vortices in the superconducting state of underdoped high-t c superconductors a. j. millis department...

Vortices in the Superconducting State of Underdoped High-Tc Superconductors

A. J. MillisDepartment of PhysicsColumbia University

Support: NSF-DMR-0338376;and Rutgers Center for Materials Theory

L. B. Ioffe Center for Materials Theory

Rutgers

References:PRB66 094513 2002; JPCS (cond-mat/0112509)

CIAR Oct 2003

Quantal Vortex Liquid??


Hc2(0)

T

H

Tc

Schematic High-Tc H-T phase diagram


Hc2(0)

T

H

Tc


Tc2(H) thermal melting of vortex lattice (esp in underdoped materials)


Hc2(0)

T

H

Tc


Tc2(H) thermal melting of vortex lattice (esp in underdoped materials)

Why not Hc2(0) quantal melting of vortex lattice

Outline

•H=0: Tc, S: which states carry the current

•Vortex statics: length scales

•Tc(H) and S(H): thermal melting of vortex lattice and ‘Volovik’ depairing

•Hc(T=0): quantal melting of vortex lattice: vortex viscosity and effective magnetic field

•Summary and confusions.

High Tc phase diagram: Superconductivity + other phases…

Tem

pera

ture

Doping x

Assume: homogeneous superconducting phase with properties which depend smoothly on doping. Ignore other phases.

Pseudogap crossover scale

Superconducting phase

Low energy theory of d-wave SC state

Two coupled degrees of freedom:•order parameter phase φ

•Fermions (quasiparticles) c+p

=> Leading low energy behavior from Lagrangian

(Transverse) phase fluctuations:T=0 superfluid stiffness

ρS0 describes H=T=0 ‘bosonic’ superfluid properties (no quasiparticles) Expect ρS0 ~x E4 indep of x

Bosonic length scale: limit to length on which supercurrent can vary:

L = S0 d2r Q2(r) +E4Q4 + …

Q=+ieA

QB2 ~ S0 /E4 ~ x

Fermions in d-SC: Dirac spectrum

vF q.p. fermi velocity (change in energy as move normal to fermi surface)

vΔ =dΔ/d(pFθ) fixed by shape of s.c. gap near nodes (change in energy as move along fermi surface)

Δ

θ

θ : position around fermi surface

vΔ

Mixing TermQuasiparticle (ψ) – phase (φ) coupling

• involves vF only (not vΔ)


• involves vF only (not vΔ) •new parameter Z quasiparticle charge



Z->0 =>quasiparticles turn into ’spinons’ as approach Mott phase. Small Z :‘footprint’ of spin-charge separation



Z->0 =>quasiparticles turn into ’spinons’ as approach Mott phase. Small Z :‘footprint’ of spin-charge separation

Corrections: when p,q ~ Δ/vF indep of x

Summary: parameters

***Well established values***

ρS0: T=0 superfluid stiffness vF: usual fermi velocityroughly ~ doping x 1.8eV-A (indep of x)

***Less well established***

vΔ: opening angle of gap node indep of x Z: q.p. charge renormalization near gap node indep of x

StaticsIntegrate out fermions

Note: fermions excited by T or supercurrent H- field (Volovik)

T-linear penetration depth

Q~B1/2 so this gives ‘Volovik effect’

Implication: ‘Fermionic bounds’

Tc = 2 S(Tc, H) /

At H=0: Tc = 2 S0 / ( + ln(2) Z2vF/2v ~x

AT T=0, must have Q < QF ~ 4S0 v/ Z3vF ~x

If Z~1, fermionic bounds more stringent than bosonic bounds (x vs x1/2)

Determining vΔ

Specific heat counts # excited quasiparticles: density of States ~E=> C~T2

Direct measurments—not yet clearly interpretableIndirect: ‘thermal conductivity’ +’universal limit’ analysis

=>vΔ increases as doping goes down(Harris et al PRB 64 06509 ’01; Chiao PRL 82 2943 ’00)

