Abstract
Full Text
PHYSICS
M. I. KLINGER
THEORY OF LOW-TEMPERATURE TRANSPORT PHENOMENA IN SEMICONDUCTORS WITH LOW MOBILITY
(Presented by Academician A. A. Lebedev on 10 VII 1962)
- In papers ((^1,{}^6)) a theory was proposed for high-temperature transport phenomena in semiconductors with low mobility, i.e., in a certain approximation, a solution of the problem of non-Boltzmann transport in an electron–phonon system with strong interaction, whose parameter is (\Phi_0 \gg 1) (see below the criterion (\hbar \Gamma \gg \Delta)). To calculate the transport coefficients (\sigma_{MN}), their general expressions ((^2)) of the type
[
D_{MN}=\int_{-\infty}^{\infty}\frac{dl}{2\pi}\,\operatorname{Re}\operatorname{Sp} e^{\beta F-\beta H}N(\mathcal H-l+i\varepsilon)^{-1}M(\mathcal H-l-i\varepsilon)^{-1}\bigg|_{\varepsilon\to+0},
\tag{1}
]
are used; for these, effective expansions are constructed in the basis of localized small-polaron functions (|sn\rangle) ((^4,{}^{6\mathrm{b}})) ((s) is the cell number, (n \equiv (\ldots N_{fj}\ldots)), (N_{fj}) are phonon numbers), where (\mathcal H), (M), (N) are the Hamiltonian and the current operators for the “one-particle” system, and
[
(sn|\mathcal H|sn)=\varepsilon_n-\varepsilon_{\mathrm{pol}},
]
where ((^{6\mathrm{a}})), p. 1.4)
[
\varepsilon_n=\sum_{fj}\hbar\omega_{fj}N_{fj};\qquad
\varepsilon_{\mathrm{pol}}=\varepsilon_0+(s|\delta V|s)-\delta\varepsilon,\qquad
\delta\varepsilon=\sum_{fj}|V^{ss}{fj}|^2\omega.}^{-1
]
According to ((^{6\mathrm{b}})), for (\operatorname{Re}D_{MN}) the first correction (\delta_1) and the second correction (\delta_2) are smaller than the principal term (\operatorname{Re}D^0_{MN}(00;00)) (respectively for (\operatorname{Im}D_{MN})), if (below (\hbar=c=k=1))*
[
\Delta \ll \Gamma,\qquad \Gamma \ll \omega_p;
\tag{2}
]
[
\Delta_e^2(\omega_p\Phi_0T)^{-1}\ll 1,\qquad
\Delta_e^2(\omega_p\Phi_0)^{-2}\underset{\chi<1}{\lesssim}\exp(-\beta\hbar\mathcal E_0)\ll 1,
\tag{3}
]
for we have**
[
|\delta_1|\cdot|\operatorname{Re}D_{MN}|^{-1}\sim \Delta_e(\omega_p\Phi_0)^{-1};
\qquad
|\delta_2|\cdot|\operatorname{Re}D_{MN}|^{-1}\in
\left{\Delta_e^2(\omega_p\Phi_0T)^{-1};
\Delta_e^2(\omega_p\Phi_0)^{-2}\exp(\beta\hbar\mathcal E_0)\right}.
]
In (2), (3):
[
\Delta=\Delta_0\exp[-\Phi(T)],
]
where usually (\Delta_0\simeq\Delta_e), (\Delta_e) is the width of the electronic band;
[
\Phi(T)\equiv \frac12\sum_{fj}\lambda_{fj}^{ss'}\operatorname{cth}\left(\frac{\beta\omega_{fj}}{2}\right);
]
* For (\Delta\ll\Gamma), the method for estimating the corrections in ((^{6\mathrm{b}})) is equivalent to taking account of the successive subtraction, at (\Delta\ll\Gamma), of small contributions of “singular” matrix elements (in the resolvent expansion) in which sets (n) are repeated (the account of the operation (ir) is essential, and footnote ((^4)) in ((^{6\mathrm{b}})) refers only to the first correction).
