PHYSICS

M. I. KLINGER

THEORY OF NONSTATIONARY CONDUCTIVITY OF SEMICONDUCTORS WITH LOW MOBILITY*

(Presented by Academician A. A. Lebedev, 24 V 1963)

1. In the present communication, within the framework of the theory developed in \((^{1-3,\,6})\), the tensor of nonstationary conductivity \(\sigma_{\mu\nu}(\omega)\) and of mobility \(u_{\mu\nu}(\omega)=|eN_c|^{-1}\operatorname{Re}\sigma_{\mu\nu}(\omega)\) is calculated for a semiconductor with low mobility, in which a current carrier of the small-polaron type is described mainly by local-type functions \(|sn\rangle=|s\rangle|n^{(s)}\rangle\) (a polaron packet at the \(s\)-th site \((^1)\)) of the electron–phonon system with energy \(\varepsilon_n=\sum_{fj}\hbar\omega_{fj}N_{fj}\), where \(n\equiv(\ldots N_{fj}\ldots)\); the binding energy of the small polaron \(\delta\varepsilon \gtrsim \hbar\omega_p\Phi_0 \gg \Delta_e(\sim\Delta_{ss'})\) (\(\Delta_e\) is the width of the band of the Bloch electron); \(\omega_p\) (and \(\Delta\omega_p\)) is the frequency (and branch width) of essential phonons; \(\Phi_0\equiv \Phi(T=0)\), see (6). In the notation of \((^{1-3})\), in the absence of a magnetic field \(H\) \((E_\beta(\omega)=\frac{\hbar\omega}{2}\operatorname{cth}\frac{\beta\hbar\omega}{2};\ \beta\equiv(kT)^{-1})\)

\[ u_{\mu\nu}(\omega)=\frac{|e|}{E_\beta(\omega)}\lim_{\varepsilon\to+0}\int_0^\infty dt\, e^{-\varepsilon t}\cos\omega t\,\operatorname{Re}\langle v_\nu v_\mu(t)\rangle \tag{1} \]

(in the absence of degeneracy \(N_c\approx N_0\exp\left[\frac{e\alpha}{k}-\beta\delta\varepsilon\right]\), where \(\alpha\) is the thermo-e.m.f.; \(N_0\) is the number of cells per \(1\ \mathrm{cm}^3\)). The criteria of the theory are:

\[ \Delta_e\ll \hbar\omega_p\Phi_0;\qquad \Delta_e<\left\{\Delta_e^0\equiv(\hbar\omega_p\Phi_0 kT_1)^{1/2};\;(\hbar\omega_p\Phi_0 kT_0)^{1/2}\ \text{for } T>T_1\right\}(^{1,8}), \]

\[ \Delta\equiv \Delta_e\exp[-\Phi(T)]\ll \hbar\Gamma_0;\qquad \omega<\omega_p\Phi_0 . \tag{2} \]

In the general case \(\Gamma_0\approx \Gamma_h+\Gamma'\) (\(\Gamma_h\) see in (5)), and for \(kT\ll \hbar\omega_p\Phi_0\)

\[ \Gamma'\sim(\Delta_e/\hbar\omega_p\Phi_0)^4\,\omega_p^2(\Delta\omega_p)^{-1}\operatorname{sh}^{-1}(\beta\hbar\omega_p/2); \]

if \(\Delta(T_0')=\hbar\Gamma_0(T_0')\), then \(\Delta\ll\hbar\Gamma_0\) for \(T>T_0'\), with

\[ T_0'=T_0=\gamma\,\frac{\hbar\omega_p}{2k}\bigg|_{\gamma\sim1} \]

for \(\Gamma_0\approx\Gamma_h\), and \(T_0'<T_0\) for \(\Gamma_0\gg\Gamma_h\); in (2)

\[ T_1=\gamma'\frac{\hbar\omega_p}{2k}(\operatorname{Ar\,sh}\Phi_0)^{-1}\bigg|_{\gamma'\sim1}, \]

(see \((^{1,7,8})\))**.

