Transcript Document
High Temperature Superconductivity:
Outline:
S. Kivelson
V. Emery
E. Carlson
M. Granath
V. Oganesyan
X-J. Zhou
Z-X. Shen
Basic facts concerning the cuprates
Stripes: What are they and why do they occur
Experimental signatures of stripes
Are stripes good or bad for superconductivity ?
Consequences of stripe formation:
• Fractionalization
• Confinement
D. Orgad
Racah Institute, Hebrew University, Jerusalem
The Cuprates: Basic Structure
La(2 - x)SrxCuO4
• Universal element – CuO planes
• Parent (undoped) compounds – Heisenberg antiferromagnets
• Hole doping by chemical substitution / Oxygen doping
The Cuprates: Typical Phase Diagram
Renner et al.
Harris et al.
Warren et al.
Takagi et al.
ARPES
NMR
DC resistivity
UD Bi2212
T
tunneling
Puchkov et al.
AF
Pseudogap
SC
under
optimal
doping
over
x
Optical conductivity
Neutron scattering, Specific heat …
The Central Question: What happens to an AF upon doping with holes?
Holes in an AF : Why Do Stripes Occur?
t J x t
Coulomb Interactions
PHASE SEPARATION
Kinetic Energy
H t
Frustration
( cis c js
ij s
STRIPES
ni n j
h.c.) J ( Si S j
)
4
ij
Stripes in Other Systems:
Competing Interactions
Ferrofluid between glass plates
Ferromagnetic garnet film
l~1cm
l~10mm
l~10mm
l~400A
Ferromagnetic garnet film
Block copolymers film
Stripe Signatures in S(k,w)
Real Space
Momentum Space
ky
kx
lss
l
ls
l
lccc
2
ls
Experimental Evidence for Stripes:
Neutron Scattering
k
y
Static stripe
order (LNSCO)
kx
0.25
E=24.5meV
Dynamic stripes
(YBCO)
Mook et al.
Tranquada et al.
0.12
Experimental Evidence for Stripes:
ARPES
Angle Resolved PhotoEmission Spectroscopy measures
the single hole spectral function A (k ,w ) dx dt ei ( kxw t ) ( x, t )(0,0)
n(k ) dw A (k ,w )
LNSCO
Experimental Evidence for Stripes:
Tunneling Microscopy
B=5T
B=0
Howald et al.
Hoffman et al.
Consequences of Stripe Formation:
Spin-gap and Enhanced SC Correlations
Doped Spin Ladders: known to be spin-gapped
s Je w
L R
RL
T
cos( 2 s )e
i 2c
s e
i 2 c
AF
Stripes
PG
SC
x
• The spin-gap creates an amplitude of the SC order parameter
• Provides high pairing scale (avoid Coulomb repulsion)
A Problem …
Good News:
In 1D a spin-gap enhances pairing:
divergent for Kc>1/2
(Kc<1 for repulsive interactions)
1 /( 2 K C 1 )
T c ~ E F ( g SC / E F )
Bad News:
It also enhances CDW correlations:
more divergent !
sc sT
( 2 Kc1 )
CDW
sT
( 2 K c )
1 /( 2 K C )
T c ~ E F ( g CDW / E F )
Old problem from search for organic superconductors
… And Its Resolution
T
Stripe fluctuations
(quantum, thermal or quenched)
are necessary for high Tc!
y2
y1
Nematic?
L
L2
1
Phase
Phase
Stiffness
PG
Stiffness
AF
x
SC
y
static
fluctuating
x
dissolved
Stripe fluctuations dephase CDW coupling: e
Yamada et al.
2ik F ( L1 L2 )
Stripe fluctuations enhance phase coupling: e
| y1 y2 |
e
2 k F L2
2 y 2
e2
Consequences of Stripe Formation:
Electron Fractionalization Above Tc
In a Fermi liquid the elementary
w excitations have the quantum numbers
of an electron
Mo surface
k
w v F k
state
multi-qp
background
Valla et al.
qp peak
In a Luttinger liquid the excitations come in
4 flavors RL cs
w
EDC
k
MDC
MDC (w 0) EDC ( k 0)
w vc | k |
c 0
c 0.3
w
w vs | k |
c 0.5
( L, c )
w v sk
w v ck
( L, s )
| w | v c k
| w | v s k
k
Evidence for Fractionalization
ARPES in La1.25Nd0.6Sr0.15CuO4
Breakdown of W-F Law
2 kB
L0
T 3 e
2
1DEG
s 0 , c 0.5
v s 0.7 eV A
in Pr1.85Ce0.15CuO4
v c 3.5 eV A
Orgad et al.
