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Study of Collective Modes in Stripesby Means of RPA
E. Kaneshita, M. Ichioka, K. Machida
1. Introduction
3. Collective excitations in stripes
Stripes in High Tc cuprates
5. Summary
4. Phonon anomaly due to stripes
Random phase approximation
Results of RPA
2. Mean-field approach to stripes
Self-consistent calculation
Phonon anomalies in High Tc cuprates
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High Tc cuprates
La2-xAxCuO4 (A=Ba, Sr, Ca) Tc ~ 40K
Bi2Sr2CaCu2O8 Tc ~ 80K
Bi2Sr2Ca2Cu3O10 Tc ~ 110K
YBa2Cu3O6+x Tc ~ 95K
Tl2Ba2Ca2Cu3O10 Tc ~ 120K
Hg2Ba2Ca2Cu3O8 Tc ~ 135K
Nd2-xCexCuO4 Tc ~ 25K
CuO2 plane
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structure
La2-xSrxCuO4 YBa2Cu3O6+x
CuO2 plane
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Before doping After doping
Anti Ferro
holehole doping
Hole doping
2-D square lattice(Cu site only)
CuO2 plane
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Phase diagram
B. Keimer, et al,Phys. Rev. B 46, 14034 (1992)
Spin and charge ordered structure
La2-xSrxCuO4
Considering the underdoped region
Underdoped
AF
SC T
Hole concentration
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Stripe
Spin ordering vector Q
Charge ordering vector 2 Q
filled : up spinopen : down spin
AF domain
hole
spin-charge ordering
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Vertical stripe Diagonal stripe
Superconductor Insulator
In the neutron scattering experiments for LSCO,these stripe structures are observed .
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Elastic neutron scattering experiment
Incommensurate peak
H. Yamada, et al.,Phys. Rev. B 59 (1998) 6165
S. Wakimoto, et al.,Phys. Rev. B 61 (2000) 3699
Diagonal stripe Vertical stripe
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Incommensurability
M. Matsuda, et al.,Phys. Rev. B 62 (2000) 9148
diagonal stripe vertical stripe
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Anomaly appears in phonon spectrum observed by neutron inelastic scattering.
R. J. McQueeney, et al.,Phys. Rev. Lett. 82, 628 (1999)
H. A. Mook, and F. Dogan,Nature 401, 145 (1999)
La1.85Sr0.15CuO4 YBa2Cu3O7-x
Phonon spectrum
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Formulation
stripe order
self-consistent mean-field calculation
collective mode
random phase approximation
①
②
phonon spectrum
renormalize the collective stripe mode to the phonon spectrum
③
Hubbard model Assumingthe spin order
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Mean field approximation
Hubbard model
Mean field approximation
t : nearest neighbor hoppingt’ : next nearest neighbor hoppingU: on-site coulomb
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Fourier transformation
Periodicity
(k0 : within a reduced zone)
Assuming the spin order : (N-site periodicity)
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diagonal stripe case vertical stripe case
path path
reduced zone
(N: periodicity)
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diagonalization
self-consistent condition
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diagonal stripe (insulator)
vertical stripe (insulator)
vertical stripe (metal)
charge density
spin density
charge density
spin density
spin density
charge density
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Collective excitations in stripes
Stripe stateself-consistent mean-field calculation
RPA
single-particle Green function
HF susceptibilities
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single-particle Green function
HF susceptibilities
= k2 k1
=
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Dynamicalsusceptibilities
: charge excitation (phason mode)
: spin longitudinal excitation (phason mode)
: spin transverse excitation (spin wave mode)
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We calculate these values by means of RPA:
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RPA
i for spin flip
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Spin wave excitation
direction (A): effective J is smaller
Comparing the spin velocities
direction A direction B
path anisotropy of spin wave
A
B
exchange coupling
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path
period :8 site
spin density excitation (phason)
sliding mode
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meandering mode compression mode
Meandering mode has lower energy.
Anisotropy of phason mode
sliding mode of stripe(Just Q point)
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sliding mode of stripe
Charge collective mode at 2Q
sliding mode of stripe
Charge excitation
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phonon green function
k-k’ = l ×(l : integer)
Umklapp process
2Q
Spectral function
phonon self energy
electron-phononinteraction
Of Frohlich type
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Results 1
unperturbedphonon H. A. Mook, and F. Dogan,
Nature 401, 145 (1999)
log-plot
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charge order
( : energy of free phonon )
Effect of stripe
sliding mode
charge order band foldingsliding mode gap
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oscillation mode
below the gap ( in-phase)
above the gap ( out of phase )
difference of oscillation mode above and below the gap
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Results 2
R. J. McQueeney, et al.,Phys. Rev. Lett. 82, 628 (1999)
unperturbedphonon
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dynamical susceptibility by RPA
collective modes in the stripe
Summary
Phonon anomaly
•Anisotropy of collective excitations•sliding mode of the stripe
Coupling with the sliding mode
referencesE. Kaneshita, et al., J. Phys. Soc. Jpn. 70 (2001) 866E. Kaneshita, et al., Phys. Rev. Lett. 88 (2002) 115501