Bardeen-Cooper-Schrieffer theory

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Leon Cooper and Dmitri Feldman (2009), Scholarpedia, 4(1):6439. doi:10.4249/scholarpedia.6439 revision #91028 [link to/cite this article]
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Curator: Leon Cooper

The BCS theory or Bardeen-Cooper-Schrieffer theory is the theory of superconductivity developed by John Bardeen, Leon N Cooper and John R. Schrieffer in 1957.

Contents

Electron correlations that produce superconductivity

The superconducting phase exhibits correlations absent in the normal metal. It is believed that any attractive interaction between fermions in a many-fermion system can produce a superconducting-like state and this is thought to be the case, in addition to metals, in nuclei, neutron stars and He3. Thus the BCS theory focuses on the consequences of an attractive two-body interaction without enquiring too much further about its origin.

The fundamental difference between the superconducting and normal ground state wave functions is produced when the large degeneracy of single-particle electron levels in the normal state is removed. If one visualizes the Hamiltonian matrix that results from an attractive two-body interaction in the basis of normal metal configurations, one finds sub-matrices in which all single-particle states except for one pair of electrons remain unchanged. These two electrons can scatter via their interaction to all states of the same total momentum. Such pair may be thought to “wend its way” over all states unoccupied by other electrons. It is known as a Cooper pair and plays the central role in BCS.

Since every such state is connected to every other, one is presented with submatrices of the entire Hamiltonian corresponding to an \(M\)-dimensional space of two-particle excited states on top of the Fermi see. The origin of Cooper pairs can be understood from the following simplified example: all off-diagonal elements are set equal to \(-V\) and all diagonal terms are set equal to 0 as though all the initial electron levels were completely degenerate,

\[ \left( \begin{array}{cccccccccccc} 0 & -V & -V & -V & . &. & .& .& .& .& .& -V\\ -V & 0 & -V & . & . &. & & & & & .& . \\ -V & -V & 0 & -V & .&.& & & .& .& & . \\ . & . & .& .& .& . & & & & & & . \\ . & & .& & & &. & & & & & . \\ . & . & & & & & & & & & & . \\ . & & & & .& & & . & .&. & & .\\ . & & & .& & & & &. & & & .\\ . & & & & & & & .& & .& & .\\ . & . & . & & & &. & .& .& .&. & .\\ . & . & . & & & & . &. & .& -V& 0& -V \\ -V & -V & . & .& .& . &. &. & .& -V& -V&0 \end{array} \right) \]


Diagonalizing this matrix results in an energy level structure with \(M-1\) levels with energy \(V\) and one level with the energy \(E=-(M-1)V\ .\) The latter level is a superposition of all original levels. The number of levels \( M \) is proportional to the volume of the system while the scattering matrix element \( V \) is inversely proportional to the volume. Hence, \(E\) is independent of the volume. In other words, the removal of the degeneracy produces a single level separated from the others by a volume independent gap.

To incorporate this into a solution of the full Hamiltonian, one must devise a technique by which all electron pairs can scatter while obeying the exclusion principle. In the BCS wave function that accomplishes this, an inspired guess of Robert Schrieffer, each pair gains an energy due to the removal of the degeneracy as above and one obtains the maximum correlation of the entire wave function if all pairs have the same total momentum. For a combination of statistical and dynamical reasons, this gives a coherence to the wave function with a preference for momentum zero, singlet spin correlations. Formation of the superconducting state is a phase transition, and is characterized by an order parameter. Gor'kov showed that the superconducting symmetry parameter is proportional to the wave function of the Cooper pair.

Ground state

Figure 1: The ground state of the superconductor is a linear superposition of states in which pairs \((k\uparrow,-k\downarrow)\)are occupied or unoccupied.


