Superconductivity

Abstract

Superconductors are materials that behave as normal metals at high temperatures ($T > T_\mathrm{c}$; however, below $T_\mathrm{c}$ they have the following properties:

• (i) the dc resistivity vanishes.

• (ii) they are perfect diamagnets; by this we mean that any magnetic field that is present in the bulk of the sample when $T > T_\mathrm{c}$ is expelled when T is lowered through the transition temperature. This is called the Meissner effect.

• (iii) the electronic properties can be understood by assuming that an energy gap $2\varDelta$ exists in the electronic spectrum at the Fermi energy.

Some common superconducting elements and their transition temperatures are given in Table 15.1.

Appendices

Problems

15.1

Demonstrate that the electronic contribution to the specific heat of common intrinsic semiconductors shows the exponential temperature behavior at low temperatures.

15.2

Let’s consider the equation of motion of a super fluid electron, which is dissipationless, in an electric field $\mathbf E$ that is momentarily present in the superconductor. That is, $m\frac{dv_\mathrm{S}}{dt} = -eE$, where $v_\mathrm{S}$ is the mean velocity of the super fluid electron caused by the field $\mathbf E$. In order to explain the Meissner effect, London proposed the London equation written as $\nabla \times \mathbf j + \frac{n_\mathrm{S}e^2}{mc}\mathbf B=0.$ Show that $\nabla ^2 \dot{\mathbf j} = \frac{1}{\varLambda _\mathrm{L}^2}\mathbf j$ where $\varLambda _\mathrm{L} = \sqrt{\frac{mc^2}{4\pi n_\mathrm{S}e^2}}$ is the so-called the London penetration depth.

15.3

Assume $c_{\mathbf k\sigma }$ and $c_{\mathbf k^{\prime }\sigma ^{\prime }}^\dag$ satisfy standard Fermion anticommutation relations. Show that $\left[ \alpha _\mathbf k,\alpha _{\mathbf k^\prime }^\dag \right] _+$ and $\left[ \beta _\mathbf k,\beta _{\mathbf k^\prime }^\dag \right] _+$ each equal $\delta _{\mathbf k\mathbf k^\prime }$ for $\alpha _\mathbf k$ and $\beta _\mathbf k^\prime$ defined by the Bogoliubov–Valatin transformation

$$ \begin{array}{ll} \alpha _\mathbf k &{}= u_\mathbf k c_{\mathbf k\uparrow } -v_\mathbf k c_{-\mathbf k\downarrow }^\dag \\ \beta _\mathbf k^\dag &{}= u_\mathbf k c_{-\mathbf k\downarrow }^\dag +v_\mathbf k c_{\mathbf k\uparrow } \end{array} $$

with real c-number coefficients $u_\mathbf k$ and $v_\mathbf k$.

15.4

Let us consider the interaction Hamiltonian $H_1$ given by

$$ H_1=-V\sum _{\mathbf k\mathbf k^\prime }^\prime c_{\mathbf k^\prime \uparrow }^\dag c_{-\mathbf k^\prime \downarrow }^\dag c_{-\mathbf k\downarrow } c_{\mathbf k\uparrow }. $$

Use the Bogoliubov–Valatin transformation to show that $H_1$ can be written as

$$ \begin{array}{lll} H_1&{}=-V\sum _{\mathbf k\mathbf k^\prime } (u_{\mathbf k^\prime } \alpha _{\mathbf k^\prime }^\dag +v_{\mathbf k^\prime } \beta _{\mathbf k^\prime })(u_{\mathbf k^\prime } \beta _{\mathbf k^\prime }^\dag -v_{\mathbf k^\prime } \alpha _{\mathbf k^\prime })(u_\mathbf k \beta _\mathbf k -v_\mathbf k\alpha _\mathbf k^\dag )(u_\mathbf k \alpha _\mathbf k+v_\mathbf k \beta _\mathbf k^\dag )&{} \\ &{}=-V\sum _{\mathbf k\mathbf k^\prime } \bigl [u_{\mathbf k^\prime }v_{\mathbf k^\prime }u_{\mathbf k}v_{\mathbf k} (1-\alpha _{\mathbf k^\prime }^\dag \alpha _{\mathbf k^\prime }-\beta _{\mathbf k^\prime }^\dag \beta _{\mathbf k^\prime })(1-\alpha _{\mathbf k}^\dag \alpha _{\mathbf k}-\beta _{\mathbf k}^\dag \beta _{\mathbf k})&{}\\ &{}\qquad \qquad \qquad +u_{\mathbf k^\prime }v_{\mathbf k^\prime } (1-\alpha _{\mathbf k^\prime }^\dag \alpha _{\mathbf k^\prime }-\beta _{\mathbf k^\prime }^\dag \beta _{\mathbf k^\prime }) (u_\mathbf k^2-v_\mathbf k^2)(\alpha _\mathbf k^\dag \beta _\mathbf k^\dag +\beta _\mathbf k\alpha _\mathbf k)&{}\\ {} &{}\qquad \qquad \qquad + 4\mathrm{th} \text{ order } \text{ off-diagonal } \text{ terms } \bigr ].&{} \end{array} $$

