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Heuristic Derivation of the BCS Energy Gap Equation∗

F. Zuniga Frias, I. Chavez and M. de Llano

The superconducting energy gap of BCS (Bardeen, Cooper and Schrieffer) is one of the foun-dational concepts of conventional superconductivity. Here we show an heuristic, yet correct, ap-proximation to the energy gap of BCS. This is done using a variational method through which thesuperconductor energy is minimized with respect to the occupation probability of electrons. Theminimization yields the well known self-consistent BCS gap equation for T = 0K , which is easily

extended to the finite T > 0 regime using heuristic arguments.

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PACS numbers:

Structure:

I. INTRODUCTION

Superconductivity may be regarded as one of the mostfascinating quantum phenomena acting on a macroscopicscale. This occurs when a superconducting material iscooled below a critical temperature, T c, where a sec-

ond order phase transition takes place. Phenomenologi-cally, when this transition takes place, the electrical re-sistance of the material disappears completely and it be-comes perfectly diamagnetic. Moreover, it is stronglyrepelled by and expels a magnetic field. Superconduc-tivity along the zero-resistance phenomenon was discov-ered by Kamerlingh Onnes in 1911; the perfect diamag-netism phenomenon was found by Meissner and Ochsen-feld in 1933, and is called the Meissner-Ochsenfeld ef-fect. Some years after, in 1950, the isotope effect wasdiscovered. Experimentally it showed how the T c var-ied inversely with the square root of the isotopic mass of certain materials. This suggested that superconductiv-ity should arise from the attractive interaction of elec-

trons and the lattice vibrations. Theoretically, however,the phenomenon of superconductivity remained unclearuntil 1957 when the first microscopic theory of super-conductivity, called BCS (J. Bardeen, L. Cooper and R.Schrieffer) [? ], explained correctly the zero electrical re-sistance, the Meissner-Ochsenfeld effect, and the isotopeeffect.

The BCS theory of superconductivity is in a secondquantization formalism and is based on two foundationalconcepts: i) Cooper pairs . For electron pairing to occurit is supposed an effective attractive interaction betweenelectrons near the Fermi surface. This is basically thesum of the attractive phonon-induced interaction and

a repulsive screened Coulomb interaction between twocharge-carrier electrons. This bound state of the two elec-trons is formed even when the interaction is weak, thiswas shown by Leon Cooper in 1956 [2]. This perspectivelead to the correct deduction of ii) an energy gap. Theidea of an energy gap had been suggested several years

∗ A footnote to the article title

before BCS [3][4] as a promising explanation of supercon-ductivity. Bardeen himself, by 1950 suggested that “if one could find the reason for the energy gap, one wouldvery likely have the explanation of superconductivity”[5].The existence of a superconducting gap was experimen-tally shown by Corak et al. in 1946 [7]. But it was notcorrectly explained until the arrival of the BCS, which

derived it using a variational method and gave an ex-ceptionally good picture of an electron spectrum with agap.

The approach given here is heuristic as opposed to theoriginal detailed derivation published in 1957 by BCS.Here it is shown that an energy gap can be obtained witha variational method through which the superconductingenergy is minimized with respect to the occupation prob-ability of electrons.

II. SECOND SECTION

As stated before, superconductivity is a second orderphase transition. This phase transition is, however, of a special kind which takes place in momentum space k(or velocity space), where k is the wave vector associ-ated to the linear momentum p ≡ k. This means thatwhen a material becomes superconducting, from the to-tal number of electrons, a fraction of them (of the orderof 106 electrons) assume the same velocities. When thiscondensation occurs the energy of the material in normalstate diminishes by a small amount, leading to the super-conducting state. This lowering of energy is believed tocome from the interaction of electrons and the phononfield, which in turn, lead to the attractive interaction

between electrons, and hence, to a negative sign in thepotential interaction.

The best way to model the attractive interaction is byoccupying individual particle states in pairs in such a waythat if one element of the pair is occupied the other oneis also occupied. Additionally, transitions between pairsmust be possible, which implies that all pairs must havethe same total momentum K.

