Abstract
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PHYSICS
S. V. TYABLIKOV and V. A. MOSKALENKO
THE METHOD OF QUANTUM GREEN’S FUNCTIONS IN THE THEORY OF OPTICAL BANDS IN CRYSTALS
(Presented by Academician N. N. Bogolyubov on 1 IV 1961)
Let us consider an optical multiphonon transition in a polaron or an \(F\)-center. We shall assume an adiabatic character of the coupling between the electron of the polaron or \(F\)-center and the lattice vibrations, and use the corresponding representations of the adiabatic method, paying attention here to the consistent inclusion, in this scheme, of the effects of changes in phonon frequencies.
Denote by \(H_a\) and \(H_b\) the phonon Hamiltonians corresponding to the initial electronic state \(a\) and the final one \(b\). In the representation in which \(H_a\) is diagonal, we have
\[ H_a=\sum_\omega \hbar\Omega_\omega b_\omega^+ b_\omega+E_a,\qquad H_b=E_i+H_a+V, \tag{1} \]
where
\[ V=\sum_\mu \left(V_\mu b_\mu+V_\mu^* b_\mu^+\right) +\sum_{\mu\mu'}\left[A_{\mu\mu'} b_\mu^+ b_{\mu'} +\frac{1}{2}B_{\mu\mu'} b_\mu b_{\mu'} +\frac{1}{2}B_{\mu\mu'}^* b_\mu^+ b_{\mu'}^+\right]. \tag{2} \]
In (2), \(b_\omega^+\), \(b_\omega\) are the phonon creation and annihilation operators; \(E_i\) is a \(c\)-number. The coefficients \(V_\mu\) are, by their origin, associated with the displacement of the equilibrium position of the phonons in the electronic transition. The action of the mean field of the electron on the motion of the phonons leads to a renormalization of their frequencies. This mean field depends on the state of the electron, and therefore, when it changes, the renormalized frequencies also change. The coefficients \(V_\mu\) are partly connected with the process of the emergence of renormalized frequencies in state \(b\), whereas the coefficients \(A_{\mu\mu'}\) and \(B_{\mu\mu'}\) are caused by the change of the renormalized phonon frequencies.
The purpose of the present note is to take consistent account of the above-mentioned features of phonon behavior and to clarify their influence on the shape of the absorption and emission bands of light in crystals with adiabatic electron–phonon coupling.
The shape of the optical bands is characterized by the first spectral moments, for the calculation of which, in the works of Lax \((^1)\) and Kubo and Toyozawa \((^2)\), a method is proposed based on the use of certain characteristic functions. Thus, for example, in Lax’s work the moment of order \(r\), reckoned from some arbitrary frequency \(\nu_1\), has the form
\[ \left\langle (\nu-\nu_1)^r\right\rangle =\frac{(2\pi i)^r}{I(0)}\, \frac{d^r}{dt^r}\left(I(t)e^{2\pi i\nu_1 t}\right)\bigg|_{t=0}, \tag{3} \]
where the characteristic function \(I(t)\) for the process of light absorption is defined by the equality
\[ I(t)=\left\langle M_{ba}^+ e^{-iH_b t/\hbar} M_{ba} e^{iH_a t/\hbar}\right\rangle . \tag{4} \]
In (4), \(M_{ba}\) is the electronic matrix element, which for simplicity is taken to be a \(c\)-number. Here and below, \(\langle \ldots \rangle\) is understood to mean the statistical average over the states of the Hamiltonian \(H_a\).
In our paper (³) it was shown that, under the assumptions made about the character of the operators \(H_a\), \(H_b\), and \(I(t)\), the function \(I(t)\) can be represented in the form
\[ I(t)=|M_{ba}|^2 \exp\left(f(t)-\frac{iE_i t}{\hbar}\right), \tag{5} \]
where
\[ \begin{aligned} f(t)=&-\frac{i}{\hbar}\int d\alpha \int_0^t d\tau \Biggl\{ \sum_{\mu}\bigl(V_\mu \varphi(\mu\tau\|)+V_\mu^{*}\varphi(\|\mu\tau)\bigr) \\ &\quad+\sum_{\mu\mu'} \bigl[A_{\mu'\mu}D(\mu\tau|\mu'\tau) +\tfrac12 B_{\mu'\mu}D(\mu'\tau;\mu\tau\|) +\tfrac12 B_{\mu'\mu}^{*}D(\|\mu'\tau;\mu\tau)\bigr] \Biggr\}. \end{aligned} \tag{6} \]
The quantum Green’s functions \(D\) and the functions \(\varphi\) contained in (6) are defined as follows:
\[ \begin{aligned} D(\mu_1\tau_1|\mu_2\tau_2) &=\langle P\{b_{\mu_1}(\tau_1)b_{\mu_2}^{+}(\tau_2)U_\alpha(t)\}\rangle \langle U_\alpha(t)\rangle^{-1},\\ \varphi(\mu_1\tau_1\|) &=\langle P\{b_{\mu_1}(\tau_1)U_\alpha(t)\}\rangle \langle U_\alpha(t)\rangle^{-1}. \end{aligned} \tag{7} \]
If in the first formula (7) two annihilation operators or two creation operators are used, then one obtains the functions \(D(\mu_1\tau_1;\mu_2\tau_2\|)\) and \(D(\|\mu_1\tau_1;\mu_2\tau_2)\), respectively. The function \(\varphi(\|\mu_1\tau_1)\) has the form of the second formula (7), in which the operator \(b_{\mu_1}\) is replaced by \(b_{\mu_1}^{+}\).
