Abstract Generated abstract
This paper develops a thermodynamic treatment of superconductivity starting directly from the Fröhlich electron phonon Hamiltonian. Using Feynman’s operator calculus to eliminate phonon variables and applying a variational principle for the grand statistical sum, the author introduces Bogolyubov quasiparticle amplitudes and derives self-consistent equations for the excitation spectrum and superconducting gap. The resulting formulation yields an integral gap equation, a linearized equation for the critical temperature, and a simplified estimate of that temperature in terms of phonon and coupling parameters. The analysis also indicates that the transition at the critical point is not first order, while the heat capacity has a finite jump.
Full Text
PHYSICS
V. A. MOSKALENKO
ON THE THERMODYNAMICS OF SUPERCONDUCTIVITY
(Presented by Academician N. N. Bogolyubov on 24 VI 1958)
In a recent work by N. N. Bogolyubov, D. N. Zubarev, and Yu. A. Tserkovnikov \((^1)\), a theory of the phase transition of superconductors was considered on the basis of a model Hamiltonian. The thermodynamics of superconductivity was also constructed by Bardeen, Cooper, and Schrieffer \((^2)\) by means of an approximate variational method. In papers \((^3)\) the possibility was investigated of applying the variational principle to a model Hamiltonian of a more general form, and results were obtained confirming \((^1,\,^2)\). Below it will be shown that the variational principle makes it possible to obtain, in a more general form, the results of the works mentioned, if one starts directly from the Fröhlich Hamiltonian.
Bearing in mind the calculation of the statistical sum for an ensemble described by the grand Gibbs distribution, we introduce the chemical potential \(\lambda\) and consider the Hamiltonian
\[ H=\sum_{k,\sigma}(E(k)-\lambda)a_{k,\sigma}^{+}a_{k,\sigma} +\sum_q \hbar\omega_q b_q^{+}b_q+ \]
\[ +g\sum_{\substack{k,q,\sigma\\ q=k'-k}} \left(\frac{\hbar\omega_q}{2V}\right)^{1/2} a_{k,\sigma}^{+}a_{k',\sigma}b_q^{+} +\text{complex conjugate}, \tag{1} \]
where \(E(k)\) is the electron energy; \(\hbar\omega_q\) is the phonon energy; \(g\) is the coupling constant, \(V\) is the volume of the system.
For calculating the statistical sum of the system
\[ Z=\operatorname{Sp} e^{-\beta H},\qquad \beta=\frac{1}{k_0T}, \tag{2} \]
we shall use Feynman’s operator calculus \((^4)\) and eliminate the phonon amplitudes, after which we obtain
\[ Z=X\,\operatorname{Sp}\exp\left\{ -\int_0^\beta H_e\,ds +\sum_{q\ne0}(1+\bar n_q)\int_0^\beta ds\int_0^s dp\, \gamma_{qs}^{+}\gamma_{qp}e^{-\hbar\omega_q(s-p)} \right. \]
\[ \left. +\sum_{q\ne0}\bar n_q\int_0^\beta ds\int_0^s dp\, \gamma_{qs}^{+}\gamma_{qp}e^{\hbar\omega_q(s-p)} \right\}. \tag{3} \]
Here \(H_e\) is the first term of operator (1),
\[ \gamma_q=g\left(\frac{\hbar\omega_q}{2V}\right)^{1/2} \sum_{k,\sigma} a_{k,\sigma}^{+}a_{k+q,\sigma}, \]
\[ X=\prod_{q\ne0}(1+\bar n_q) \]
is the statistical sum of the free lattice;
\[ \bar n_q=(e^{\beta\hbar\omega_q}-1)^{-1}; \]
\(s,p\) are ordering indices.
Let us note that if in (3) one drops the ordering indices \(s, p\) on the operators \(H_e, \gamma_q, \gamma_q^+\) and carries out the integration, then in the exponent of this expression there appears the fourth form of the model Hamiltonian, derived earlier in paper \((^5)\).