Formally stated, let k ≡ 1

2(k1 − k2) be the relative

wavevector of two electrons in a pair state, and let k1 and

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2

k’

k

-k’

-k

FIG. 1. Feynman diagram of the interaction of two electronsin a vibrating crystal lattice through the virtual exchange of a phonon.

k2 be the wave vectors of each electron, where k1 = k↑and k2 = −k↓. That is, two electrons with the samevelocity but opposite direction and spin. This is shown infigure 1. Further, it must be noted that the total center-of-mass momentum K = k1 + k2 of any pair is assumedby BCS to be zero, avoiding any other pair state withnonzero total momentum.

We now formalize the BCS assumption of an attrac-tive phonon-electron interaction in the gas of electronsto model the electron-electron interaction, along with arepulsive coulombic interaction which is overwhelmed bythe electron-phonon attraction. This leaves a net attrac-tion needed to electrons by pairs into Cooper pairs whichare virtually universally recognized as essential to pro-duce superconductivity.

V kk =

−V, if − ωD < k − k < ωD

0, otherwise (1)

In other words, the interaction between electrons nearthe fermi energy and in the interval − ωD < k − k < ωD, is a net attraction and, hence, the minus sign in V .Here εk ≡

2k2/2m is the single particle kinetic energyand ωD is the maximum (Debye) energy of a phononabsorbed or emitted by an electron. It must be noted thatV is taken as a mean value of the potential interactionvalue in order to neglect anisotropic effects.

As we are dealing with a great number of electron pairsin the superconducting state, intuition tells us to try witha statistical approach and a variational method. To ac-complish this we set the probabilities of occupation asfollows: Let v2

k be the probability of the two states char-

acterized by k being occupied, and (1 − v2k

) as the prob-ability of states being unoccupied. This allows electronpairs to hop from state to state and therefore to modelthe probability transition from k → k, which only re-quires that initially k must be full and k must be empty .

Then the transition probability v2

k(1 − v2

k) is propor-tional to the k→ k transition, and the opposite transi-tion probability v2

k(1−v2k

) is proportional to the k → k

transition. Our next goal is to show that the minimiza-tion of the superconducting energy with respect to theoccupation probability yields an energy gap equation for

the case T = 0K . We have formulated the potentialinteraction in (1), it only remains to add the kinetic en-ergy of the set of pairs, E kin = 2

k

εkv2k

to obtain thesuperconducting energy

E S = 2k

εkv2k−V k

k

[v2k(1−v2k)v2k(1−v2k)]1/2.

(2)

Note that when V → 0

E S → E N = 2k

εknk

with nk = θ(|kF − k|) at T = 0 and θ the Heaviside stepfunction.

Now, we choose v2k

such that E S [v2

k] is minimum, but

subject to the constraint N = 2

k v2k

= constant. Andusing the definition of a functional derivative

δ

δv2kv2l ≡ δ kl, δ kl ≡

1 if k = l0 if k = l

In order to simplify the notation, we set xk ≡ v2k, then

δ

δxl{E S [xk] − 2µ

k

xk} = 0 (3)

where µ is a Lagrange multiplier. Substituting equation(2) into (3) gives the expression to minimize

δ

δxl{2k

εkxk − V k

k

[xk(1− xk)×

×xk(1 − xk)]1/2 − 2µk

xk} = 0(4)

It is straightforward to show that

0 = 2(εk − µ)

− 1

2V k

k

[xkδ kl + xkδ kl − 2xkxkδ kl − x2

kδ kl −

[xk(1 − xk

Defining εk ≡ εk − µ and reducing terms lead to

2εk − 1

2V k

k

[xk(1− xk)]1/2(1− 2xk)

[xk(1 − xk)]1/2

δ kl+

[xk(1− xk]1/2(1− 2xk)

[xk(1− xk)]1/2

δ kl

= 0

Since we are dealing with a symmetric process anddummy indices, the sum may depend only on one index,

2εk − V k

[xk(1 − xk)]1/2(1 − 2xk)

[xk(1− xk)]1/2

= 0.

