UDC 539.2

PHYSICS

M. I. KLINGER

ON HOPPING TRANSPORT IN A DISORDERED LATTICE*

(Presented by Academician S. V. Vonsovskii, 31 X 1967)

§ 1. Statement of the problem. In this paper a quantum theory is developed (the basic relations) for the even (with respect to the magnetic field \(H\)) coefficients \(\sigma_{AB}^{(s)}\) of hopping transport in certain disordered lattices. For definiteness, the theory is set forth for the electrical conductivity \(\sigma_{xx}\equiv\sigma_p^{(l)}\) in the ohmic (and in a certain non-ohmic) region of electric fields \(E\), and for the thermopower \(\gamma_p^{(l)}\). The basic relations describe, at least for \(l=1\), the transport of nonadiabatic small polarons (below, s.p.) in a doped host lattice of a semiconductor; for \(l=2\), impurity conduction (in a more general approach, cf. \((^{3,4})\)), in particular, for strong-coupling polarons of both large and small radius.

A. Retaining essentially the model (and notation) of \((^{1,2})\), we consider a quantum system of interacting second-quantized electrons (holes) and phonons in the “random” field of a disordered lattice of the proper “ions” (host impurity centers—for \(l=2\)); the corresponding states \(|s\rangle\) of Wannier type are basic for the electrons. For brevity, the discussion concerns mainly the case of strong electron-phonon coupling, when its parameter \(\Phi\equiv\Phi^{(l)}(T)\geq\Phi_0\equiv\Phi^{(l)}(0)\gg1\). Taking into account here the smallness of Bloch electronic overlaps \(\Delta_e(s_{12})\) (formulas (2), (3)) and the pair repulsion \(I\) of polarons on an individual “ion,” the Hamiltonian here (see also \((^{14})\)) may be taken in the form \((\hbar\equiv1)\)**:

\[ \hat{\mathcal H}\equiv\hat{\mathcal H}^{(l)}=\hat{\mathcal H}_0+\hat{\mathcal H}_1;\qquad \hat{\mathcal H}_0=\sum_{s\sigma}\varepsilon(s)\hat n_{s\sigma} +\sum_{\Lambda}\omega_{\Lambda}c_{\Lambda}^{+}c_{\Lambda} +I\sum_s \hat n_{s,1/2}\hat n_{s,-1/2}; \]

\[ \hat{\mathcal H}_1=\sum_{s_1s_2\sigma}\sum_{\Lambda} \Delta_e(s_{12})\left(\hat T_{\Lambda}^{s_1}+\hat T_{\Lambda}^{s_2}\right) a_{s_1\sigma}^{+}a_{s_2\sigma}, \tag{1} \]

where

\[ \hat n_{s\sigma}\equiv a_{s\sigma}^{+}a_{s\sigma};\qquad c_{\Lambda}\equiv \hat T_{\Lambda}^{+}b_{\Lambda}\hat T;\qquad \sigma=\pm \tfrac12;\qquad \Lambda\equiv\{fj\}\ \text{or}\ \{\lambda\}; \]

\(j=1,2,\ldots\), and \(\hat T_{\Lambda}\equiv \prod \hat T_{\Lambda}^{s}\) is a renormalizing unitary transformation \((^2)\); \(\lambda\) describes possible local phonons \((^5)\). Below the actual case meant is \(I\gg I_0\equiv kT+D_{cl}\). In calculating \(N_c(\zeta)\) and other

* The main results (§ 2 A, B and § 3) were contained in the author’s review report at the International Conference on Magnetic Oxides (Czechoslovakia, 1966) \((^{16})\); see also \((^{1a})\).

** In what follows the following notation is also used: \(c\equiv N_iN_0^{-1}<1\), where \(N_0\equiv N^{(i=1)}\) and \(N_i\equiv N^{(i=2)}\); \(N\equiv N^{(l)}=3/4\pi r_0^3\) and \(N_c\equiv N_c^{(l)}\) are the concentrations of “ions” and current carriers, respectively, with \(r_0\equiv r_0^{(l)}\); \(K\) is the degree of compensation of the host impurity, \(0\leq K\leq1\).

macroscopic quantities \((^{5,1})\) require an adequate averaging \(\langle \ldots \rangle_{AV}\) over “random” configurations \(\{s\}\) of “ions,” with allowance for concentration broadening due to the fluctuations \(\{\Delta_e(s)\}\) and \(\{u\}\), \(u \equiv \varepsilon(s)-\bar{\varepsilon} \equiv u_s\) (energy measured from the bottom of the conduction band at \(c=0\)); it turns out that the chemical potential of the charge carriers (hereafter called polarons) \(\bar{\zeta}\equiv \zeta-\bar{\varepsilon}\ll I/2\) for \(p\equiv p^{(l)}\equiv N_cN^{-1}\ll 1\), with \(p^{(1)}\ll 1\) for \(c(1-K)\ll 1-K_0\) and \(p^{(2)}\ll 1\) for \(K\gg K_0\sim \exp(-I/2kT)\ll 1\);

