Ginzburg-landau Theory

$F\; =\; F\_n\; +\; \backslash alpha\; |\backslash psi|^2\; +\; \backslash frac\{\backslash beta\}\{2\}\; |\backslash psi|^4\; +\; \backslash frac\{1\}\{2m\}\; \backslash left|\; \backslash left(-i\backslash hbar\backslash nabla\; -\; 2e\backslash mathbf\{A\}\; \backslash right)\; \backslash psi\; \backslash right|^2\; +\; \backslash frac$>

/dd> where Fn is the free energy in the normal phase, α and β are phenomenological parameters, A is the electromagnetic vector potential, and H is the magnetic field. By minimizing the free energy with respect to fluctuations in the order parameter and the vector potential, one arrives at the Ginzburg-Landau equations $\backslash alpha\; \backslash psi\; +\; \backslash beta\; |\backslash psi|^2\; \backslash psi\; +\; \backslash frac\{1\}\{2m\}\; \backslash left(-i\backslash hbar\backslash nabla\; -\; 2e\backslash mathbf\{A\}\; \backslash right)^2\; \backslash psi\; =\; 0$ $\backslash mathbf\{J\}\; =\; \backslash frac\{2e\}\{m\}\backslash left(\; \backslash psi^*\; \backslash left(-i\backslash hbar\backslash nabla\; -\; 2e\; \backslash mathbf\{A\}\; \backslash right)\; \backslash psi\; \backslash right)$ where J denotes the electrical current. The first equation, which bears interesting similarities to the time-independent Schrdinger equation, determines the order parameter ψ based on the applied magnetic field. The second equation then provides the superconducting current. The Ginzburg-Landau equations produce many interesting and valid results. Perhaps the most important of these is its prediction of the existence of two characteristic lengths in a superconductor. The first is a coherence length ξ, given by $\backslash xi\; =\; \backslash sqrt\{\backslash frac\{\backslash hbar^2\}\{2\; m\; |\backslash alpha$

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$\backslash lambda\; =\; \backslash sqrt\{\backslash frac\{m\}\{4\; \backslash mu\_0\; e^2\; \backslash psi\_0^2\}\}$

Selected papers

• V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950)
• A.A. Ginzburg, Zh. Eksp. Teor. Fiz. 32, 1442 (1957)
• L.P. Gor'kov, Zh. Eksp. Teor. Fiz. 36, 1364 (1959)

Books

• D. Saint-James, G. Sarma and E. J. Thomas, Type II Superconductivity Pergamon (Oxford 1969)

• M. Tinkham, Introduction to Superconductivity, McGraw-Hill (New York 1996)

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