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
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