Ginzburg-Landau theory of supercondutivity

Transcription

1 Ginzburg-Landau theory of supercondutivity

2 Ginzburg-Landau theory of superconductivity Let us apply the above to superconductivity. Our starting point is the free energy functional Z F[Ψ] = d d x [F(Ψ) + 12 ] K( Ψ)2 In addition, the system is charged - there are electrons - and thus we must transform the free energy functional into a gauge invariant form. In practice: 1. Include the vector potential A 2. replace the momentum term a gauge invariant momentum. Let s put all that together: F [ ] Ψ, A = Z { h 2 ( d x 2m e i 2e ) hc A Ψ 2 + α Ψ 2 + β 2 Ψ 4 } + (8π) 1 Z d x( A) 2

3 ...but shouldn t we have particles Propagation of an electron polarizes (i.e. deforms) the lattice. Ions are positive and hence the deformation creates an attractive potential for the other electron.!!!!!

4 Ginzburg-Landau: Order parameter In analogy to the definition of the wave function in quantum mechanics, the superconducting state is characterized by a complex valued orderparameter Ψ Ψ( x,t) representing the underlying quantum mechanical coherence of the system. The order-parameter can be related to the density of the superconducting electrons via n s Ψ( x,t) 2 It should be noted that Ψ Ψ( x,t) is not the wavefunction of the whole system, but can be considered as an effective wavefunction for the superconducting condensate. Important: This is a description for the whole condensate, not for individual atoms!

5 Ginzburg-Landau theory of superconductivity } Let s take a look at the different terms F [ ] Ψ, A = Z { h 2 ( d x 2m e i 2e ) hc A electron mass gauge invariant kinetic energy Ψ 2 due to Cooper pairs absolute value since the order parameter is complex valued + α Ψ 2 + β 2 Ψ 4 } Landau free energy + (8π) 1 Z d x( A) 2 } magnetic field B = A The transition to the superconducting state is characterized by a change of sign in α(t ) such that α(t ) < 0for the superconducting state. The other expansion coefficient is assumed to be temperature independent. As all the microscopics is buried in the coefficients, it is the overall symmetry properties that are most important. Analogy: Universality classes in equilibrium critical phenomena.

6 Ginzburg-Landau theory of superconductivity The so-called GL equations are obtained from the free energy functional in by requiring that variations δf/δψ and vanish. These are the stationarity conditions for the free energy. The latter condition is due to the fact that the electromagnetic fields interact with the superconducting condensate. A straightforward calculation yields [ h2 i 2e ] 2 2m e hc A Ψ + αψ + βψ Ψ 2 = 0 ( A) = 4π c J δf/δ A with J = e h im e (Ψ Ψ Ψ Ψ ) 4e2 m e c A Ψ 2 The GL equations provide the two characteristic length scales of superconductors, i.e., the magnetic penetration length and the coherence length.

7 Ginzburg-Landau theory of superconductivity The GL equations are nonlinear, we can only solve them in certain special cases. 1. Ψ = 0 the high temperature solution (normal state). No information about length scales. Let s look for a nontrivial solution. 2. Assume that electromagnetic fields are absent and fix the gauge to A = 0. With these assumptions and conventions! is real. Look for a solution in the case of a semi-infinite superconductor filling the area x>0. Boundary conditions Ψ(0) = 0 and Ψ( ) = 1. Using the 1st GL equation gives Ψ(x) = n s tanh(x/ξ) where n s = α/β and ξ 2 (T ) = h2 2m e α temperature dependent measure of the variations of the order parameter & measure of the thickness of the surface layer in a superconductor.

8 Ginzburg-Landau theory of superconductivity The second GL equation provides a measure for the range of variations of the magnetic field. Using ( A) = 4π c J and and taking a curl of both sides gives B(x) = B(0) exp( x/λ) J = e h im e (Ψ Ψ Ψ Ψ ) 4e2 m e c A Ψ 2 with λ(t ) = [ me c 2 16πe 2 Ψ 2 0 ] 1/2 determines the range of the variation for the magnetic field inside a superconductor: the magnetic penetration length discussed. In addition, J = e h im e (Ψ Ψ Ψ Ψ ) 4e2 m e c A Ψ 2 shows that variations in the supercurrent are due to variations in the phase of the order parameter, since the gradients of the amplitude cancel out.