2. For \(\omega\ge 0\), in the basic approximation the Ohmic mobility has the form (see (1) and \((^{1,3,6})\))

\[ u_{\mu\nu}^0(\omega)\equiv u_{\mu\nu}^h(\omega)+u_{\mu\nu}^d(\omega) = \frac{|e|}{E_\beta(\omega)} \sum_{(s',0)} \{v_\nu^{0s'},v_\mu^{s'0}\}\times \]

\[ \times\lim_{\varepsilon\to+0}\operatorname{Re}\int_0^\infty dt\,\cos\omega t \sum_{nn'}\exp(\beta F_0-\beta\varepsilon_n)\, |\langle n^{(0)}|n'^{(s')}\rangle|^2\times \]

\[ \times \exp\left[\frac{it}{\hbar}(\varepsilon_n-\varepsilon_{n'})\right]\varphi_{nn'}(\varepsilon,t;\omega), \tag{3} \]

where \(\varphi_{nn'}=\varphi_{nn'}^h+\varphi_{nn'}^d\) (or \(\approx \exp(-|t|\widetilde{\Gamma}_{0s'}(\omega))\),

\[ \varphi_{nn'}^h=(1-\delta_{nn'})e^{-\xi t}; \]

\[ \varphi_{nn'}^d=\delta_{nn'}\exp(-t\widetilde{\Gamma}_{0s'}(\omega)),\qquad \delta_{nn'}\equiv\prod_{fj}\delta_{N_{fj}N'_{fj}} . \]

* The principal results of the article were reported at the Fifth Conference on Semiconductor Theory in Baku, 30 X 1962.

** The criterion \(\Delta\ll\hbar\Gamma_h\) is sufficient \((^1)\), but for \(\Gamma_0\gg\Gamma_h\) it is not necessary (also not necessary is the criterion \(\Delta_e^2<\mathscr{G}^2\exp(-\varkappa\beta\mathscr{G})_{\varkappa<1}\) from \((^{6,8})\), i.e., in \((^{6,8})\) here one should put \(\varkappa=0\)).

In (3), in the nearest-neighbor approximation, \(\widetilde{\Gamma}_{0s'}=\Gamma_0-W_0(s')\), where \(\Gamma_0\approx \nu_c\Gamma_{(0)s'}\) (\(\nu_c\) is the number of nearest neighbors) and \(W_0(s')\) is the mean Fourier component of the probability of “scattering” \(\mathbf{k}\to\mathbf{k}'\) (in the main approximation), and usually \(W_0(s')\sim \Gamma_0\sim \Gamma_{0s'}\);

\[ v_\nu^{ss'}\approx \frac{i}{\hbar}\Delta_{ss'}(s'_\nu-s_\nu); \qquad \{v_\nu^{0s'},v_\mu^{s'0}\} = \frac{|\Delta_{0s'}|^2}{\hbar^2}s'_\mu s'_\nu \sim \frac{\Delta_\varepsilon^2}{\hbar^2}a^2, \]

(see (1)) and substituting (13) into (6). Calculating multiple sums over \(n,n'\) \((1\text{--}3,{}^{5,6})\), we obtain, for \(\omega<\omega_p\Phi_0,\ T>T_1\),

\[ u_{\mu\nu}^{h}(\omega) = u_{\mu\nu}^{h}(0) \frac{\operatorname{sh}\beta\hbar\omega/2}{\beta\hbar\omega/2} \exp\left(-\frac{\omega^2\tau_l^2}{4}\right); \tag{4} \]

\[ u_{\mu\nu}^{h}(0) = |e|\beta \sum_{(s'|0)} s'_\mu s'_\nu \Gamma_{h;s'}; \tag{5} \]

\[ \Gamma_{h;s'} = \frac{|\Delta_{0s'}|^2}{\hbar^2} \int_0^\infty dt\,\cos\omega t \left\{ \exp\left[\Psi\left(t-i\frac{\hbar\beta}{2}\right)\right]-1 \right\} \exp(-2\Phi(T)); \]

