Abstract Generated abstract
This paper derives generalized microscopic equations for superconductivity in an inhomogeneous fermion superfluid at arbitrary temperature, allowing the superconducting energy gap to depend on position as well as temperature. Starting from temperature Green function equations with an external vector potential, the authors formulate an exact integral equation for the gap and obtain an approximate differential equation under assumptions of slow spatial variation and small gap magnitude. The resulting equation has the form of the Ginzburg Landau equation but includes explicit microscopic temperature dependence, the homogeneous BCS gap, and the effective charge of a Cooper pair, with calculable coefficients. Near the critical temperature it reduces to Gor’kov’s result, while in zero vector potential it is interpreted as describing quantum vortices in a superfluid fermion system.
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Reports of the Academy of Sciences of the USSR
- Volume 145, No. 4
PHYSICS
L. P. RAPOPORT and A. G. KRYLOVETSKII
GENERALIZED EQUATIONS OF SUPERCONDUCTIVITY
(Presented by Academician N. N. Bogolyubov, 13 III 1962)
In addition to the well-known temperature dependence, the energy gap in the spectrum of a superconductor may in the general case also depend on the coordinates. Such inhomogeneity is in principle not necessarily connected with the presence of external fields and may be inherent in the equation itself. For example, a bound Cooper pair of fermions in many respects behaves like a boson, and for it one may expect the existence of solutions of vortex type. On the other hand, it is known that the inhomogeneous phenomenological Ginzburg–Landau equation gives a qualitatively good description of the behavior of superconductors in a magnetic field near the critical temperature (¹). Recently L. P. Gor’kov made an attempt to derive the latter equation from the modern microscopic theory of superconductivity (²). However, the expansion he used of the Green functions \(G(x,x')\) and \(F^{+}(x,x')\) in powers of \(\Delta(\mathbf r)\) leads to the correct result only at temperatures close to the transition temperature.
In the present work an equation is derived which describes a superfluid system of fermions in the inhomogeneous case at an arbitrary temperature. In the limiting case of temperatures close to the critical one, the equation obtained coincides with the equation for the function \(\Delta(\mathbf r)\) in work (²).
We shall start from the following system of equations for the Fourier components of the temperature Green functions \(G_\omega(\mathbf r,\mathbf r')\) and \(F_\omega^{+}(\mathbf r,\mathbf r')\) (³):
\[ \left\{ i\omega+\frac{1}{2m}\left(\frac{\partial}{\partial \mathbf r}-ie\mathbf A(\mathbf r)\right)^2+\mu \right\} G_\omega(\mathbf r,\mathbf r') +\Delta_T(\mathbf r)F_\omega^{+}(\mathbf r,\mathbf r') =\delta(\mathbf r-\mathbf r'); \]
\[ \left\{ -i\omega+\frac{1}{2m}\left(\frac{\partial}{\partial \mathbf r}+ie\mathbf A(\mathbf r)\right)^2+\mu \right\} F_\omega^{+}(\mathbf r,\mathbf r') -\Delta_T^{*}(\mathbf r)G_\omega(\mathbf r,\mathbf r') =0; \tag{1} \]
\[ \Delta_T^{*}(\mathbf r)=Tg\sum_{\omega}F_\omega^{+}(\mathbf r,\mathbf r'); \qquad \omega=\pi T(2n+1); \qquad n=\ldots,-1,0,+1,\ldots \tag{2} \]
Introducing \(\widetilde G_\omega(\mathbf r,\mathbf r')\)—the Fourier component of the Green function of an electron for a normal metal—we obtain, instead of (1), the system of integral equations
\[ G_\omega(\mathbf r,\mathbf r') = G_\omega(\mathbf r,\mathbf r') -\int \widetilde G_\omega(\mathbf r,\mathbf s)\Delta_T(\mathbf s) \widetilde G_{-\omega}(\mathbf l,\mathbf s)\Delta_T^{*}(\mathbf l) G_\omega(\mathbf l,\mathbf r')\,d^3l\,d^3s; \tag{3} \]
\[ F_\omega^{+}(\mathbf r,\mathbf r') = \int \widetilde G_{-\omega}(\mathbf s,\mathbf r)\Delta_T^{*}(\mathbf s) \widetilde G_\omega(\mathbf s,\mathbf r')\,d^3s - \]
