 # 1 A. Derivation of GL equations macroscopic magnetic field Several standard definitions: -Field of “external” currents -magnetization -free energy II.

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1 A. Derivation of GL equations macroscopic magnetic field Several standard definitions: -Field of “external” currents -magnetization -free energy II. TYPE I vs TYPE II SUPERCONDUCTIVITY 1.Macroscopic magnetostatics

2 In equilibrium under fixed external magnetic field the relevant thermodynamic quantity is the Gibbs energy: Inserting the GL free energy: one obtains:

3 2. Derivation of the LG equations By variation with respect to order parameter  one obtains the nonlinear Schrodinger equation and by variation with respect to vector potential A - the supercurrent equation out of five equations only four are independent (local gauge invariance)

4 while the magnetization is parallel to it: (**) (*) The equations should be supplemented by the boundary conditions. The covariant gradient is perpendicular to the surface Note that the external magnetic field enters boundary conditions only – magnetic field is a “topological charge”.

5 Details of the derivation of the set of GL equations and boundary conditions We have to vary with respect to five independent fields: and Two components of the complex order parameter field are varied independently. One of them is:

6 Integration by parts of the first term gives (*)

7 If the full variation  G is to vanish, one has to require that both the nonlinear Shroedinger eq. and the boundary condition (*) are satisfied. Variation with respect to  (x) just gives the corresponding complex conjugate equation.

8 The variation and supercurrent

9 This defines supercurrent The covariant derivative representation makes its gauge invariance obvious. The variation of is identical to that used in derivation of Maxwell equations.

10 This leads to the supercurrent equation and the boundary condition (**).

11 The boundary condition (*) Supercurrent therefore cannot leave the superconductor through the boundary and therefore circles inside the sample. after multiplication by the order parameter field leads to

12 The degenerate minima are at B. Homogeneous and slightly perturbed SC solution with free energy density 1. Zero magnetic field. Homogeneous solutions.

13 In addition to the degenerate SC solution (global minima) there exists a nondegeneratetrivial normal solution (a local maximum): The condensation energy The free energy density difference between the normal and the superconducting ground states (the condensation energy) is: where Hc is defined as the “thermodynamic critical field”. As will be clear later at this field nothing special happens in type II superconductor.

14 Assume that variations are along the x direction only and the magnetic field contributions are small : 2. A small inhomogeneity near the SC state Is real Deviations of the order parameter Defining the normalized order parameter

15 one is left with a single scale the coherence length one linearizes the (anharmonic oscillator type) equation with For small deviations of from

16 Deviations of from decay exponentially on the scale of correlation length This corresponds to the “harmonic approximation”.

17 This is the Londons’ equation, valid beyond GL theory, everywhere close to deep inside the superconductor. Taking a curl, one obtains: The magnetic field penetration profile In the supercurrent equation the magnetic field cannot be neglected. However in this case one can neglect setting

18 The solution of the linear Londons’ eq. is also exponential: The magnetic field decays exponentially inside superconductor on the scale of magnetic penetration depth. The relevant scale here is the penetration depth

19 In unitary gauge (and absence of topological charge=flux) order parameter Y can be made real Anderson – Higgs mechanism

20 In harmonic approximation one expands to second order around the SC state and obtains (up to a constant) following quadratic terms (linear terms generally vanish due to eq. of motion or GL eqs.):

21 In the normal phase one has three massless excitation fields: two transverse polarizations of photon (use, for example the Colomb gauge In the SC phase the situation changes dramatically: due to “mixing” all the excitations become massive. In the unitary gauge this is seen as a three component massive vector field A. and the phase of order parameter The situation is sometimes termed “spontaneous gauge symmetry breaking”.

22 Of course when deviations are not small like in the SC-N junction one has to consider both the order parameter and the magnetic field simultaneously and go beyond the perturbation theory. Type I small    interface > 0  N SC Type II large    interface < 0  N SC Beyond perturbation theory

23 C. The SC-normal domain wall surface energy. 1.Extreme type II case: the energy gain due to magnetic field penetration into SC In the SC region but Assume first normal superconductor Gn Gs

24 The energy gain is therefore: On the SC side assume that one still can use the Londons asymptotics with : The Gibbs free energy density in the N part (assuming ) is the same as is homogeneous SC Highly unusual!

25 In the junction region but The energy loss of the condensation energy naively is: In the opposite case 2. Extreme type I limit: the energy loss due to order parameter depression near N Less naively one solves the anharmonic oscillator type equation exactly: Gn Gssuper normal

26 Multiplying the eq. by and integrate over x with boundary conditions one obtains Details of solution

27 The energy per unit area

28 Therefore in type I SC the behavior is as expected: one has to pay energy in order to create interfaces. For the domain wall energy changes sign. Type II SC unlike any other material, likes to create domain walls. Summing up naively the two contribution we obtain the interface energy

29 3. General case Set of GL equations (the solution  (x) is real) is A convenient choice of gauge for the 1D problem: or using dimensionless functions and

30 The boundary conditions still are: Using  as a unit of length this becomes

31 A simplified expression for the domain wall energy Nonlinear Schrodinger equation simplifies the expression: Exercise 1: solve the GL equations for S-N numerically using the shooting method for 

32 4. For what  the interface energy vanishes? Obviously  in (*) vanishes if the integrand vanishes It turns out that for the exact solution (which is not known analytically) obeys it! For this particular value of  the GL equations (with x in units of takes a form:

33 Substituting the zero interface requirement (*) into the second eq.(2) one gets: Differentiating it and using (*) again one gets eq.(1): The value therefore separates between type I and type II.

34 Summary 1. Order parameter changes on the scale of coherence length , while magnetic field on the scale of the penetration depth. The only dimensionless quantity is  2. The interface energy between the normal and the superconducting phases in type II SC is negative. This leads to energetic stability of an inhomogeneous configuration. 3. The critical value of the Ginzburg parameter  at which a SC becomes type II is

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