The time dependence of the operators \(b_\mu(\tau)\) and \(b_\mu^{+}(\tau)\) is determined by the factors \(e^{-i\Omega_\mu\tau}\) and \(e^{i\Omega_\mu\tau}\), respectively.
The evolution operator has the form
\[ U_\alpha(t)=e^{-iE_\alpha t/\hbar}\, P\exp\left(-\frac{i\alpha}{\hbar}\int_0^t V(\tau)\,d\tau\right). \tag{8} \]
The function \(D(\mu_1\tau_1|\mu_2\tau_2)\) in formula (6) for coinciding time arguments is defined as the limit as \(\tau_2\) tends to \(\tau_1+0\).
The Green’s functions (7) differ from the commonly used temperature-time quantum Green’s functions (⁴) in that, in (7), instead of the \(S\)-matrix the evolution operator \(U_\alpha(t)\) is used, and in that the statistical averaging is performed over the states of the unperturbed Hamiltonian \(H_a\). These features of the definition of the functions (7) deprive them of the property of time homogeneity.
At \(t=0\), four of the introduced functions vanish. Only the function \(D(\mu_1\tau_1|\mu_2\tau_2)\) is nonzero; in this case it coincides with the free Green’s function:
\[ D_0(\mu_1\tau_1|\mu_2\tau_2) =\delta_{\mu_1\mu_2}e^{i\Omega_{\mu_1}(\tau_2-\tau_1)} \left[\theta(\tau_1-\tau_2)(\bar n_{\mu_1}+1)+\bar n_{\mu_1}\theta(\tau_2-\tau_1)\right], \tag{9} \]
where
\[ \bar n_\mu=(e^{\beta\Omega_\mu}-1)^{-1}. \]
Owing to the quadratic character of the operator \(V\), the equations for the functions \(D\) and \(\varphi\) are closed. We give some of them (³):
\[ \begin{aligned} D(\mu_1\tau_1|\mu_2\tau_2) &=D_0(\mu_1\tau_1|\mu_2\tau_2) -\frac{i\alpha}{\hbar}\int_0^t d\tau\, D_0(\mu_1\tau_1|\mu\tau) \\ &\quad\times \left[ V_{\mu_1}^{*}\varphi(\|\mu_2\tau_2) +\sum_{\mu}\left( A_{\mu_1\mu}D(\mu\tau|\mu_2\tau_2) +\bar B_{\mu_1\mu}^{*}D(\|\mu_2\tau_2;\mu\tau) \right) \right]; \end{aligned} \tag{10} \]
\[ \varphi(\mu_1\tau_1\|) =-\frac{i\alpha}{\hbar}\int_0^t d\tau\, D_0(\mu_1\tau_1|\mu\tau) \left[ V_{\mu_1}^{*} +\sum_{\mu}\left( A_{\mu_1\mu}\varphi(\mu\tau\|) +\bar B_{\mu_1\mu}^{*}\varphi(\|\mu\tau) \right) \right], \]
where \(\bar B_{\mu\mu'}=\frac{1}{2}(B_{\mu\mu'}+B_{\mu'\mu})\). The equations for the remaining functions have an analogous form.
To determine the spectral moments it is sufficient to know the behavior of the characteristic function \(I(t)\) in the region of values of \(t\) close to zero. For this region of values of the parameter \(t\), solutions of equations (10) can be obtained by the iteration method, and thus an approximate determination of the function \(f(t)\) is possible. This method at the same time makes it possible to compute the exact values of a number of derivatives of the function \(f(t)\) at \(t=0\), provided that a sufficient number of terms of the iteration series is retained. Thus, for example, to determine the first three derivatives of \(f(t)\) it is sufficient to carry out two iterations in solving equations (10).
Having found the function \(f(t)\) in the indicated manner and computed its first derivatives at \(t=0\), it is not difficult, from formulas (3) and (5), to find the first moments of the spectrum of light absorption by a polaron or by an \(F\)-center.