Returning to the consideration of (3), we shall use the variational principle for the statistical sum \((^6)\)
\[ Z \geqslant Z_0 e^{-\beta S}, \tag{4} \]
where
\[ Z_0=\operatorname{Sp} e^{-\beta H_0},\qquad S=\frac{1}{Z_0\beta}\operatorname{Sp}\left[\int_0^\beta(\widetilde H-H_0)_s\,ds\, \exp\left(-\int_0^\beta H_0\rho\,d\rho\right)\right]. \tag{5} \]
By \(-\int_0^\beta \widetilde H_s ds\) is meant the expression contained in the braces in (3); \(H_0\) is the trial Hamiltonian operator. (5) takes into account that the trace (3) is taken of a \(T\)-ordered exponential.
It is easy to see that in the case when \((\widetilde H-H_0)_s\) depends only on a single ordering index, (5) goes over into the corresponding expression of paper \((^{6a})\). At the same time, the variational theorem (4), (5) is equivalent to the general variational theorem \((^{6b})\), presented in Lagrangian form.
For the choice of \(H_0\), following the basic idea of N. N. Bogoliubov \((^7)\), we pass to new Fermi amplitudes \(\alpha_{f0}, \alpha_{f1}\) by means of the canonical transformation*
\[ \alpha_{f0}=u_f a_{f,\,1/2}-v_f a^+_{-f,\,-1/2},\qquad \alpha_{f1}=u_f a_{-f,\,-1/2}+v_f a^+_{f,\,1/2},\qquad u_f^2+v_f^2=1. \tag{6} \]
As \(H_0\) we take the expression
\[ H_0=\sum_k \Omega(k)\left(\alpha^+_{k0}\alpha_{k0}+\alpha^+_{k1}\alpha_{k1}\right), \tag{7} \]
where \(\Omega(k)\) is the energy of the new elementary excitations. The functions \(\Omega(k), u_k, v_k\) will be determined with the aid of the variational principle from the condition of a maximum of the right-hand side of (4), or of a minimum of the thermodynamic potential of the system. As a result of the calculations we obtain for \(S\) the expression
\[ \begin{aligned} S={}&2\sum (E(k)-\lambda)v_k^2 +2\sum_k \left[(E(k)-\lambda)(u_k^2-v_k^2)-\Omega_k\right]N_k \\ &-\frac{g^2}{2V}\sum_k\sum_{k'} \hbar\omega_q\Bigg\{ (u_k v_{k'}+u_{k'}v_k)^2 \left[ \frac{(1+\bar n_q)(1-\bar N_{k'}-\bar N_k)+\bar N_k\bar N_{k'}} {\hbar\omega_q+\Omega_k+\Omega_{k'}} \right.\\ &\left.\qquad\qquad\qquad\qquad +\frac{\bar N_k\bar N_{k'}-\bar n_q(1-\bar N_{k'}-\bar N_k)} {\hbar\omega_q-\Omega_{k'}-\Omega_k} \right]\\ &\qquad\qquad +(u_k u_{k'}-v_k v_{k'})^2 \left[ \frac{\bar n_q(\bar N_{k'}-\bar N_k)+\bar N_{k'}-\bar N_k\bar N_{k'}} {\hbar\omega_q+\Omega_k-\Omega_{k'}} \right.\\ &\left.\qquad\qquad\qquad\qquad +\frac{\bar n_q(\bar N_k-\bar N_{k'})+\bar N_k-\bar N_k\bar N_{k'}} {\hbar\omega_q+\Omega_{k'}-\Omega_k} \right]\Bigg\}, \end{aligned} \tag{8} \]
where
\[ \bar N_k=\left(1+e^{\beta\Omega_k}\right)^{-1}. \]
The thermodynamic potential of the system has the form
\[ \Psi=S-\frac{1}{\beta}\ln Z_0, \tag{9} \]
where \(S\) is defined by formula (8),
\[ Z_0=\prod_k\left(1+e^{-\beta\Omega_k}\right)^2. \]
* The transformation (6) is effected by the unitary operator
\[ S=\exp\left[ \sum_k w_k\left(a^+_{k,\,1/2}a^+_{-k,\,-1/2} -a_{-k,\,-1/2}a_{k,\,1/2}\right) \right], \qquad u_k=\cos w_k,\quad v_k=\sin w_k. \]
