On the Mobility of Electrons in Pure Non-Polar Insulators (1948)

It is pointed out that Peierls' condition for the validity of time-dependent perturbation theory in the determination of the mobility $\ensuremath{\mu}$ of electrons in crystals can be transformed to the form $\ensuremath{\mu}>30 \frac{{\mathrm{cm}}^{2}}{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}\ensuremath{-}\mathrm{s}\mathrm{e}\mathrm{c}.} (\mathrm{at}\mathrm{room}\mathrm{temperature})$ in the case in which the electrons are distributed classically. Since the recent investigations of the mobility of electrons in diamond, silicon, and germanium indicate that the mobility in these materials is at least several times greater than the foregoing limit, it would appear that perturbation methods may be used to discuss the mobility in these materials and possibly in other non-polar insulating materials, such as sulfur.An expression is derived for the collision frequency for conduction electrons having velocity $v$ as a result of collissions with the acoustical modes of oscillation. It is pointed out that these modes will be the only ones of interest in diamond at room temperature because the characteristic temperature is in the vicinity of 1800\ifmmode^\circ\else\textdegree\fi{}K. The collision time $\ensuremath{\tau}$ is found to satisfy the equation $\frac{1}{\ensuremath{\tau}}=(\frac{4}{9\ensuremath{\pi}})(\frac{{C}^{2}{k}_{0}T}{{\ensuremath{\hbar}}^{3}{c}^{2}{n}_{0}})(\frac{{m}^{*2}}{M})k$ (A) where $C$ is a constant, having the dimensions of energy. that measures the interaction between the lattice and the electron, ${k}_{0}$ is Boltzmann's constant, $T$ is the absolute temperature, $c$ is the acoustical velocity, ${n}_{0}$ is the density of atoms, ${m}^{*}$ is the effective electron mass, $M$ is the mass of the atoms in the crystal, and $k$ is the wave number vector of the electron. Since the mean free path is proportional to $k$, it follows that the mean free path is independent of velocity in the approximation in which (A) is valid. The conditions for validity of the equation, all of which are normally well satisfied in diamond at room temperature, are: (1) The temperature be sufficiently low that the principal inelastic collisions involve only one lattice vibrational quantum; (2) only the acoustical modes of vibration be excited; (3) the temperature of the electrons be sufficiently high that their mean energy be large compared with ${m}^{*}{c}^{2}$, in which $c$ is the acoustical velocity; (4) the electrostatic field be sufficiently low that only linear terms are important. The second of these conditions is usually not satisfied in materials such as silicon, germanium, or sulfur near room temperature, although it will be satisfied at lower temperatures. The temperature $\frac{{m}^{*}{c}^{2}}{{k}_{0}}$ is near 10\ifmmode^\circ\else\textdegree\fi{}K for diamond, but is much less than this for most other materials, so that (A) should be applicable for determining the mobility of conduction electrons in a large number of pure non-polar insulators at low temperatures not too close to 1\ifmmode^\circ\else\textdegree\fi{}K.In the case of diamond, the mobility at room temperature is found to have the value $\ensuremath{\mu}=(\frac{1.46}{{C}^{2}}){10}^{5} \frac{{\mathrm{cm}}^{2}}{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}\ensuremath{-}\mathrm{s}\mathrm{e}\mathrm{c}.}$ if ${m}^{*}$ is taken as the free electron mass. This leads to a mobility of about 156 ${\mathrm{cm}}^{2}$/volt-sec. if $C$ is assumed to have a value of 30.6 ev obtained from the relation $C(\mathrm{ev})=1.7\ifmmode\cdot\else\textperiodcentered\fi{}{10}^{\ensuremath{-}2}\ensuremath{\Theta}$ that seems to be obeyed for the electrons in the simpler metals. In a range of temperature in which only acoustical modes of vibration are excited the relative mobilities of two different substances ${\ensuremath{\mu}}_{a}$ and ${\ensuremath{\mu}}_{b}$ should satisfy the relation $\frac{{\ensuremath{\mu}}_{a}}{{\ensuremath{\mu}}_{b}}={(\frac{{\ensuremath{\Theta}}_{a}}{{\ensuremath{\Theta}}_{b}})}^{2}{(\frac{{n}_{a}}{{n}_{b}})}^{\frac{1}{3}}{(\frac{{C}_{b}}{{C}_{a}})}^{2}(\frac{{M}_{a}}{{M}_{b}})$ in which the subscripts refer to the two different substances, $\ensuremath{\Theta}$ is the characteristic temperature, $n$ is the atomic density, $C$ is the interaction constant appearing in (A), and $M$ is the atomic mass. Since $C$ and $\ensuremath{\Theta}$ are probably roughly proportional to one another, this relationship suggests that the mobilities of various insulators should have similar values in a range of temperature where only the acoustical modes cause excitation.The influence of non-acoustical modes in non-polar materials are discussed. It is pointed out that two interesting cases can occur and the case of diamond is discussed and clarified with use of group theory. It is found that the scattering from non-acoustical modes contributes to the collision frequency to about the same degree as the non-acoustical modes when the crystal temperature is not too far below the characteristic temperature.The implications of the foregoing work for the problem of crystal counters is discussed. It is pointed out that the mobility is probably sufficiently large at room temperature in any of the non-polar crystalline insulators that they would make satisfactory counters if mobility were the only determining factor. It is shown that the concentration of trapping centers in a give specimen is probably the greatest limitation on the use of the material for purposes of a counter.