A Quarter Century of High Temperature Superconductors

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Nov 15, 2013 (3 years and 9 months ago)

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A Quarter Century of High Temperature Superconductors





T. Maurice Rice






ETH Zurich,






Hong
Kong U
., K.
-
Y. Yang,
F.C
.
Zhang

& W.
-
Q. Chen





Brookhaven
Natl. Lab
. R.
Konik
, A.
Tsvelik

&
A.James





Introduction to the
Cuprates

and the exotic
Pseudogap

Phase



Lower sights
-
> Phenomenological YRZ propagator in
Pseudogap

Phase



Comparison to
E
xperiment ARPES ,


Andreev
T
unneling Spectra


etc.






Excitations in
Pseudogap

Phase


Spin Response


PSI 16 March 2012


High Temperature Superconductivity

CuO
2

plane

Copper
-
oxide compounds

1986:

J.G. Bednorz & K.A. Müller

La
2
-
x
Ba
x
CuO
4

T
c

=35 K

AF

SC

T

x

T
N

T
c

T*

Doped antiferromagnetic

Mott insulator

under optimally over

doped

pseudogap

strange

metal


Tc up to 133K Schilling & Ott ‘93

They are
unconventional d
-
wave
superconductors


and materials with many anomalous properties

Generic Phase Diagram

Cu
2+

spin S = 1/2

CuO
2

plane electronically relevant

Parent compound: La
2
CuO
4


e
g

t
2g

x
2
-
y
2

3z
2
-
r
2

yz

zx

xy

1 hole in 3d
-
e
g

Cu
2+

3d
9

Single ½
-
filled Band of Cu
-
O hybridized States


Strong Onsite Coulomb
interaction


Single Band Hubbard Model on a 2D square lattice

CuO
2
-
plane

O

Cu

2D square
lattice of Cu
-
ions

O
-
octahedra

Mott limit
-

electrons localized in real space

Example: Lattice of H
-
Atoms
:

a
B

<< d


Onsite e
-
e
-

repulsion:
U

= E(H
+
) + E(H
-
)

Electrons localized:
Mott Insulator at 1 el/site

Low
-
energy physics purely due

to electron spins




H
Heisenberg

J

S
i


S
j
i
,
j

antiferromagnetic spin order


generally
at low T

H
+

H
-

H

2a
B

d

-
t

S=1/2



U
STRONG

Fundamentally different from a band
insulator

No breaking of translational symmetry involved

U > 2zt

Hubbard Model Kin.
Energy

+ U




U
MODERATE

U < 2zt
z
:
nn

number



Electrons

itinerant
:
metallic
state

at

1el./
site



-

-

-

-

-

-

-

-

-

-

-

-

-



General Remarks


Cuprates

described by a ‘ simple’ model


Hubbard model
2D
square lattice


Cuprates

are
highly quantum
with only a single relevant orbital e.g. Cu
2+
& Cu
3+


-

Favors Superconductivity ?


Contrast to
--

d
1
-

oxides (Ti2O3, VO2
etc

forming singlet dimer lattices ),





--

Nickelates

[
Jahn
-
Teller
polarons

when doped Ni
2+
-
> Ni
3+
] . .



--

Also Fe
-
pnictides

less symmetric [leading to a lower
Tc

? ]




Is this why the
cuprates

are such special superconductors ?

Full

Metal

with

Large Fermi
Surface

Mott Insulator

No

Fermi
Surface


Charge Gap


Pseudogap

Phase : Hall
effect

as

a
doped

Mott
insulator






Translational

s
ymmetry

preserved

in



clean
underdoped

samples

: 124 & YBCO6.5


unlike

Cr
-
alloys
:



Comm
. AF


Incomm

AF


para.metal






v
F
|
nodal


>
const
.
as

n
h

-
>
0
unlike

3
He






AF

D
-
SC

T

X

: doping

T
N

T
c

T*

How

do
cuprates

crossover

from

full

metal

to

Mott
insulator

?

