BCS Theory and Superconductivity

1. Introduction of the superconductor (7) (11). This information based on experiments on naturally occurring mercury and its Superconductivity discovered in 1911 by Onnes (9), is the isotope, observed a decrease in Tc with an increase in quantum phenomena that certain materials exhibit un- isotopic mass. This allows for the assumption that the der particular magnetic and temperature regimes. There basis of superconductivity relies on electron-phonon in- existed no consistent microscopic theory that described teractions, an assumption that would later lead to the why superconductivity arose, from the time it was dis- formation of the BCS theory. covered until the 1950’s, only macroscopic theories that allowed you to calculate certain thermodynamic and elec- 2.2 In search of a microscopic theory trodynamic quantities. In 1957, two papers released by Bardeen, Cooper, and Schrieffer (1) (2) described the Because superconductivity was found in materials before conceptual and mathematical foundation for conventional the physics community predicted the phenomena, theo- superconductivity, the Bardeen-Cooper-Schrieffer (BCS) ries were formed to attempt to explain, match and pre- theory, for which they later received the Nobel Prize for dict the characteristics of these materials that undergo in 1972. the phase transition. London and London were inter- We will look at features of superconductors before the ested in superconductivity but did not attempt to de- discovery of the BCS theory, and examine the assump- scribe the reason for it. Instead, they derived an equa- tions and methods used to develop the theory. We will tion for the penetration depth, λ, of the superconductor then calculate and study interesting quantities of the su- (4), but their results consistently overestimated the ex- perconducting system, and finally describe how the re- perimentally found values, and so their assumptions were sults predicted by the BCS theory fare against experi- discarded. mental evidence obtained about superconductors. The Ginzburg-Landau theory in 1950 (6), was a phe- nomenological theory using physical intuition and the 2. Before BCS Theory variational principle of quantum mechanics. It allowed the calculation of macroscopic quantities of the material 2.1 Aspects of Superconductivity in the superconducting state if one assumed the phase transition to be of second order. His results were able Onness discovery of superconductivity came when he wit- to accurately match the experimental results of the time, nessed a sudden drop in the resistance of solid mercury and were later shown to be a specific form of the BCS at 4.2 K. All superconductors show this drop of resis- theory. While useful and accurate for macroscopic quan- tance, either gradually or suddenly, at a particular tran- tities, like the London-London attempt, it did not explain sition temperature, Tc. Infinite conductivity implies that the foundation for superconductivity in these materials. if a current were passed through the material during its superconducting phase, the current would follow forever 3. Foundations of BCS Theory without any dissipation. Another characteristic, later found by Meissner and In this section, we lay out the theoretical grounds for BCS Ochsenfeld (8), is that all superconductors are diamag- theory. All derivations may be referenced in “Theory of nets. Diamagnetism occurs when an external magnetic Superconductivity” by Bardeen, Cooper, and Schrieffer field penetrates only a finite, amount of the material, and or Tinkham’s Introduction to Superconductivity. does not hinder the remaining inner parts of the material. The penetration depth is usually small compared to the 3.1 Cooper Pairs width of the material. Also known as the Meissner effect, this event also indicates that a particular magnetic field The BCS theory relies on the assumption that supercon- would destroy the superconductivity of a material. ductivity arises when the attractive Cooper pair interac- The isotope effect discovered by Maxwell and Reynolds tion dominates over the repulsive Coulomb force (2). A describes the relation between Tc and the isotopic mass Cooper pair is a weak electron-electron bound pair medi-