In the simplest model only singlet zero-momentum pairs interact and scatter. They can be conveniently described by the pair operators \[ b_k=c_{-k\downarrow}c_{k\uparrow} \] \[ b_k^\dagger = c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger, \] where \(c_{k\uparrow,\downarrow}\) are fermion annihilation operators, the arrow shows the spin projection. We expect that in a good approximation the wave function can be constructed from the ground state of a non-interacting electron gas entirely with combinations of the pair operators. Using this one extracts from the full Hamiltonian the so-called reduced Hamiltonian

\[ H_{\rm reduced}=\sum_{k<k_f}2|\epsilon_k|b_kb_k^\dagger+2\sum_{k>k_f}\epsilon_k b^\dagger_kb_k +\sum_{k,q}V_{kp}b^\dagger_kb_p, \] where \(k_f\) is the Fermi momentum, \(\epsilon_k\) are single electron energy levels (the energy is zero at the Fermi level) and \(V_{kp}\) is the scattering matrix element between the pair states \(k\) and \(p\ .\) The reduced Hamiltonian has correct matrix elements in the subspace built with pair operators.


The ground state of the superconductor is a linear combination of pair states in which the pairs \((k\uparrow,-k\downarrow)\) are occupied (state \(O_k\)) or unoccupied (state \(O_{(k)}\)) as indicated in Figure 1:

\[ \psi_0= u_kO_{(k)}+v_kO_k. \] The probability amplitude that the pair state is (is not) occupied is then \(v_k(u_k)\ .\) Normalization requires that \(|u|^2+|v|^2=1\ .\) The phase of the ground state may be chosen so that with no loss of generality \(u_k\) is real. We can then write

\[ u=\sqrt{1-h}; v=\sqrt{h}\exp(i\phi), \] where \(0\le h\le 1\ .\)

Figure 2: A decomposition of the ground state of the superconductor into states in which the pair states \(k\) and \(p\) are either occupied or unoccupied.

A further decomposition of the ground state wave function in which pair states \(k\) and \(p\) are occupied or unoccupied ( Figure 2) is

\[ \psi_0= (u_kO_{(k)}+v_kO_k) (u_pO_{(p)}+v_p O_p). \] This is a Hartree-like approximation in the probability amplitudes for the occupation of pair states. The decomposition procedure should be continued to include all \(k\ ,\) so that the complete BCS ansatz for the ground state is given by

\[ \psi_0= \prod_k ( u_kO_{(k)}+v_kO_k). \]

This wave function does not have a fixed number of particles \(N\ ,\) but the relative fluctuations of \(N\) are of the order \(1/\sqrt{N}\) in the thermodynamic limit, so negligible for macroscopic samples. Minimizing the energy

\[ W=\langle\psi_0|H_{\rm reduced}|\psi_0\rangle \] with respect to the variational parameters \(h_k\) and \(\phi_k\) gives

\[ h=(1-\epsilon/E)/2, \] where \[\tag{1} E(k)=(\epsilon_k^2+|\Delta(k)|^2)^{1/2} \]

and \(\Delta(k)\) satisfies the equation

\[ \Delta(k)=-\frac{1}{2}\sum_qV_{kq}\frac{\Delta(q)}{E(q)} \]

If a non-zero solution of this integral equation exists, \(W<0\ ,\) then the normal Fermi sea is unstable under the formation of Cooper pairs. In the wave function that results there are strong correlations between pairs of electrons with opposite spin and zero total momentum. These correlations are built from normal excitations near the Fermi surface and extend over spatial distances typically of the order of \(10^{-4}\) cm. They can be constructed due to the large wave numbers available because of the exclusion principle. Thus, with a small additional expenditure of kinetic energy there can be a greater gain in the potential energy term.

Single-particle excitations

In considering the excited states of the superconductor it is useful to make a distinction between single-particle and collective excitations: it is the single-particle excitation spectrum whose alteration is responsible for superfluid properties. For the superconductor, excited (quasiparticle) states can be defined in one-to-one correspondence with the excitations of the normal metal. One finds, for example, that the expectation value of \(H_{\rm reduced}\) for the excitation in Fig. 3 is given by

\[ E=\sqrt{\epsilon_k^2+|\Delta|^2}. \]

Figure 3: A single particle excitation of the superconductor in one-to-one correspondence with an excitation of the normal fermion system.


