Phase Transitions and Superconductivity

RQEMP 2005 (Hilke@physics.mcgill.ca)

• Temperature driven Phase transitions (normal-to-superconductor transitions) • Phase transitions of vortices • Quantum Phase Transitions: – Disorder induced transitions – Interaction induces transitions

"Taste" of a phase transition

1 Chocolate Water-ice

2 3 Superconductivity perfect conductivity

Kamerlingh Onnes, 1911

4 Gap determination by tunneling

Nobel 1973 normal SC Al thin insulator

Al - Al 2O3 Al - Al 2O3 - Pb

Tunneling experiments

I Al Pb

normal SC

thin insulator ∆ Pb normal dI/dV

Pb superconducting

V V

5 What’s the order ∆ parameter?

Analogy to Ferromagnetic transition: Order Parameter=Magnetization r M 1. T=0 well ordered r r M ≡ S > 0 2. 0Tc r r disordered M = S = 0

T

6 is the order parameter: → continuous phase SC transition

Mean-field (Landau) theory of continuous phase transitions

Approximate free energy close to phase transition:  r r b(T )  F = ∫ d D x ∇φ * ∇φ + a (T )φ * φ + (φ * φ ) 2  2  Higher order terms (∇φ *∇φ)2 (, φ *φ)3,.... are expected to be smaller close enough to Tc . Expand around Tc :

aT( )=α ( TT −c ) + ..., b( T )=β + ... F F

φ φ T< T c T> T c

7 Ginzburg –Landau: Order parameter: The complex order parameter is “amplitude of the Cooper pair center of mass”: Ψ()x ∝∆ (), x ∆ () k ∝ ckck − ↑()() ↓ which is “the gap function”. iφ( x ) Ψ()x ≡ ns *() xe n( x ) n*( x ) = s density of the Cooper pairs s 2

the superconductor U(1) φ(x ) phase

For B=0 , the GL free energy is:  h2 r r β  F[]Ψ= dx3 ∇Ψ∇Ψ+ *α ()* TT − ΨΨ+ (*) ΨΨ 2 ∫  c  2m * 2 

m*= 2 m e

For B≠0: Invariance under local gauge transformations:

Ψ()x → eiχ ( x ) Ψ () x   r r hc r e* = 2 e  Ax()→ Ax () + ∇ χ () x  e *

8 To ensure local gauge invariance one makes the “minimal substitution”, namely replaces any derivative by a covariant derivative: r re * r  DxΨ() =∇− iAx  Ψ () hc 

The local gauge invariance of the gradient term: r rr r ie* c h   iχ( x ) DxΨ→∇− ( ) A +∇χ ( xxe )   Ψ ( ) hc e *   r rie * r r  = ∇Ψ +Ψ⋅∇iχ −A ⋅Ψ − i ∇⋅Ψ χ e iχ( x ) hc  r = DΨ⋅ e iχ( x ) r r |Dx Ψ ()|2 → | Dx Ψ ()| 2

Ginzburg – Landau equations:

Minimizing the free energy with covariant derivatives one arrives at the set of GL equations: the nonlinear Schrödinger equation (variation with respect to ΨΨΨ)

2 2 h r e * r  2 − ∇−iA  Ψ+α( TT −Ψ+c ) β Ψ Ψ = 0 2m * h c 

and the supercurrent equation (variation of A):

crrr ie*h r r e * 2 r ∇×BJ = ≡−( Ψ∇Ψ−Ψ∇Ψ* * ) − Ψ 2 A 4π s 2*m m * c

9 Two characteristic length scales

Coherence length ξ characterizes variations of Ψ ( x ) , while the penetration depth λλλ characterizes variations of B(x) h ξ (T )= , 2*(mα Tc − T ) c m * β λ(T ) = e* 4 πα () Tc − T Both diverge at T=Tc.