1 ated by a phonon interaction. Although somewhat am- for N0V 1. Thus, the energy of the pair satisfies biguous, one can visualize this pairing by the following E = 2 − 2 ω e−2/N0V < 2 . (1) explanation. Imagine an electron moving within a mate- F ~ c F rial. The Coulomb attraction between the electron and 3.2 The model the positively charged cores of ions in the material will leave a net positive charge in the vicinity. A “paired” We now proceed to write down the model Hamiltonian electron is one with opposite momentum and spin that is for the theory. This is most easily done in the language † attracted to this force. of second quantization. Let ckσ and ckσ to be electron This heuristic explanation is somewhat incomplete, annihilation and creation operators of momentum k and because at the heart of the phonon-mediated interaction spin σ =↑ or ↓. The usual commutation relations are: is a long range attraction and thus, requires quantum † (3) 0 {c , c } = δ (k − k )δ 0 mechanics for a full explanation. Cooper’s monumental kσ k0σ σσ 1956 work showed that due to the fermi statistics of the and − − † † electron, this paired e − e state can have energy less {ckσ, ck0σ} = 0 = {ckσ, ck0σ}. than that of the Fermi-energy of the material(3). Thus, The proposed Hamiltonian is taken to be at adequately low temperatures, when thermal energy is − − 0 X † X † † not a factor, bound e − e states can form. H = kckσckσ + Vklck↑c−k↓c−l↓cl↑. (2) We give a short, simplified argument of for this fact. k,σ k,l Suppose we have two electrons interacting with this at- The first term is the usual kinetic energy of the electrons. tractive Cooper force with a background Fermi sea at The second term is the translation of the phonon me- T = 0 by which these electrons only interact via Pauli- diated electron-electron interaction into this framework. exclusion. We look for a zero-momentum wave-function The matrix element Vkk0 may be taken to be general, but of the form: we shall simplify it by using the mean field approximation X mentioned earlier in later calculations. Ψ(r , r ) = g eik·(r1−r2) (|↑↓i − |↓↑i) . 0 1 k Now, in a normal state we would expect no formation k of Cooper pairs, hence the operator ck↑c−k↓ should aver- Antisymmetry demands gk = g−k. Placing this in the age out to zero. It is then natural to define the quantity Schrödingerequation HΨ = EΨ yields the following con- b = hc c i . dition: k k↑ −k↓ X (E − 2k)gk = Vkk0 gk0 The so-called gap energy is then defined to be 0 k >kF X ∆k = − Vkk0 bk0 . (3) 1 0 R 3 i(k−k )·r 0 where Vkk0 = vol vol d r V (r)e . Now the follow- k ing mean field approximation is made: To allow the exchange of particles it makes sense to consider the new Hamiltonian H = H0 + µN where µ is the −V for F < k < F + ~ωc V 0 = chemical potential. kk 0 else The Hamiltonian can be diagonalized following the method of Bogoliubov. We defined the linearly trans- where F is the Fermi energy and ωc is a cutoff frequency. This indicates that we only consider interactions that are formed states γk0 and γk1 by allowed by the metal’s frequency range, similar to the ∗ † ck↑ = ukγk0 + vkγ assumptions made in the Debeye model. We then have k1 † ∗ † c = −v γk0 + ukγ 1 1 k↓ k k1 X 2 2 = such that |uk| + |vk| = 1. In practice, we can let one of V 2k − E k>k F uk or vk be real. To complete the diagonaization we look Z F +~ωc d at the the Hamiltonian in this basis: → N0 2 − E 0 X n 2 2 † † F H = ξk (|uk| − |vk| )(γ γk1 + γ γ−k0) k1 −k0 N0 2F − E + 2~ωc k = ln 2 2F − E 2 ∗ ∗ † † +2|vk| + 2ukvkγ−k0γk1 + 2ukvkγk1γ−k0

Simplifying: ∗ ∗ ∗ ∗ † † (∆kukvk + ∆kukvk)(γk1γk1 + γ−k0γ−k0 − 1) 1 1 ∗ ∗ ∗ (2 − E) = ≈ e−2/N0V +(∆k(vk) − ∆k(uk) )γ−k0γk1 F 2/N V 2~ωc e 0 − 1 ∗ 2 2 † † ∗ +(∆kvk − ∆kuk)γk1γ−k0 +∆kbk)} .

2 This is analogous to the raising and lowering operators acting on the ground state of the one-dimensional simple harmonic oscillator.