** In paper ((^{6\mathrm{b}})) it should read: in III, 11, instead of (\Delta_e(\omega_p\Phi_0)^{-1}\ll 1), read
[
\Delta_e^2(\omega_p\Phi_0)^{-2}\lesssim \exp(-\beta\hbar\mathcal E_0)\ll 1;
]
in Appendix 2.v, P.2.9, instead of (\operatorname{Re}D^0_{MN}(00;10)), read (\operatorname{Im}D^0_{MN}(00;10)); in P.2.10, instead of (\exp(-\beta V_0)), read (\exp(+\beta\mathcal E_0)); in the last estimate, instead of
[
|\delta_2|\cdot|\operatorname{Re}D^0_{MN}(00;10)|^{-1}\lesssim(\Omega/\omega_p)^{1/2},
]
read
[
|\delta_2|\cdot|\operatorname{Re}D_{MN}|^{-1}\in
\left{\Delta_e^2(\omega_p\Phi_0T)^{-1};
\Delta_e^2(\omega_p\Phi_0)^{-2}e^{\beta\hbar\mathcal E_0}\right}
]
(i.e., the second condition (3) is sufficient). In P.38 ((^{6\mathrm{b}})) the correction must have the form:
[
\delta_1D_{MN}=D^1_{MN}(00;00)-D^0_{MN}(00;00)\sim(\mathcal H')^3,
]
since
[
\delta_0G\sim(\mathcal H')^3.
]
[
\Phi(0) \equiv \Phi_0;\quad \lambda_{fj}^{ss'} \text{ are the lattice-deformation parameters; at } T \gg \omega_p \text{ and } \Phi_0 \gg 1
]
[
\Gamma \simeq \Gamma_b' \equiv 2\pi \sum_{k;\, n,n'(\ne n)}
\exp(\beta F_0-\beta \varepsilon_{nk}) |(kn|\mathcal H'|kn')|^2
\delta(\varepsilon_{nk}-\varepsilon_{n'k}) \simeq \Gamma_0 e^{-\beta \mathcal E_0},
\tag{4}
]
[
\Gamma_0 \equiv \frac{\sqrt{\pi}}{2}\Delta_e^2(\mathcal E_0 T)^{-1/2},\qquad
\mathcal E_0 \equiv \sum_{fj}\frac{1}{4}\lambda_{fj}^{ss'}\omega_{fj}.
\tag{5}
]
Above, (\omega_p) is the characteristic frequency of the essential phonons. Let us note that (\Gamma \simeq \Gamma' \gg \Delta) at (T>T_0) and for not too small (\Delta_e), (T_0 \sim \omega_p) [6] (in the general case (T_0 \gtrsim \omega_p)).
2. Let us consider here the basic relations of the theory of transport phenomena in the same semiconductors, but at low (T<T_0' \equiv \omega_p/\Phi_0<T_0). In this range of (T), considering for simplicity the static case, we have for (\sigma_{MN}^{(s,a)} \equiv \frac12(\sigma_{MN}\pm\sigma_{NM})):
[
\sigma_{MN}^{(a)} =
N_c \int_0^\infty dt\, e^{-\varepsilon t}
(\Psi_{MN}-\Psi_{NM})
\int_0^t d\tau\, \frac{2}{\pi\hbar}\ln \operatorname{cth}\frac{\pi\tau}{2\beta\hbar},
]
[
\sigma_{MN}^{(s)} =
\frac12 \beta N_c \int_0^\infty dt\, e^{-\varepsilon t}
(\Psi_{MN}+\Psi_{NM})
\equiv \frac{N_c}{2}(\varphi_{MN}+\varphi_{NM}),
\tag{6}
]
where
[
\Psi_{MN}(t) \equiv
\operatorname{Re}\operatorname{Sp}\exp(\beta F-\beta\mathcal H)\,
N\exp(i\mathcal H t)M\exp(-i\mathcal H t).