9 Two types of superconductors The surface energy per unit area for the normal-superconducting interface is! " Z 1 1 σ= F d!xhc2 Area 8π free energy condensation energy Rescale everything using the Ginzburg-Landau parameter κ(t ) = λ(t )/ξ(t ) Lengthy, left as homework excercise. In any case, the result has two limits } κ(t )! 1 positive surface energy. The amount of surface is minimized. Type I κ(t )! 1 negative surface energy. Creation of vortices is preferred. Type II!!!!! # # #"!!! #!" "!!!!! # # "#"# " "!!! "#! "#!! "!! " " "# "#!!! # $"

10 Topological defects - the Kosterlitz- Thouless transition (or Kosterlitz-Thouless-Nelson-Halperin-Young-Berezhisnky)

11 Kosterlitz-Thouless transition As discussed, phase transitions from a high-temperature phase to a lowtemperature phase are usually characterized by the appearance of long-range order. However, there exist some two-dimensional systems with continuous symmetry that exhibit a transition to a low-temperature phase without ordering. This transition is called the Kosterlitz-Thouless transition. Customarily the long-range order is described by the (two-point) correlation function which tends to a constant with the appearance of long-range order. However, this is not the case with the KT-transition, and thus the correlation function is inadequate to describe the transition. Kosterlitz and Thouless used an overall property of the system called topological order. Then, topological defects can be interpreted as dislocations in planar solids, and in superfluid films and in the classical XYmodel it is interpreted as vorticity. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973), J.M. Kosterlitz, J. Phys. C 7, 1046 (1974), V. L. Berezinskii. Zh. Eksp. Teor. Fiz. 59, 907 (1970), B.I. Halperin, D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978), A.P. Young, Phys. Rev. B 19, 1855 (1979).

12 Observation related to the KT-transition Originally, Kosterlitz and Thouless used a 2D Coulomb gas model and renormalization group approach. Here, we will use the 2D XYmodel instead and not go into RG. Systems that exhibit the KT-transition interact through logarithmic interactions. Those will also arise in the 2D XY-model. In Coulomb gas. Solving the 2D Green function gives the follwing for interactions, U( r i r j ) = { 2qi q j ln r i r j r 0 + 2µ, r > r0 0, r < r 0

13 Mermin-Wagner theorem (1966/68) In the absence of external fields the fluctuations due to lowtemperature excitations (phonons in crystals, spin waves in magnets), i.e., Goldstone modes, prevent long-range order in twodimensional systems provided they exhibit continuous symmetry. spin wave - the destroy local order for T>0 (rigidity) compare to Ising spins the Mexican hat potential negative vortex in 2D positive vortex in 2D N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966), N.D. Mermin, Phys. Rev. 176, 250 (1968), V.L. Berezinskii, Sov. Phys. JETP 32, 493 (1971)

14 Classical spins restricted to x-y -plane XY-model H = J S i S j = J cos(φ i φ j ) <i j> strength of interaction S i = 1 In a low enough temperature, the neighboring spins tend to align so that the difference between the angles is very small and we can expand (since φ i φ j 2π ) the cosine using cosx 1 x 2 /2! + x 4 /4! x 6 /6! +... H znj 2 J 2 (φ i φ j ) 2 the angle of the spin with respect to some arbitrary axis 2D XY model By taking only the first two terms and by factorizing the first term we obtain coordination number number of sites this is also called the harmonic approximation The first term on the right-hand side is the ground state energy.

15 XY-model Let s now divide the field into vortex free and vortex field ψ po φ(r)= +φ. We also need to impose to boundary conditions: 1. For all closed uum notation curves around ψ a dr position = 0. For the vorte φ(r) dr = 2πn, where n ±1, ± Charge neutrality ψ dr = 0. H E 0 H= The energy of an isolated vortex can be written now as since φ = n/r. d 2 r φ(r) φ(r) = πjn ln(r/a) vortex strength system size Using the above we can obtain free energy and conditions for the existence of vortices.

16 H E 0 = XY-model Let s write the Hamiltonian in terms of the vortex and vortex free fields ( φ(r)) 2 d 2 r + ( ψ(r)) 2 d 2 r. Let ρ(r) stand for the vortex density field. The interaction between vortices can now be solved. Let s start with 2 φ(r) = 2πρ(r) The energy of the system can now be written (using a Green function) as H E 0 = J Z d 2 r( ψ) 2 4π 2 J Z Z d 2 rd 2 r ρ(r)g(r r )ρ(r ) + 2πJ Z d 2 rd 2 r ρ(r)ρ(r )ln R r 0 The vortex free term is responsible for the absence of long-range order in 2D and below. The second term is the interaction term and third imposes charge neutrality. Solving the Green function gives G(r) 1 2π ln r r 0

17 XY-model Let s look at the spin-spin correlation function in the absence of vortices < S i S j >=< e i(φ i φ j ) > Now, let s move to a lattice and write the harmonic Hamiltonian as Next, we Fourier transform the field: H E 0 = J 4 [φ(r) φ(r + a)] 2 R a φ(r) = 1/ N k φ k e ik R If we now write the Hamiltonian in terms using the transformed variables, we notice that they are independent and have a Gaussian distribution. Then, The solution in 2D is < S i S j >=< e i(φ(r) φ(0)) >= e 1 2 <(φ(r) φ(0)2 > < S i S j > r i r j k B T /4πJ r 0 = r i r j η(t ) r 0 There us no single critical exponent! Long wavelegth excitations destroy order and the transition is related to binding and dissociation of vortices.