Sharp in Momentum Broad in Energy
Hill et al.
Below Tc: A Coherent Peak
Optimally Doped BSCCO (Tc=91K)
Not a Conventional QP
• Not present above Tc
• Intensity grows below Tc
• Energy and lifetime not
temperature dependent
Fedorov et al.
Josephson Coupling Confines 1D Solitons
The electronic operator L e
s and c
i
2
c c s s )
s , c
creates kinks in
2
x
Charge and spin solitons create phase shift in pair field
cos( 2 s )e
i 2 c
s
c
Frustrated Josephson Coupling H ijJosephson J SC [ i j h.c.]
between solitons
Bound spin-charge soliton pair
<
A (k,w) in the Superconducting Phase
A (k ,w ) Z (w E ) incoherent
• Quasiparticle weight depends on superfluid density:
Z (T , x ) (T , x )
( 2 c 1 / 2 )
Feng et al.
Conclusions
• Stripes are ubiquitous in the cuprate high temperature
superconductors
• They are important for high temperature
superconductivity
• There is evidence that the normal state of the cuprates
is fractionalized
• In a quasi-one-dimensional superconductor Tc also
marks a confinement transition
Landau Theory of Stripe Phases
Coupled charge (CDW) order k and spin (SDW) order SQ q
,
a a
* 2
1
2
4 1
2
4
F r | k | U | k | rS | SQ q | U S | SQ q | U x | SQ q SQ q |
2
2
l1[( SQ q SQ q ) k h.c.] l2 | SQ q |2 | k |2
k 2q
Stripes are “charge
driven” :
0
S 0
Spin order is secondary and may be absent
Zachar et al.
Spin-gap Proximity Effect
~
kF kF
Single particle tunneling irrelevant “system” “environment”
Pair tunneling K K
F
When pair
F
K~ K~
F
1 ~
~
~
Ks Ks 1
4 Kc Kc
F
possible
tunneling
kF
~
kF
~
~
H pair t cos( 2 s ) cos( 2 s ) cos[ 2 ( c c )] is relevant.
The spin modes and the relative charge phase mode are gapped.
The only gapless mode involves the total SC phase c ~c
• Kinetic energy driven pairing
• Repulsive interactions within system and environment increase
• Repulsive interactions between system and environment decrease
• Pre-existing spin-gap in environment decreases
ARPES and Stripes
Angle Resolved PhotoEmission Spectroscopy measures
the single hole spectral function A (k ,w ) dx dt ei ( kxw t ) ( x, t )(0,0)
n(k )
dw A ( k , w )
LNSCO
LNSCO
LSCO
m
dw A ( k , w )
m 30 meV
Zhou et al.
Disordered Stripe Array: Spectral Weight
Low Energy Spectral Weight
( )
m
1
dw eik ( r r ') n ( r )n ( r ' ) (w En )
S r ,r '
n
m 0.2
( )
Granath et al.
( )
Disordered Stripe Array:
Interacting Spectral Function
Granath et al.
A Model:
Quasi-one-dimensional Superconductor
Charge: Gapless
Spin: Gapped
Weak Pair Tunneling
(Couples charge and spin)
Prediction: New Magnetic Resonance
Neutron scattering measures the spin-spin correlation function:
dx dt e
i ( k x w t )
S2 k
F
( x, t ) S2k (0,0)
F
1
S2k R ' L ' creates 2 spin solitons and 2 charge solitons
F
2 , '
Treat more massive spin solitons as static and solve for the charges:
s
s
2
c (x )
Hc
vc
[K c ( x c )2 ( xc )
2
Kc
] c ( x) cos(
2 c )
Get effective Schrodinger equation for spins:
H
eff
vs2 2 2
2 s
V ( x1 x2 )
2
2 s j 1 x j
•Spin 1 mode that exists below 0.4 Tc
,0
•2kF mode: should appear around
2
•Threshold at 2s