Transcript Local density of states and response functions for the disorder

Disorder and Zeeman Field-driven
superconductor-insulator transition
Nandini Trivedi
The Ohio State University
Karim
Bouadim
Yen Lee
Loh
Mohit
Randeria
See Poster
“Exotic Insulating States of Matter”, Johns Hopkins University, Jan 14-16, 2010
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SUPERCONDUCTOR-INSULATOR TRANSITION
amorphous quench
condensed films
QPT
T
*
I
I
disorder
What kind of insulator?
Exotic?
Unusual?
Trivial?
Band?
Anderson?
Mott?
Wigner?
Topological?
Quantum Hall?
Bose Glass? Fermi glass? Vortex glass?
“disorder”
SC
SC
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)
Outline of talk:
Focus on three puzzling pieces of data:

Adams: Origin of low energy states in tunneling
DOS in h field-tuned SIT
||


Sacépé: Disappearance of coherence peaks
in density of states above Tc
Armitage: Origin of states within the SC gap
observed in Re()

Model: Attractive Hubbard + disorder + field
H
Kinetic energy
ci c j  h.c.

2D
i , j 
+
Attraction
(U controls size of Cooper pairs)
P(V)
t
+
Random potential

-V 0 V
 |U |  nini
i
(Vi    h )ni
i
Zeeman Field

V=0 s-wave SC
|U|=0 localization problem of non-interacting electrons
* Ignore Coulomb interactions
Methods
Bogoliubov-de Gennes-Hartree-Fock MFT


Local expectation values
Solve self consistently
n (r)  c r c r
 
(r) |U | c r
c r |U | F(r)
BdG keeps only amplitude fluctuations

Determinantal Quantum Monte Carlo
No sign problem for any filling
Keeps both amplitude and phase fluctuations
Maximum entropy method for analytic continuation
1
N ( )   A(k ,  )
N k

 e  
G(k , )   Tck ( )c (0)    d 
A(k ,  )
  
1  e 

5
DOS and LDOS

k
Part I:
Superconducting Film in Zeeman Field:
Soft Gaps in DOS
Where do the states at
zero bias come from?
Magnetic impurities?
Orbital pair breaking?
??
states
Experiment
in gap
Tunneling conductance into
exchange-biased
superconducting Al films
Catelani, Xiong, Wu, and Adams,
PRB 80, 054512 (2009)
Also Adams, private communication
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Part I:
Superconducting Film in Zeeman Field:
Soft Gaps in DOS
N(E)

states
Experiment
in gap
Tunneling conductance into
exchange-biased
superconducting Al films
Catelani, Xiong, Wu, and Adams,
PRB 80, 054512 (2009)
Theory
Disordered LO states
provide spectral
signatures at low energy
Loh and Trivedi, preprint
7
SC + Zeeman field
−k
↑↑ ↓
k
BCS
FL
Δ0
pairing
Δ
polarization m
1
hc 
 0  0.707 0
2
h
Zeeman field

Chandrasekhar, Appl. Phys. Lett., 1, 7 (1962); Clogston, PRL 9, 266 (1962)
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Modulated (LO) SC order parameter
Δ
m
Δ
m
BCS
Δ0
pairing Δ
Strong LO
Δ
Weak LO
x
FL
Microscale
phase
separation =
polarized
domain walls
polarization m
hc1
Y-L. Loh and Trivedi, arxiv
0907.0679
m
m
Δ
hc
hc2
h
9
V  2t
U  4t
Disorder + Zeeman field
Local magnetization
h = 0.8
Spin resolved DOS
N ( )

Local
Pairing
amplitude

N()
DOS


I. Paired unpolarized SC

V  2t
U  4t
Disorder + Zeeman field
h = 0.95
N ( )


N()
+ and −
domains


Disordered LO

soft
gap
V  2t
+
pairing
pairing
m>0.05
F>0.05
F<−0.05
magnetization in
domain walls
Close-up view of a disordered LO state (h=1)
12
Disorder + Zeeman field
V  2t
h = 1.5
N ( )
 
+ and −
domains
N()

 
Non-superconducting
Magnetization Pairing
Spectrum
N ( )
hard
gap
BCS
+ and domains
Disordered
LO

N()

soft
gap
gapless
Normal
state
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Part II:
Local and Total Density of States
Previous Results:Self consistent mean field theory
Bogoliubov de-Gennes (BdG)
T=0
Pairing amplitude
n (r)  c r c r
 
(r) |U | c r
c r |U | F(r)
Ghosal, Randeria, Trivedi PRL 81, 3940 (1998); PRB 65, 14501 (2002)
DOS
Previous Results:Self consistent mean field theory
Gap in single particle DOS
persists in insulator
Bogoliubov de-Gennes (BdG)
Pairing amplitude
T=0
GAP
S
n (r)  c r c r
 
(r) |U | c r
c r |U | F(r)
Ghosal, Randeria, Trivedi PRL 81, 3940 (1998); PRB 65, 14501 (2002)
DOS
Why is the gap finite? Where do excitations live?
Lowest excited states
Pairing
amplitude
map (r)
~0
~0
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?
Phase Diagram
U=-4t
n=0.88
Pairing scale
N
PG (generated
by disorder)
SC
Coherence scale
INS
Determining T*: peak in spin susceptibility
N
T*
T
SC
INS
Disorder V

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Determining Tc:
Vanishing of Superfluid stiffness
N
S
T*
T
SC
Tc
INS

Disorder V
S

Twisted Boundary Condition
1
E  s ( )2
2
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Spectral properties
N
0.33
0.2
T SC
0
INS
1.6V
Disorder

N
0.33
0.2
T SC
0
INS
1.6V
Disorder

N
0.33
0.2
T SC
0
INS
1.6V
Disorder

N
0.33
0.2
T
INS
SC
0
1.6V
Disorder

QMC DOS for SC: T dependence
U=-4t
V=1
Gapless
N
0.33
PG
T*(QMC) ~ 0.6
0.2
T
SC
0
Disorder
INS
1.6
Pseudogap
Coh peaks destroyed
V
T < Tc

T = Tc
T > Tc
Tc(QMC) ~ 0.12
SC gap
Coh peaks
28
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
U=-4t
T=0.1
N
0.33
Ins gap
No coh peaks
0.2
T
SC
0
Disorder
INS
1.6
Vc~1.6t
V
BdG gap

V 0
E gap,QMC ~ 0.7t
SC gap
Coh peaks
E gap,BdG ~ 1.4t
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DOS: Summary
T
T
N
SC
N
N
INS
V
Gap survives
Coh peaks die
SC
T
SC
INS
INS
V
V
SC gap closes
Coh peaks die
INS gap closes31
No coh peaks
Local Density of States
Site (5, 4)
Pairing survives with V
Coherence peak survives
Site (5, 1)
Pairing destroyed by V
Coherence peak destroyed;
incoherent weight builds up
33
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*.
34
Local DOS: T dependence
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
“Coherence peaks” disappear at every site at the same temperature
Pseudogaps still exist above Tc
35
Main Results:
1. Disordered LO states
provide spectral signatures
at low energy for Zeemanfield tuned superconductors
2. Coherence peaks disappear at
every site at the same T~Tc
Pseudogaps disappear at every site at T ~ T*
In disorder tuned transition the
gap survives BUT coherence peaks
die at V~Vc
Disorder
Paired Insulator
Phase disordered
The End
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