Phenomenological theory and k −ψ k ) 0 symmetry aspects of ...

Phenomenological theory and

symmetry aspects of superconductivity

Generalized gap functions

Even parity, spin singlet:

€

ˆ Δ ! k =0 ψ(

! k )

−ψ(! k ) 0

%

& '

(

) * = iσ yψ(

! k )

€

ˆ Δ ! k

=−dx + idy dz

dz dx + idy

$

% &

'

( ) = i! d (! k ) ⋅ ! σ σ yOdd parity, spin triplet:

Scalar wave function:

Vector wave function:

€

ψ! k ( )

€

! d ! k ( )

with

€

ψ −! k ( ) =ψ

! k ( )

with

€

! d −! k ( ) = −

! d ! k ( )

|�~k|2 = | (~k)|2

|�~k|2 = |~d(~k)|2

Linearized gap equation and gap symmetry Eigenvalue equation

Solutions classified according to the irreducible representations of symmetry group

�representation

example angular momentum

l

degenerate solutions for � = N(0)V�

�m(~k)

~d�m(~k)m = 1, . . . , dim�

Symmetry operations Symmetries of normal phase:

orbital rotation

G = Go x Gs x K x U(1)

spin rotation time reversal gauge

orbital rotation

spin rotation

time reversal

U(1) gauge

symmetry operation

€

€

ˆ D = ei! θ ⋅! ˆ σ / 2

€

ˆ R o orbital rotation

€

! A '=! A + "c

2e! ∇ φ

presence of strong spin-orbit coupling spin and lattice rotation go together

go

c~

ks

= cR

o

~

ks

gsc~ks =X

s0

Dss0c~ks0

Kc~ks =X

s0

(�i�y)ss0c�~ks0

�c~ks = ei�/2c�~ks0

Symmetry operations Symmetries of normal phase: G = Go x Gs x K x U(1)

orbital rotation

spin rotation

time reversal

U(1) gauge

symmetry operation spin singlet spin triplet

presence of strong spin-orbit coupling spin and lattice rotation go together

spin triplet pairing:

€

g! d ! k ( ) = ˆ R s

! d ˆ R o! k ( ) identical 3D rotations

€

ˆ R oˆ R s

go

(~k) = (Ro

~k)

gs (~k) = (~k)

K (~k) = ⇤(~k)

inversion I (~k) = (~k)

� (~k) = ei� (~k) �~d(~k) = ei� ~d(~k)

I ~d(~k) = �~d(~k)

K ~d(~k) = ~d⇤(~k)

gs ~d(~k) = Rs~d(~k)

go

~d(~k) = ~d(Ro

~k)

Symmetry operations Symmetries of normal phase: G = Go x Gs x K x U(1)

orbital rotation

spin rotation

time reversal

U(1) gauge

symmetry operation spin singlet spin triplet

go

(~k) = (Ro

~k)

gs (~k) = (~k)

K (~k) = ⇤(~k)

inversion I (~k) = (~k)

� (~k) = ei� (~k) �~d(~k) = ei� ~d(~k)

I ~d(~k) = �~d(~k)

K ~d(~k) = ~d⇤(~k)

gs ~d(~k) = Rs~d(~k)

go

~d(~k) = ~d(Ro

~k)

Possible broken symmetries: U(1)-gauge superconductivity Orbital rotation crystal structure Spin rotation magnetism Time reversal magnetism

Order parameter relevant order parameter in represenation with highest Tc (instability) �

restrict to set (vector space basis) { �m}{~d�m}

(~k) =X

m

⌘m �m(~k) ~d(~k) =X

m

⌘m ~d�m(~k)

⌘m = ⌘m(~r, T )order parameter

trivial representation � conventional pairing / SC phase

other representations � unconventional pairing / SC phase

e.g.: l = 0 non-degenerate

e.g.: l > 0 � 1degeneracy

complex

Order parameter relevant order parameter in represenation with highest Tc (instability) �

restrict to set (vector space basis) { �m}{~d�m}

(~k) =X

m

⌘m �m(~k) ~d(~k) =X

m

⌘m ~d�m(~k)