In contrast to normal systems, for the superconductor even as \(\epsilon\) goes to zero, the excitation energy \(E\) remains larger than zero, its lowest positive value is \(|\Delta|\ .\) One can therefore produce single particle excitations from the superconducting ground state only with the expenditure of a small but finite amount of energy. This is called the energy gap (in gapless superconductors \(\Delta( k)\) is zero at special directions of the momentum). In the ideal superconductor, the energy gap appears because not a single pair can be broken nor a single element of the phase space be removed without a finite energy cost. If a single pair is broken, one loses its correlation energy \(W\ ;\) if one removes an element of phase space from the system, the number of possible transitions of all pairs is reduced resulting in both cases in an increase in the energy which does not go to zero as the volume of the system increases.

The excitation spectrum of the superconductor can be conveniently treated by introducing a linear combination of the fermion creation and annihilation operators. This is known as the Valatin-Bogoliubov transformation:

\[ \gamma_{k\uparrow}^\dagger=u_kc^\dagger_{k, \uparrow}-v^*_kc_{-k,\downarrow}; \] \[ \gamma_{k\downarrow}^\dagger=v_k^* c_{k, \uparrow}+u_kc^\dagger_{-k,\downarrow}. \] It follows that \[ \gamma_{k\sigma}\psi_0=0 \] so that the \(\gamma_{k\sigma}\) play the role of annihilation operators, while \(\gamma^\dagger_{k\sigma}\) create excitations:

\[ \gamma_{ki}^\dagger\dots\gamma_{qj}^\dagger\psi_0=\psi_{ki,\dots,qj}. \]

The \(\gamma\) operators satisfy Fermi anti-commutation relations so that they produce a complete set of fermionic excitations in one-to-one correspondence with the excitations of a normal metal. In the ground state of the superconductor all the electrons are in singlet-pair correlated states of zero total momentum. In an \(m\)-electron excited state the excited electrons are in a ‘quasipartilce’ states, very similar to Fermi liquid excitations and not strongly correlated with any other electrons. The other electrons are still correlated much as they were in the ground state. The excited electrons behave in a manner similar to normal electrons; they can be easily scattered or excited further. But the background electrons – those which remain correlated – retain their special behavior; they are difficult to scatter or excite.

Thus, one can identify two almost independent fluids. The correlated portion of the wave function shows the resistance to change and the very small specific heat characteristic of superfluid, while the excitations behave very much like normal electrons, displaying an almost normal specific heat and resistance. When a steady electric field is applied to the metal, the superfluid electrons short out the normal ones, but with higher frequency fields the resistive properties of the excited electrons can be observed.

Thermodynamic properties of the ideal superconductor

The thermodynamic properties of the superconductor follow from the excitation spectrum discussed in the previous section. The free energy of the superconductor is given by

\[ F[h,\phi,f]=W(T)-TS, \] where \(T\) is the temperature and \(S\) is the entropy; \(f(k)\) is the probability that the state \(k\) is occupied by a quasiparticle (a superconducting Fermi function). The entropy of the system comes entirely from the excitations as the correlated portion of the wave-function is not degenerate. Since a portion of the phase space is occupied by excitations at finite temperatures, making it unavailable for the transitions of bound pairs, the energy is a function of the temperature \(W(T)\ .\) As \(T\) increases, \(|W(T)|\) and at the same time \(\Delta\) decrease until the critical temperature is reached and the system reverts to the normal phase.