The condition of a minimum of the thermodynamic potential leads to the equations
\[ \frac{\partial \Psi}{\partial v_k}=0; \tag{10} \]
\[ \frac{\partial \Psi}{\partial \Omega_k}=0; \tag{10a} \]
\[ u_k^2+v_k^2=1. \tag{10б} \]
From (10) we obtain the following expressions for \(u_k, v_k\):
\[ u_k^2=\frac12\left(1+\frac{\xi(k)}{\sqrt{\xi^2(k)+c^2(k)}}\right), \qquad v_k^2=\frac12\left(1-\frac{\xi(k)}{\sqrt{\xi^2(k)+c^2(k)}}\right), \tag{11} \]
where
\[ \xi(k)=E(k)-\lambda-\operatorname{cth}\frac{\beta\Omega_k}{2} \sum_{k'}\bigl(J(kk')-H(kk')\bigr)(u_{k'}^2-v_{k'}^2), \]
\[ c(k)=2\operatorname{cth}\frac{\beta\Omega_k}{2} \sum_{k'}\bigl(J(k,k')-H(k,k')\bigr)u_{k'}v_{k'}, \]
\[ J(k,k')= \]
\[ =\frac{g^2}{2V}\hbar\omega_q \frac{ \hbar\omega_q[1+2\overline N_k\overline N_{k'}-\overline N_k-\overline N_{k'}] +(\Omega_k+\Omega_{k'})\operatorname{cth}\frac{\beta\hbar\omega_q}{2} (\overline N_k+\overline N_{k'}-1) }{ [\hbar^2\omega_q^2-(\Omega_k+\Omega_{k'})^2] }, \tag{12} \]
\[ H(k,k')=\frac{g^2}{2V}\hbar\omega_q \frac{ \hbar\omega_q[\overline N_k+\overline N_{k'}-2\overline N_k\overline N_{k'}] +(\Omega_{k'}-\Omega_k)\operatorname{cth}\frac{\beta\hbar\omega_q}{2} (\overline N_{k'}-\overline N_k) }{ [\hbar^2\omega_q^2-(\Omega_k-\Omega_{k'})^2] }. \]
Turning to relation (10a), let us note that in calculating the derivatives of \(\Psi\) with respect to \(\Omega_k\) one should not alter the energy denominators of this expression, since this may lead to a substantial overestimation of the accuracy of the theory. It is not difficult to see that condition (10a), together with the above restriction, is equivalent to the equations
\[ \frac{\partial \Psi_0}{\partial \Omega_k}=2\overline N_k, \qquad \frac{\partial}{\partial \overline N_k}\frac{1}{\beta} \frac{ \operatorname{Sp}\left[\displaystyle\int_0^\beta \widetilde H_s\,ds\, \exp\left(-\displaystyle\int_0^\beta H_0\,dt\right)\right] }{ Z_0 } =2\Omega_k, \tag{10в} \]
where
\[ \Psi_0=-\frac{1}{\beta}\ln Z_0. \]
The right-hand sides of equations (10в) are doubled, since the energies of elementary excitations of the two kinds of quasiparticles 0 and 1 are identical. After some simplifications, (10в) leads to the following results:
\[ \Omega_k=\sqrt{(1-\eta(k))^2\xi^2(k)+c^2(k)}, \tag{13} \]
where
\[ \eta(k)=-\frac{g^2}{2V}\sum_{k'}\hbar\omega_q \left\{ \frac{ 2\hbar\omega_q\Omega_{k'} \left[ \operatorname{cth}\frac{\beta\Omega_k}{2}(u_k^2-v_k^2)(u_{k'}^2-v_{k'}^2) +\operatorname{th}\frac{\beta\Omega_{k'}}{2} \right] }{ [\hbar\omega_q^2-(\Omega_k+\Omega_{k'})^2][\hbar\omega_q^2-(\Omega_k-\Omega_{k'})^2] } -\right. \]
\[ \left. -\frac{ [\hbar\omega_q^2-\Omega_k^2+\Omega_{k'}^2] \left[ \operatorname{cth}\frac{\beta\hbar\omega_q}{2} \operatorname{th}\frac{\beta\Omega_{k'}}{2} \operatorname{cth}\frac{\beta\Omega_k}{2} (u_k^2-v_k^2)(u_{k'}^2-v_{k'}^2) +\operatorname{cth}\frac{\beta\hbar\omega_q}{2} \right] }{ [\hbar\omega_q^2-(\Omega_k+\Omega_{k'})^2][\hbar\omega_q^2-(\Omega_k-\Omega_{k'})^2] } \right\}. \tag{14} \]