Returning to our old notation v2

k we have

2εk − V k

v2k

(1− v2k

)(1− 2v2k)

[v2k(1 − v2

k)]1/2

= 0. (5)

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Defining V

k

v2k

(1 − v2k

) ≡ ∆ ≥ 0, then (5) is rewrit-ten as

2εk = ∆ (1− 2v2

k)

[v2k

(1 − v2k

)]1/2

or rearranging

2εk[v2

k(1 − v2

k)]1/2

= ∆(1 − 2v2

k). (6)

Squaring (6) we obtain

4ε2k[v2k(1− v2

k)] = ∆2(1− 2v2k)

2

. (7)

And noticing that

(1− 2v2k)2

=[(1 − v2k)− (v2k)]2

=[(1 − v2k)

2

+ (v2k)2 − 2(1− v2

k)v2k]

=[(1 − v2k) + v2

k]2 − 4v2k(1 − v2

k)

we arrive at

4(ε2k + ∆2)[v2k(1− v2k)] = ∆2[(1 − v2k) + v2k ] = ∆2.

Then

[v2k(1− v2k)]1/2 = ∆

2

(ε2k

+ ∆2). (8)

Therefore, when we substitute (8) in (6) we obtain

v2k = 1

2

1 −

εk

ε2k + ∆2

(9)

the distribution function. Thus, when we substitute (8)in the definition of ∆ we obtain the self-consistent BCSenergy gap equation at T = 0

1 = V

2

k

1 (εk − µ)2 + ∆2

. (10)

We can substitute the sum by an integral

k

1

ε2k

+ ∆2−→

ωD

−ωD

dε g(ε)

ε2k

+ ∆2

g(E F ) ωD−ωD

dε 1 ε2k

+ ∆2

(11)

provided that ωD E F where g(ε) is the electronicdensity of states for a single spin. Then, defining λ ≡V g(E F ),

1

λ ≡

1

g(E F )V =

ωD

0

dε ε2k− ∆2

= sinh−1

ωD

∆

.

(12)

Finally

∆ = ωD

sinh(1/λ) −−−→λ→0

2 ωD e−1/λ. (13)

FIG. 2. Temperature dependence of the BCS energy gap.The energy gap is a good approximation to experimental dataof many conventional superconductors. As can be seen, thisuniversal curve fits correctly to the data of Sn, Nb and Ta.[8]

III. THIRD SECTION

For T > 0 we include the probability that either |01or |10 can occur.

ωD

0

dε ε2k−∆2

[1 − 2f k] (14)

where f k = 1/(exp

ε2k−∆2 + 1).Thence

1

λ =

ωD

0

dε ε2k−∆2

[1 − 2f k]

= ωD

0

dε

ε2k

−∆2tanh

β

ε2k−∆2

2 (15)

=

ωD

0

dε ε2k− ∆2

tanh

β

ε2k−∆2

2

ln

2 eγ β c ωD

π

(16)Remembering that β = 1/kT c, then we have

kT c = β −1

c = 2 eγ

π 1.13 ωD e−1/λ (17)

where γ 0.57721 is the Euler constant.

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[1] J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 108,1175 (1957).

[2] L.N. Cooper, Phys. Rev. 104, 1189 (1956).[3] J.G. Daunt and K. Mendelssohn, Proc. Roy. Soc. (London)

A185, 225 (1946)[4] J. Bardeen, Theory of superconductivity in S. Flugge (ed.),

Handbuch der Physik , vol XV, Springer-Verlag, Berlin(1956)

[5] C. Slichter, http://www.aip.org/history/mod/superconductivity/links.html(accessed 12/5/2007) Center for History of Physics.

[6] M. Tinkham, Introduction to Superconductivity , McGraw-Hill, New York, in series: International series in pure andapplied physics. (1975).

[7] W.S. Corak, B.B. Goodman, and C.B. Satterthwaite,Phys. Rev., 96, 225 (1946).

[8] J.M. Blatt, Modern Physics , McGraw-Hill, New York.

(1992)

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