\tag{7}
]
It is convenient to compute the trace in the basis of orthonormalized functions
[
(kn)\equiv u_{nk}e^{i\mathbf{kx}}
\equiv N_0^{-1}\sum_s e^{-i\mathbf{k s}}|sn\rangle
=|\mathbf{k}+2\pi\mathbf{g}, n\rangle,
\tag{8}
]
describing the state of the carrier (polaron) in the band and the phonon system ((\mathbf{k}) is the total quasimomentum of the system; (\mathbf{g}) is a reciprocal-lattice vector). The energy of the system unperturbed in the (kn)-basis is
((kn|\mathcal H|kn)\equiv \varepsilon_{nk}=\varepsilon_n+\varepsilon_n(\mathbf{k})), and in the nearest-neighbor approximation
[
\varepsilon_n(\mathbf{k}) = E(\mathbf{k}_n)\exp[-\Phi(n)] = \varepsilon_n(\mathbf{k}+2\pi\mathbf{g}),
]
[
\Phi(n)\equiv \frac12\sum_{fj}\lambda_{fj}^{ss'}(1+2N_{fj}),
]
where (E(\mathbf{k})) is the dispersion law in the corresponding electron band, (\mathbf{k}n=\mathbf{k}-\sum fj N)) is (\Delta); for (\Phi_0\gg1), (\Delta\ll\omega_p). The perturbation matrix is}) is the carrier quasimomentum (to within (2\pi\mathbf{g})), and the mean width of the bands (\varepsilon_n(\mathbf{k
((kn|\mathcal H'|k'n')\equiv(kn|\mathcal H'|k'n')_{n\ne n'}). The transport under consideration is determined by scattering of band carriers between (|kn\rangle)-states by means of multiphonon processes. Since (\mathcal H) is translationally invariant, in the reduced-(\mathbf{k}) scheme
[
(kn|\mathcal H'|k'n')=\delta_{kk'}(kn|\mathcal H'|kn').
\tag{9}
]
In the principal approximation of the theory one should set:
[
(kn|\rho_0(\mathcal H)|k'n')\equiv
(kn|\exp(\beta F-\beta\mathcal H)|k'n')
\simeq f_0(kn)\delta_{kk'}\delta_{nn'},
\tag{10}
]
[
f_0(kn)=\exp(\beta F_0-\beta\varepsilon_{nk});\qquad
\delta_{nn'}\equiv\prod_{fj}\delta_{N_{fj},N'_{fj}},
\tag{11}
]
and, at (H=0) ((H) is the external magnetic field), for ((kn|M|k'n'))
[
(kn|M|k'n')=
\delta_{kk'}\bigl[(kn|M^0|kn)\delta_{nn'}+M_{nn'}(\mathbf{k})\bigr]
\simeq \delta_{kk'}\delta_{nn'}M^0(kn).
]
The operator defined in (9) satisfies, as is easy to show, the basic Van Hove relation (5), and below the consequence of (9, 10) is used. We find that
[
\varphi^{a}{MN}\equiv \sum(kn)\, f'} N^{0{M}(kn)
= \sum(\varepsilon;kn),} N^{0}(kn)\, f'_{M
\tag{12}
]
[
f'{M}(\varepsilon;kn)=\beta\,\operatorname{Re}\int}^{\infty} dt\, e^{-\varepsilon t
(kn|\rho_{0}e^{i\mathcal H t}Me^{-i\mathcal H t}|kn)
=\beta\,\operatorname{Re}(kn|\rho_{0}f'_{M}(\varepsilon)|kn).
\tag{13}
]
Further (10),
[
Y f'{M}(\varepsilon)
=(\varepsilon-iL(\varepsilon)\right)}-iYL')^{-1}\left(M'+iYL'D f'_{M
]
and
[
I_{\varepsilon}D f'{M}(\varepsilon)
\equiv [\varepsilon+DL'(\varepsilon-iL(\varepsilon)}-iYL')^{-1}YL']D f'_{M
=
]
[
=Q_{M}(\varepsilon)\equiv D[1+iL'(\varepsilon-iL_{0}-iYL')^{-1}Y]M;
\tag{14}
]
[
(\alpha|D\hat f|\alpha')\equiv(\alpha|\hat f|\alpha')\delta_{\alpha\alpha'};
\qquad
(\alpha|Y\hat f|\alpha')\equiv(\alpha|\hat f|\alpha')_{\alpha\ne\alpha'},
\tag{15}
]
[
DM\equiv M^{0},\qquad
L_{0}\hat f\equiv[\mathcal H_{0},\hat f];\qquad
L'\hat f\equiv[\mathcal H',\hat f],\qquad
YM\equiv M'.