X


0.2


QCP ?

OVERDOPED

UNDERDOPED

Vignolle

et al Nature 2008

G


Breakdown
of

Landau Fermi Liquid in
Overdoped

Cuprates





Recent

Experiments on Single Layer
Overdoped

Tl
2
Ba
2
CuO
6+x


N.Hussey

and

collaborators




Angular
Dependent

Magnetoresistance

(ADMR) in High
Magnetic

Fields




Superconductivity

Suppressed

( B = 45T )
---
> Normal State
at

low

T



Strongly

Anisotropic

Scattering

rate
around

the

Fermi
surface



Largest

between

antinodal

regions

:
Grows

as

doping

decreases



Agrees

with

results

of

RG
calculations

for

2D Hubbard
model


Honerkamp

et al PRB 2001,
Ossadnik

et al PRL 2009





Single

2D
Band






Full

Metallic Fermi
Surface

Also
seen

in Quantum
Oscillations


Fermi
Surface

agrees

with

LDA

M

antinodal

nodal

G

Full

Metal

with

Large Fermi
Surface

Mott Insulator

No

Fermi
Surface


Charge Gap


Pseudogap

Phase : Hall
effect

as

a
doped

Mott
insulator









Translational

s
ymmetry

preserved

in clean



underdoped

samples
: YBa
2
Cu
4
O
8
& YBCO6.5






AF

D
-
SC

T

X

: doping

T
N

T
c

T*


Pseudogap

Phase
between

full

metal

&
Mott
insulator

?

X


0.2


QCP ?

OVERDOPED

UNDERDOPED

Vignolle

et al Nature 2008

G

^

T
c


Spin Gap
=>Singlet
Pairing of Cu
2+
-
Spins in the
Pseudogap

Phase

Well Ordered and


Underdoped




Continuous
Onset of Spin Pairing in Normal
Phase


no kinks




Spin
Susceptibility well below AF value at T ~
T
c

indicating



singlet pairing rather than AF fluctuations

YBa
2
Cu
4
O
8

Knight Shift ~

Spin
Susceptibility


Not Pauli like at T
C
< T < T*

Bankay,Mali

et al
PRB (‘94)

2D RVB State which is a


superposition of configurations


with Singlet Pairs


can be written as a projected


BCS
-

State.

singlet

Resonating Valence Bond Theory

Doping allows
singlets

to move as electron
pairs



Elegant idea but difficult to develop a microscopic theory



for this strongly interacting
fermionic

state with


strong short range correlations.

Proposed
by P.W.
Anderson 1987

Singlet energy gain is 3x Classical energy



|

>

S = 0


Fermi surface in the
pseudogap

phase







ARPES shows full Fermi Surface 4 nodal arcs



Norman,
Campuzano
, H. Ding . . ‘98





Ca
2
-
x
Na
x
CuCl
2
O
2
-
Shen

et al ‘05

G

ARPES signal within
±

10
meV

of Fermi energy shows no weight at
antinodal

energy gap



Decrease

Doping
thru
‘ QCP
=>
Pseudogap

Phase
with

an
Antinodal

Gap



??



Full

Fermi
Surface

=>
Nodal

Fermi Pockets [
Arcs
]



1
-

Gap Scenario
: Cooper d
-
wave Pairing Gap opens on full Fermi Surface at T*


Tc

<< T*
:

strong phase fluctuations ( n
s
~ x (hole density) )




=> near
antinodal


sc

>>
Tc

=> Fermi arcs in ARPES at T>
Tc




2
-

Gap Scenario
:
Pseudogap

is an insulating gap opening near
antinodal

k
-

points


=> RVB charge gap opens on AFBZ

a fixed surface in
k
-

space



due to
ph

&
pp

umklapp

scattering


-

Precursor to Mott insulator at zero doping




=>

Various Experiments at T <
Tc

support insulating character of
pseudogap


e. g. Andreev & Giaever Tunneling Scales in over
-

&
underdoping


[
Deutscher

RMP 2005, Yang et al PRL 2010]







Mott Insulating State viewed in k
-
space


Real space
k
-
space


Underlying lattice
Umklapp

scattering processes allowed



-
> Momentum conserved modulo {
G
}



Band filling 1 el./site Surface in
k
-
space enclosing



an area of ½
-

Brillouin

zone.