On the basis of (11) and (12) we obtain the following integral equation for determining the magnitude of the gap \(c(\mathbf{k})\) in the spectrum of elementary excitations of the superconductor (13):
\[ c(\mathbf{k})=\operatorname{cth}\frac{\beta\Omega_k}{2} \sum_{\mathbf{k}'}\bigl(J(\mathbf{k},\mathbf{k}')-H(\mathbf{k},\mathbf{k}')\bigr) \frac{c(\mathbf{k}')}{\sqrt{\xi^2(\mathbf{k}')+c^2(\mathbf{k}')}} . \tag{15} \]
Equation (15) always has the trivial solution \(c(\mathbf{k})=0\). A nontrivial solution of (15) exists only at temperatures \(T<T_c\). The critical temperature \(T_c\) can be determined from the following exact equation, obtained by linearizing (15):
\[ \gamma(k)=\operatorname{cth}\frac{\beta\Omega_n(k)}{2} \sum_{k'}\bigl(J_n(\mathbf{k},\mathbf{k}')-H_n(\mathbf{k},\mathbf{k}')\bigr) \frac{\gamma(k')}{|\xi_n(k')|}, \tag{15a} \]
where the index \(n\) means that the corresponding quantities are taken in the normal state of the metal, when \(c=0\).
A very simplified equation for estimating \(T_c\) is obtained by setting the trace of the kernel of equation (15a) equal to unity. As a result we have (for \(k_m\leqslant 2k_F\)):
\[ k_0T_c=1.14\hbar\omega_{\mathrm{cp}}e^{-1/\rho}, \qquad \rho=\frac{mk_Fg^2}{2\pi^2\hbar^2}\left(\frac{k_m}{2k_F}\right)^2, \tag{16} \]
\[ \omega_{\mathrm{cp}}=0.303\,\omega_D, \tag{17} \]
where \(k_F\) is the Fermi momentum, \(\omega_D=\omega(k_m)\) is the Debye frequency. (It is possible that the true value of \(T_c\) differs somewhat from the one given.)
On the basis of equations (10) it is easy to show that at the critical point there is no first-order phase transition; however, at this point there is a nonzero jump in the heat capacity of the metal, \(\Delta\sigma\):
\[ \Delta\sigma = -\beta^3 \sum_p \left[ \frac{\partial v_p}{\partial\beta} \frac{\partial}{\partial v_p} \left(\frac{\partial\Psi}{\partial\beta}\right) + \frac{\partial\Omega_p}{\partial\beta} \frac{\partial}{\partial\Omega_p} \left(\partial\Psi\right) \right]_{T_c}. \tag{18} \]
Both terms in (18) give a nonzero contribution to \(\Delta\sigma\); however, in modeling the second term proves to be the more significant.
Note. The thermodynamics of superconductivity in the Fröhlich model was considered independently by D. N. Zubarev and Yu. A. Tserkovnikov by means of the thermodynamic theory of perturbations, taking account of the renormalization of the velocity of sound \((^8)\).
The author takes this opportunity to express deep gratitude to Acad. N. N. Bogolyubov, and also to D. N. Zubarev, I. A. Kvasnikov, V. V. Tolmachev, and Yu. A. Tserkovnikov for valuable advice and discussion of the results.
Moscow State University
named after M. V. Lomonosov
Received
20 VI 1958
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