1

Disorder and Zeeman Field-driven superconductor-insulator

transition

Nandini TrivediThe Ohio State University

“Exotic Insulating States of Matter”, Johns Hopkins University, Jan 14-16, 2010

Karim Bouadim

Yen Lee Loh

Mohit Randeria

See Poster

SC

I “disorder”

*SC I

disorder

amorphous quench condensed films

Haviland et.al. PRL 62, 2180 (’89)Valles et.al. PRL 69, 3567 (’92)Hebard in “Strongly Correlated Electronic Systems”, ed. Bedell et. al. (’94)Goldman and Markovic, Phys. Today 51, 39 (1998)

SUPERCONDUCTOR-INSULATOR TRANSITION

What kind of insulator?Exotic?Unusual?Trivial?Band?Anderson?Mott?Wigner?Topological?Quantum Hall?Bose Glass?Fermi glass? Vortex glass?

QPT

T

Outline of talk:

Focus on three puzzling pieces of data:

Adams: Origin of low energy states in tunneling DOS in field-tuned SIT

Sacépé: Disappearance of coherence peaks in density of states above Tc

Armitage: Origin of states within the SC gap observed in €

h||

€

Reσ (ω)

H

€

−|U | ni↑ni↓

i

∑

€

+ (Vi − μ − hσ )ni

i

∑

..,

chcct jji

i

Kinetic energy

+Attraction

+

(U controls size of Cooper pairs)

Random potential

P(V)

-V 0 V

V=0 s-wave SC|U|=0 localization problem of non-interacting electrons

* Ignore Coulomb interactions

Zeeman Field

Model: Attractive Hubbard + disorder + field

2D

5

Determinantal Quantum Monte Carlo

No sign problem for any fillingKeeps both amplitude and phase fluctuations

Bogoliubov-de Gennes-Hartree-Fock MFT

Maximum entropy method for analytic continuation

€

nσ (r) ≡ crσ+ crσ

Δ(r) ≡|U | cr↑+ cr↓

+ ≡|U | F(r)

BdG keeps only amplitude fluctuations

Local expectation values Solve self consistently

),(1

)0()(),(

kAe

edcTckG kk

k

kAN

N ),(1

)(

Methods

DOS and LDOS

6

Part I: Superconducting Film in Zeeman Field:

Soft Gaps in DOS

ExperimentTunneling conductance into

exchange-biased superconducting Al films

states in gap

Catelani, Xiong, Wu, and Adams, PRB 80, 054512 (2009)

Where do the states at zero bias come from?

Magnetic impurities?Orbital pair breaking???

Also Adams, private communication

7

Part I: Superconducting Film in Zeeman Field:

Soft Gaps in DOS

ExperimentTunneling conductance into

exchange-biased superconducting Al films

TheoryDisordered LO states

provide spectral signatures at low

energy

Loh and Trivedi, preprint

states in gap

Catelani, Xiong, Wu, and Adams, PRB 80, 054512 (2009)

€

N(E)

8

FLBCS

h

polarization mΔ0

pairing Δ

k−k ↑↑ ↓

SC + Zeeman field

Zeeman field

€

hc =1

2Δ0 = 0.707Δ0

Chandrasekhar, Appl. Phys. Lett., 1, 7 (1962); Clogston, PRL 9, 266 (1962)

9

FLBCS Weak LOStrong LO

Δm

Δm Δ

m

xΔ

m

Modulated (LO) SC order parameter

hc

h

Δ0

hc1 hc2

polarization m

pairing Δ

Microscale phase separation = polarized domain walls

Y-L. Loh and Trivedi, arxiv 0907.0679

h = 0.8

I. Paired unpolarized SC

Local magnetization

Local Pairing amplitude

Spin resolved DOS

DOS

Disorder + Zeeman field

€

V = 2t

U = −4 t

€

Nσ (ω)

€

N(ω)

€

€

Disorder + Zeeman fieldh = 0.95

Disordered LO

+ and − domains

softgap

€

V = 2t

U = −4 t

€

Nσ (ω)

€

N(ω)

€

€

12

Close-up view of a disordered LO state (h=1)

+ pairing

- pairing

magnetization in domain walls

F<−0.05F>0.05 m>0.05

€

V = 2t

Disorder + Zeeman fieldh = 1.5

Non-superconducting

+ and − domains

€

V = 2t

€

Nσ (ω)

€

N(ω)

€

€

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Magnetization Pairing Spectrum

softgap

BCS

DisorderedLO

+ and - domains

Normalstate

hard gap

gapless

€

N(ω)