\[ \Phi(T) = \sum_{fj}\frac{1}{2}\lambda_{fj}^{0s'}\operatorname{cth}\frac{\beta\hbar\omega_{fj}}{2}; \qquad \Psi(t) = \sum_{fj}\lambda_{fj}^{0s'}\cos\omega_{fj}t/\operatorname{sh}\frac{\beta\hbar\omega_{fj}}{2}, \tag{6} \]

\[ \lambda_{fj}^{0s'}\equiv \frac{\omega_{fj}}{2\hbar}(q_{fj}^{s'}-q_{fj}^{0})^2; \qquad \tau_l^2\equiv - \left(\frac{d^2\Psi(t)}{dt^2}\right)^{-1}_{t=0}; \]

\[ u_{\mu\nu}^{d}(\omega) = \frac{|e|}{E_\beta(\omega)} \frac{1}{2} \sum_{\pm}\sum_{s'} \widetilde{\Gamma}_{0s'}(\pm\omega) \left(\widetilde{\Gamma}_{0s'}^2(\pm\omega)+\omega^2\right)^{-1} \Delta_{0s'}^2 e^{-2\Phi(T)}*. \tag{7} \]

For \(T>T_1\), when \(\exp[\Psi(i\hbar/2kT)]>\exp[\Psi(i\hbar/2kT_1)]=1\),

\[ \Gamma_{h;s'}\sim \Gamma_h\approx \frac{\sqrt{\pi}}{\hbar^2}\Delta_{0s'}^2\tau_l \exp[-\beta\mathcal{E}(T)], \]

where

\[ \mathcal{E}(T) = \beta^{-1}\sum_{fj}\lambda_{fj}^{0s'}\operatorname{th}\frac{\beta\hbar\omega_{fj}}{4}; \]

for \(T<T_1\)

\[ \Gamma_{h;s'} \approx \frac{2\pi}{\hbar^2}\Delta_{0s'}^2 \sum_{f_1 f_2} \prod_{r=1,2} \lambda_{f_r}^{0s'}\operatorname{sh}^{-1} \left(\frac{\beta\hbar\omega_{f_r}}{2}\right) \delta(\omega_{f_1}-\omega_{f_2}). \]

In (4)—(7), \(u_{\mu\nu}^{h}\) and \(u_{\mu\nu}^{d}\) are contributions to \(u_{\mu\nu}^{0}\) from “jumps” of the polaron packet and its “spreading”—the analogue of band transport. For \(T=T_q,\ T'_0\lesssim T_q\leq T_0\),

\[ \text{* More precisely, from (1) we have:} \]

\[ u_{\mu\nu}^{d}(\omega) = E_\beta^{-1}(\omega)\frac{1}{N_0} \sum_{s,s';\,n} (sn|v_\mu|s'n)\, \psi_{n\nu}^{0}(s-s') = \sum_{kn} v_\mu(kn)\psi_\nu^{0}(kn), \]

where

\[ \psi_\mu^{0}(kn) = |e|\,\operatorname{Re}\,\frac{1}{N_0} \sum_{ss'} e^{i\mathbf{k}(s-s')} \int_0^\infty dt\, e^{-\varepsilon t}\cos\omega t\, (s'n|v_\mu(t)|sn)_{\varepsilon\to+0} \]

can be obtained by the method of (8), which for equilibrium phonons also gives (7).

\(u_{\mu\mu}^{0}(0)\) has a minimum. For \(T<\mathcal{E}/2k\) and \(\hbar\omega<\mathcal{E}\), \(u_{\mu\mu}^{n}(\omega)\geq u_{\mu\mu}^{h}(0)\), and \(u_{\mu\mu}^{h}(\omega)\) increases with \(\omega\)*; for \(T\ll \mathcal{E}/k\) the “hops” of a carrier are activated by the fluctuational deformation of the lattice the more effectively, the closer the energy \(\hbar\omega\) of the photons dissipated in multiphonon processes is to the carrier binding energy \(\sim\mathcal{E}\).