\[ -\int \widetilde G_{-\omega}(\mathbf s,\mathbf r)\Delta_T^{*}(\mathbf s) \widetilde G_\omega(\mathbf s,\mathbf l)\Delta_T(\mathbf l) F_\omega^{+}(\mathbf l,\mathbf r')\,d^3s\,d^3l. \tag{4} \]
As noted by L. P. Gor'kov in Ref. \((^{2})\), the Green function \(\widetilde{G}_{\omega}(\mathbf r,\mathbf r')\) in a magnetic field differs from the Green function of a normal metal in the absence of a magnetic field, \(\widetilde{G}_{\omega}^{0}(\mathbf r-\mathbf r')\), only by a phase factor and can be represented in the form
\[ \widetilde{G}_{\omega}(\mathbf r,\mathbf r') = \exp\{ie\mathbf A(\mathbf r)(\mathbf r-\mathbf r')\} G_{\omega}^{0}(\mathbf r-\mathbf r'). \tag{5} \]
Using further (2) and (4), one can obtain the following exact equation for the gap \(\Delta(\mathbf r)\):
\[ g^{-1}\Delta_T^{*}(\mathbf r) = T\sum_{\omega}\int \widetilde{G}_{-\omega}(\mathbf s,\mathbf r)\, \Delta_T^{*}(\mathbf s)\, \widetilde{G}_{\omega}(\mathbf s,\mathbf r)\,d^{3}s - \]
\[ - T\sum_{\omega}\int \widetilde{G}_{-\omega}(\mathbf s,\mathbf r)\, \Delta_T^{*}(\mathbf s)\, \widetilde{G}_{\omega}(\mathbf s,\mathbf l)\, \Delta_T(\mathbf l)\, F_{\omega}^{+}(\mathbf l,\mathbf r)\, d^{3}s\,d^{3}l . \tag{6} \]
We shall now solve equation (6) approximately. In doing so, in order not to lose the “superconducting” solution, we choose the first iteration of equation (6), bearing in mind the known expression for \(F_{\omega}^{+}\) in the homogeneous case, in the form
\[ F_{\omega}^{+}(\mathbf l,\mathbf r) = \Delta_T^{+}\!\left(\frac{\mathbf l+\mathbf r}{2}\right) (2\pi)^{-3} \int \frac{e^{i\mathbf p(\mathbf l-\mathbf r)}\,d^{3}p} {\omega^{2}+\varepsilon_{p}^{2}+\Delta_T^{2}} \equiv \Delta_T^{+}\!\left(\frac{\mathbf l+\mathbf r}{2}\right) F_{\omega}^{+}(\mathbf l-\mathbf r). \tag{7} \]
If we now assume that \(\Delta_T(\mathbf r)\) and \(\mathbf A(\mathbf r)\) are slowly varying functions of the coordinates and that \(\Delta_T(\mathbf r)\) is small, then, expanding the exponents in the Green functions (5) and \(\Delta_T(\mathbf r)\), we can bring equation (6), with account of (7), to the form
\[ \left\{ \frac{1}{2m} \left( \frac{\partial}{\partial \mathbf r} + 2ie\mathbf A(\mathbf r) \right)^{2} + \frac{1}{C} \left[ D-g^{-1}+B|\Delta_T(\mathbf r)|^{2} \right] \right\} \Delta_T^{*}(\mathbf r)=0, \tag{8} \]
where
\[ B = - T\sum_{\omega}\int \widetilde{G}_{-\omega}^{0}(\mathbf r-\mathbf s)\, \widetilde{G}_{\omega}^{0}(\mathbf s-\mathbf l)\, F_{\omega}^{0}(\mathbf l-\mathbf r)\, d^{3}s\,d^{3}l, \]
\[ C = T\sum_{\omega}\int \widetilde{G}_{-\omega}^{0}(\mathbf r-\mathbf s)\, (\mathbf r-\mathbf s)^{2} \widetilde{G}_{\omega}^{0}(\mathbf s-\mathbf r)\, d^{3}s, \tag{9} \]
\[ D = T\sum_{\omega}\int \widetilde{G}_{-\omega}^{0}(\mathbf r-\mathbf s)\, \widetilde{G}_{\omega}^{0}(\mathbf s-\mathbf r)\, d^{3}s. \]
In evaluating the integrals (9), it is necessary to take into account the momentum cutoff that enters into the definition of the Hamiltonian in the Bardeen–Cooper–Schrieffer model \((^{4})\), since one must take into account only the interaction of electrons lying in a layer of the Fermi surface of thickness \(\widetilde{\omega}\). Therefore it is more convenient to carry out the calculation in the momentum representation. In this case
\[ \widetilde{G}_{\omega}^{0}(\mathbf p) = (i\omega-\varepsilon)^{-1}; \qquad \varepsilon=v_F(p-p_F), \]
and we have:
\[ D = T\sum_{\omega}(2\pi)^{-3} \int d^{3}p\, \widetilde{G}_{\omega}^{0}(\mathbf p)\, \widetilde{G}_{-\omega}^{0}(\mathbf p) = \frac{N(T)}{2} \int_{-\widetilde{\omega}}^{\widetilde{\omega}} \frac{\operatorname{th}(\varepsilon/2T)}{\varepsilon}\, d\varepsilon . \tag{10} \]
The last integral was evaluated in Ref. \((^{4})\). Using this result, we obtain:
\[ D=N(T)\ln\frac{2\widetilde{\omega}\gamma}{\pi T}. \]
Here \(N(T)\) is the density of distribution of electrons at the Fermi surface at temperature \(T\); \(\gamma\) is the logarithm of Euler’s constant.
