2.1.1 The London Penetration Depth

Let us first consider the magnetic penetration depth in the context of the conventional London theory [7] at $T \! = \! 0$. If a magnetic field is applied to a superconductor which is initially in zero field, the magnetic field is a function of time. According to the Maxwell equation ${\bf \nabla} \! \times \! {\bf E} \! = \! \frac{-1}{c} \partial {\bf B}/ \partial t$,the time-varying magnetic field gives rise to an electric field. In a normal metal this will induce eddy currents, but in a superconductor the ${\bf E}$-field will give rise to persistent currents (i.e. supercurrents). The induced supercurrents will in turn generate a magnetic field of their own which opposes the applied magnetic field. If the applied magnetic field is weak, the flux is totally screened from the bulk of the superconductor. This phenomenon is often described as ``perfect diamagnetism''. From Newton's law, the equation of motion for a superconducting carrier with mass m and charge -e in the presence of an electric field ${\bf E}$ is  

$${\bf F} = m \frac{d{\bf v_s}}{dt} = -e {\bf E} \, ,$$ (6)

where ${\bf v_s}$ is the velocity of the superconducting carrier. The field-induced supercurrent density is given by  

$${\bf J_s} = -e n_s {\bf v_s} \, ,$$ (7)

where n_s is the local density of superconducting carriers. Substituting Eq. (2.7) into Eq. (2.6) gives  

$$\frac{d {\bf J_s}}{dt} = \frac{n_s e^2}{m} {\bf E} \, ,$$ (8)

which is known as the ``first London equation''. Taking the curl of both sides of Eq. (2.8) gives  

$$\frac{m}{n_s e^2} \left( {\bf \nabla} \times \frac{d {\bf J_s}}{dt} \right) = {\bf \nabla} \times {\bf E} \, ,$$ (9)

which can be rewritten using the Maxwell equation ${\bf \nabla} \! \times \! {\bf E} \! = \! \frac{-1}{c} d{\bf B}/dt$ to give  

$$\frac{m c}{n_s e^2} \left( {\bf \nabla} \times \frac{d {\bf J_s}}{dt} \right) + \frac{d {\bf B}}{dt} = 0 \, .$$ (10)

In order to obtain the Meissner effect (i.e. ${\bf B} \! = \! 0$ in the bulk of the superconductor) the London brothers removed the time derivative in Eq. (2.10) and postulated the new equation  

$$\frac{m c}{n_s e^2} \left( {\bf \nabla} \times {\bf J_s} \right) + {\bf B} = 0 \, .$$ (11)

Equation (2.11) is commonly referred to as the ``second London equation''. Since the supercurrent density is related to the field ${\bf B}$ by another Maxwell equation  

$${\bf J_s} = \frac{c}{4 \pi} ( {\bf \nabla} \times {\bf B}) \, ,$$ (12)

substitution of Eq. (2.12) into Eq. (2.11) gives  

$$\lambda_L^2 ( {\bf \nabla} \times {\bf \nabla} \times {\bf B} ) + {\bf B} = 0 \, ,$$ (13)

where  

$$\frac{1}{\lambda_L^2} = \frac{4 \pi n_s e^2}{m c^2} \, .$$ (14)

The variable $\lambda_L$ is called the London penetration depth. Note that $\lambda_L^{-2}$ is proportional to the superfluid density n_s.

The relationship between $\lambda_L$ and ${\bf B}$ is best realized by a simple example. Consider the vacuum-superconductor interface illustrated in Fig. 2.1. If a field ${\bf B}(0)$ is applied parallel to the surface of the superconductor, the solution of Eq. (2.13) is  

$${\bf B}(x) = {\bf B}(0) \exp (-x/\lambda_L) \,.$$ (15)

Thus, both the magnetic field B(x) and the supercurrent density J_s(x) decay exponentially with distance inside the superconductor over the length scale $\lambda_L$.

 

Equation (2.14) was derived for $T \! = \! 0$.At nonzero temperature the behaviour of $\lambda_L$ can be approximated by incorporating the two-fluid model of Gorter and Casimir [8]. In this model the electron system is assumed to contain a superconducting component with electron density n_s, and a normal component with an electron density n_n. The total electron density $n \! = \! n_s + n_n$ becomes $n \! = \! n_s$ at $T \! = \! 0$, and $n \! = \! n_n$ for $T \! \gt \! T_c$, where T_c is the superconducting transition temperature. For arbitrary temperature, Gorter and Casimir found that good agreement with early experiments could be obtained if one assumes that $n_s (T) \! = \! n[1-(T/T_c)^4]$.When this expression is combined with Eq. (2.14) the penetration depth is given by

$$\lambda_L(T) = \frac{\lambda_L(0)}{\left[ 1-(T/T_c)^4 \right]^{1/2}} \, .$$ (16)

As the temperature T increases, n_s is reduced and the magnetic penetration depth increases with T, such that $\lambda_L \! \rightarrow \! \infty$ as $T \! \rightarrow \! T_c$.Although this relation gives a good first-order description of many of the experimental results for conventional superconductors, the temperature dependence is not a prediction from the two-fluid model. Better agreement is obtained with the behaviour predicted from the microscopic theory developed by Bardeen, Cooper and Schrieffer (BCS) [9].

Before considering the BCS theory, let us work backwards to determine the kernel $Q ({\bf k})$ in Eq. (2.2) for the London theory. The second London equation [i.e. Eq. (2.13)] can be rewritten using the Maxwell relation in Eq. (2.12) and the relation ${\bf B} \! = \! {\bf \nabla} \times {\bf A}$ to give

$$\frac{4 \pi \lambda_L^2}{c} ( {\bf \nabla} \times {\bf J_s}) + {\bf \nabla} \times {\bf A} = 0 \, ,$$ (17)

which is easily solved for ${\bf J_s}$ to give  

$${\bf J_s}({\bf r}) = - \frac{c}{4 \pi \lambda_L^2} {\bf A}({\bf r}) \, .$$ (18)

The ${\bf k}$th Fourier component of ${\bf J_s}$ is thus  

$${\bf J_s}({\bf k}) = - \frac{c}{4 \pi \lambda_L^2} {\bf A}({\bf k}) \, .$$ (19)

Comparing Eq. (2.19) with Eq. (2.2) gives the following simple result for the London kernel

$$Q_L ({\bf k}) = \frac{1}{\lambda_L^2} \, .$$ (20)

Clearly, Q_L is independent of ${\bf k}$. This implies that the London theory is a local theory, where the value of ${\bf J_s}$ at a point ${\bf r}$only depends on the value of ${\bf A}$ at that same point ${\bf r}$.Generally, the value of ${\bf A}$ in the vicinity of the point where ${\bf J_s}$ is being measured is also important. In a more general nonlocal theory, $Q ({\bf k})$ depends on ${\bf k}$.

The current density due to the normal electrons is

$${\bf J_n} = \sigma_n {\bf E} = -e n_n {\bf v_n} \, ,$$ (21)

where $\sigma_n$ is the finite conductivity of the normal fluid. Thus, the total current density at arbitrary temperature below T_c is simply