1 The life is not so simple One can imagine also collective excitations which

1 the life is not so simple one can imagine also

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1 The life is not so simple. One can imagine also collective excitations which may have no gap. As has been shown, it is not the case because of the finite electron charge.
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360 CHAPTER 18. MICROSCOPIC THEORY 18.4 Temperature Dependence of the Energy Gap Now we are prepared to discuss the temperature dependence of the gap. From Eq. (18.10) we get 1 = λ 2 Z ( dk ) 1 - 2 n 0 ( ε k ) ε k = λg ( F ) 2 Z ~ ω D 0 tanh( p ξ 2 + ∆ 2 / 2 k B T ) p ξ 2 + ∆ 2 . (18.13) At T 0 tanh( p ξ 2 + ∆ 2 / 2 k B T ) 1 and 1 = λg ( F ) 2 ln 2 ~ ω D ∆(0) ∆(0) = 2 ~ ω D exp ( - 2 /λg ( F )) (18.14) (note the extra factor 2). At T T c 0, and 1 = λg ( F ) 2 Z ~ ω D 0 tanh( ξ/ 2 k B T c ) ξ = λg ( F ) 2 ln 2 ~ ω D γ πk B T c , γ = 1 . 78 . Thus k B T c = 2 ~ ω D γ π exp ( - 2 /λg ( F )) (18.15) and ∆(0) = π γ k B T c = 1 . 76 k B T c . (18.16) To discuss the temperature dependence one can simply calculate the integral in (18.13). It is interesting to show the analytical formula, so we analyze briefly this integral. At low temperatures we can rewrite the formula as ln ∆(0) = Z 0 1 - tanh( p ξ 2 + ∆ 2 / 2 k B T ) p ξ 2 + ∆ 2 = 2 f k B T where f ( x ) = Z 1 dy (1 + e yx ) p y 2 - 1 , y = p ξ 2 + ∆ 2 . Here we have used the expression (18.14) and expanded the integration region to infinity (the important region is ∆). We are interested in large values of x, f ( x ) = Z 1 dy X n =1 ( - 1) n +1 e - nyx p y 2 - 1 . Then we use the integral representation for the McDonald function K ν ( z ) = Γ(1 / 2) Γ( ν + 1 / 2) z 2 ν Z 1 e - yz ( y 2 - 1) v - 1 / 2 dy
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18.4. TEMPERATURE DEPENDENCE ... 361 and get f ( x ) = X n =1 ( - 1) n +1 K 0 ( nx ) r π 2 x e - x . Thus, at T T c ∆( T ) = ∆(0) - p 2 π ∆(0) k B T e - ∆(0) /k B T | {z } . number of quasiparticles (18.17) At T T c it is convenient to expand over ∆ 0 . To do this we also divide the expression (18.13) by ( λg ( F ) / 2) and subtract its limit at ∆ = 0 . We get ln T c T = Z 0 " tanh ξ ξ - tanh( p ξ 2 + ∆ 2 / 2 k B T ) p ξ 2 + ∆ 2 # . Then we use the formula tanh πx 2 = 4 x π X k =0 1 (2 k + 1) 2 + x 2 . Substituting this formula into the previous one, expanding over ∆ and integrating over ξ we get ln T c T = 2 X n =1 ( - 1) n +1 (2 n - 1)!! (2 n )!! πk B T 2 n X k =0 1 (2 k + 1) 2 n +1 . The first item leads to 3 . 06 r T c T c - T . The graph of the dependence ∆( T ) is given in Fig. 18.3. Figure 18.3: Temperature dependence of the energy gap.
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362 CHAPTER 18. MICROSCOPIC THEORY 18.5 Thermodynamics of a Superconductor According to statistical physics, the thermodynamic potential Ω depending on the variables T, V , and chemical potential is Ω = - k B T ln Z, with the partition function Z = X s,N exp - E sN - N k B T = Tr exp - H- N k B T . Splitting the Hamiltonian as H 0 + H int and differentiating with respect to the interaction constant λ we get ∂λ = 1 λ Tr h H int exp - H- N k B T i Tr h exp - H- N k B T i = 1 λ hH int i . Substituting the expression for the interaction Hamiltonian 2 H int = - λ X k 1 + k 2 = k 0 1 + k 0 2 k 1 6 = k 0 1 a k 0 1 a k 0 2 a k 2 a k 1 and expressing the electron operators through the quasiparticle ones as a k = u k α k + v k α - k , a k = u k α k - v k α - k we get ( Problem 18.2 ) ∂λ = - 2 λ 2 . (18.18)
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