4. Thermodynamic Calculations

4.1 Evaluating Tc

By definition of ∆k and γkσ, we have X ∆k = − Vkk0 bk0 k0 X ∗ D † † E = − Vkk0 ukvk0 1 − γk00γk00 − γk01γk01 . k0 Figure 1: The energy gap seen between the superconduct- † 0 † Now, γ γ = δ(k − k ) = γ γ 0 thus, ing state (top) and the normal state (bottom) produces k0 k0 k01 k 1 the order parameter of the system. X ∗ D † † E ∆k = − Vkk0 ukvk0 1 − γk00γk00 − γk01γk01 k0 X ∗ We determine the u and v terms by demanding the coef- = − Vkk0 uk0 vk0 (1 − 2(f(Ek0 )) ficients of γ γ and γ† γ† . This is equivalent to k0 −k0 k1 k1 −k0 X ∆k0 βEk ∆∗ v q = − Vkk0 tanh k k 2 2 2Ek0 2 = ξk + |∆k| − ξk ≡ Ek − ξk k0 uk V X ∆k0 βEk = tanh . where 2 E 0 2 q 0 k 2 2 k Ek ≡ ξ + |∆k| . k In the last step, we have made the usual mean field ap- From this we observe that the gap energy ∆k is the proximation Vkk0 = −V . In this approximation we know order parameter for this interacting theory as seen in Fig. then that ∆k is independent of k. Thus, we have the 1. relation The graph also indicates with the order parameter tanh βEk that the superconducting state still has energy greater 1 1 X 2 βEk = tanh than zero with no kinetic energy, unlike the normal state. V 2 Ek0 2 k0 The role of this order parameter will become even more p 2 2 apparent with further calculations of thermodynamic quan- We know that generally Ek = ξk + ∆ , and at the tities. transition temperature the energy gap, ∆, vanishes leav- Our objective now is to find a BCS ground state. That ing the relation: Ek = |ξk|, which is symmetric about the is, one that the transformed Bogoliubov operators act on. Fermi energy. Then, If |0i is the free ground state, the most general candidate tanh βξk for the BCS ground state would be a wave function of the 1 1 X 2 βξk = 2 · tanh form V 2 ξk0 2 Y k0 |Ψ i ∼ γ |0i 0 kσ Z c 1 βξ k,σ → N0 dξ tanh . 0 ξ 2 as this state is killed by γkσ for any σ because γkσγkσ ≡ 0. By substituting our original electron operators, this says We will evaluate the integral from ξ = 0 to the Cooper that the BCS ground state has the form energy, c = ~ωc, where the transition to superconductivity begins. Note that we expect ωc ' ωD the Debye Y † † frequency. This is due to the dependence of the Cooper |BCSi = uk + vkck↑c−k↓ |0i . (4) k formation having phonons exist within the material. To compute this integral, we make the substitution Once we have this ground state, we may obtain super- βξ x = 2 to get, conducting excited states by 1 Z βcc/2 tanh(x) γ† γ† ··· γ† |BCSi . = dx. k1σ1 k2σ2 knσn N0V 0 x

3 (a) Conventional superconductors Figure 2: Comparing the entropy dependence on temperature (12), we see that the superconducting state, Ses, is more ordered than the normal state Sen.