B. We apply Kubo’s general formulas for \(\sigma^{(s)}_{AB}\), which also include \(\langle \ldots \rangle_{AV}\) (see above), and the special perturbation theory from \((^{2,1})\) in the present case of classical concentration broadening \((D_{cl})\), with effective width \(D_{cl}\ll (2\mathcal{E}\omega_p)^{1/2}\simeq \Phi_0^{1/2}\omega_p\) (and \(D_{cl}\ll \delta_0\equiv |\varepsilon^{(2)}_{c=0}|\)). From an analysis of the series of terms in the expansion of \(\sigma_p\) (in \(\Delta_e\)) it follows that the principal contribution \(\sigma_p=\sigma^{(hop)}_p{}_0\) is due to uncorrelated two-site hopping involving incoherent (multi-)phonon processes \((^2)\). For \(p\ll 1\), the inequalities \((^1)\) were adopted as criteria of the theory (for \(\omega_\Lambda\simeq \mathrm{const}\equiv \omega_p\)): \(\Delta_0\ll (\mathcal{E}kT)^{1/2}(<\mathcal{E})\) for \(\mathcal{E}/k>T>T_0\equiv \hbar\omega_p/2k\), or \(\Delta_0\ll \xi D_{cl}\) for \(T<T_0(\operatorname{Ar\,sh}2\Phi_0)^{-1}\) for m.p.; moreover \(\mathcal{E}\equiv \mathcal{E}^{(l)}\simeq \tfrac12\omega_p\Phi_0\) and \(\xi\equiv \xi^{(l)}\), \(\xi^{(1)}\approx 1\) and \(\xi^{(2)}>1\); \(\Delta^{(2)}_0\ll \xi^{(2)}D^{(2)}_{cl}\) for \(\xi^{(2)}\gtrsim 1\) for continuum polarons of strong coupling. Here \(\Delta_0\equiv \Delta^{(l)}_0(r_0)\equiv \Delta_e[r_0a_l(r_0)]\), with \(a_l(r_0)\approx 1\), while \(\Delta^2_0(r_0)\) can be estimated from the values \(\sigma_\ell{}_0\propto \Delta_0^2\) (see (2) and below). The criterion \(\Delta^{(2)}_0\ll \xi^{(2)}D^{(2)}_{cl}\) can be satisfied for small \(N_i<N_{cr}\equiv 3/4\pi r^3_{cr}\) and, tentatively, apparently \(r^3_{cr}(r^{imp}_B)^{-3}\sim (10^2\div 10^3)\) (cf. \((^{3,7})\)).

§ 2. Electrical conductivity. Optical absorption.

A. The explicit formula obtained for \(\bar{\sigma}_p{}_0(\omega)\equiv \operatorname{Re}\sigma^{(l)}_p{}_0(\omega)\) (in a weak field \(E\)) takes into account the “self-consistent” character of the hops of charge carriers under their mutual repulsion \(I\) and the Pauli correlation in Fermi degeneracy. In particular, for \(\eta\equiv \mu_BH\sigma\ll kT\) and \(p\ll 1\) (cf. \(\bar{\sigma}^h_{xx}(\omega)\) in \((^2)\))

\[ \bar{\sigma}_p{}_0(\omega)\equiv |e|N_c\mu_p(\omega) = \frac{2e^2}{\omega}\operatorname{th}\frac{\beta\omega}{2}\cdot \varphi^{(0)}_p(\omega), \tag{2} \]

where

\[ \varphi^{(q)}_p(\omega)\equiv \sum_{s_1s_2}\sum_{u_1u_2} \left\langle \sigma_p(s_{12};u_1,u_2;\omega) \left(\frac{u_1+u_2}{2}\right)^q \right\rangle_{AV}; \]

\[ \sigma_p(s_{12};u_1,u_2;\omega)\simeq (s_{12})_x^2 W^{12}_h(\omega) Z^{(12)}_{cc}; \]

\[ W^{(12)}_h(\omega) = \frac12|\Delta_e(s_{12})|^2 \sum_{\pm}\operatorname{Re}\int_0^\infty dt\, \left(e^{\Psi(t)}- \right. \]