⌘m = ⌘m(~r, T )order parameter

order parameters describe spontaneous symmetry breaking Ginzburg-Landau theory

complex

Ginzburg-Landau theory for teneral order parameters Landau�s recipe: order parameters belong to irreducible representations

of the normal state symmetry group

free energy functional as a scalar function of

€

ηmtransform according to the representation

€

F ηm[ ] = d3∫ r[a ηm2

+ bm1 ,...m4ηm1* ηm2

* ηm3ηm4

m1 ,...m4

∑m∑

€

+ Km1m2n1n2Dn1

ηm1( )*

Dn2ηm2( )

n1 ,n2

∑m1 ,m2

∑ +18π! ∇ ×! A ( )

2]

invariant under all symmetry operations of rotations, time reversal and U(1)-gauge

€

! D =! ∇ + i 2e

"c! A

€

a = a'(T −Tc )

€

bm ,Kmm 'nn', real constants, gradient: gauge-invariance

(~k) =X

m

⌘m �m(~k)

T Tc

GL free energy functional:

€

! D =! ∇ + i 2e

"c! A

€

a(T) = a'(T −Tc )

€

a',b,K > 0

€

! B =! ∇ ×! A

|Ψ|

F

uniform phase:

spontaneously chosen phase "

broken U(1) gauge symmetry

T>TC

T<Tc

Ginzburg-Landau theory of conventional superconductor

trivial representation (~k) = ⌘(~r, T ) 0(~k)

Invariant under all symmetry operations

F [⌘, ~A] =

Zd3r

"a|⌘|2 + b|⌘|4 +K| ~D⌘|2 +

~r⇥ ~A

8⇡

#

|⌘(T )|2 =a0(Tc � T )

2b|⌘|

⌘ = |⌘|ei↵↵

GL free energy functional:

€

! D =! ∇ + i 2e

"c! A

€

a(T) = a'(T −Tc )

€

a',b,K > 0

€

! B =! ∇ ×! A

Ginzburg-Landau theory of conventional superconductor

trivial representation (~k) = ⌘(~r, T ) 0(~k)

Invariant under all symmetry operations

F [⌘, ~A] =

Zd3r

"a|⌘|2 + b|⌘|4 +K| ~D⌘|2 +

~r⇥ ~A

8⇡

#

variational GL equations for inhomogeneous order parameter: n

a+ 2b|⌘|2 �K ~D2o

⌘ = 0spatial structures vortices, domain walls, boundary …

~r⇥ ~B =4⇡

c~J

~J =eK

2~i

n

⌘⇤( ~D⌘)� ⌘( ~D⌘)⇤o

supercurrent ~r2 ~B = ��2 ~B

magnetic properties London equation

��2 =32⇡e2

c2K|⌘|2

Specific example:

Superconductor with tetragonal crystal structure

Example of a tetragonal crystal with spin orbit coupling

Point group: D4h 4 one-dim., 1 two-dim. representation

Γ E C2 2C4 2C2’ 2C2” "A1 1 1 1 1 1 "A2 1 1 1 -1 -1 B1 1 1 -1 1 -1 B2 1 1 -1 -1 1 E 2 -2 0 0 0 "

Character table for D4

D4h contains inversion

even and odd representations


A1g

A2g

B1g

B2g

Eg

A1u

A2u

B1u

B2u

Eu

€

ψ! k ( )

€

! d ! k ( )

€

1

€

kxky kx2 − ky

2( )

€

kx2 − ky

2

€

kxky

€

kxkz,kykz{ }

€

ˆ x kx + ˆ y ky

€

ˆ x kx − ˆ y ky

€

ˆ y kx − ˆ x ky

€

ˆ y kx + ˆ x ky

€

ˆ z kx, ˆ z ky{ }

€

ˆ x kz, ˆ y kz{ }

Γ" Γ"

Point group: D4h 4 one-dim., 1 two-dim. representation even (g) / odd (u) parity

Conventional: A1g Unconventional: everything else

only one representation is relevant for the superconducting phase transition

Ginzburg-Landau free energy functionals: 1-dimensional representations:

€

ψ! k ( ) =ηψ0

! k ( )

€

! d ! k ( ) =η

! d 0! k ( )