15.5

Consider the condition given by

$$ 2u_\mathbf kv_\mathbf k\tilde{\varepsilon }_\mathbf k =(u_{\mathbf k}^2-v_{\mathbf k}^2)V\sum _{{\mathbf k}^\prime }u_{{\mathbf k}^\prime }v_{{\mathbf k}^\prime }. $$

• (a) Determine $u_{\mathbf k}$ and $v_{\mathbf k}$ satisfying the condition given above. Note that $u_{\mathbf k}^2+v_{\mathbf k}^2=1$.

• (b) Obtain the expression $\varDelta$ defined by $\varDelta = V\sum _{{\mathbf k}^\prime } u_{{\mathbf k}^\prime }v_{{\mathbf k}^\prime }.$

Summary

In this chapter we first briefly review some phenomenological observations of superconductivity and discuss a phenomenological theory by London. Then we introduce ideas of electron–phonon interaction and Cooper pairing to discuss microscopic theory by Bardeen, Cooper, and Schrieffer. The BCS ground state and excited states are discussed through Bogoliubov–Valatin transformation, and condensation energy and thermodynamic behavior of the superconducting energy gap are analyzed. Finally type I and type II superconductors are compared in terms of coherence length and London penetration depth.

The Meissner effect indicates that any magnetic field that is present in a bulk superconductor when $T > T_\mathrm{c}$ is expelled when T is lowered through the transition temperature $T_\mathrm{c}$. In a type I superconductor, the magnetic induction B vanishes in the bulk of the superconductor for $H<H_\mathrm{c}(T)$. In a type II superconductor, the magnetic field starts to penetrate the sample at an applied field $H_\mathrm{c1}$ lower than the $H_\mathrm{c}$. Between $H_\mathrm{c2}$ and $H_\mathrm{c1}$ flux penetrates the superconductor giving a mixed state consisting of superconductor penetrated by threads of the material in its normal state or flux lines. The mixed state consists of vortices each carrying a single flux $\varPhi =\frac{hc}{2e}$.

The London equation is written as $\nabla \times \mathbf j + \frac{n_\mathrm{S}e^2}{mc}\mathbf B=0,$ which implies that, in stationary conditions, a superconductor cannot sustain a magnetic field in its interior, but only within a narrow surface layer: $\nabla ^2\mathbf B =\frac{4\pi n_\mathrm{S}e^2}{mc^2}\mathbf B \equiv \frac{1}{\varLambda _\mathrm{L}^2}\mathbf B.$ Here the quantity $\varLambda _\mathrm{L} = \sqrt{\frac{mc^2}{4\pi n_\mathrm{S}e^2}}$ is called the London penetration depth.

The Hamiltonian of the electrons in a metal is written as

$$ H=\sum _{\mathbf k\sigma }\varepsilon _{\mathbf k\sigma }c_{\mathbf k\sigma }^\dag c_{\mathbf k\sigma } +\sum _{\mathbf k\sigma ,\mathbf k^\prime \sigma ^\prime ,{\mathbf q}}W(\mathbf k,{\mathbf q})c_{\mathbf k+{\mathbf q}\sigma }^\dag c_{\mathbf k^\prime -{\mathbf q}\sigma ^\prime }^\dag c_{\mathbf k^\prime \sigma ^\prime }c_{\mathbf k\sigma }, $$

where $W_{\mathbf k{\mathbf q}}$ is defined by $W_{\mathbf k{\mathbf q}}=\frac{\mid M_{\mathbf q}\mid ^2 \hbar \omega _{\mathbf q}}{\left[ \varepsilon _{\mathbf k+{\mathbf q}}-\varepsilon _{\mathbf k}\right] ^2-\left( \hbar \omega _{\mathbf q}\right) ^2}.$ Here $M_{\mathbf q}$ is the electron–phonon matrix element.