\[ \left. -1\right) \exp[-2\Phi+it(\eta_1-\eta_2)+i(t-i\beta/2)(u_{12}\pm \omega)]; \]

\[ Z^{(12)}_{cc}\equiv f_\infty(u_1)(1-f_\infty(u_2)); \]

\[ f_\infty(u)\equiv f(u;\beta I\gg 1) =\{1+\tfrac12\exp(\beta u-\beta\bar{\zeta})\}^{-1}; \]

\[ \beta\equiv 1/kT;\qquad s_{12}=s_1-s_2;\qquad u_{12}\equiv u_1-u_2; \]

\[ \Psi(t)= \sum_\Lambda |\gamma^{s_1}_\Lambda-\gamma^{s_2}_\Lambda|^2 \omega_\Lambda^{-2}\cos\omega_\Lambda t \left(\operatorname{sh}\frac{\beta\omega_\Lambda}{2}\right)^{-1}. \]

\(W_h\) is the “one-particle” (without allowing for \(I\)) hopping probability (per 1 sec.), with \(\mu_p\ll \mu_0\equiv |e|r_0^2\hbar^{-1}\). For sufficiently small \(p<p_0(\ll 1)\), when \(\zeta(p)<\zeta(p_0)<0\) (absence of degeneracy), the contribution of \(I\) is insignificant and \(\sigma_p{}_0\propto p\simeq 2Z_1\exp(-\beta|\bar{\zeta}|)\) for \(Z_1\equiv \langle \exp(-\beta u)\rangle_{AV}\). For \(p\ll 1\) and \(T>T_0\)

\[ \sigma_p^{(l)}(\omega=0)\sim \mu_0\cdot \beta z\Delta_0^2/(\mathscr E kT)^{1/2}\cdot \exp[-\beta \mathscr E^{(l)}-\beta W^{(l)}], \]
with \(W^{(l)}=W^{(l)}(c,K;T)\); here
\[ N_c^{(l)}\propto \exp(-\beta W^{(l)}). \]
And, in general,
\[ \sigma_p^{(2)}(N_c^{(2)}W_h^{(2)}(0))^{-1}\propto \exp(-\beta W^{(2)}). \]

B. In general, the characteristic shape of the absorption band associated with \(\bar\sigma_p(\omega)\) by a standard specimen (for \(\omega>2kT\) and \(\omega\gg D_{cl}\), both for \(T>T_0\) and for \(T<T_0\)) is obtained from the expansion of the \(\omega\)-“distribution” \(W_h(\omega)\) in its semi-invariants \(\lambda_\nu\), \(\nu=1,2,\ldots\) (for \(D_{cl}\ll \{(\mathscr E kT_0)^{1/2}\}\))\(^{13}\)

\[ \bar\sigma_p(\omega)\propto \omega^{-1}\chi(x),\quad \chi(x)\simeq G-\frac{\lambda_3}{6\lambda_2^{3/2}}\frac{d^2G}{dx^2} +\frac{\lambda_4}{24\lambda_2^2}\frac{d^4G}{dx^4} +\frac{\lambda_3^2}{72\lambda_2^3}\frac{d^6G}{dx^6}, \tag{3} \]

where \(G\equiv G(x)=e^{-x^2/2}\) and \(x\equiv(\omega-\bar\omega_0)D^{-1}\), with
\[ \bar\omega_0\equiv \omega_0+\chi D_{cl}\big|_{\chi\sim 1}\simeq \omega_0; \]
\[ \lambda_3\lambda_2^{-3/2}\simeq (2\Phi_0)^{-1/2}(\operatorname{th}T_0/T)^{3/2},\quad \lambda_4\lambda_2^{-2}\simeq (2\Phi_0)^{-1}\operatorname{th}T_0/T; \]
\[ \lambda_2=D^2\simeq 4\mathscr E\omega_p\operatorname{cth}T_0/T, \]
and for \(\Phi_0\gg 1\) formula (3) is adequate, at least in the region of width \(2D\) of the band peak for \(|x|\lesssim 1\)—at \(T\leq \mathscr E/2k\) (cf. \(^{1,8}\)) \(\bar\sigma_p(\omega)\) has a broad \((2D>kT+\omega_p+D_{cl})\), but distinct \((2D<\omega_0)\), peak at \(\omega\simeq \omega_0\equiv 4\mathscr E\) with a shape the closer to Gaussian the higher \(T\) is (similarly for m.p., see \(^{1,2,8,9,17}\)). Owing to concentration broadening, for \(l=1\) the peak frequency (and effectively its width) may depend on \(c\), while
\[ \bar\sigma_p^{(2)}(\bar\omega_0)=\bar\sigma_p^{(2)}(\bar\omega_0;c,K)\propto (\Delta_0^{(2)}(c))^2. \]