€

F η,! A [ ] = d3r aη 2

+ bη 4+ K! D η

2+18π! ∇ ×! A ( )

2&

' ( )

* + ∫ like conventional SC

,

2-dimensional represenations:

€

ψ =ηxψx +ηyψy

€

! d =ηx

! d x +ηy

! d y,

€

+K1 Dxηx2

+ Dyηy

2{ } + K2 Dxηy

2+ Dyηx

2{ } + K3 Dzηx2

+ Dzηy

2{ }

Possible homogeneous superconducting phases Higher-dimensional order parameters are interesting:

b2 / b1

b3 / b1

A

B

C

A

B

C

K magnetism

D4h D2h crystal deformation

Degeneracy: 2 domain formation possible

Phases A-phase:

€

ψ! k ( ) =η kz kx + iky( ), η kz kx − iky( )

kx ky

kz �axial symmetric�

time reversal

B-phase:

€

ψ! k ( ) =η kz kx + ky( ), η kz kx − ky( )

C4

C-phase:

€

ψ! k ( ) =η kzkx, η kzky

C4

kz

ky

kx

kz

ky

kx

Weak-coupling condensation energy: less nodes more stable

A-phase more stable than B/C-phase

magnetic

nematic

nematic

Anisotropy B- and C-phase violated crystal symmetry: tetragonal orthorhombic

spontaneous crystal deformation Tiny !

Diamagnetic screening: supercurrents

tensorial London equation:

€

∇2! B = ˆ Λ

! B

tetragonal orthorhombic

Important for vortex lattice structure!

Volovik-Gor'kov classification

orbital angular momentum:

"ferromagnetic" (chiral)

"antiferromagnetic"

e.g. -wave

e.g. -wave

spin:

analogous Volovik-Gor'kov classification

non-unitary states, e.g. A1-phase of superfluid 3He in a magnetic field

Volovik-Gor'kov classification


analogous Volovik-Gor'kov classification

non-unitary states, e.g. A1-phase of superfluid 3He in a magnetic field

"ferromagnetic" (chiral)

"antiferromagnetic"

e.g. -wave

e.g. -wave

spin:

Ferromagnetic or chiral phase A: For pairing state within representation Γ"decomposition of includes pseudovector representation

e.g.: chiral p-wave in Eu

mz

Orbital angular momentum and magnetic moment chiral p-wave phase:

with


electron charge: magnetic moment

Moment

Effective magnetic moment Cooper pairs with angular momentum

magnetic moment of superconductor

M

-M

U(1)-gauge

symmetry transformations:

rotation

conserved "charge":

N: electron charge

??

Volovik

Bz

q j

E

anomalous electromagnetism

Goryo & Ishikawa Volovik & Yakovenko

Bintern

Luke, Uemura et al (1998) Superconductivity generates

magnetism (0.1 - 1 Gauss)

Intrinsic magnetism µSR zero-field relaxation"

depolarization in random internal field:

σKT2 : 2nd moment of field distribution"

PKT 1

1/3

σKT2

Kubo-Toyabe

€

PKT (t) =131+ 2 1−σKT

2 t 2( )exp(−σKT2 t 2 /2)[ ]

polarization of spin of trapped muon

σKT2

Sr2RuO4

U1-xThxBe13, Re6Zr, PrOs4Sb12 UPt3 , SrPtAs, …. other SC showing intrinsic magnetism:

Surface states - spontaneous supercurrents for chiral phase surface scattering detrimental interference effects:

y x

specular scattering Eu :

destructive constructive

Supercurrent:

jy Bz

ξ" λ" Screening length London penetration depth

surfa

ce cu

rrent

scre

ening

curre

nt

€

! j = −c ∂F

∂! A

|ηx|"

|ηy|"

x

ξ" coherence length surface

η0"

�extension of Cooper pairs�

~dx

(~k) = zkx

~dy(~k) = zky~d = ⌘

x

(~r)~dx

+ ⌘y

~dy

~dx

! �~dx

~dy

! +~dy

Tests for the pairing symmetry


A1g

A2g

B1g

B2g

Eg

A1u

A2u

B1u

B2u

Eu

€

ψ! k ( )

