Why is high-Tc superconductivity exciting?

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Hole
-
Doped Antiferromagnets:

Relief of Frustration

Through Stripe Formation

John Tranquada




International Workshop on Frustrated Magnetism

September 13
-

17, 2004


Montauk, New York

Outline


Early ideas about La
2
CuO
4
: quantum spin liquid



Reality: La
2
CuO
4

is a good antiferromagnet



Hole doping

frustrates commensurate Néel order



Formation of charge stripes reduces magnetic frustration
(and lowers KE)



Are stripe correlations relevant to superconducting cuprates?

Anderson’s RVB proposal for La
2
CuO
4

PW Anderson, Science
235
, 1196 (1987)

“The oxide superconductors, particularly those … base on La
2
CuO
4
, … tend …
to occur near a metal
-
insulator transition … . This insulating phase is proposed
to be the long
-
sought ‘resonating
-
valence
-
bond’ state or ‘quantum spin liquid’
hypothesized in 1973. This insulating magnetic phase is favored by
low spin
,
low dimensionality
, and
magnetic frustration
.”

PW Anderson, Mat. Res. Bull.
8
, 153 (1973)

“Resonating Valence Bonds: A New Kind of Insulator”


Proposal for S=1/2 on a triangular lattice

Local RVB singlets

Kivelson, Rokhsar, and Sethna,

PRB
35
, 8865 (1987)


Existence of a spin gap leads to

Bose condensation of doped holes

Requires dynamic modulation of

superexchange by phonons

Reality
: Cu
-
O bonds are stiff

Frustration by AF next
-
nearest
-
neighbor exchange

Sachdev and Read, Int. J. Mod. Phys. B
5
, 219 (1991)

spin
-
Peierls order

Reality: An isolated CuO
2

plane would order at T = 0

S(
q
2D
) ~ 1 / [(
q
2D
)
2

+

-
2
]




= spin
-
spin correlation length



-
1

~ exp(
-

J/T)


J = 135 meV ~ 1500 K

Theory:


Chakravarty, Halperin,+Nelson,

PRB
39
, 2344 (1989)


Hasenfratz+Niedermayer,

PL B
268
, 231 (1991)

Expt:
Birgeneau
et al
., JPCS
56
, 1913 (1995)







as T


0

Spin waves in La
2
CuO
4
: No sign of frustration

J = 146 meV

J
c

= 61 meV at T = 10K

J’ = J’’ = 2 meV


Coldea
et al
., PRL
86
, 5377 (2001)

Typical Phase Diagram: La
2
-
x
Sr
x
CuO
4

Doping kills LRO but not SRO

Phase diagram for La
2
-
x
Sr
x
CuO
4

and

Y
1
-
2x
Ca
2x
Ba
2
Cu
3
O
6



p
sh

= x

Local magnetic field at T = 1 K

measured by muon spin rotation

Niedermayer, Budnick, et al.

PRL
80
, 3843 (1998)

Magnetic dilution

Destruction of LRO

requires 40% dilution!

Experimental results

for
La
2
Cu
1
-
z
(Zn,Mg)
z
O
4


Vajk
et al
., Science
295
, 1691 (2002)

Competing Interactions

Motion of hole

lowers kinetic energy



but

costs superexchange energy

One hole in an antiferromagnet

Dispersion measured by angle
-
resolved photoemision in Sr
2
CuO
2
Cl
2

Wells
et al
., PRL
74
, 964 (1995).

Bandwidth for occupied states is ~ 2J << 4t

Hole segregation to antiphase domain walls

1D

model

2D

extrapolation

Charge and spin stripe order

Early stripe predictions

Zaanen and Gunnarson

Phys. Rev. B
40
, 7391 (1989)


Hubbard model

Mean
-
field solution

White and Scalapino,


PRL 80, 1272 (1998)


t
-
J model

Density matrix renormalization group

Alternative: Frustrated Phase Separation

Löw, Emery, Fabricius, and

Kivelson, PRL
72
, 1918 (1994)

Competing interactions result in striped and checkerboard phases

Analysis of t
-
J model by Emery and Kivelson:


Holes tend to phase separate!



t
-
J model lacks long
-
range part of Coulomb interaction


Long
-
range Coulomb repulsion frustrates phase separation

Stripe ORDER seen only in special cases

1/8 problem

LTT

LTO

Antiferromagnetic “resonance” in SC cuprates


T
-
dependent resonance observed by Keimer
and coworkers in


YBa
2
Cu
3
O
6+x


bilayer


Bi
2
Sr
2
CaCu
2
O
8+



bilayer


Tl
2
Ba
2
CuO
6+



single layer


(But not in La
2
-
x
Sr
x
CuO
4
)



YBa
2
Cu
3
O
7

Mook et
al
., PRL
70
, 3490 (1993)

Spin fluctuations in YBCO do not look like spin waves

Bourges et al., Science
288
, 1234 (2000)

YBa
2
Cu
3
O
6.85

Bourges et al., PRL
90
, 147202 (2002)

La
1.79
Sr
0.31
NiO
4

Large crystals of La
1.875
Ba
0.125
CuO
4

studied on MAPS

Diameter = 8 mm

Length = 140 mm

Mass > 40 g

MAPS spectrometer


at ISIS

Crystals grown at BNL

by Genda Gu

Constant
-
energy slices through magnetic scattering


Stripe
-
ordered

La
1.875
Ba
0.125
CuO
4



T = 12 K



T
c

< 6 K

24 meV

34 meV

66 meV

105 meV

h

k

La
2
-
x
Ba
x
CuO
4




x = 1/8


Normal state


with


Stripe order


YBa
2
Cu
3
O
6.6



Superconducting


state

Hayden et al.,

Nature
429
, 531 (2004)

Comparison of LBCO and YBCO


Magnetic excitation spectra look the same!
(
E
LBCO

~ 1.5
E
YBCO
)


Implies same mechanism at work in both



Excitations in LBCO associated with stripes


Suggests stripe correlations present in YBCO



“Resonance peak” is just the most visible part of the spectrum


Present even in non
-
superconducting LBCO

How can we understand the stripe excitation spectrum?

Comparison with ladder model

2
-
leg, AF

spin ladder


J = 100 meV


two domains

Evidence for spin gap

Better theoretical models

Weakly
-
coupled stripes


Vojta and Ulbricht

cond
-
mat/0402377


Uhrig, Schmidt, and Grüninger

cond
-
mat/0402659


included 4
-
spin cyclic exchange

Mean
-
field stripe order + fluctuations


Seibold and Lorenzana

cond
-
mat/0406589



dispersion is more 2D
-
like

Universal Spectrum + Spin gap

LSCO(?)

YBCO(?)

Conclusions


Stripes form due to competing interactions (frustration)


Magnetic excitation spectrum of a stripe
-
ordered cuprate is
same as in good superconductors


Suggests a universal spectrum


Quantum spin gap of two
-
leg ladders may be important for
hole pairing

LBCO results:



Nature
429
, 534 (2004)





Collaborators

BNL


Hyungje Woo


Genda Gu


Guangyong Xu

IMR, Tohoku Univ.


Masa Fujita


Hideto Goka


Kazu Yamada

ISIS


Toby Perring

“Resonance” effects can be incommensurate

LSCO x = 0.16

Christensen et al.

cond
-
mat/0403439

Superconducting

Normal state

Effect of magnetic field in LSCO x=0.18

PRB
69
, 174507 (2004)

Expected scattering patterns in reciprocal space

Single
-
domain YBa
2
Cu
3
O
6.85

Hinkov
et al
., Nature
430
, 650 (2004)

E

= 35 meV


E
res

= 41 meV