# The Solution to the BCS Gap Equation for Superconductivity and Its Temperature Dependence

Authors: Shuji Watanabe
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

From the viewpoint of operator theory, we deal with the temperature dependence of the solution to the BCS gap equation for superconductivity. When the potential is a positive constant, the BCS gap equation reduces to the simple gap equation. We first show that there is a unique nonnegative solution to the simple gap equation, that it is continuous and strictly decreasing, and that it is of classC2 with respect to the temperature. We next deal with the case where the potential is not a constant but a function. When the potential is not a constant, we give another proof of the existence and uniqueness of the solution to the BCS gap equation, and show how the solution varies with the temperature. We finally show that the solution to the BCS gap equation is indeed continuous with respect to both the temperature and the energy under a certain condition when the potential is not a constant.


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## Body

## 1. Introduction

We use the unitkB=1, where kB stands for the Boltzmann constant. Let ωD>0 and k∈ℝ3 stand for the Debye frequency and the wave vector of an electron, respectively. Let h>0 be Planck’s constant, and set ℏ=h/(2π). Let m>0 and μ>0 stand for the electron mass and the chemical potential, respectively. We denote by T(≥0) the absolute temperature, and by x the kinetic energy of an electron minus the chemical potential; that is, x=ℏ2|k|2/(2m)-μ. Note that 0<ℏωD≪μ.In the BCS model [1, 2] of superconductivity, the solution to the BCS gap equation (1) is called the gap function. The gap function corresponds to the energy gap between the superconducting ground state and the superconducting first excited state. Accordingly, the value of the gap function (the solution) is nonnegative. We regard the gap function as a function of both T and x and denote it by u; that is, u:(T,x)↦u(T,x)(≥0). The BCS gap equation is the following nonlinear integral equation  (0<ɛ≤x≤ℏωD):
(1)u(T,x)=∫ɛℏωD‍U(x,ξ)u(T,ξ)ξ2+u(T,ξ)2×tanhξ2+u(T,ξ)22Tdξ,
where U(·,·)>0 is the potential multiplied by the density of states per unit energy at the Fermi surface and is a function of x and ξ. In (1) we introduce ɛ>0, which is small enough and fixed (0<ɛ≪ℏωD). In the original BCS model, the integration interval is [0,ℏωD]; it is not [ɛ,ℏωD]. However, we introduce very small ɛ>0 for the following mathematical reasons. In order to show the continuity of the solution to the BCS gap equation with respect to the temperature and in order to show that the transition to a superconducting state is a second-order phase transition, we make the form of the BCS gap equation somewhat easier to handle. So we choose the closed interval [ɛ,ℏωD] as the integration interval in (1).The integral with respect toξ in (1) is sometimes replaced by the integral over ℝ3 with respect to the wave vector k. Odeh [3] and Billard and Fano [4] established the existence and uniqueness of the positive solution to the BCS gap equation in the case T=0. For T≥0, Vansevenant [5] determined the transition temperature (the critical temperature) and showed that there is a unique positive solution to the BCS gap equation. Recently, Frank et al. [6] gave a rigorous analysis of the asymptotic behavior of the transition temperature at weak coupling. Hainzl et al. [7] proved that the existence of a positive solution to the BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator to show the existence of a transition temperature. Moreover, Hainzl and Seiringer [8] derived upper and lower bounds on the transition temperature and the energy gap for the BCS gap equation.Since the existence and uniqueness of the solution were established for each fixedT in the previous literature, the temperature dependence of the solution is not covered. It is well known that studying the temperature dependence of the solution to the BCS gap equation is very important in condensed matter physics. This is because, by dealing with the thermodynamical potential, this study leads to a mathematical proof of the statement that the transition to a superconducting state is a second-order phase transition. So, in condensed matter physics, it is highly desirable to study the temperature dependence of the solution to the BCS gap equation.When the potentialU(·,·) in (1) is a positive constant, the BCS gap equation reduces to the simple gap equation (3). In this case, one assumes in the BCS model that there is a unique nonnegative solution to the simple gap equation (3) and that the solution is of class C2 with respect to the temperature T (see e.g., [1] and [9, (11.45), page 392]). In this paper, applying the implicit function theorem, we first show that this assumption of the BCS model indeed holds true; we show that there is a unique nonnegative solution to the simple gap equation (3) and that the solution is of class C2 with respect to the temperature T. We next deal with the case where the potential is not a constant but a function. In order to show how the solution varies with the temperature, we then give another proof of the existence and uniqueness of the solution to the BCS gap equation (1) when the potential is not a constant. More precisely, we show that the solution belongs to the subset VT (see (12)). Note that the subset VT depends on T. We finally show that the solution to the BCS gap equation (1) is indeed continuous with respect to both T and x when T satisfies (20) when the potential is not a constant.Let(2)U(x,ξ)=U1atall(x,ξ)∈[ɛ,ℏωD]2,
where U1>0 is a constant. Then the gap function depends on the temperature T only. So we denote the gap function by Δ1 in this case; that is, Δ1:T↦Δ1(T). Then (1) leads to the simple gap equation
(3)1=U1∫ɛℏωD‍1ξ2+Δ1(T)2tanhξ2+Δ1(T)22Tdξ.
The following is the definition of the temperature τ1>0.Definition 1 (see [1]).
Consider(4)1=U1∫ɛℏωD‍1ξtanhξ2τ1dξ.


