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Landau Theory of Phase Transitions As a reminder of Landau theory, take the example of a ferromagnetic to paramagnetic transition where the free energy is expressed as F(T,M) F(T,0) a(T TCM )M2 bM4 c M 2 M is the magnetisation - the so-called order parameter of the magnetised ferromagnetic state and M gradM is associated with variations in magnetisation (or applied field) F(T,M) The stable state is found at the minimum of the free energy, ie when T>TCM F(T,M) 0 M We find M=0 for T>TCM M0 for T<TCM Any second order transition can be described in the same way, replacing M with an order parameter that goes to zero as T approaches TC Lecture 5 T=TCM M T<TCM Superconductivity and Superfluidity The Superconducting Order Parameter We have already suggested that superconductivity is carried by superelectrons of density ns ns could thus be the “order parameter” as it goes to zero at Tc However, Ginzburg and Landau chose a quantum mechanical approach, using a wave function to describe the superelectrons, ie (r ) (r ) ei(r ) This complex scalar is the Ginzburg-Landau order parameter (i) its modulus * is roughly interpreted as the number density of superelectrons at point r (ii) The phase factor (r ) is related to the supercurrent that flows through the material below Tc (iii) 0 in the superconducting state, but 0 above Tc Lecture 5 Superconductivity and Superfluidity Free energy of a superconductor The free energy of a superconductor in the absence of a magnetic field and spatial variations of ns can be 4 2 written as Fs Fn 2 and are parameters to be determined,and it is assumed that is positive irrespective of T and that = a(T-Tc) as in Landau theory Fs-Fn >0 2 Assuming that ns the equilibrium value of the order parameter is obtained from (Fs Fn ) 2 0 ns Fs-Fn <0 we find: 2 for >0 minimum must be when 0 2 for <0 minimum is when where is defined as in the interior of the sample, far from any gradients in 2 Lecture 5 Superconductivity and Superfluidity Free energy of a superconductor In the superconducting state we have 4 2 Fs Fn 2 with 2 2 Fs-Fn <0 changes sign at Tc and is always positive for a second order transition also at equilibrium 2 Fs Fn 2 1 2 H o c 2 But we have already shown that 1 Fs Fn oHc2 2 2 2 so oHc We will use this later Lecture 6 Superconductivity and Superfluidity The full G-L free energy If we now take the full expression for the Ginzburg-Landau free energy at a point r in the presence of magnetic fields and spatial gradients we have: 4 Fs Fn 2 1 oH2 (r ) 2 2 1 i e * A 2 2m * the term we have already discussed the magnetic energy associated with the magnetisation in a local field H(r) A kinetic energy term associated with the fact that is not uniform in space, but has a gradient e* and m* are the charge and mass of the superelectrons and A is the vector potential We should look at the origin of the kinetic energy term in more detail. Lecture 6 Superconductivity and Superfluidity A charged particle in a field Consider a particle of charge e* and mass m* moving in a field free region with velocity v1 when a magnetic field is switched on at time t=0 The field can only increase at a finite rate, and while it builds up there is an induced electric field which satisfies Maxwell’s equations, ie curlE B If A is the magnetic vector potential (B=curl A) then d (curlA ) dt Integration with respect to spatial coordinates gives curlE E dA dt So the momentum at time t is t m * v 2 m * v1 e * E dt m * v1 e * 0 and m * v 2 m * v1 e * A Lecture 6 or t dA dt 0 dt m * v1 m * v 2 e * A Superconductivity and Superfluidity A charged particle in a field If m * v 2 m * v1 e * A and m * v1 m * v 2 e * A the vector p m * v e * A must be conserved during the application of a magnetic field The kinetic energy, , depends only upon m*v so if = f(m*v) before the field is applied we must write = f(p-e*A) after the field is applied Quantum mechanically we can replace p by the momentum operator -iħ So the final energy in the presence of a field is: 1 i e * A 2 2m * Lecture 6 Superconductivity and Superfluidity Back to G-L Free Energy - 1st GL Equation Remember that the total free energy is 2 Fs Fn 4 1 1 i e * A 2 oH2 (r ) 2 2 2m * This free energy, Fs((r), A(r)), must now be minimised with respect to the order parameter, (r) , and also with respect to the vector potential A(r) To do this we must use the Euler-Lagrange equations: 1 F j F 0 x j ( j ) 2 1 Is easy to evaluate - we only need 2 Lecture 6 F A j F 0 x j (A j x j ) F 0 ie 1 i e * A 2 0 2m * This is the First G-L equation Superconductivity and Superfluidity The second G-L equation Evaluation of the second derivative in 2 F A j F 0 x j (A j x j ) 1 curl curl A gives o Remember that B=curl A, and that curl B = oJ Therefore 2 where J is the current density gives J e* 2 (i e * A ) m* This is the Second G-L equation This is the same quantum mechanical expression for a current of particles described by a wavefunction Lecture 6 Superconductivity and Superfluidity Magnetic penetration within G-L Theory e* 2 Taking the second GL equation: J (i e * A ) m* spatial variations of : and neglecting e *2 2 J A m* So, if curl B = oJ and using ns 2 o e * 2 n * s curl(o J) curlcurlB curlA m* and as curlcurlB grad divB 2B 2B This gives and finally with Lecture 6 o e * 2 n * s B B m* 2 B *2 2B 0 m* *2 oe * 2 n * s Compare these equations directly with the London equations Superconductivity and Superfluidity A comparison of GL and London theory We will now pre-empt a result we shall derive later in the course and recognise that superconductivity is related to the pairing of electrons. (This was not known at the time of Ginzburg and Landau’s theory) If electrons are paired in the superconducting state then: m* = 2me e* = 2e n*s = ns/2 and hence Lecture 6 *2 2m m 2 L o 4e2ns / 2 oe2ns Superconductivity and Superfluidity The coherence length We shall now look at how the concept of the coherence length arises in the G-L Theory Taking the 1st G-L equation in 1d without a magnetic field, ie: 1 2 i e * A 2 0 2m * 2 d2 2 Eq 1 0 becomes 2 2m * dx Earlier we showed that the square of the order parameter can be written 2 with <0 However we believe that the order parameter can vary slowly with distance, so we shall now change variables and use instead a normalised order parameter f 1 2 where f varies with distance Lecture 6 Superconductivity and Superfluidity The coherence length Substituting the normalised order parameter f into equation 1 on the previous slide, and noting that 2 , we obtain a “non linear Schrodinger equation” 1 1 1 2 2 2 2 2 df 3 “-” disappears as <0 f f 0 2 2m * dx and we introduce | | 2 d2f 3 f f 0 2 2m * dx 2 d2f 3 hence f f 0 Canceling gives 2 2m * dx 2 2 Making the substitutions f=1+ f´ where f´ is small and negative, and 2m * 2 d f' we have 2 2 1 f '(1 3f '.....) 0 dx d2 f ' 2f ' 2 hence 2 dx the solution of which is f ' ( x ) exp( x 2 ) is therefore the coherence length, characteristic distance over which the order parameter varies Lecture 6 Superconductivity and Superfluidity Relationship between Bc, * and To summarise we have 2 2m * 2 m* ns o e * 2 * 2 2 1 Solving for and Finally, using 1 o2Hc2e *2 *2 m* from 3oHc2e * 4 * 4 m *2 by substituting in 2 2 and 2 3 3 we have H o c 2 1 * cons tan t 2e* 2 2 or 2 and oHc2 2 Bc22 *2 cons tan t So, although Bc, * and are all temperature dependent, their product is not although experimentally it is found to be not quite independent of T Lecture 6 Superconductivity and Superfluidity