A pair of electrons interacting in the presence of a Fermi sea of ‘spectator electrons’ is described by $H=\sum _{\mathbf {\ell },\sigma }\varepsilon _{\mathbf {\ell }}c_{\mathbf {\ell }\sigma }^\dag c_{\mathbf {\ell }\sigma } -\frac{1}{2}V\sum _{\mathbf {\ell }{\mathbf {\ell }}^\prime \sigma }c_{{\mathbf {\ell }}^\prime \sigma }^\dag c_{-{\mathbf {\ell }}^\prime \bar{\sigma }}^\dag c_{-\mathbf {\ell }\bar{\sigma }} c_{\mathbf {\ell }\sigma },$ where $\varepsilon _{\mathbf {\ell }}=\frac{\hbar ^2\ell ^2}{2m}$ and the strength of the interaction, V, is taken as a constant for a small region of $\mathbf k$-space close to the Fermi surface. A variational trial function $\varPsi =\sum _\mathbf k a_\mathbf k c_{\mathbf k\sigma }^\dag c_{-\mathbf k\bar{\sigma }}^\dag \mid G>$ gives us $\frac{1}{V}=\sum _\mathbf k\frac{1}{2\varepsilon _\mathbf k-E}.$ Here $\mid G\rangle$ is the Fermi sea of spectator electrons, $\mid G\rangle =\prod _{|\mathbf k|,\sigma }^{k<k_\mathrm{F}}c_{\mathbf k\sigma }^\dag c_{-\mathbf k\bar{\sigma }}^\dag \mid 0\rangle$. Approximating the sum by an integral over the energy $\varepsilon$, we have $E\simeq 2E_\mathrm{F} -2\hbar \omega _\mathrm{D} \mathrm e^{-2/gV}.$ The quantity $2\hbar \omega _\mathrm{D} \mathrm e^{-2/gV}$ is the binding energy of the Cooper pair.

In the BCS theory, H is rewritten as $H=H_0 +H_1,$ where

$$ H_0=\sum _\mathbf k\varepsilon _\mathbf k \left( c_{\mathbf k\uparrow }^\dag c_{\mathbf k\uparrow } +c_{-\mathbf k\downarrow }^\dag c_{-\mathbf k\downarrow }\right) \text{ and } H_1=-V\sum _{\mathbf k\mathbf k^\prime }^{\prime } c_{\mathbf k^\prime \uparrow }^\dag c_{-\mathbf k^\prime \downarrow }^\dag c_{-\mathbf k\downarrow } c_{\mathbf k\uparrow }. $$

Introducing $\alpha _{\mathbf k\sigma }^\dag = u_\mathbf k c_{\mathbf k\sigma }^\dag + v_{-\mathbf k} c_{-\mathbf k\bar{\sigma }} \text{ and } \alpha _{\mathbf k\sigma } = u_\mathbf k c_{\mathbf k\sigma } + v_{-\mathbf k} c_{-\mathbf k\bar{\sigma }}^\dag$ the noninteracting Hamiltonian becomes $H_0=E_0+\sum _{\mathbf k,\sigma }\mid \tilde{\varepsilon }_\mathbf k\mid \alpha _{\mathbf k\sigma }^\dag \alpha _{\mathbf k\sigma }.$ The ground state of the noninteracting electron gas (filled Fermi sphere) is given by $\mid \mathrm{GS}\rangle = \prod _{\mathbf k\sigma }\alpha _{\mathbf k\sigma }\alpha _{-\mathbf k\bar{\sigma }}\mid \mathrm{VAC}\rangle ,$ where $\mid \mathrm{VAC}\rangle$ is the true vacuum state.