C. It turns out that a strong (constant) electric field \(E\) may here (for \(\Phi_0\gg 1\)) play a role analogous to that of the absorbed quantum \(\omega\), which activates (for \(T<\mathscr E/2k\) and \(\omega\simeq \omega_0\)) the jumps. This is connected with the fact that, when the interaction \(\hat V\) of local polarons with the field \(\bar E\equiv E_x\) is taken into account, \(\hat V=eE\hat x\) and \(\hat{\mathscr H}\to \hat{\mathscr H}+\hat V\), the Hamiltonian retains its structure, with
\[ \varepsilon(s)\to \varepsilon(s)+eEs_x,\quad \Delta_e(s_{12})\to \Delta_{12}(E)\equiv \Delta_e(s_{12})+eEx_{12} \]
for \(x_{12}=\langle s_1|x|s_2\rangle(1-\delta_{s_1s_2})\). In this case one can in an analogous way (see § 1 and \(^{1,2}\)) explicitly calculate \(\sigma_p(E)\) also in a certain range of non-ohmic fields
\[ E\equiv \nabla\varphi(\mathbf r)\quad (E<E_m\equiv E_m^{(l)}), \]
if it is assumed that \(\sigma_p(E)\) is determined by the Kubo formula for \(\sigma_{xx}\), in which the current correlator \(\operatorname{Re}\langle \hat j_x\hat j_x(t)\rangle\) is modified in the sense that
\[ \hat j_x(t)=\exp[it(\hat{\mathscr H}+\hat V)]\hat j_x \times \exp[-it(\hat{\mathscr H}+\hat V)] \]
with the contribution of \(\hat V(\propto E)\) taken into account. This apparently follows if, in considering
\[ \sigma_p(E)\equiv \sigma_p^{(l)}(E), \]
one applies method \(^{15}\) for not too strong “non-ohmic” fields
\[ E<E_m\equiv E_m^{(l)}\ ( \equiv \delta\varphi/r_m), \]
for which the characteristic length
\[ \Lambda_0(\sim |\Delta\varphi/\varphi|)>r_m>r_0. \]
In this case (allowing for the replacements mentioned in (1)) the inequalities from § 1,B are still adopted as criteria of the theory (for \(E<E_m\)), and in the formula for \(\sigma_p(E)\), analogous to (2), \(eE(s_{12})_x\) plays the role of the “external” frequency \(\omega\).

The region of non-ohmic behavior occurs for
\[ E>E_0\equiv E_0^{(l)}=2kT/|e|r_0, \]
where \(E_0^{(1)}\gg E_0^{(2)}\) for \(c\ll 1\), and one may assume that for \(kT\ll \mathscr E\) usually \(E_m\gg E_0\). The principal result is that, for all the \(T\) considered and for \(E_0<E(<E_m)\), at least for
\[ |x_E|\equiv |x_E^{(l)}|\lesssim 1, \]
\[ \sigma_p(E)\equiv \sigma_p^{(l)}(E)\propto \Delta_0^2(E)\chi(x_E)E^{-1}, \tag{4} \]

and for \(T>T_0\) and \(0<E\lesssim \omega_0 T(|e|r_0T_0)^{-1}(<E_m)\)

\[ \sigma_p^{(1)}(E)\simeq \sigma_p^{(1)}(E=0)\, \frac{\operatorname{sh}\left(\tfrac12\beta |e|Er_0^{(1)}\right)} {\tfrac12\beta |e|r_0^{(1)}E} \exp\left[-\,\frac{(eEr_0^{(1)})^2}{16\mathscr{E}kT}\right], \]

\[ \left.\sigma_p^{(1)}(E)\right|_{E>E_0^{(1)}}\propto \frac{\Delta_0^2}{E}\,G(x_E), \tag{5} \]