Practically, for \(T>\{T_{1},T_{0}'\}\) (and certainly for not too small \(\omega\)) \(u_{\mu\mu}^{0}(\omega)\approx u_{\mu\mu}^{h}(\omega)\), while for \(\omega\gg\mathcal{E}/\hbar\), apparently, \(u_{\mu\mu}^{0}(\omega)\) decreases with \(\omega\), so that the mobility \(u_{\mu\mu}^{0}(\omega)\) at \(\omega\sim \mathcal{E}/\hbar\) probably has a noticeable maximum (with width \(\Delta\omega<\mathcal{E}/\hbar\)).

1. In an analogous way one can estimate the Faraday angle \(\theta_{\mathrm F}\) and \(u_{\mu\nu}^{(a)}(\omega)\equiv \tfrac12\bigl(u_{\mu\nu}(\omega)-u_{\nu\mu}(\omega)\bigr)\) for \(H\ne0\). For \(H\parallel OZ\), in the simplest case and for not too small \(\omega\) \((\ll\mathcal{E}/\hbar)\), for crystals (I), in which three appropriate sites \(s_{1},s_{2},s_{3}\) can be mutual nearest neighbors \((^{6})\), the formula for \(\theta_{\mathrm F}^{\mathrm I}(\omega)\equiv\theta_{\mathrm F}^{\mathrm I}(u_{xy}^{(a)};u_{\mu\mu};\omega)\) can be represented in the form (for the most essential \(T<\mathcal{E}/k\))

\[ \theta_{\mathrm F}^{\mathrm I}(\omega)\approx \frac{eN_{c}}{2cv_{0}(\omega)}\,u_{xy}^{\mathrm I}(\omega)\approx \]

\[ \approx \frac{eN_{c}}{2cv_{0}(\omega)} \left\{ u_{xy}^{0\mathrm I}\, \frac{1-(\hbar\omega/4\mathcal{E})^{2}}{\beta E_{\beta}(\omega)} + O\left( \delta_{2}u_{xy}^{\mathrm I}\, \frac{\operatorname{ch}\beta\hbar\omega/2}{\beta E_{\beta}(\omega)} \right) \right\}, \tag{8} \]

where \(v_{0}(\omega)\) is the refractive index at \(H=0\). In (8), \(u_{xy}^{0\mathrm I}\) and \(\delta_{i}u_{xy}^{\mathrm I}\) are determined for \(T>T_{0}'\) by the general expression (6) (in (6) \(\tau_{0}(\beta)\approx\hbar\beta\); see also \(\Psi_i(t)\)):

\[ u_{xy}^{\mathrm I}\equiv u_{xy}^{\mathrm I}(0)= \frac{|e|\beta}{2\hbar^{3}}\, \frac{1}{N_{0}} \sum_{s_{1}s_{2}s_{3}} \bigl(Q^{0}+\delta_{1}Q+\delta_{2}Q\bigr)\times \]

\[ \times \left\{ \frac{\partial}{i\,\partial H} \left[\Delta_{s_{1}s_{2}}\Delta_{s_{2}s_{3}}\Delta_{s_{3}s_{1}}\right] \right\}_{H=0} \equiv u_{xy}^{0\mathrm I}+\delta_{1}u_{xy}^{\mathrm I}+\delta_{2}u_{xy}^{\mathrm I}; \]

\[ Q^{0}= \exp\left[-\sum_{p,r=1}^{3}\Phi_{s_{p}s_{r}}(T)\right] \int_{0}^{\beta\hbar/2}d\tau' \left\{ 2\operatorname{Im}\int_{0}^{\infty}dt\,e^{-\Gamma_{0}|t|}\times \right. \]

\[ \left. \times \exp\left[ \Psi_{1}(t)+ \Psi_{2}\left(i\tau'+i\frac{\beta\hbar}{2}\right)+ \Psi_{3}\left(t-i\tau'+i\frac{\beta\hbar}{2}\right) \right] - \right. \]

\[ \left. - \int_{0}^{\beta\hbar/2}d\tau\, \exp\left[ \Psi_{1}\left(i\tau+i\frac{\beta\hbar}{2}\right)+ \Psi_{2}\left(i\tau'+i\frac{\beta\hbar}{2}\right)+ \Psi_{3}(i\tau-i\tau') \right] \right\}; \tag{9} \]