18 ξ(t ) exp Correlation length Solving for the correlation length gives an unusual form: ( const. (T T KT ) 1/2 What is going on here, no algebraic decay?! ) for T > T KT That has its origin in the creation of unbound vortices as T->TKT. The phase is distorted, there are screening effects and so on. One can also argue that the origin of the exponential is related to the density of the vortices which has the Boltzmann form exp(evortex/kbt). To see this better we should resort to renormalization group calculation. The exponential has also experimental relevance: since it decays very rapidly it is virtually impossible to see that experimentally (we will see that in a moment).

19 XY-model in various dimensions Here is how the XY-model behaves in various dimensions: S(r)S(0) e const.t for d > 2 ( ) η r for d = 2 L exp( T r) for d = 1. 2Ja Homework exercise (use the same approach as above).

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21 Vorticity. Simulations of the XY-model T= spins T=0.85 T=0.90 all pairs are tightly bound first pairs with distance greater than the lattice size appear clusters with many vortices appear. Also d~r appears. T=0.95 T=1.00 T=1.05 unbound vortices ~40% of vortices in large clusters J. Tobochnik and G. V. Chester, Phys. Rev. B 20, 3761 (1979)

22 Vorticity. Simulations of the XY-model Specific heat Vortex pair density Tc=0.89 peak at T= spins 3600 spins spins 3600 spins Tc=0.89 number of vortex pairs ν e 2µ/T fit energy to a create vortex pair In the 2d XY-model T KT /J ± J. Tobochnik and G. V. Chester, Phys. Rev. B 20, 3761 (1979) 1/T 2µ = 9.4 ± 0.3 KT prediction: 10.2

23 KT-melting in a colloidal suspension angular average translational order orientational order C. A. Murray and D. H. van Winkle, Phys. Rev. Lett. 58, 1200 (1987).

24 KT-melting in smectics R. Bruinsma and D.R. Nelson, Phys. Rev. B 23, 402 (1981)

25 Superfluids Other systems The same symmetry possessed by the phase field of the XY-model is present in the Ginzburg-Landau free energy of superfluids and of superconductors. In a superfluid the topological excitations are of vortices. Using superfluid He films it has been experimentally found that the superfluid phase is destroyed by the KT-transition. Superconductors Since the condensate in superconductors is charged, screening effects play a role. However, for thin superconducting films the effective screening length can easily become macroscopic. When that occurs, superconductivity is destroyed by unbinding of vortex pairs according to the KT-transition. The broken pairs can move freely when they respond to an applied electric current. As they move they cause phase slips in the superconducting order parameter. These phase slips induce a voltage drop according to the Josephson relation. Solid state The case of dislocations in solids in particularly interesting since it is related to melting of solids and the presence of dislocations has a profound effect on the strength of solids. In fact, Kosterlitz and Thouless suggested that the unbinding of dislocation pairs corresponds to melting transition: the response of a solid to shear stress changes from elastic to viscous since the dislocations are able to move. In addition the crystal structure of solids can be very naturally used to explain the concept of topological long-range order. In solids there exists also directional order, i.e. the crystallographic axes tend to remain parallel. The directional order can be present even if positional long-range order is lost. Using this fact Nelson and Halperin refined the KT-interpretation of melting of solids. Nelson and Halperin suggested that there exists in fact two transtions, the first one as suggested by Kosterlitz and Thouless from solid to an intermediate, a liquid crystal, state, and second transition that leads to a liquid state.

26 Summary 1. I have tried to tell you about various aspects in phase transitions. The presentation has been very qualitative - leaves homework exercises for you :-) 2. The aim was to demonstrate how some intuitively different approaches are connected together. The Landau theory has a connecting role. 3. The concept of order parameter and symmetry breaking are of fundamental importance. The latter: Collective excitations, rigidity, defects, discreteness of phase transitions. 4. A lot was left out. Hopfully you will be inspired to take read more about these topics.