The Bogoliubov–Valatin transformation defined by

$$ \begin{array}{ll} \alpha _\mathbf k &{}= u_\mathbf k c_{\mathbf k\uparrow } -v_\mathbf k c_{-\mathbf k\downarrow }^\dag \text{; } \beta _\mathbf k^\dag = u_\mathbf k c_{-\mathbf k\downarrow }^\dag +v_\mathbf k c_{\mathbf k\uparrow }\\ \alpha _{\mathbf k}^\dag &{}= u_\mathbf k c_{\mathbf k\uparrow }^\dag - v_{\mathbf k} c_{-\mathbf k\downarrow } \text{; } \beta _\mathbf k = u_\mathbf k c_{-\mathbf k\downarrow } +v_\mathbf k c_{\mathbf k\uparrow }^\dag \end{array} $$

gives

$$ H=H(0) + H(2) + H(4), $$

where

$$ H(0)=2\sum _{\mathbf k} \tilde{\varepsilon }_\mathbf k v_\mathbf k^2 -V\sum _{\mathbf k\mathbf k^\prime } u_\mathbf k v_\mathbf k u_{\mathbf k^\prime } v_{\mathbf k^\prime } \text{; } H(2) =\sum _{\mathbf k} E_{\mathbf k}(\alpha _{\mathbf k}^\dag \alpha _{\mathbf k}+\beta _{\mathbf k}^\dag \beta _{\mathbf k}). $$

Here $E_{\mathbf k}= \tilde{\varepsilon }_{\mathbf k} (u_{\mathbf k}^2-v_{\mathbf k}^2)+2\varDelta u_{\mathbf k}v_{\mathbf k}$ and H(4) contains interactions between the elementary excitations. The equation for the energy gap $\varDelta$ is given by

$$ 1 = \frac{V}{2}\sum _{\mathbf k} \frac{1}{\sqrt{\tilde{\varepsilon }_{\mathbf k}^2+\varDelta ^2}} \text{ and } \varDelta =2\hbar \omega _q\mathrm e^{-\frac{2}{g(E_\mathrm{F})V}}. $$

The ground state wave function $\varPsi _0$ of the superconducting system is

$$ \mid \varPsi _0\rangle =\prod _{{\mathbf k}}(u_{\mathbf k}+v_{\mathbf k} c_{\mathbf k\uparrow }^\dag c_{-{\mathbf k}\downarrow }^\dag )\mid \mathrm VAC\rangle . $$

The energy of a quasiparticle is $E_{\mathbf k}= \sqrt{\tilde{\varepsilon }_{\mathbf k}^2+\varDelta ^2},$ where $\tilde{\varepsilon }_{\mathbf k}=\frac{\hbar ^2 {\mathbf k}^2}{2m}-E_\mathrm{F}$. The density of quasiparticle states in the superconductor is given by

$$ g_\mathrm{S}(E) = \frac{m k_\mathrm{F}}{\pi ^2\hbar ^2}\frac{E}{\sqrt{E^2-\varDelta ^2}}. $$

The type I and type II superconductors are distinguished by whether the ratio of the coherence length $\xi _0$ to the London penetration depth $\varLambda _\mathrm{L}$ is large or small compared to unity. The local relation $\mathbf j (\mathbf r) =- \frac{n_\mathrm{S}e^2}{mc}\mathbf A(\mathbf r)$ is valid only for type II materials where $\varLambda _\mathrm{L} \gg \xi _0$ and $\mathbf A(\mathbf r)$ varies slowly on the scale of $\xi _0$. For the case $\varLambda _\mathrm{eff} \ll \xi _0$, the vector potential $\mathbf A(\mathbf r)$ is finite only in a surface layer and we have

$$ \mathbf j (\mathbf r) =- \frac{n_\mathrm{S}e^2}{mc}\frac{\varLambda _\mathrm{eff}}{\xi _0}\mathbf A(\mathbf r) $$

leading to $\varLambda _\mathrm{eff} \approx \varLambda _\mathrm{L} \left( \frac{\xi _0}{\varLambda _\mathrm{L}}\right) ^{1/3}.$ The intermediate state is a mixture of normal and superconducting regions due to geometric factors and it has a domain structure. In the extreme dirty limit of $\xi _0 \gg l$, we have

$$ \mathbf j(\mathbf r) = -\frac{n_\mathrm{S}e^2}{mc}\frac{l}{\xi _0}\mathbf A(\mathbf r), $$

and the effective penetration depth $\varLambda _\mathrm{eff} = \varLambda _\mathrm{L} \sqrt{\xi _0/l}$ is increased greatly.