The minimization of the free energy with respect to the variational parameters \(h\) and \(f\) gives the Fermi-Dirac expression for \(f\ :\)

\[ f=\frac{1}{1+\exp(E/kT)}, \] where \(E\) is given by Eq. (1) and the gap \(\Delta\) satisfies the fundamental integral equation of the theory

\[\tag{2} \Delta_k(T)=-\frac{1}{2}\sum_{q}V_{k q}\frac{\Delta_q(T)}{E_q(T)}\tanh\frac{E_q(T)}{2kT}. \]


This equation has a nonzero solution for the gap \(\Delta(T)\) below a critical temperature \(T_c\ .\) At \(T<T_c\) the properties of the system are qualitatively different from the normal metal.

In the simplified model used in the original formulation of the BCS theory, the scattering potential was approximated as

\[ V_{kq}=-V, |\epsilon|<\hbar\omega_D \] \[ V_{kq}=0, |\epsilon|>\hbar\omega_D. \]

where \(\omega_D\) is the Debye frequency of the material. The energy-dependent density of states was replaced by its value at the Fermi surface \(N(0)\ .\) In this approximation the second-order transition temperature in zero magnetic field

\[ kT_c=1.14\hbar \omega_D\exp(-1/N(0)V). \] At zero temperature the gap

\[ \Delta(0)=1.76kT_c. \] The Cooper pair size can be estimated as \(\xi=hv_F/[\pi \Delta]\ ,\) where \(v_F\) is the Fermi velocity in the normal state. The gap approaches zero as

\[ \Delta(T)=1.74\Delta(0)\sqrt{1-T/T_c} \] near the critical temperature. The specific heat is discontinuous at the transition point. Right below the transition it is 2.43 times greater than immediately above it. At low temperatures the specific heat is exponentially small as a function of the inverse temperature.

Microscopic interference effects

In its interaction with external perturbations the superconductor displays remarkable interference effects which result form the paired nature of the wave function and are not at all present in similar normal metal interactions. Neither would they be present in any ordinary two-fluid model of superconductivity. This ‘coherence effects’ are in a sense manifestations of interference in spin and momentum space on a microscopic scale, analogous to the macroscopic quantum effects due to interference in ordinary space. They depend on the behavior under time reversal of the perturbing fields.

Near the transition temperature these coherence effects produce quite dramatic contrasts in the behavior of coefficients which measure interactions with the conduction electrons. Historically, the comparison with theory of the behavior of the relaxation rate of nuclear spins and the attenuation of longitudinal ultrasonic waves in clean samples as the temperature is decreased through \(T_c\) provided an early test of the detailed structure of the theory.

The attenuation of longitudinal acoustic waves due to their interaction with the conduction electrons in a metal undergoes a very rapid drop as the temperature drops below \(T_c\ .\) Since the scattering of phonons from ‘normal’ electrons is responsible for most of acoustic attenuation, a drop is to be expected both in BCS in a two-fluid model but the rapidity of the decrease is difficult to reconcile with theoretical estimates within a two-fluid model.

Figure 4: Comparison of observed nuclear spin relaxation rate with theory. The circles represent experimental data of Hebel and Slichter, the crosses data by Anderson and Redfield.

The rate of relaxation of nuclear spins was measured by Hebel and Slichter in zero magnetic field in aluminum (\(T_c=1.18\)K) from 0.92K to 4.2K just at the time of the development of the BCS theory in 1957, Figure 4. The dominant relaxation mechanism is provided by interaction with the conduction electrons so that one would expect, on the basis of a two-fluid model, that this rate should decrease below the transition temperature due to the diminishing density of ‘normal’ electrons. The experimental results however show just the reverse. The relaxation rate does not drop but increases by a factor of more than two just below the transition temperature in agreement with BCS but contrary to the predictions of other approaches to the theory of superconductivity.