We define the frequency \(\nu_1\) in the following way:
\[ h\nu_1=E_i+\sum_\mu A_{\mu\mu}\bar n_\mu . \tag{11} \]
Then for the second-order moment, which determines the half-width of the light-absorption band, we obtain
\[ \left\langle |h(\nu-\nu_1)|^2\right\rangle = \sum_\mu |V_\mu|^2(2\bar n_\mu+1)+ \frac{1}{2\hbar^2}\sum_{\mu\mu'} \left\{ |A_{\mu\mu'}|^2(\bar n_\mu+\bar n_{\mu'}+2\bar n_\mu\bar n_{\mu'}) + |\bar B_{\mu\mu'}|^2(1+\bar n_\mu+\bar n_{\mu'}+2\bar n_\mu\bar n_{\mu'}) \right\}. \tag{12} \]
The third moment, which determines the asymmetry of the spectral curve, has the form:
\[ \begin{aligned} \left\langle [h(\nu-\nu_1)]^3\right\rangle ={}& \hbar\sum_\mu |V_\mu|^2\Omega_\mu + \sum_{\mu\mu'} \Biggl\{ A_{\mu\mu'}V_\mu V_{\mu'}^{*} \bigl(1+3\bar n_\mu+3\bar n_{\mu'}+6\bar n_\mu\bar n_{\mu'}\bigr) \\ &\quad +\frac{1}{2}\bigl(\bar B_{\mu\mu'}^{*}V_\mu V_{\mu'} +\bar B_{\mu\mu'}V_\mu^{*}V_{\mu'}^{*}\bigr) \bigl(2+3\bar n_\mu+3\bar n_{\mu'}+6\bar n_\mu\bar n_{\mu'}\bigr) \\ &\quad +\frac{\hbar}{2}\sum_{\mu\mu'} \left[ |A_{\mu\mu'}|^2(\Omega_\mu-\Omega_{\mu'})(\bar n_{\mu'}-\bar n_\mu) + |\bar B_{\mu\mu'}|^2(\Omega_\mu+\Omega_{\mu'})(1+\bar n_\mu+\bar n_{\mu'}) \right] \Biggr\} \\ &\quad +\sum_{\mu\mu'\mu_1} \Biggl\{ \frac{1}{3!}A_{\mu'\mu}A_{\mu\mu_1}A_{\mu_1\mu'} \bigl(\bar n_\mu+\bar n_{\mu'}+\bar n_{\mu_1} +3\bar n_\mu\bar n_{\mu_1} \\ &\qquad +3\bar n_{\mu'}\bar n_\mu +3\bar n_\mu\bar n_{\mu_1} +6\bar n_\mu\bar n_{\mu'}\bar n_{\mu_1}\bigr) \\ &\qquad +\bar B_{\mu'\mu_1}\bar B_{\mu\mu_1}^{*}A_{\mu'\mu} \bigl(1+\bar n_{\mu_1}+2\bar n_{\mu'}+2\bar n_\mu +3\bar n_\mu\bar n_{\mu'} \\ &\qquad\qquad +3\bar n_\mu\bar n_{\mu_1} +3\bar n_{\mu'}\bar n_{\mu_1} +6\bar n_\mu\bar n_{\mu'}\bar n_{\mu_1}\bigr) \Biggr\}. \tag{13} \end{aligned} \]
Within the limits of the assumption made about the character of the operators \(H_a\), \(H_b\), and \(M_{ba}\), formulas (11)—(13) are exact. They take into account both the displacement of the oscillators from the equilibrium position and the change in their frequencies.
According to (11), the maximum of the absorption band is displaced when the temperature changes. This displacement follows a linear law in the region of high temperatures. The temperature displacement is absent if one neglects the effect of the change in phonon frequencies.
Next we note that the half-width of the light-absorption band turns out to be larger if the change in the renormalized frequencies during the electronic transition is taken into account than in the case when this effect is neglected.
Finally, let us note that when light emission is considered, the quantities \(V_{\mu\nu}'\), \(\Omega_{\mu}\), \(A_{\mu\nu}\), and \(B_{\mu\nu}\) will, generally speaking, be different from those in the case of light absorption, and therefore such quantities as the half-width of the spectral curve may prove to differ from their values for the light-absorption curve.*
More detailed information can be obtained after choosing a specific model of the light-absorbing system and determining the functions of the theory.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Laboratory of Theoretical Physics
Moldavian Branch of the Academy of Sciences of the USSR
Received
28 III 1961
CITED LITERATURE
- M. Lax, J. Chem. Phys., 20, 1752 (1952).
- R. Kubo, Y. Toyozawa, Progr. Theor. Phys., 13, 160 (1955).
- S. V. Tyablikov, V. A. Moskalenko, Uch. zap. Kishinevsk. gos. univ., 55, 143 (1960).
- N. N. Bogolyubov, S. V. Tyablikov, DAN, 126, 53 (1959).