## 2. The Simple Gap Equation

Set(5)Δ=(ℏωD-ɛe1/U1)(ℏωD-ɛe-1/U1)sinh(1/U1).Proposition 2 (see [10, Proposition 2.2]).
LetΔ be as in (5). Then there is a unique nonnegative solution Δ1:[0,τ1]→[0,∞) to the simple gap equation (3) such that the solution Δ1 is continuous and strictly decreasing on the closed interval [0,τ1]:
(6)Δ1(0)=Δ>Δ1(T1)>Δ1(T2)>Δ1(τ1)=0,0<T1<T2<τ1.
Moreover, the solution Δ1 is of class C2 on the interval [0,τ1) and satisfies
(7)Δ1′(0)=Δ1′′(0)=0,limT↑τ1Δ1′(T)=-∞.Proof.
SettingY=Δ1(T)2 turns (3) into
(8)1=U1∫ɛℏωD‍1ξ2+Ytanhξ2+Y2Tdξ.
Note that the right side is a function of the two variables T and Y. We see that there is a unique function T↦Y defined by (8) implicitly, that the function T↦Y is continuous and strictly decreasing on [0,τ1], and that Y=0 at T=τ1. We moreover see that the function T↦Y is of class C2 on the closed interval [0,τ1].Remark 3.
We setΔ1(T)=0 for T>τ1.Remark 4.
In Proposition2, Δ1(T) is nothing but f(T) in [10, Proposition 2.2].We introduce another positive constantU2>0. Let 0<U1<U2. We assume the following condition on U(·,·):
(9)U1≤U(x,ξ)≤U2atall(x,ξ)∈[ɛ,ℏωD]2,U(·,·)∈C([ɛ,ℏωD]2).
When U(x,ξ)=U2 at all (x,ξ)∈[ɛ,ℏωD]2, an argument similar to that in Proposition 2 gives that there is a unique nonnegative solution Δ2:[0,τ2]→[0,∞) to the simple gap equation
(10)1=U2∫ɛℏωD‍1ξ2+Δ2(T)2×tanhξ2+Δ2(T)22Tdξ,0≤T≤τ2.
Here, τ2>0 is defined by
(11)1=U2∫ɛℏωD‍1ξtanhξ2τ2dξ.
We again set Δ2(T)=0 for T>τ2. A straightforward calculation gives the following.Lemma 5 ([11, Lemma 1.5]).
(a) The inequalityτ1<τ2 holds. (b) If0≤T<τ2, then Δ1(T)<Δ2(T). If  T≥τ2, then Δ1(T)=Δ2(T)=0.Note that Proposition2 and Lemma 5 point out how Δ1 and Δ2 depend on the temperature and how Δ1 and Δ2 vary with the temperature; see Figure 1.Figure 1
The graphs of the functionsΔ1 and Δ2.Remark 6.
On the basis of Proposition2, the present author [10, Theorem 2.3] proved that the transition to a superconducting state is a second-order phase transition under the restriction (2).