Measuring Z

“Ferrel-Glover-Tinkham” (f) sum rule: ρtotal = ρS + ρN

Total (low energy) charge response Superfluid part of

charge response

Normal fluid (quasiparticle) part of charge response

ρtotal is conserved as T changes =>decrease in ρS implies increase in ρN

(has been checked in optimal YBCO: Hosseini et al PRB 01)

Z II

Quasiparticle charge response depends on BOTH number (vF/vΔ) and on ‘charge’ Ze

If vΔ known then ρS(T)=>Ze

In pictures

Δ

θ

Small T>0:

# q.p. ~ T2/(vF vΔ )

current per q.p. ~Z vF/T

Coupling to field: Z

ρn ~ Z2T(vF /vΔ )

qp here excited from condensate

‘Conventional’ d-superconductor

ρS0~vFpF: all carriers near fermi surface condense into s.c. state and contribute to supercurrent

Z~1: No significant charge renormalization

Δ

θ

qp near nodes excited out of condensate

ρS

T




Δ

θmore qp excited out of cond.

ρS

T




Δ

θ almost all qp out of condensate

ρS

T

Gap begins to decrease


ρS0~vFpF: all carriers near fermi surface condense into s.c. state and contribute to supercurrentZ~1: No significant charge renormalization=> Superfluidity dies when particles all over FS are excited above the gap

Δ

θ

ρS

TT=Tc no particles left in condensateTc

ΔInitial slope =>ρS (T)= ρS0 [1-T/Δ]

It’s different in underdoped high-Tc ! ρS0 strongly renormalized

From freq. dep conductivity(Bonn, Czech Jnl Phys 46 3195 ‘96)

Optimally doped YBCO:

ρS0 = ρS,band/7= vFpF/3

Underdoped YBCO:


Note Tc~S=> transition driven by loss of phase stiffness (Uemura; Kivelson)

Proximity to Mott phase reduces low frequency conductivity and therefore ρS0

(S. Uchida et al PRB43 7942 ’91) (J Orenstein et al PRB42 6342 ’90)

More or less: low frequency peak condenses into ρS0

Two classes of theory

‘Brinkman-Rice’

Quasiparticles get slow:vF->0 as x->0

‘RVB/Gauge theory’

Quasiparticle lose charge(turn into ‘spinons’)Z->0 as doping->0

In both cases, ρS->0 as doping->0

AND dρS/dT ->0 as doping->0

‘Conventional’ Mott insulating superconductor

ρS0~vF*

pF: all carriers near fermi surface condense into s.c. state and contribute to supercurrentZ(x): Doping dependent charge renormalizationvF

*(x): Doping dependent fermi velocity

Δ

θ almost all qp out of condensate

ρS

T/Tc

ρS(T) ->0 when excite all fermi surface over the gapInitial slope =>

ρS (T)= ρS0 [1-T/Δ]

It’s different in underdoped high-Tc ! ρS0 strongly renormalized

Initial slope dρS/dT does not change much (P. Lee)

From freq. dep conductivity(Bonn, Czech Jnl Phys 46 3195 ‘96)

Optimally doped YBCO:


Underdoped YBCO:


Underdoped high Tc superconductorρS0 small but slope not smallvF does not change much Nor does Z??

ρS

Tc


Δ

θ

ρS

Tc ρS reduced almost to 0 after exciting only q.p. near nodes!?


Δ

θ

ρS


=>shaded regions do not carry (much) supercurrent


Δ

θ

ρS


=>shaded regions do not carry (much) supercurrent

?Supercurrent carried only by near-node states?

Crucial issue: v

Δ

θ

Big gap: density wave (charge, spin, stripe, …)Extra gap—superconducting origin

vΔ : unrelated to gap max; doping dependent

vΔ

Example: density wave => doping-dependent ‘conducting arc’

If v doping independent, then must assume angle-dependent ‘Z’

Δ

θ

Possibility: Z->0 as move away from diagonal: range where Z~1 decreases as reduce doping??Angle-dependent spin-charge separation??

Z(θ)

Half width of Z~1 region: ~doping??

Summary: Part IHigh Tc superconductors <--> doped Mott insulators

Three classes of incipient Mott physics:

•‘Brinkman-Rice’ vF->0: disagrees with photoemission

•‘Slave boson’ Z->0 uniformly: disagrees with S(T)

•Conducting Arc: no theory of angle-dependent Z—but apparently supported by data

Rest of talk: mainly assume fermionic bounds dominate.