€

! d ! k ( )

€

1

€

kxky kx2 − ky

2( )

€

kx2 − ky

2

€

kxky

€

kxkz,kykz{ }

€

ˆ x kx + ˆ y ky

€

ˆ x kx − ˆ y ky

€

ˆ y kx − ˆ x ky

€

ˆ y kx + ˆ x ky

€

ˆ z kx, ˆ z ky{ }

€

ˆ x kz, ˆ y kz{ }

Γ" Γ"

Point group: D4h 4 one-dim., 1 two-dim. representation even (g) / odd (u) parity

cuprate high-Tc superconductor: B1g B1g (~k) = k2

x

� k2y

d x2-y2 - wave pairing

+

+

Internal phase structure

0

π"

node

different phase in different directions

C4 ψ(k) = - ψ(k) = eiπ ψ(k)

Gap structure

4 line nodes on basically cylindrical Fermi surface

nodes (~k) = �0(k

2x

� k2y

)

|�~k| = | (~k)|

Low-temperature properties

powerlaws in other quantities depending on gap topology

line nodes point nodes

specific heat C(T)

London penetration depth λ(T)

NMR 1/T1

heat conductivity κ(T)

T2 T3

T (T3) T2 (T4)

T3 T5

T2 T3

quantity

London penetration depth

YBa2Cu3O7

Hardy et al.

high-temperature superconductors with line nodes in the gap

NMR 1/T1 YBa2Cu3O7 Martindale et al.

1/T1 T3

Probing the phase structure

Phase probed by interference using Josephson effect

ψ1 = |ψ1| eiα" ψ2 = |ψ2| eiβ"

Supercurrent between SC1 and SC2 due to phase coherent Cooper pair tunneling

J = Jc sin(β - α)

Current determined by phase difference

SQUID (Superconducting QUantum Interference Device)

Jc

2πΦ/Φ0"0 -π +π -2π +2π

J 1

2

Φ"

magnetic flux

Aharonov-Bohm effect for Cooper pairs



ψ1 = |ψ1| eiα" ψ2 = |ψ2| eiβ"


J = Jc sin(β - α)



0 0

J Φ"

+ Conventional SC

Jc

2πΦ/Φ0"0 -π +π -2π +2π



ψ1 = |ψ1| eiα" ψ2 = |ψ2| eiβ"


J = Jc sin(β - α)



Jc

0 -π +π -2π +2π

Conventional SC

J Φ"

+ + -

-

π"

0

Wollman, van Harlingen et al. (1993), Brawner & Ott (1994), Mathai et al. (1995), Iguchi & Wen (1995)

π-shift"

Superconducting loops

Single-piece loop

€

J ∝! A − Φ0

2π! ∇ φ

(

) *

+

, - ∫ ⋅ d! s = 0

Single valued order parameter

Ψ = |Ψ | eiφ" φ �> φ + 2π n"

€

Φ = Φ0n

Φ0 = hc/2e"unit-flux:

n: integer

Φ"

0

0

π = Δφi"

€

Φ =Φ0 n +12

#

$ %

&

' (

half-integer flux quantization

note: smallest flux |Φ| = Φ0 / 2

Segmented loop

€

! A − Φ0

2π! ∇ φ

'

( )

*

+ , ∫ ⋅ d! s = Δφi

i∑

odd number of π�shifts Δφi

Φ"

Tsuei, Kirtley et al. (1995)

Tri-cristall-configuration

Superconducting loop YBa2Cu3O7 Tc = 92 K

= 60 µm

Tsuei-Kirtley frustrated loops

SQUID-scanning-microscope

magnetic field

unfrustrated loops

magnetic field

even number of π-shifts

no signal

Tsuei, Kirtley et al. (1995)

Tri-cristall-configuration

Superconducting loop YBa2Cu3O7 Tc = 92 K

SQUID-scanning-microscope

magnetic field

spontenous current and magnetic flux

Φ0/2"

π"

= 60 µm

Tsuei-Kirtley frustrated loops

frustrated loop

odd number of π-shifts

Phenomenological theory and k −ψ k ) 0 symmetry aspects of ...

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