Chapter 9 Ginzburg-Landau theory

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1 Chapter 9 Ginzburg-Landau theory

2 The liit of London theory The London equation J µ λ The London theory is plausible when B 1. The penetration depth is the doinant length scale ean free path λ ξ coherent length λ l. The field is sall and can be treated as a perturbation 3. n s is nearly constant everywhere The coherent length should be included in a new theory

3 Ginzburg-Landau theory 1. A acroscopic theory. A phenoenological theory 3. A quantu theory London theory is classical Introduction of pseudo wave function Ψ) r Ψ) r is the local density of superconducting electrons Ψ ) r n s ) r

4 The free energy density The difference of free energy density for noral state and superconducting state can be written as powers of and Ψ Ψ potential energy Kinetic energy Ginzburg-Landau free energy density at zero field g s β 4 1 gn + α Ψ + Ψ + Ψ i nd order phase transition Quantu echanics

5 nd order phase transition Potential energy U β α Ψ + Ψ 4 A reasonable theory is bounded, i. e. U β > U ) Ψ U Classical solutions Ψ Ψ Ψ α > Single well α < double well

6 Spontaneous syetry breaking U Ψ The phase syetry of the ground state wave function is broken i Ψ Ψe ϕ α Ψ Ψ α > α α < Ψ Ψ Noral state Critical point superconducting state β Ψ density of superconducting electrons

7 The eaning of α The superconducting critical point is α α α > T > T c α α < T T c T < T c t1 t Near the critical point, If β is regular near T c then t 1) α α α Ψ β c t 1) t T T c The London penetration depth is 1 λ 1 1 λl ns L 1 t) 1 µ ne s Consistent with the observation λl T ) 1 λ ) 4 L 1 t ) 1

8 Magnetic field contribution at non zero field, there are two odifications p p e A The vector potential B A 1 g µ H For perfect diaagnetis ) H a g Ha µ MdH a

9 The canonical oentu The first odification is to include the hailtonian of a charged particle in a agnetic field E φ A t B A For a charged paticle, v) t v) + q Εdt v) qa t v) t + qa v) is conserved in the agnetic field The canonical oentu is chosen as pcanonical v+ qa The kinetic energy is 1 1 q ) v pcanonical A

10 Gauge transforation A A A+ χ E φ t φ φ φ χ B A t A The physics is unchanged The phase of the particle wave function will be changed by a phase factor ) ) )exp ie Ψ r Ψ r Ψ r χ ie p ea Ψ ) r i ea Ψexp χ ie exp χ i ea ) Ψ+ χ) Ψ ie exp χ i ea) Ψ ) ) { } H 1 p ea) + U HΨ H Ψ Coent: not all theory are gauge-invariant, the theory keeps gauge-invariance is called a gauge theory

11 Energy density The eaning of Ψ 1 1 ea ea e Ψ ϕ Ψ + Ψ Ψ i i I. part Real part with 1 { ) ) } Ψ + ϕ ea Ψ iϕ i ΨΨe ϕ The first ter arises when the nuber density n s has a nonzero gradient, for exaple near the N-S boundary the length scale is coherent length ξ, and in type I SC, ξ<<λ) The second ter is the kinetic ter associated with the supercurrent. If the phase is constant of position, it gives e A Ψ

12 Penetration near the N-S boundary Near the surface, the agnetic induction is ) ) B x z x Bz e λ y for x< B A Choose the gauge ) λ ) A x B x z B z x) We found the energy density A x e A Ψ e λ B Ψ Should be equal to the field energy density λ e µ Ψ y super z Bx) B µ Fro London s theory λ e y H a x noral µ n S

13 1 Kinetic energy density ns vs The supercurrent velocity v p e A For ϕ S S ϕ e A 1 e A ns vs ns While in GL theory, the energy density e A Ψ n S Ψ The eaning of the wavefunction Ψ

14 GL theory and London theory In bulk superconductors µ gs gn HC β 4 α Ψ + Ψ α β In previous discussion, we havein bulk) Ψ with λ we have e µ n S n S α β HC e HC S µ µ λ α n β e H C µ λ

15 The teperature dependences near critical point Near the critical point 1 4 S e µλ T λ 4 1 t TC Ψ n 1 t 1 t C C ) 1 ) H H t t ε 4 ε) ε) ε t) 4 1 t t ) ε ) α λ HC 1 t t ε 4 C t ) 4 1 t ) ε ) 4ε ) 1 β λ H constant of t Paraeters in GL theory can be deterined by λt) and H C T)