Ginzburg – Landau parameter: Temperature independent dimensionless parameter:

λ(T ) m * c β κ = = ξ()T e *h 2 π Properties depend on the GL parameter:

1 1 κ < κ > 2 2 Type I Type II

Abrikosov (1957)

10 Continuous Phase transition and the Ginzburg – Landau Theory ‘61 ∆

∆

Vortices

A.A. Abrikosov 1957 2003

1967 1989 Bitter STM Decoration

U. Essmann H.F. Hess H. Trauble Bell labs

11 Abrikosov vortices in type II superconductors as seen by electron beam tomography. KT pair

Tonomura et al Tonomura et al PRL66,2519 (1993) PRB43,7631 (1991)

Phase diagram

Type I Type II

H Hc2 (T)

H (T) Normal H c Normal c metal H mixed state metal H (T) Meissner state c1 Meissner state

T Tc T Tc

12 Overview of properties of vortices and systems of vortices (vortex matter) (Adapted from Zeldov) Inter-vortex repulsion and the Abrikosov flux line lattice Line energy To create a vortex, one has to provide energy per unit length ( line tension ) Φ  2 λ  ε ≈ 0  log   4πλ   ξ  Therefore vortices enter an infinite sample only when field exceeds certain value

Interactions between vortices They interact with each other via a complicated vector- vector force. Parallel straight vortices repel each other forming highly ordered structures like flux line lattice (as seen by STM and neutron scattering).

S.R.Park et al (2000)

Pan et al (2002)

13 Two critical fields

As a result the phase diagram of type II SC is richer than that of the two-phase type I

Φ H H ≈ 0 first vortex c1 4πλ 2 penetrates.

Hc2 Normal Φ H = 0 cores overlap c2 2 Mixed 2πξ

H H 2λ 2 c1 c2 ≈ =2κ 2 > 1 Meissner T 2 Hc1 ξ

Tc

Two theoretical approaches to the mixed state

Just below H c 2 vortex cores almost overlap. Instead of lines one just sees array of superconducting “islands”

H c2 Lowest Landau level appr. for constant B Mixed

Hc1 Meissner Tc

London appr. for Just above H c 1 vortices are well separated and have very thin cores infinitely thin lines

14 Thermal fluctuations and the vortex liquid

In high Tc SC due to higher Tc , smaller ξξξ and high anisotropy thermal fluctuations are not negligible. Thermally induced vibrations of the flux lattice can melt it into a “vortex liquid”. The phase diagram becomes H more complicated.

H Normal c2 FLL Vortex liquid H c1 Meissner T Tc

First order melting of the Abrikosov lattice

Magnetization Specific heat

Zeldov et al Schilling et al Nature (1995) Nature (1996,2001)

15 Thermal expansivity:

Vortex (solid-liquid) transition

SC transition

Vortex “cutting” and entanglement

Vortices can entangle around each other like polymers, however due to vectorial nature of their interaction they can also “disentangle” or “cut each other”.

There are therefore profound differences compared to the physics of polymers

16 Disorder and the vortex glass

point Columnar

Vortices are typically pinned by disorder. For vortex systems pinning create a glassy state or viscous entangled liquid. In the glass phase material becomes superconducting (zero resistance) below certain critical current Jc .

17 18 Vortex dynamics

Vortices move under influence of external current (due to the Lorentz force).

Field driven flux motion probed by STM on NbSe2 A.M.Troianovski (2004)

The motion is generally friction dominated. Energy is dissipated in the vortex core which is just a normal metal. The resistivity of the flux flow is no longer zero.

19 20 21 22 1st Order triple point

23 Transverse Motion

3 2

1 ]

Ω

m 0 [

y

x

R al rm no 6 5 depinning 2 10 pinning 9 4 1 8 ing 7 B 3 inn g 6 ep tin5 [T d uc ] 2 nd 4 ] rco 3 A 1 pe 2 [m su 1 I 0 0 J. Lefebvre ‘05

24 1st order phase transition (TCP)

Strong transverse motion (smectic, depinning 2) Pinned disordered phase

Quasi-ordered phase (Bragg glass)

(2003)

25 Classical Phase Transitions: Temperature driven (Variations of Melting transition, like chocolate)

James Bay

Winter Summer

Ice Water Quantum Classical (Ground State) (Higher temperature phases)

26 Classical vs. Quantum

(Ref: Vojta)

27 Typically power law behaviour in critical (Ref: Vojta) region

Quantum Phase Transition : change of Ground state phase as a function of some parameter g.

gc

28 Disorder driven Quantum Phase Transition (superconductor-to-insulator transition)

(Ref: Goldman)

29 Vojta

Sondhi

Interaction driven Quantum Phase Transition (insulator-to-superconductor-to-metal transition)

Dobrosavljevic

30 Other SC-QPTs

Thank you and bon appétit

31