To understand how such effects come about in theory, one needs to consider the transition probability per unit time of a process involving electron transitions from the excited electron state \(k\) to the state \(p\) with the emission or absorption of energy from the interacting field. What is to be calculated is the rate of transition between an initial state \(\langle i|\) and a final state \(\langle f|\) with an absorption or emission of the energy \(\hbar \omega_{|{\vec k}-{\vec p}|}\) (a phonon for example in the interaction of sound waves with the superconductor). All of this properly summed over final states and averaged with statistical factors over initial states may be written:

\[ \Omega=\frac{2\pi}{\hbar}\frac{\sum_{i,f}\exp(-W_i/kT)|\langle f|H_{\rm int}| i\rangle|^2\delta(W_i-W_f)}{\sum_i \exp(-W_i/kT)}, \] where \(W_l\) are the energies in different states and \(H_{\rm int}\) is the interaction Hamiltonian. The interaction Hamiltonian can typically be represented as

\[ H_{\rm int}=\sum_{K P}B_{P K}c^\dagger_{P}c_{K}, \] where the operator \(B\) is the electronic part of the matrix element between the full final and initial states \(\langle f|H_{\rm int}| i\rangle=m_{fi}\langle f|B| i\rangle \ ,\) the indices \(K, P\) contain both the momentum \(k,p\) and spin; the spins of the states \(K\) and \(-K\) are opposite.

In the normal system, scattering from single-particle electron states \(K\) to \(P\) is independent of scattering from \(-P\) to \(-K\ .\) But the superconducting states are linear combinations of \((K,-K)\) occupied and unoccupied. Because of this, states with excitations \(k\uparrow ,p\uparrow\) are connected not only by \(c^\dagger_{p\uparrow}c_{k\uparrow}\) but also by \(c^\dagger_{-p\downarrow}c_{-k\downarrow}\ :\) if the state \(|f\rangle\) contains the single-particle excitation \(p\uparrow\) while the state \(|i\rangle\) contains \(k\uparrow\ ,\) as a result of the superposition of occupied and unoccupied pair states in the coherent part of the wave function, these are connected not only by \(B_{PK}c^\dagger_{P}c_{K}\) but also by \(B_{-K-P}c^\dagger_{-K}c_{-P}\ .\)

Many operators \(B\) (e.g., the electric current, or the charge density operator) have a well-defined behavior under the operation of time reversal so that

\[ B_{PK}=\pm B_{-K-P}=B_{p,k}, \] where the last expression does not contain spin indices. Then \(B\) becomes

\[ B=\sum_{k p}B_{pk}(c^\dagger_{p\uparrow}c_{k\uparrow}\pm c^\dagger_{-k\downarrow}c_{ -p\downarrow}), \] where the upper (lower) sign results for operators even (odd) under time reversal. As a result the matrix element squared \(|\langle f|B| i\rangle|^2\) contains terms of the form \(|B_{pk}|^2|u_{p}u_k\mp v_{p}v_k^*|^2\ .\)

Applied to processes involving the emission or absorption of boson quanta such as phonons or photons, the squared matrix element above is averaged with the appropriate statistical factors over initial and summed over final states; subtracting emission from absorption probability per unit time, one typically obtains

\[ a=\frac{4\pi}{\hbar}|m|^2\sum_{kp}|u_{p}u_k\mp v_{p}v_k^*|^2(f_{p}-f_k) \delta(E_{p}-E_k-\hbar\omega_{|{\vec k}-{\vec p}|}), \] where \(f_k\) is the occupation probability for an excitation with the momentum \(k\ .\)

For the ideal superconductor there is isotropy around the Fermi surface and symmetry between particles and holes; therefore sums of the form \(\sum_k\) can be converted to integrals over the superconducting excitation energy, \(E\ :\) \(\sum_k\rightarrow 2N(0)\int_\Delta^\infty E/\sqrt{E^2-\Delta^2}dE\ ,\) where \(N_s(E)=N(0) E/\sqrt{E^2-\Delta^2}\) is the density of excitations in the superconductor, Figure 5.