\[ \delta_{1}Q= \exp\left[-\sum_{p,r=1}^{3}\Phi_{s_{p}s_{r}}(T)\right] \operatorname{Re}\int_{0}^{\infty}dt\,e^{-\Gamma_{0}|t|} \int_{0}^{t}dt'\times \]

\[ \times \left\{ \exp\left[ \Psi_{1}(t)+ \Psi_{2}\left(t'+i\frac{\beta\hbar}{2}\right)+ \Psi_{3}\left(t-t'+i\frac{\beta\hbar}{2}\right) \right]+ \right. \]

\[ \left. + \exp\left[ \Psi_{1}\left(t+i\frac{\beta\hbar}{2}\right)+ \Psi_{2}\left(t'+i\frac{\beta\hbar}{2}\right)+ \Psi_{3}(t-t') \right] \right\}; \tag{10} \]

* For \(T<T_{0}'\) the mobility \(u_{\mu\mu}(\omega)\), as \(\omega\) increases, generally speaking decreases at small \(\omega\) (as also for \(T<T_{q}\)), has a minimum at \(\omega=\omega_{q}\) \(\bigl(u_{\mu\mu}^{h}(\omega_{q})=u_{\mu\mu}^{d}(\omega_{q})\bigr)\), and increases for \(\omega>\omega_{q}\) up to \(\omega\sim\mathcal{E}/\hbar\) (decreasing after the maximum). On p. 287 in (8), instead of \(T<T_{0}'\equiv\omega_{p}/\Phi_{0}<T_{0}\) one should read \(T<T_{0}'(\ll T_{0})\), \(T<\dfrac{\hbar\omega_{p}}{2k}\,(\operatorname{Arsh}\Phi_{0})^{-1}\); the contribution of acoustic phonons to \(\Omega_{b}\equiv\Omega_{0}(\sim\Gamma)\), in addition to the contribution of the form (22) from (8), may contain a contribution of the form \(AT^{l>0}\).

\[ \delta_2 Q = - \exp\left[-\sum_{p,r=1}^{3}\Phi_{s_p s_r}(T)\right] \operatorname{Im}\int_{0}^{\beta\hbar/2}d\tau'\int_{0}^{\infty}dt\, \times \]

\[ \times e^{-\Gamma_0|t|} \exp\left[ \Psi_1\left(t+i\frac{\beta\hbar}{2}\right) +\Psi_2\left(i\tau'+i\frac{\beta\hbar}{2}\right) +\Psi_3(t+i\tau') \right]. \tag{11} \]

In particular, for \(T<\mathscr{E}/k\),

\[ |\overset{0}{u}{}^{\mathrm I}_{xy}| \sim u_0\,\frac{H}{H_0}\,\frac{\beta\Delta_e^3}{\mathscr{E}^2}; \qquad \delta_2 u_{xy}^{\mathrm I} \sim u_0\,\frac{H}{H_0}\, \Delta_e^3\beta^2\mathscr{E}^{-1}e^{-\beta\tilde{\mathscr{E}}(T)}; \]

\[ |\delta_1 u_{xy}^{\mathrm I}| \sim u_0\,\frac{H}{H_0}\, \beta^2\mathscr{E}^{-1}\Delta_e^3 e^{-\beta\tilde{\mathscr{E}}(T)} +\Delta u_{xy}^{d}; \qquad \Delta u_{xy}^{d} \sim u_0\,\frac{H}{H_0}\, \beta\Delta^3(\hbar\Gamma_0)^{-2}, \]