Conclusion ; look at U
-
surface which is






a) spanned by elastic U
-

scattering processes



b) encloses an area of ½
-
Brillouin

zone.

-
t

2
-
Leg Hubbard Ladders
-
> A simple model with a Mott State at weak coupling


½
-
Filling
-
> Unique Insulating D
-
Mott
Groundstate

:
Dagotto

& TMR Science

96



Balents
, Fisher et al PRB . . .





Charge & Spin Gap due to
Umklapp

Scattering Processes at
E
f




e.g. K
f1R

& K
f2R

-
>
K
f1R

& K
f2R

:

momentum change 2π




Translational Symmetry along legs preserved







Short Range AF & d
-
wave Pairing correlations






Single Particle Propagator takes a BCS form: G(
k
x
,
w
) =



but without an anomalous component: F = 0.



Konik

& Ludwig PRB (2001)

+..

-
t


YRZ
Ansatz

for Green
`s Fn. in analogy with coupled ladders





K.
-
Y. Yang, Rice
& F. C.
Zhang PRB ‘06
see
R.Konik
, Rice
& A.
Tsvelik

PRL
`05



RVB Gap

R
(
k
)

opens

on p
-
p
Umklapp

Surface

( = AF
Brillouin

Zone in 2D)




F
ixed
line

of

zeros

in G
RVB
(
k
,0)
on
Umklapp

Surface

:

no

change

with

doping




U
-
Surface

encloses

1
-
el./
site

and

is

spanned

by

U
-

scattering

processes

Gutzwiller

Renorm.Factors




nn

nnn

nnnn

hopping

t
(x) =
g
t
(x)t
0
+(3/8)
g
s
(x)
J
c


t’(x)=
g
t
(x)t’
0



0
(x)

> 0
at

x =x
c
( = 0.2) : RVB Gap
from

Renorm
.
Mean

Field
Theory



-
F. C. Zhang
et al
`88

Note: t
-
J model
-
a
spinon
-
holon

boundstate

=> el.
G(
k,
w
)
with
YRZ form

P.A.Lee

2011

Pairing Self Energy in a normal Green’s Fn.


but with energy fixed on the U
-
surface

infinities

zeros

G(
k
,0) > 0 in shaded area bounded by zeros and infinities


Luttinger
-
Dzyaloshinskii

Sum Rule => Lines of
zeros
:

1
el./ site

Fermi Pockets
evolve

into

Full

Fermi
Suface

as

doping

x
increases

G

Special Form of Nodal Pocket

QP Spectral Weight


very anisotropic


looks like an Arc








Front Large

Back Small

Pocket arises due to back
-
bending of the



Bogoliubov


Quasiparticle

dispersion

l
eading to particle


hole asymmetry in the pocket






N.B. Pocket ends in a Dirac point

Particle
-
Hole
Asymmetry

in ARPES

-
BNL Group

Peter Johnson





H.B.Yang

et al Nature
`08

I(
w)
= A(
w
)f(
w
)

ARPES with
E
nhanced Resolution


H.
-
B. Yang et al PRL 2011

QP dispersion extrapolated form maximum


Evolution of Nodal Pockets with doping

QP dispersion &


spectral weight in YRZ

QP dispersion

spectral

weight

( 0,0)

(π,π)

Angle Integrated PES
-

Hashimoto et al PRB (2009)

Experiment
-
0.2
-
0.1 E
F

+0.1(
eV
)

Theory

K.
-
Y. Yang et al EPL (2009)

Hole Pocket ends in a Dirac Point



on the nodal line.