Implication for Vortices Two definitions of core size

1 2 3 4 5

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Δ(x

)/Δ

()

r/

From quasiparticle gap

1 2 3 4 5

0.2

0.4

0.6

0.8

1

r/current

From supercurrent

j(x)

In conventional superconductors—dfns are equivalent

Generically in doped Mott insulator, expect current

diverges as doping x->0 while remains finite.

Vortex: circulating current Q~1/r

=>current-induced pairbreaking cuts off J as r->0

Current maximal at r=current

Ex: at T=0 if fermionic bound relevant (neglect E4)

S~x => current~1/x. For opt YBCO: current~100Z3A

Sensitivity to Z unfortunate

Experimental Evidencemuon spin rotation

Sonier et al PRL 79 2875 ‘97

YBCO-6.6 current ~50Å

YBCO-6.95 current ~20Å

Sonier et al PRL 83 4165 ‘99

Note! Strong field dependence not yet understood

Importance of current

Many ‘transport’ phenomena (HcII(T=0); boundary of ‘paraconductivity’ region of strong superconducting fluctuations are determined by condition

“When vortex cores overlap”




=>Important question??What do you mean by core??




=>Important question??What do you mean by core??

I will argue: should use current

Dissipation caused by moving vortex(after Tinkham, Superconductivity)

•Vortex—center position Xvort , (slow) velocity V •Supercurrent j~1/(R- Xvort) => dj/dt ~ V/(R- Xvort)2

•But j=-A is a vector potential so•dj/dt=dA/dt is an electric field

=> In presence of ‘normal fluid conductivity’ n , power P dissipated by moving vortex is

P=d2r n E2= d2r n V 2 /(R- Xvort)4 ~ (n / 2current) V 2

Bardeen-Stephen Result!! Note!! Use current-defined core size

Z->0: U(1) RVB Gauge theoryL B Ioffe and A I Larkin PRB39 8988 ’89L B. Ioffe and AJM PRB66 094513 ‘02

Separate electron into charge e ‘holon’ b charge 0 ‘spinon’ f

Assume ‘d-RVB’ pairing of spinons

Introduce ‘internal gauge field ‘a’ to project out unphysical degrees of freedom

‘spinon’ velocity v1 set by superexchange J constant as x->0

H in superconducting state 3 terms + 1 constraint

Boson stiffness B(r) length scale x-1/2 size x

Spinon stiffness F(r) length scale v1/ size pFv1

Mixing term: spinons feel a which couples to boson

Constraint: boson, spinon currents equal, opposite

Long distance theoryeliminate a

Physical superfluid stiffness ~x

Z ~x(q.p. -> spinons)

Main unphysical feature

Structure of Hphase => conventional 20 vorticesIoffe/Larkin PRB ‘98; Han & Lee PRL 85 1100 (‘00)

Physical s.c. phase

Structure of vortex:

Length scale associated with current: ~x-1/2 (see also Franz/ Tesanovic PRB63 064515 ’01)

Short length scale boson physicssolve for boson amplitude (r) and gauge field a(r)

Vortex Equation of Motion

j

Viscosity: = (/ 2current) vortex density nvort

tXvort=(z x j)

Evort=nvort j/

Current j

Etot=(1/n +nvort / )j =nvort 2

current

(1+ nvort 2current)

j

n

Big, Fast Vortices

•Large 2current (big vortex) => small (fast vortex)

•Fast vortex => no s.c. contribution to conductivity

•Current-defined cores overlap: --normal

•Quasiparticle-defined cores overlap: DOS-normal

Nernst Effect Wang PRL 88 257003 ‘02

T > HcII(T) (melt vortex lattice): resistivity quickly reverts to normal state value; Nernst coeff ey (V /T) more slowly

T

Idea: vortex has entropy =>thermal gradient T drives vortices => V

Nernst Effect

T

Idea: vortex has entropy =>thermal gradient T drives vortices => V

Speculation:

vortex entropy associated with quasiparticles, hence with quasiparticle-defined core

Vortex conductivity associated with current-defined core

=> Naturally ey(T), (T) differ

Subtleties: cf I Ussiskin et. al.

