field. These data confirm the point of view we have expressed, namely that the mobility of electrons in dielectrics should decrease with increasing field strength in the region of strong fields according to the laws \(u \sim \frac{1}{\sqrt{E}}\) and \(u \sim \frac{1}{E}\) [9].
I express my gratitude to Professor A. A. Vorob’ev for a useful discussion of the questions raised.
References
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