where
\[ \tilde{\mathscr{E}}(T)=\mathscr{E}(T)+U(T);\quad U(T)\leqslant \mathscr{E}/3; \qquad \text{for } T>T_0 \quad \tilde{\mathscr{E}}\simeq \frac{4}{3}\mathscr{E}; \]
\(u_0=|e|a^2/\hbar;\ H_0=\hbar c/|e|a^2\). The expression \(\Delta u_{xy}^{d}\) in \(u_{xy}^{\mathrm I}\) is the analog of \(u_{xx}^{d}\) in \(u_{xx}^{0}\), and
\[ |\Delta u_{xy}^{d}|\,|u_{xy}^{\mathrm I}|^{-1} \sim (\Delta/\hbar\Gamma_0)^2(\mathscr{E}/\Delta_e)^2e^{-2\Phi(T)} \leqslant 1 \]
for \(T\geqslant T_c\), and usually \(T_c\leqslant T_0'\). The Hall mobility \(u_H=cH^{-1}|u_{xy}|u_{xx}^{-1}\) decreases with increasing \(T\) for \(T>T_q\) and for \(T<T_c(<T_0')\), see \((^6,^8)\) (but, apparently, it may increase for \(T_c<T<T_q<T_0\), if \(\nu\equiv u_H(T_q)u_H^{-1}(T_c)>1\), or decrease at all \(T\), if \(\nu<1\)).

The expression for \(u_{xy}^{\mathrm I}\) is due to the contribution of phase-correlated transitions of the carrier over three sites \(s_1,s_2,s_3\); \(\delta_1u_{xy}^{\mathrm I}\) corresponds to the contribution of real correlated transitions; the leading term at \(T<\mathscr{E}/k\), \(\overset{0}{u}{}^{\mathrm I}_{xy}\), is rather the contribution of (two out of three) such virtual multiphonon transitions and, unlike \(\delta_1u_{xy}^{\mathrm I}\), contains no activation temperature dependence, describing an essentially quantum and non-Markovian transport process (with nonactivation redeformation of the lattice). For crystals (II), which do not belong to class (I), the effects odd in \(H\) (Hall, Faraday, etc.) may be due mainly to the contribution of four-site correlated transitions, and then (the formula for \(u_{xy}^{\mathrm{II}}(\omega)\) is derived analogously)*

\[ |u_{xy}^{\mathrm{II}}| \sim |u_{xy}^{\mathrm I}|\Delta_e\mathscr{E}^{-1} \ll |u_{xy}^{\mathrm I}| \quad\text{and}\quad |\theta_F^{\mathrm{II}}| \sim |\theta_F^{\mathrm I}|\frac{\Delta_e}{\mathscr{E}}. \]

The sign of \(u_{xy}^{\mathrm I}(\omega)\) and \(\theta_F^{\mathrm I}(\omega)\) is usually \((^6)\) determined by the sign of the carrier charge \(e\)**. As is seen from (8)—(12), usually, for \(T<\mathscr{E}/k\) and \(\hbar\omega\ll\mathscr{E}\), \(\theta_F^{\mathrm I}(\omega)\) decreases with increasing \(\omega\) and with \(T\) (for asymptotically large \(\omega\), \(\theta_F^{\mathrm I}(\omega)\propto u_{xy}^{\mathrm I}(\omega)\propto\omega^{-2}\)).

Note added in proof. In a recent article \((^{10})\) it is assumed that \(u_{xy}^{\mathrm I}\) is determined by the probability of three-site hops, the result of whose estimate makes a contribution analogous to the term \(\delta_1u_{xy}^{\mathrm I}\) from (12) (see \((^6)\)).

Semiconductor Institute
Academy of Sciences of the USSR

Received
7 V 1963

REFERENCES

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\[ \text{* The expression for the contribution of four-site transitions for crystals (I) determines the first correction } |\Delta_1u_{xy}^{\mathrm I}|\sim |u_{xy}^{\mathrm I}|\Delta_e\mathscr{E}^{-1}. \]

** In (7), within the formulation of the problem and of the general approach used in (1, 3), the static \((\omega=0)\) mobility of a semiconductor with low mobility was calculated in a basis of polaron-band “waves” and in the technique \((^9)\), and for the principal contribution formulas were obtained analogous to the relations from \((^1,^6,^8)\). In view of the special features of the problem, one may think that for \(T>\hbar\omega_p/2k\), when the “mean free path” of such “waves” \(l\ll a\), their use as a basis is artificial.