Summary II: Vortex statics

Generically expect: current-defined vortex core size diverges as Mott insulator phase is approached. (Lee & Wen PRL 78 4111 97; PRB64 224517 ’00)

???x-1/2 or x-1??? ??Fermionic or bosonic??

Diverging size: implications for dissipation; paraconductivity; Nernst

Experimental evidence??

Tc2(H) (resistive) is thermal melting of vortex lattice.

H

Raise T at fixed H

Simple argument

•Abrikosov lattice,spacing b

•Intervortex forceK~S

•Position fluctuations <XV

2>~T/K•‘Lindemann criterion’ <XV

2>=cL2 b2

cL2 = 0.01—0.1b

<XV2>

Blatter: Tc2(H) S(H, Tc2)/14

H

Here T<H so use S(H, 0)

Low T resistive critical field line maps out H-dependence of S

(A~1.5-1.7)

Quantal Melting of Vortex Lattice

‘Semiquantum’ argument:

•Abrikosov lattice, T=0spacing b

•Position fluctuations <XV

2>•‘Lindemann criterion’ <XV

2>=cL2 b2

cL2 = 0.01—0.1b

<XV2>

Need quantum fluctuations of vortices <--> vortex dynamics

•Add to previous action the term

i n t

•Treat dissipation properly

Vortex Equation of Motion:2 terms

Dissipation: coupling of vortex to spinon continuum

Vortex hall effect—from term it in superfluid action(cf Geshkenbein, Ioffe Larkin, PR55 3173 ’97)

Vortex Hall Effect

Superfluid action, condensate density n Ssuperfluid= i nt + ….

Move vortex around circle of area A

phase of point inside winds by

Action increases by An=>vortex feels field Beff=n

=>Measure by vortex Hall effect

Value of i nc t acceleration of vortex

•Galilean invariance, T=0—accelerating vortex drags all particles with it =>nc=ntotal =1

•Doped Mott insulator. T=0: effective Galilean invariance at low energy => nc=x =1

•Conventional sc, near Tc: ‘2 fluid’: ~(Tc/EF) •RVB: EF J and Tc set by phase fluct=> use not Tc .•D-wave nodes=>quasiparticles even at low T (high B) suggests /J except perhaps as T->0, B->0 Crossover not well understood

Data (Ong): <<1 (but perhaps increasing as x->0)

Estimate of fluctuations

Equation of motion

Force—from other vortices in lattice. Proportional to superfluid stiffness S

Constant K0<<1

FV=K XV

K=K0 nV S

2 limits: “fermion”—dissipative () term dominates “boson”—nondissipative () term dominates

Quantal Melting of Flux Lattice:Boson Limit

Boson limit (x->0; ->0)

Vortices: dissipationless particles in high Beff.

Wigner crystal to ‘vortex FQHE’ transition

cL2 = 0.01 => nV 0.1 (Beff/0) <<x

=>melting when nv2current 0.1

Quantal Melting of Flux Lattice:Fermion Limit

<XV2> =(1/ ) ln(1+/K)

Note crucial role of dissipation

Lindemann estimate

~Q

H2

0

F(H2/0)

F: function describing decrease in dissipation for r< Expect F(y) ~ ya with a>1 => lattice melts when current-defined cores touch

Nature of melted phase??

•‘Boson’ limit: QHE of vortices=>insulator (duality)

•Fermion limit: ‘soup’ of overdamped vortices. Likely to be almost classical (quantum diffusion for overdamped vortices: R~ln[t] so it takes ridiculously long for them to entangle).

Note also: quantal transition is first order!!

