# Presentation: 642093

## 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 La2CuO4: quantum spin liquid
- Reality: La2CuO4 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 La2CuO4

- PW Anderson, Science **235**, 1196 (1987)

- “The oxide superconductors, particularly those ... base on La2CuO4, ... 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 CuO2 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 La2CuO4: No sign of frustration

- J  = 146 meV
- Jc =   61 meV              at T = 10K
- J’ = J’’ = 2 meV
- Coldea *et al*., PRL **86**, 5377 (2001)

## Typical Phase Diagram:  La2-xSrxCuO4

## Doping kills LRO but not SRO

- Phase diagram for La2-xSrxCuO4 and
- Y1-2xCa2xBa2Cu3O6
-      psh = 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 La2Cu1-z(Zn,Mg)zO4
- 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 Sr2CuO2Cl2
- Wells *et al*., PRL **74**, 964 (1995).

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

## Hole segregation to antiphase domain walls

- 1D
- model

## 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
- YBa2Cu3O6+x		bilayer
- Bi2Sr2CaCu2O8+		bilayer
- Tl2Ba2CuO6+ 		single layer
-      (But not in La2-xSrxCuO4)

- YBa2Cu3O7
- Mook et *al*., PRL **70**, 3490 (1993)

## Spin fluctuations in YBCO do not look like spin waves

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

- YBa2Cu3O6.85

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

- La1.79Sr0.31NiO4

## Large crystals of La1.875Ba0.125CuO4 studied on MAPS

- Large crystals of La1.875Ba0.125CuO4 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
- La1.875Ba0.125CuO4
-      T = 12 K
-      Tc <  6 K

## La2-xBaxCuO4

- 24 meV

- 34 meV

- 66 meV

- 105 meV

- h

- k

- La2-xBaxCuO4
-      x = 1/8
- Normal state
-       with
-  Stripe order

-   YBa2Cu3O6.6
- 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 YBa2Cu3O6.85

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

*E* = 35 meV
*E*res = 41 meV