We can evaluate this integral approximately, 1 ≈ ln(1.13 βc~ωc) N0V which gives

−1/N0V kBTc ≈ 1.13 ~ωce . (5) (b) Aluminum 4.2 Entropy Figure 3: Indication that the energy gap is temperature A thermodynamic quantity that may be calculated is dependent (5). (a) Indium, Tin and Lead may then be the entropy of the system in different phases. We have considered conventional superconductors because their previously derived that for a Fermi gas, in this case the properties may be predicted by BCS theory. (b) Alu- free electrons of the material, the entropy Sen is propor- minum deviates from the predictions of BCS, and there- tional to T . For the superconducting state with Fermi fore may not be considered a conventional superconduc- function 1 tor. fk := f(Ek) = 1 + eβEk the entropy is given by using their mean field approximation, and the assump- X tion that the superconductors were weakly coupled. This Ses = −2kB {(1 − fk) ln(1 − fk) + fk ln fk} . assumption did not hold in the case of aluminum super- k conductors as seen by Fig. 3 from the same study. Fig. 2 indicates that the superconducting phase is The energy gap, the order parameter of the system, more ordered compared to its normal phase (12). manifests itself when calculating the heat capacities of the superconducting state and the normal state. Recall, dS 5. Experimental Verification CV = T dT . Therefore, ∆C = Ces − Cen at Tc gives 2 After the theory was proposed, many experiments were d∆ ∆C = −N0 . dT designed to test the predictions of superconductors. One T =Tc such experiment measured the temperature dependence of the energy gap (5), with indium, tin, and lead super- This indicates that the magnitude of the discontinuity of conductors as shown in Fig. 3. the heat capacity is explicitly dependent on the order parameter ∆. This is best shown by the heat capacities of The theory predicted that near Tc superconducting aluminum (10) in Fig. 4. Note, Tc for ∆(T ) T 1/2 aluminum is 1.163 K. ≈ 1.74 1 − ∆(0) Tc

4 REFERENCES

[1] J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer. Mi- croscopic Theory of Superconductivity. Phys. Rev., 106(1):162–164, Apr 1957. [2] J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer. The- ory of Superconductivity. Phys. Rev., 108(5):1175– 1204, Dec 1957.

[3] Leon N. Cooper. Bound electron pairs in a degen- erate fermi gas. Phys. Rev., 104(4):1189–1190, Nov 1956. [4] F. and H. London. Proc. Roy. Soc., (London):A149, 1935. Figure 4: Comparison of heat capacities of superconduct- [5] Ivar Giaever and Karl Megerle. Study of Super- ing and normal aluminum states (10). For the normal conductors by Electron Tunneling. Phys. Rev., state, the dependence is linear, while it is nonlinear for 122(4):1101–1111, May 1961. the superconducting state. The order parameter of the system can be seen clearly at Tc = 1.163 K. [6] V. L. Ginzburg and L. D. Landau. Zh. Eksperim. i Teor, 20(1064), 1950. 6. Conclusion [7] Emanuel Maxwell. Isotope effect in the superconductivity of mercury. Phys. Rev., 78(4):477, May BCS has given the ability to describe microscopically 1950. what is occurring in the lattice and Fermi system, and has [8] W. Meissner and R. Ochsenfeld. Natur- been verified by many experiments. While we did not ad- wissenchaften, 21(787), 1933. dress any of the electrodynamics of the superconducting system, the theory does support and is agreement with [9] H. Kamerlingh Onnes. Leiden Comm., 120b, 122b, experimental findings. The flexibility of the BCS theory 124c, 1911. has allowed for derivative theories that are dependent on the electron-phonon interaction. [10] Norman E. Phillips. Heat Capacity of Aluminum There are drawbacks of the theory, as seen by the de- between 0.1 K and 4.0 K. Phys. Rev., 114(3):676– viation by the data obtained on the aluminum supercon- 685, May 1959. ductors. The theory best approximates only conventional [11] C. A. Reynolds, B. Serin, W. H. Wright, and L. B. weakly coupled superconductors, which aluminum can- Nesbitt. Superconductivity of isotopes of mercury. not be. From the publication of this theory, there have Phys. Rev., 78(4):487, May 1950. been discoveries of high Tc superconductors (Tc > 100 K), which cannot be explained by BCS. [12] Michael Tinkham. Introduction to Superconductiv- Many have criticized the theory for being unable to ity. Dover, second edition, 2004. explain the inverse isotope effect, where the Tc is inversely proportional to the isotopic mass, and for not predicting which materials are superconducting. While the theory is a starting point, any deviations from its assumptions, most notably that it is weakly coupled, would not be supported and predict incorrect results. While the theory claims to understand what causes superconductivity, it gives no procedure in choosing which materials would undergo the phase transition. Instead, the results could point to what new types of interactions are occurring within the superconducting system, and use an alterna- tive theory to explain the onset of superconductivity.