€

Nσ (ω)

Part II:

Local and Total Density of States

Ghosal, Randeria, Trivedi PRL 81, 3940 (1998); PRB 65, 14501 (2002)

Previous Results:Self consistent mean field theoryBogoliubov de-Gennes (BdG)

€

nσ (r) ≡ crσ+ crσ

Δ(r) ≡|U | cr↑+ cr↓

+ ≡|U | F(r)

Pairing amplitude DOST=0

Ghosal, Randeria, Trivedi PRL 81, 3940 (1998); PRB 65, 14501 (2002)

Previous Results:Self consistent mean field theoryBogoliubov de-Gennes (BdG)

Gap in single particle DOS persists in insulator

T=0

€

nσ (r) ≡ crσ+ crσ

Δ(r) ≡|U | cr↑+ cr↓

+ ≡|U | F(r)

Pairing amplitude DOS

GAP

€

ρS

Pairing amplitude map r)

Why is the gap finite? Where do excitations live?

Low

est e

xcit

ed s

tate

s

high hills: empty

deep valleys: trapped pairs no number fluctuations

SC islands formed where |V(r)-| is small

Lowest excited states live on SC blobs

GAP PERSISTS

Ghosal, Randeria, Trivedi PRL 81, 3940 (1998); PRB 65, 14501 (2002)

What happens when phase fluctuations are included?

U=-4tn=0.88

SC

N

PG (generated by disorder)

INS

Phase Diagram

Pairing scale

Coherence scale

21

T

Disorder

€

V

SC

N

INS

Determining T*: peak in spin susceptibility

T*

22

T

Disorder

€

V

SC

N

INS

Twisted Boundary Condition

€

E =1

2ρ s(∇φ)2€

ρS

€

ρS

Determining Tc:Vanishing of Superfluid stiffness

T*

Tc

Spectral properties

T

Disorder

€

V

SC

N

INS0 1.6

0.2

0.33

T

Disorder

€

V

SC

N

INS0 1.6

0.2

0.33

T

Disorder

€

V

SC

N

INS0 1.6

0.2

0.33

T

Disorder

€

V

N

INS0 1.6

0.2

0.33

SC

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QMC DOS for SC: T dependence

Tc(QMC) ~ 0.12

T*(QMC) ~ 0.6

SC gapCoh peaks

PseudogapCoh peaks destroyed

Gapless

T < Tc

T > Tc

T = Tc

V=1

T

Disorder

€

V

SC

N

INS0 1.6

0.2

0.33

U=-4t

PG

29

Temperature Dependence of DOS

Experiments:Scanning tunneling spectroscopy

(B. Sacépé et al.)

Theory:Bogoliubov-de Gennes-Hartree-Fock,determinant quantum Monte Carlo

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QMC DOS: V dependence

T=0.1Ins gap

No coh peaks

SC gapCoh peaks

BdG gap

Vc~1.6tT

Disorder

€

V

SC

N

INS0 1.6

0.2

0.33

U=-4t

€

V = 0

Egap,QMC ~ 0.7t

Egap,BdG ~ 1.4 t

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DOS: Summary

V

N

V

T

SC INS

N

V

T

SC INS

SC gap closesCoh peaks die

INS gap closesNo coh peaks

Gap survivesCoh peaks die

N

T

SC INS

Local Density of States

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Coherence peak destroyed;incoherent weight builds up

Coherence peak survives

Site (5, 4)Pairing survives with V

Site (5, 1)Pairing destroyed by V

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Local DOS: T dependence

Pairing FBdG(r, T) disappears at every site at the same temperature, T=TBdG.

“Coherence peaks” in LDOS NQMC(r, ω, T) disappear at every site at the same T ~ Tc.

Pseudogap remains on every site up to T*.

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c.f. Experiment (Sacépé):Scanning tunneling spectroscopy on an amorphous InOx film

(thickness 15 nm, on Si/SiO2 substrate)with Tc ~ 1.7 K, at two different locations at various T

Local DOS: T dependence

“Coherence peaks” disappear at every site at the same temperaturePseudogaps still exist above Tc

Main Results:

1. Disordered LO states provide spectral signatures at low energy for Zeeman-field tuned superconductors

In disorder tuned transition the gap survives BUT coherence peaks die at V~Vc

2. Coherence peaks disappear at every site at the same T~Tc

Pseudogaps disappear at every site at T ~ T*

Disorder Paired InsulatorPhase disordered

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The End