]
Using (10)—(15) and the expansion (in (\mathcal H')) (\rho_{0}(\mathcal H_{0}+\mathcal H')) and ((\varepsilon-iL_{0}-iY')^{-1}=(\varepsilon-iL_{0})^{-1}+(\varepsilon-iL_{0})^{-1}iYL'(\varepsilon-iL_{0})^{-1}+\cdots), one can obtain an equation for (f'(kn)\equiv\sum_{M}' f'{M}(kn)F+f'{j}(kn)E). Neglecting the contribution (QD f'}(\varepsilon)-M^{0}) and the higher-order correction terms in (\mathcal H'), indicated below, in (I_{\varepsilon{M}(\varepsilon)|), we obtain, in the basic approximation so defined, the transport equation*
[
\left(\frac{\partial f}{\partial t}\right){\rm dyn}
+
\left(\frac{\partial f}{\partial t}\right)
=0;
\qquad
\left(\frac{\partial f}{\partial t}\right){\rm dyn}
=-\beta f(kn)Q(kn),
]
[
\left(\frac{\partial f}{\partial t}\right){\rm coll}
\equiv If'(kn)
=\Omega_{kn}f'(kn)
-\sum_{n''}\omega_{kn;kn''}\delta(\varepsilon_{nk}-\varepsilon_{n''k})\,f'(kn'');
\tag{16}
]
[
Q(kn)=\sum_{M}' M^{0}(kn)F_{M}+eEv^{0}(kn),
\tag{17}
]
[
\Omega_{kn}\equiv\sum_{n''}\omega_{kn;kn''}\delta(\varepsilon_{kn}-\varepsilon_{kn''});
\qquad
\omega_{kn;kn}=0;
\qquad
\langle \Omega_{kn}\rangle_{0}\equiv\Omega_{b}.
]
The expression for (\omega_{kn,kn''}), including terms (\sim(\mathcal H')^{2+l}|{l=1,2,\ldots}), is analogous to the corresponding expression in § IV (9), if (l\equiv(k,n)) and (9) is taken into account (and therefore it is not written out explicitly; item 4 in (6b) refers only to the possible case when, for (T<T')). For}), one may restrict oneself to the lowest approximation in (\mathcal H'), (\omega^{0}_{kn;kn''}=2\pi |(kn|\mathcal H'|kn'')|^{2
[
\frac{e\hbar}{m^{*}c}H\ll\Delta,
]
i.e.
[
H\ll H_{0}=\hbar c/|e|a^{2},
]
we again have (16), but
[
\left(\frac{\partial f}{\partial t}\right){\rm dyn}
=-\beta ff'(kn),}(kn)Q(kn)-e(\mathbf v\times\mathbf H]\cdot\nabla_{k
\tag{18}
]
(m^{*}=\hbar^{2}(\Delta a^{2})^{-1}) is the mean effective mass, (a) is the lattice constant.
It is not difficult to take into account the contribution (\delta\sigma_{MN}=\sum f'{M}(n,n';\mathbf k)N'(\mathbf k)) of the nondiagonal elements (N'{nn'}(\mathbf k)) in (\sigma}), and in the lowest approximation in (\mathcal H'), (\sigma^{nd{MN}\equiv\sigma^{(s)}\beta\Gamma'}\sim M^{e}N^{e}\Delta_{e}^{-2{b}N\equiv |(s|M|s')|).}), (M^{e
* (F_{M}) is the external force conjugate to the flux (M); to the charge flux (M=\mathbf j=e\mathbf v) there is conjugate the electric field (\mathbf E) ((\mathbf v) is the velocity operator); incidentally, (v^{0}(kn)=(kn|\mathbf v|kn)=\partial\varepsilon_{n}(\mathbf k)/\partial\mathbf k), (v'{nn'}(\mathbf k)=(nk|\mathbf v|n'k)).
Corrections to the principal approximation (for (\sigma_{MN})) defined by formulas (10)—(12), (16)—(18), due to the contribution of (\delta \sigma_{MN}) and to the higher, in (\mathscr H'), approximations in (Q_M(\varepsilon)) and (I_{\varepsilon f'_M}(\varepsilon)) not taken into account in the principal definition, are small for (\Phi_0 \gg 1), (\beta\Delta \ll 1), if
[
\left|\langle \delta v_{nk}\rangle_0\right| \ll \langle v_{nk}\rangle_0,\qquad
\Omega_b \ll \Delta,\quad \text{i.e.}\quad
\Delta_e<\Delta_e^0(\omega_p,\Phi_0)\ll \omega_p\Phi_0,
\tag{19}
]
where (\delta v_{nk}) is the correction to (v_{nk}=|\mathbf v^0(nk)|). Relations (10)—(12), (16)—(18) make it possible to calculate the kinetic coefficients (\sigma_{MN}) under (19) and at low (T\lesssim T_0').