## 3. The BCS Gap Equation

Let0≤T≤τ2 and fix T, where τ2 is that in (11). We consider the Banach space C[ɛ,ℏωD] consisting of continuous functions of x only and deal with the following subset VT:
(12)VT={u(T,·)∈C[ɛ,ℏωD]:Δ1(T)≤u(T,x)≤Δ2(T)atx∈[ɛ,ℏωD]}.Remark 7.
The subsetVT depends on T. So we denote each element of VT by u(T,·); see Figure 1.As it is mentioned in the introduction, the existence and uniqueness of the solution to the BCS gap equation were established for each fixedT in the previous literature, and the temperature dependence of the solution is not covered. We therefore give another proof of the existence and uniqueness of the solution to the BCS gap equation (1) so as to show how the solution varies with the temperature. More precisely, we show that the solution belongs to VT.Theorem 8 (see [11, Theorem 2.2]).
Assume condition (9) on U(·,·). Let T∈[0,τ2] be fixed. Then there is a unique nonnegative solution u0(T,·)∈VT to the BCS gap equation (1)  (x∈[ɛ,ℏωD]):
(13)u0(T,x)=∫ɛℏωD‍U(x,ξ)u0(T,ξ)ξ2+u0(T,ξ)2×tanhξ2+u0(T,ξ)22Tdξ.
Consequently, the solution is continuous with respect to x and varies with the temperature as follows:
(14)Δ1(T)≤u0(T,x)≤Δ2(T)at(T,x)∈[0,τ2]×[ɛ,ℏωD].Proof.
We define a nonlinear integral operatorA on VT by
(15)Au(T,x)=∫ɛℏωD‍U(x,ξ)u(T,ξ)ξ2+u(T,ξ)2×tanhξ2+u(T,ξ)22Tdξ,
where u(T,·)∈VT. Clearly, VT is a bounded, closed, and convex subset of the Banach space C[ɛ,ℏωD]. A straightforward calculation gives that the operator A:VT→VT is compact. Therefore, the Schauder fixed point theorem applies, and hence the operator A:VT→VT has at least one fixed point u0(T,·)∈VT. Moreover, we can show the uniqueness of the fixed point; see Figure 2.Figure 2
For eachT, the solution u0(T,x) lies between Δ1(T) and Δ2(T).The existence of the transition temperatureTc is pointed out in the previous papers [5–8]. In our case, it is defined as follows.Definition 9.
Letu0(T,·)∈VT be as in Theorem 8. The transition temperature Tc stemming from the BCS gap equation (1) is defined by
(16)Tc=inf{T>0:u0(T,x)=0atallx∈[ɛ,ℏωD]}.Remark 10.
Combining Definition9 with Theorem 8 implies that τ1≤Tc≤τ2. For T>Tc, we set u0(T,x)=0 at all x∈[ɛ,ℏωD].


## 4. Continuity of the Solution with respect to the Temperature

LetU0>0 be a constant satisfying U0<U1<U2. An argument similar to that in Proposition 2 gives that there is a unique nonnegative solution Δ0:[0,τ0]→[0,∞) to the simple gap equation
(17)1=U0∫ɛℏωD‍1ξ2+Δ0(T)2×tanhξ2+Δ0(T)22Tdξ,0≤T≤τ0.
Here, τ0>0 is defined by
(18)1=U0∫ɛℏωD‍1ξtanhξ2τ0dξ.
We set Δ0(T)=0 for T>τ0. A straightforward calculation gives the following.Lemma 11.
(a)τ0<τ1<τ2.
(b) If0≤T<τ0, then 0<Δ0(T)<Δ1(T)<Δ2(T).
(c) Ifτ0≤T<τ1, then 0=Δ0(T)<Δ1(T)<Δ2(T).
(d) Ifτ1≤T<τ2, then 0=Δ0(T)=Δ1(T)<Δ2(T).
(e) Ifτ2≤T, then 0=Δ0(T)=Δ1(T)=Δ2(T).Remark 12.
Let the functionsΔk(k=0,1,2) be as above. For each Δk, there is the inverse Δk-1:[0,Δk(0)]→[0,τk]. Here,
(19)Δk(0)=(ℏωD-ɛe1/Uk)(ℏωD-ɛe-1/Uk)sinh(1/Uk),
and Δ0(0)<Δ1(0)<Δ2(0).We introduce another temperature. LetT1 satisfy 0<T1<Δ0-1(Δ0(0)/2) and
(20)Δ0(0)4Δ2-1(Δ0(T1))tanhΔ0(0)4Δ2-1(Δ0(T1))>12(1+4ℏ2ωD2Δ0(0)2).Remark 13.
Numerically, the temperatureT1 is very small.Consider the following subsetV of the Banach space C([0,T1]×[ɛ,ℏωD]) consisting of continuous functions of both the temperature T and the energy x:
(21)V={u∈C([0,T1]×[ɛ,ℏωD]):Δ1(T)≤u(T,x)=55≤Δ2(T)at(T,x)∈[0,T1]×[ɛ,ℏωD]}.Theorem 14 (see [12, Theorem 1.2]).
Assume (9). Let u0 be as in Theorem 8 and V as in (21). Then u0∈V. Consequently, the gap function u0 is continuous on [0,T1]×[ɛ,ℏωD].Proof.
We define a nonlinear integral operatorB on V by
(22)Bu(T,x)=∫ɛℏωD‍U(x,ξ)u(T,ξ)ξ2+u(T,ξ)2×tanhξ2+u(T,ξ)22Tdξ,
where u∈V.
Clearly,V is a closed subset of the Banach space C([0,T1]×[ɛ,ℏωD]). A straightforward calculation gives that the operator B:V→V is contractive as long as (20) holds true. Therefore, the Banach fixed-point theorem applies, and hence the operator B:V→V has a unique fixed point u0∈V. The solution u0∈V to the BCS gap equation is thus continuous both with respect to the temperature and with respect to the energy x; see Figure 3.Figure 3
The solutionu0 is continuous on [0,T1]×[ɛ,ℏωD].



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