Figure 5: Ratio of superconducting to normal density as a function of \(E/\Delta\)

The shape of Figure 5 shows that contrary to intuitive expectations, the onset of superconductivity might enhance rather than diminish electronic transitions. However, the coherence factors \(|u_{p}u_k\mp v_{p}v_k^*|^2=[1+(\epsilon(k)\epsilon(p)\mp\Delta^2)/E(k)E(p)]/2\) completely negate the effect of the increased density of state in the case of the operators even under time reversal. For the operators odd under time reversal the effect of the increase of the density of states is not cancelled and can be observed in the increase of the rate of the corresponding process. The current and spin are odd under time reversal while the charge density is even. The first two operators are responsible for electromagnetic interaction and nuclear spin relaxation interaction while the latter is relevant for the electron-phonon interaction which shows strikingly different effects.

Ultrasonic attenuation in the ideal pure superconductor for \(ql\gg 1\) (the product of the phonon wave number and the electron mean free path) depends in a fundamental way on the absorption and emission of phonons. Since the matrix elements have a very weak dependence on changes near the Fermi surface in occupation numbers other than \(k\) or \(p\) that occur in the normal to superconducting transition, calculations within the quasi-particle model can be compared in a very direct manner with similar calculations for the normal metal, as \(B_{pk}\) is the same in both states. The ratio of the attenuation in the normal and superconducting states reduces to

\[ \frac{a_s}{a_n}=\frac{2}{1+\exp(\Delta(T)/kT)}. \] This prediction was used by Morse and Bohm for a direct experimental determination of the variation of \(\Delta\) with \(T\) and obtained an excellent agreement with the BCS theory, Figure 6.

Figure 6: Comparison of ultrasonic attenuation observed by Morse and Bohm with theory.

On the other hand, the ratio of the nuclear spin relaxation rates in superconducting and normal states in the same sample can be estimated as

\[ \frac{R_s}{R_n}\sim \int N_s^2(E) dE/\int N^2_n(E)dE, \] where \(N_s(E)\) and \(N_n(E)\) stay for the normal and superconducting densities of states. Taken literally, in fact, this expression diverges logarithmically at the lower limit due to the infinite density of states. When the Zeeman energy difference between the spin up and spin down states is included, the integral is no longer divergent but the integrand is much too large. Quantitative agreement between theory and experiment was obtained by Fibich by including the effect of thermal phonons.

Interference effects manifest themselves in a similar manner in the interaction of electromagnetic radiation with the superconductor. Near \(T_c\) the absorption is dominated by quasi-particle scattering matrix elements of the type discussed above. Near \(T=0\) the number of quasiparticle excitations goes to zero and the matrix elements that contribute are those in which quasiparticle pairs are created from \(\psi_0\ .\) For absorption these latter occur only when \(\hbar\omega>2\Delta\ .\) For the linear response of the superconductor to a static magnetic field, the interference occurs in such a manner that the paramagnetic contribution goes to zero leaving the diamagnetic part which gives the Meissner effect.

It is now believed that the finite many-nucleon system that is the atomic nucleus enters a correlated state analogous to that of a superconductor. Similar considerations have been applied to many-fermion systems as diverse as neutron stars, liquid He3 and to elementary fermions. In more recent years the importance of BCS pairing was demonstrated in ultra-cold gases. It was proposed that BCS-type physics can be responsible for some of the observed quantum Hall states. In addition the idea of spontaneously broken symmetry of a degenerate vacuum has been applied widely in elementary particle theory, in particular, in the theory of electroweak interactions. The pairing which the electron-phonon interaction has produced between electrons in metals may be produced by the van der Waals interaction between atoms in He3, the nuclear interaction in nuclei and neutron stars, and the fundamental interactions in elementary fermions. Whatever the success of these attempts, for the theoretician the possible existence of this correlated paired state must in the future be considered for any many-fermion system where there is some kind of effective attraction between fermions for transitions near Fermi surface.

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Internal references

  • Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.


Recommended reading

  • A. A. Abrikosov (1988), Fundamentals of the Theory of Metals, Elsevier Science.
  • P. G. de Gennes (1966), Superconductivity of Metals and Alloys, Benjamin, New York.
  • M. Tinkham (1996), Introduction to Superconductivity, Second edition, Dover, New York.
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