- Equation (16)—(18) is solved in the usual way, and for ((H\parallel OZ))
[
f'(\mathbf{kn})=
\left{1-eHI_0^{-1}
\left(v_y(\mathbf{kn})\frac{\partial}{\partial k_x}
-v_x(\mathbf{kn})\frac{\partial}{\partial k_y}\right)\right}^{-1}
I_0^{-1}Q(\mathbf{kn}).
\tag{20}
]
From (12), (20) it follows that at (H=0) the drift (ohmic) mobility is
[
u_{ij}^{(s)}(T)\equiv |e|\frac{\beta}{2}
\left\langle
v_i(\mathbf{kn})I_0^{-1}v_j(\mathbf{kn})
+v_j(\mathbf{kn})I_0^{-1}v_i(\mathbf{kn})
\right\rangle_0,
\tag{21}
]
[
i,j\equiv x,y,z;\qquad
\langle\ldots\rangle_0\equiv
\iiint_{\text{(over the Brillouin zone)}}\frac{V(d\mathbf k)}{(2\pi)^3}
\sum_n f_0(n\mathbf k)(\ldots),
]
and approximately (cf. ((^3)))
[
u(T)\sim |e|\beta\Delta(T)\bigl(m^*(T)\Omega_b\bigr)^{-1}
=\frac{|e|a^2}{\hbar}\frac{\beta\Delta^2}{\Omega_b}.
\tag{22}
]
In the principal approximation (\Omega_b\sim \exp(-\beta\omega_p)), i.e. it increases with (T) (two-phonon scattering of a polaron in a narrow band (\Delta\ll\omega_p)). Since (\Delta(T)) decreases as (T) increases, one should expect that, in the main, (u(T)) decreases exponentially as (T) increases.
From (12), (16)—(18), for (H<H_0\Omega_b\Delta^{-1}\ll H_0) and (T<T_0'), it follows that the Hall mobility (u_H=|R_H|\sigma) has the form
[
u_H(T)=
\frac{
\left|
e\left\langle
v_y(\mathbf{kn})I_0^{-1}
\left(
v_y(\mathbf{kn})\frac{\partial}{\partial k_x}
-v_x(\mathbf{kn})\frac{\partial}{\partial k_y}
\right)
I_0^{-1}v_x(\mathbf{kn})
\right\rangle_0
\right|
}{
\left\langle v_y(\mathbf{kn})I_0^{-1}v_y(\mathbf{kn})\right\rangle_0
}
\sim \frac{|e|}{m^*\Omega_b},
\tag{23}
]
i.e. (u_Hu^{-1}\sim(\beta\Delta)^{-1}\gg 1). For (T>T_0) also usually (u_H\gg u), if (\sum_{fi}\chi_{fi}^{ss'}\omega_{fi}) are the same for all nearest (s,s'); (u_H) contains terms of the form ((^7))
[
u_H^0=|e|a^2\hbar^{-1}\Delta_e\xi_0^{-1}(\beta\xi_0)^{-1/2}\exp(\beta\xi_0)
]
and
[
\delta u_H\equiv |e|a^2\hbar^{-1}\Delta_e(\xi_0T)^{-1/2}\times
\exp(-\beta U_0)\ll u_H^0,
]
i.e. (u_H\sim u_H^0) (in the preliminary estimates ((^{6a})) only the part of (u_H) of the form (\delta u_H), the same as in (11), was included). For (T<T_0') the thermopower is
[
\gamma_0\simeq \frac{1}{eT}(\mu-\varepsilon_{\mathrm{pol}}-\delta\varepsilon),
]
just as for (T>T_0) ((^{6a})). (For (T\gg T_0) and (N_i\ll N_0),
[
\gamma_0\simeq e^{-1}\left(\ln\frac{N_i}{N_0}+\frac{\delta\varepsilon}{T}\right),
]
and usually one may put (\gamma_0\sim -e^{-1}\ln(N_i/N_0)).)
Semiconductor Institute
Academy of Sciences of the USSR
Received
19 VI 1962
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