Summary: Melting

•Key length—current-defined core size

•Key parameter: non-dissipative coefficient

•Key question: what is a ‘quantum-melted vortex lattice’

Gauge theory—unphysical features—but ‘existence proof’ that a theory exists with these features

First order

Second order

H

T

Limits of Long Wavelength Action

We wrote: S=S0Q2+fermion terms

Theory breaks down when corrections to S0 become of same order as S0 . Two kinds of corrections

•Q4 (higher order gradients short range ‘bosonic’ physics•Fermionic excitations

Thus

S => S0Q2+fermion terms + EQQ4 +…

•Bosonic corrections important when

Q2~ S0/ EQ or T~ S0

•Fermionic corrections important when

??Which happens first?? Data suggest fermionic physics more important

Conclusions I

•Simple parametrization of low-T d-superconductor: vF, v, S, Z

Experiment:• Z~1 (problem for gauge th.)

•current is only carried by near-zone-diagonal states (angle-dependent spin-charge separation??)

Data not (yet) fully consistent

Δ

θ

Current only from these

Conclusions II

•Vortex size as defined from supercurrent crucial for transport, HcII(T=0)•Explanation for Nernst vs resistivity•Quantal melting of vortex lattice•Vortex hall coefficient ; charge renormalization Z--boson vs fermion physics

5 10 15 20 25

0.25

0.5

0.75

1

1.25

1.5

1.75

2

5 10 15 20 25

0.2

0.4

0.6

0.8

1

j(x)

Δ(x

)/Δ

()

Dissipation

Dissipation

Current-defined core radius

Dissipation

(Effective charge) 2


Dissipation



Factors from detailed calculation

Dissipation



Factors from detailed calculation

Key point—dissipation ~-2

Value of : II

=>interpolation formula: higher doping, smaller : fermion physicsLower doping, larger : boson physics

Experiment (vortex hall angle):

small in all high-Tc=> no ‘boson physics’ visible??

??at very small doping, low T, clean samples??

Phase diagram—II: ‘Pseudogap’

M. Suzuki , PRL 85 4787 (2000) (tunnelling— similar observations in photoemission)

‘

Tem

pera

ture

Doping

‘Pseudogap’—gap ΔPG in electronic DOS at T>Tc Note T-scale not too high

‘Pseudogap’ vs Superconducting Gap

T

Doping

V. M.Krasnov et al PRL 86 2657 (2001)

‘

Superconducting gap—feature ΔSC appearing inside pseudogap

ΔPG

ΔSC

K Lang et al Nature p. 412 v. 425 (2001)

Inhomogeneity in SC gap

indep of x

Vortex properties(assume 2d system for rest of talk)

Length scale in ρ(r) must diverge as approach Mott phase (Lee & Wen PRL 78 4111 97; PRB64 224517 ’00)

Scale over which supercurrent can vary must diverge as doping x->0

Implication for vortex core size

Current carried by states far from nodes? Possible test: ρS(T?)

T-dependence not useful: transition is driven by thermal phase fluctuations when ρS(Tc)=2 Tc/π (Uemura; Emery/Kivelson)

‘KT line’ ρS(T)=2T/π for bilayer system

But: quantal flucts weaker than thermal flucts….

Possible better test : ρS(B,T=0)

For a 2d vortex lattice, if Z is constant (A=1.5… for square or triangle lattice)

??? does this formula predict HcII(T=0) in underdopedIf HcII(T=0) is larger than this formula—angle-dependent Z?

ρS

B1/2

???At what field does vortex lattice quantum melt???

The Paradox:

Cor

son

et. a

l. c

ond-

mat

/981

0280

Superfluid stiffness v. temp

Vortices proliferate: whole sample is ‘vortex core’ for Tc<T<<T*

•d-wave like gap•s-c transition ‘K-T’-like

‘Pseudogap’:

=>’Pseudogap==(phase) fluctuating s.c.

IF SAMPLE IS ‘ALL CORE’: WHY DOES GAP PERSIST TO HIGH T?


Fermion limit ( important)

Ratio: gap to (small # times) superfluid stiffness. Very small in conventional SC; order 1 in high-Tc




Ratio: vortex spacing to current-defined vortex size




Ratio: vortex spacing to current-defined vortex size

Conventional SC: cores touch when lattice meltsHigh-Tc: melting when current-defined cores touch

vortices in the superconducting state of underdoped high-t c superconductors a. j. millis department...

Documents

high tc phase diagram

vortex viscosity

approach mott phase

t changes

small z

z vftcoupling

fermi surface v

charge renormalization