DOS ~ E [ ≠ const. ]

-
0.2
-
0.1 E
F

+0.1



YRZ
Ansatz 2
-
Gap
Phenomenology







Antinodal

energy

gap

different
to

dSC

gap

{
mainly

on
nodal

pockets
}




YRZ
Ansatz
for

dSC

used

to

model

T & x
dependence

of

:





Raman
Scattering


Valenzuela
-
Bascones

PRB ‘07
;
PRB ‘08



Carbotte
, Nicol &
collaborators

PRB ‘10





Specific

Heat


Carbotte
, Nicol &
collaborators

PRB ‘09





Infrared
Conductivity

Carbotte
, Nicol &
collaborators

PRB ‘09





Penetration
Depth


Carbotte
, Nicol &
collaborators

PRB ‘10





Andreev Tunneling

K
-
Y. Yang et al
PRL

10



Review
:
-

TMR, K.
-
Y. Yang & F. C. Zhang
Rep.
Prog
. Phys.
75
,
01650 (2012)








Generalized Drude form for optical conductivty

Infrared
optical

conductivity



Illes,Carbotte

& Nicol PRB
`09

London
penetration

length


Carbotte

et al PRB
`10



Tunneling
Experiments
:
Transparent
Barrier

=
> Andreev
Reflection



High
Barrier

=>
Giaever [Single
Electron
]
Tunneling

BCS
S
uperconductors

Voltage

Scales
:

Andreev [ |V| <
 ]
& Giaever [|V|>
]

tunneling

are

equal




Overdoped

Cuprates

-
>
Equal

Voltages

Scales

are

observed



Pseudogap

Phase
-

> Different
Voltage

Scales

are

observed



Calculate

tunneling

conductance

using

YRZ
generalized

to

d
-
wave

sc

in
the

Keldysh

formulation





-

K
-
Y.
Yang,
K.Huang,W
-
Q Chen, TMR &
F.C.Zhang

PRL 2010



Giaever

Andreev

See Deutscher RMP (2005)

Overdoped Cuprates: SC Gap over Full Fermi Surface

Equal

Energy

Scales


Unequal

Voltage

Scales
: Andreev
conductance

s
A
s

; Giaever
pseudogap

peaks

in
s
n

s
s


Tunneling
into

Underdoped

Cuprate
,
[x
=
0.1]
: Andreev vs. Giaever


YRZ
:
SC
pairing

amplitude

and

gap




antinodal

Pseudogap



mainly

on
the

arcs

(
pockets
) in normal [
n
]
and

sc

[s]
states

Spin Response in the YRZ Model
: Andrew James, Robert
Konik

& TMR



Brookhaven Natl. Lab.
arXiv

1112.2676



Spin Response in RPA using YRZ propagators to calculate bare response :

Spin
Excitations

in
Pseudogap

Phase



>
Hourglass

Spectrum




> Triplet
magnon

in
the

pseudogap






&
qp

transitions

at

low

energies

[
suppressed

below

Tc

]


Hourglass Spectrum rescaled
expts
.

LSCO, LBCO & YBCO
-

underdoped


Gapped Region

Hole Pockets

Stock et al PRB 2004


Conclusions
: main points of YRZ phenomenological theory




Interpolates between Fermi pockets at nodal & insulating gap at
antinodal




without translational symmetry breaking



also found in DMFT calculations


Georges,
Kotliar
, Millis, Tremblay + . . . .






Superconductivity in
Pseudogap

Phase supported through coupling of


4 near
-
nodal Fermi pockets and
Cooperon

mode at
antinodal
.



[
Konik,TMR

&
Tsvelik

PRB (2010) ]






Magnetic Response has Hourglass Form in
Pseudogap

Phase.



AF order at small x thru’ softening of the Triplet
Magnon

mode






Pseudogap

Phase is an Unstable Fixed Point
[a phase with only short range



correlations

analagous

to
the Landau
Fermi
liquid in a metal]
,




-

unstable against
dSC

etc.
at low T [ Anderson
Physica

B ’02]