16 GL differential eqns The solution for iniizing g s in absence of the field, boundary and current is Ψ Ψ In general cases, the wavefunction can be written as ΨΨ r) By variational ethod gdv We have δ S V 1 αψ + β Ψ Ψ+ e A Ψ i e ) e J Ψ Ψ Ψ Ψ A Ψ i e ) ϕ e A Ψ e Ψ v S 1st eq) nd eq)

17 Derivation for GL eqns δ gdv S V g s is a function of Ψ and,, ) Ψ Ψ Ψ Ψ 1 3 With boundary conditions, i.e Ψ or Ψ gs Ψ gs Ψ g S i i iψ) g S i ) i iψ β 4 1 A Ω Ω Euler-Lagrange eq. B gs gn + α Ψ + Ψ + e Ψ + µ M H i µ gs e α β A Ψ+ Ψ Ψ+ e A Ψ Ψ i

18 i gs 1 i i i eai Ψ Ψ i i i ) i i i 1 Ψ e Ψ A e A 1 αψ+ β Ψ Ψ+ e e A Ψ+ Ψ A Ψ i i i 1 αψ+ β Ψ Ψ+ Ψ e Ψ+ e Ψ A A i i 1 αψ+ β Ψ Ψ+ e A Ψ i First GL equation

19 δ gdv S V g s is a function of Α and A Β With boundary conditions, i.e A or A g A g S S i j i i j A ) Ω Euler-Lagrange eq. β 4 1 A i B gs gn + α Ψ + Ψ + e Ψ + µ M H µ gs e e e Ψ A Ψ+Ψ A Ψ Aj i j i j e Aj Ψ Ψ Ψ Ψ + Ψ i e ) j j i Ω

20 A) ε ) ) lnεlqr An q Ar g 1 ε ε A ) A) ) ) An qar A ) S i ln lqr i i µ i j i lnqr i j 1 ε ε δ δ + δ δ µ { } A ) A ) 1 1 ε ε ε µ µ A ) ) ln lij i n ijl i l i ln i 1 µ lnqr i lnqr ln lqr i i nj q r qi rj n j A e ) e A 1 Ψ Ψ Ψ Ψ + Ψ + A i µ e ) e A J Ψ Ψ Ψ Ψ Ψ i 1 1 A Β J µ µ second GL equation

21 Boundary conditions The GL eqns are derived by assuing boundary conditions that i e A Ψ Ω and J Ω These are true for a SC-insulator boundary, but not correct for an N-SC boundary The boundary condition for N-SC is derived by de Gennes using a icroscopic theory: i i e A Ψ Ψ b Ω SC noral Thus the wavefunction will leak into the noral region with a characteristic length, b. This is called proxiity effect. b

22 J At zero field, H ϕ GL coherent length Ψ Ψ Ψ Ψ and Superconducting phase is constant of position GL eq 1) αψ+ β Ψ Ψ Ψ In 1D syste d + + Ψ dx d 3 f f f α dx + 3 f α f β f Diension[L ] f Ψ Ψ Ψ α β

23 A length scale can be defined α ξ d ξ + dx since α 1 t 3 f f f ξ 1 t 1 Consider the situation that f~1 deep in the SC) We can expand the GL eq: f 1 g g d 3 ξ g + + g + g ξ g ) ) 1 1 dx ) ± x g g x e ξ

24 Exact solution d ξ f + f 1 f ) dx u u The solution e e f tanh u u u e + e When u is large f tanh u e e with df du u 1 e ) u 1+ e ) u) u 1 e 1 e ) + 1 e 1 e When u is sall u u u x ξ f tanh u u 1 cosh x ξ u f x u ξ Ψ Ψ 1 d f du tanh u cosh u Slope 1 x ξ

25 Diensionless GL paraeter e H C α µ λ When α ξ κ > λ κ ξ 1 1 t 1 Φ ξ µ e λh πµ H λ πµ H C λ Φ 1 t 1+ t 4 type SC C Φ κ < h e 1 C The fluxoid type 1 SC

26 London penetration depth GL eq ) e ) J ϕ e A Ψ e A If ϕ for x ξ J Ψ e e 1 J Ψ A Ψ B B µλ Fro Apere s law B µ J B ) µ J B B B B 1 B B λ A λ A x Jy x) e µλ x λ x λ Bz Be e λ We get the London eq. x ξ J, A B a x

27 B z x) Effect of sall Ψ x Bz Be λ Two length scales, ξ and λ λ Ψ noral ξ SC ΨΨ tanh x ξ Type SC ξ λ J y x) Effect of sall Ψ ) x Jy x J e λ