Similarly we have
\[ B=-\left\{\frac{N(T)}{\Delta_T^2}\ln\frac{2\tilde{\omega}\gamma}{\pi T} -\frac{N(T)}{2\Delta_T^2}\int_{-\tilde{\omega}}^{\tilde{\omega}} \frac{\operatorname{th}\sqrt{\varepsilon^2+\Delta_T^2}/2T} {\sqrt{\varepsilon^2+\Delta_T^2}}\,d\varepsilon\right\}. \tag{11} \]
The integral \(C\) can be calculated by differentiating the expression for \(D\) with respect to the parameter. After simple calculations we have
\[ C=\frac{7N(T)\zeta(3)mv_F^2}{4\pi^2T^2}, \tag{12} \]
where \(\zeta(3)\) is the Riemann zeta function.
To determine the integral in (11) and to elucidate the meaning of the function \(\Delta_T\), let us substitute \(B\), \(C\), and \(D\) into equation (8) and pass to the limiting case of a homogeneous solution. In doing so it is necessary to set \(\mathbf{A}(\mathbf{r})=0\), \(\Delta_T(\mathbf{r})=\mathrm{const}\). Then we obtain \(\Delta=\Delta_T\), where \(\Delta_T\) is found from the well-known equation (4)
\[ g\,\frac{N(T)}{2}\int_{-\tilde{\omega}}^{\tilde{\omega}} \frac{\operatorname{th}\sqrt{\varepsilon^2+\Delta_T^2}/2T} {\sqrt{\varepsilon^2-\Delta_T^2}}\,d\varepsilon=1, \tag{13} \]
which determines the width of the energy gap at temperature \(T\) in the case of a homogeneous solution.
Substituting into (8) the values found for \(B\), \(C\), and \(D\), taking into account the relations \(\Delta_0=2\tilde{\omega}\exp[-2/gN(0)]\), \(\Delta_0=\pi T_c/\gamma\), and equation (13), we obtain
\[ \left\{ \frac{1}{2m}\left(\frac{\partial}{\partial\mathbf{r}}+2ie\mathbf{A}(\mathbf{r})\right)^2 +\frac{8\pi^2T^2}{7\zeta(3)\varepsilon_F}\ln\frac{T_c}{T} \left(1-\frac{|\Delta_T(\mathbf{r})|^2}{\Delta_T^2}\right) \right\}\Delta_T^*(\mathbf{r})=0. \tag{14} \]
Equation (14) can be written in a more general form if one notes that the coefficient can be expressed through the homogeneous gap \(\Delta_T\sim T_c\),
\[ \frac{8\pi^2T^2}{7\zeta(3)}\ln\frac{T_c}{T}=\Delta_{T\sim T_c}^2, \]
which is connected with the expansion of \(\Delta(\mathbf{r})\) carried out in a series. For the exact solution the coefficient will simply be equal to \(\Delta_T^2\), and instead of (14) we have an equation suitable for all temperatures down to \(T=0\),
\[ \left\{ \frac{1}{2m}\left(\frac{\partial}{\partial\mathbf{r}}+2ie\mathbf{A}(\mathbf{r})\right)^2 +\frac{\Delta_T^2}{\varepsilon_F} \left(1-\frac{|\Delta_T(\mathbf{r})|^2}{\Delta_T^2}\right) \right\}\Delta_T^*(\mathbf{r})=0. \tag{14'} \]
Equations (14) and (3) are the equations sought. They describe the superconductivity of a system of fermions in the inhomogeneous case at arbitrary temperature (in the derivation, a restriction was imposed only on the rate of change and on the magnitude of the gap).
Equation (14), in form, coincides with the phenomenological Ginzburg–Landau equation and differs from it only by the explicit temperature dependence obtained, by the presence of the energy gap, and by the doubled charge of the electron. The occurrence of all these features is clear from the physical meaning of the phenomenon of superconductivity. At the same time, the phenomenological constants can now be calculated.
As is known, for \(T\) near \(T_c\) the dependence of \(\Delta_T\) on temperature, described by equation (13), to first order in \(T-T_c\) has the form \({}^{(5)}\)
\[ \left(\Delta_{T\sim T_c}=\frac{8\pi^2}{7\zeta(3)}\right)^{1/2}T_c\sqrt{1-T/T_c}. \]
Substituting this quantity into (14), we see that our equation goes over into the equation obtained by L. P. Gor'kov in paper (²).
From the form of equation (14) it is clear that it does not lose its meaning also when \(\mathbf{A}(\mathbf{r})=0\). The inhomogeneous equation thereby obtained describes a quantum vortex in a system of fermions that are in the superfluid state at temperature \(T\).
Voronezh State
University
Received
12 III 1962
References
¹ V. L. Ginzburg, L. D. Landau, ZhETF, 20, 1064 (1950).
² L. P. Gor'kov, ZhETF, 36, 1918 (1959).
³ A. A. Abrikosov, L. P. Gor'kov, I. E. Dzyaloshinskii, ZhETF, 36, 900 (1959).
⁴ J. Bardeen, L. Cooper, J. Schrieffer, Phys. Rev., 108, 1175 (1958).
⁵ A. A. Abrikosov, I. M. Khalatnikov, UFN, 65, 551 (1958).