where \(x_E\equiv (|e|Er_0-\bar{\omega}_0)D^{-1}\), and for real \(\omega_0\equiv 4\mathscr{E}\) one may put \(E_m>\omega_0/|e|r_0\). As follows for \(T<\mathscr{E}/2k\), \(\sigma_p(E)\), from (4)—(5), has a characteristic “resonance” peak at \(eEr_0\approx \omega_0\), with a shape close to Gaussian; moreover, as for \(\sigma_p(\omega)\), this peak is broad (and it is easier to observe it for \(T\lesssim T_0\)). In particular, what has been said, together with (4), (5), for \(l=1\) describes \(\sigma_p^{(1)}(E)\) for s.p. in an ideal lattice (for \(kT\gg \Delta_e e^{-\Phi_0}\) and \(D_{cl}^{(1)}=0\)), also for \(E>E_0^{(1)}\); formula (5), for \(D_{cl}^{(1)}=0\), is analogous to the formula obtained in \((^{16})\) for \(T>T_0\) (for \(p^{(1)}<p_0^{(1)}\ll 1\)) in Holstein’s quasiclassical scheme \((^{6})\), and the “quasiclassical” region \(E\), apparently (see above), is applicable also outside this region, in particular for \(|e|Er_0^{(1)}\approx \omega_0\equiv 4\mathscr{E}\) for \(kT\lesssim \mathscr{E}\).

§ 3. For \(p\ll 1\), to calculate the thermoelectric power \(\gamma_p\) (neglecting phonon drag) from the Kubo formulas, as in \((^{10,2,1,12})\), the canonical energy-current operator of the system is used in the form \(\hat I_x=\hat I_x^{(el)}+\hat I_x^{(ph)}\), where its phonon part \(\hat I_x^{(ph)}\) has the standard form (and is not written explicitly in \((^{10,2})\)); \(I_x\) is \(\hat I_x^{(el)}\), while

\[ \hat I_x^{(el)}=(2e)^{-1}\{j_x,\mathcal H_c+\mathcal H_{int}\} \]

for \(\Phi_0\gg 1\); \(\{a,b\}\equiv \tfrac12(ab+ba)\). Similarly, energy transfer has mainly the character of convection (without the kinetic contribution), so that \(\gamma_p\) is isotropic in the general case, i.e. \((\gamma_p)_{\mu\nu}\simeq \gamma_p\delta_{\mu\nu}\), and \((^{1a})\)

\[ \gamma_p\simeq (eT)^{-1}(U_0-\bar{\zeta}),\qquad U_0=(u)+x_0 I_p\big|_{x_0\sim 1},\qquad (u)\equiv \varphi_p^{(1)}(0)/\varphi_p^{(0)}(0). \tag{6} \]

For \(D_{cl}^{(1)}\ll\{\Delta_e^{(1)},\,kT;\,\omega_p\}\), formulas (2)—(6) actually describe \(\bar\sigma_p(\omega)\), \(\sigma_p(E)\), and \(\gamma_p\) for s.p. (including degenerate ones) in an ideal lattice; for \(p<p_0^{(1)}\), i.e. in the absence of degeneracy,

\[ \gamma_{0p}^{(1)}\simeq -\,\frac{\bar{\zeta}}{eT} =\frac{k}{e}\ln\frac{2N_0}{N_i^{(1)}} \]

cf. \((^{2,1,10-12,16,17})\)*.

Institute of Semiconductors
Academy of Sciences of the USSR

Received
9 X 1968

CITED LITERATURE

1. M. Klinger, a) Rep. Progr. Phys. 31, 1 (1968); b) Czechoslovak J. Phys. 17B**, No. 4, 301 (1967).
2. M. Klinger, Phys. Stat. Sol., 11, 499 (1965).
3. N. F. Mott, D. Twose, Adv. Phys., 10, 107 (1961).
4. M. Klinger, Phys. Lett., 16, 7 (1965).
5. I. M. Lifshitz, UFN, 84, 617 (1964).
6. T. Holstein, Ann. Phys. N. Y., 8, 343 (1959).
7. P. Anderson, Phys. Rev., 109 (1958).
8. M. Klinger, Phys. Stat. Sol., 3, 820 (1963); M. Klinger, DAN, 157, 566 (1964); Phys. Lett., 7, 102 (1963).
9. H. Reik, Solid State Commun., 1, 67 (1963).
10. M. M. Klinger, Phys. Stat. Sol., 2, 1066 (1962).
11. F. Morin, Bell. Techn. J., 37, 1047 (1958).
12. K. Schotte, Zs. Phys., 196, 393 (1966).
13. H. Cramer, Random Variables, Moscow, 1947, p. 108.
14. A. G. Samoilovich, M. Klinger, V. M. Nitsovich, ZhTF, 27, No. 12 (1957); J. Hubbard, Proc. Roy. Soc., A, 271, 401 (1964).
15. D. N. Zubarev, DAN, 140, 92 (1961).
16. A. L. Efros, FTT, 7, 1152 (1967).
17. M. Klinger, Phys. Stat. Sol., 27, 479 (1968).

* See also § 11.1 of \((^{1a})\) and footnote 5 in \((^{17})\).

** These contain the relevant references on the theory of transport by small polarons in an ideal lattice.