frustration-driven magnetic order on the shastry
TRANSCRIPT
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Frustration-driven magnetic order on the Shastry-Sutherland lattice
Pinaki Sengupta Nanyang Technological University
Singapore
10th Feb., 2015
Workshop on Current Trends in Frustrated Magnetism Jawaharlal Nehru University, New Delhi
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Collaborators
• Keola Wierschem, NTU • Zhifeng Zhang, NTU • Naoki Kawashima, ISSP, U. Tokyo • Takafumi Suzuki, U. Hyogo
Theory:
Expt.: • Christos Panagopoulos, NTU • Sunku Sai Swaroop, NTU • Tai Kong, Ames Lab, Iowa • Paul Canfield, Ames Lab, Iowa
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Outline v Rare earth tetraborides – a “new” family of Shastry-
Sutherland compounds
v Generalized Shastry-Sutherland model
v Magnetization plateaus in TmB4
v Spin supersolid in the generalized SSM v Proliferation of plateaus in the generalized SSM
v Conclusion
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“Other” realizations of the SSL model
v Rare earth tetraborides: TmB4, ErB4, HoB4, DyB4, GdB4, TbB4
v R2T2M: Yb2Pt2Pb, Ce2Pt2Pb
ZF order µeff (µΒ)
Θ (K)
TN1
TN2 ΣJij (K) D(K)
TmB4 Ising-AFM (π,π) 6.6 -63 11.7 9.8 -4.8 -6.4
ErB4 Ising-AFM (π,0) 8.4 -22.7 15.3 -1.4 -4.6
HoB4 Ising-AFM (π,π) 9.2 -12.7 7.4 6.3 -1.3 0.3
q Weakly coupled layers
q R3+ ions arranged in SSL geometry
Mata’s. et.al., J. Phys: Conf. Ser., 200, 032041 (2010)
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Magnetization plateaus in RB4
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.5 1.6 1.7 1.8
0.12
0.14
2.0 2.5 3.0 3.50.49
0.50
0.51
M/M
sat
B (T)
0.1359
M/M
sat
B (T)
0.1235
0.5092
M/M
sat
B (T)
0.4962
q Ground state has long range magnetic ordering in most of the rare earth tetraborides
q Field-induced plateaus observed in all members
q Sequence of plateaus differ across the family
q Multi-step melting of magnetic order observed in many RB4 compounds (a) magnetization plateaus, (b) phase diagram
TmB4
Uniform susceptibility and field dependent magnetization in ErB4
Magnetization plateaus in HoB4
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Rare earth tetraborides
Ø Low saturation field (~ 10T) – low-T neutron scattering possible Ø Span wide parameter regime including magnetic ground state at zero field Ø Higher spins, single ion anisotropies effective exchange anisotropy Ø Additional interactions
Generic features
Extensive insight into the interplay of competing strong interactions and geometric frustration in the Shastry-Sutherland lattice
Metallic ground state – interaction of itinerant electrons with localized magnetic moments in a frustrated configuration – interesting magneto-electric effects
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Magnetization Plateaus in TmB4
Stripe superstructures modulate underlying short-range structures
K. Siemensmeyer et. al., PRL 101, 177201 (2008)
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Low energy effective model
Ø Large magnetic moment for Tm3+:
Ø Large single-ion anisotropy: −D Siz( )2,D J ≈ 5
Sz = ±6
Sz = ±5
Sz = ±4
~100K
S = 6
Effective low energy model involving the lowest 2 levels
Ising limit
Ferromagnetic exchange term – sign problem in QMC simulations alleviated
Stochastic Series Expansion QMC algorithm used for simulation
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Generalized Shastry-Sutherland model Ø S=1/2 XXZ model with large Ising anisotropy
Ø J and J’ along the SSL lattice axes
Longer range interactions mediated by itinerant electrons
Ø NNN interaction (J3) along the diagonals of the plaquettes with no J
Ø Additional 3rd neighbor interaction J4
Ø Same anisotropy for all interactions.
q Rich variety of magnetic phases
q Potentially realized in the different members of the rare-earth tetraborides
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Low energy effective model for TmB4
Start with the generalized Shastry Sutherland model with J and J’
Need additional interactions
Ø AFM J3 necessary to account for the appearance of ½ plateau
Ø FM J4 necessary to explain the suppression of 1/3 plateau
Together they explain the principal plateau structure observed in TmB4.
Ising limit: Extended 1/3 plateau followed by a direct transition to full saturation TmB4: Extended ½ plateau, no 1/3 plateau
Longer range RKKY interactions mediated by itinerant electrons
Suzuki, et. al., PRB 80, 180405 (2009); PRB 82, 214404 (2010)
inconsistency with experimental observation
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Modeling TmB4
ü Correct critical values for m/ms=1/2 plateau reproduced
ü Correct saturation field reproduced
ü No evidence of lower magnetization plateaus for the current model
ü No evidence for “hysteresis” effects.
Suzuki, et. al., PRB 80, 180405 (2009); PRB 82, 214404 (2010)
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Fractional plateau in TmB4
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.5 1.6 1.7 1.8
0.12
0.14
2.0 2.5 3.0 3.50.49
0.50
0.51
M/M
sat
B (T)
0.1359
M/M
sat
B (T)
0.1235
0.5092
M/M
sat
B (T)
0.4962
(a)
(b)
(c)
Neutron scattering data Michimura, et.al., 2009
Fractional plateau and hysteresis in TmB4
Schematic spin configuration
Modulated AFM
1/8 plateau
½ plateau
Ø Fractional plateau observed in TmB4 at m/ms~1/8
Ø Magnetic hysteresis observed at the fractional plateau
Ø Neutron scattering modulated AFM ground state with incommensurate periodicity
Ø Provides simple explanation for observed magnetic behaviour
Ø Predict modulated structure for both fractional and ½ plateaus – need to be confirmed experimentally (a) (b)
J2
J3
J1
3
2 1
4
(a)
(b)
(c)
3
1 2
4
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Magnetization in generalized SSM
q Varied sequence of magnetization plateaus as the parameters are varied
q Fertile framework for investing frustration drivn field induced magnetic phases – plateaus, spins-supersolid (?)
q Potentially realizable in the different members of the rare earth tetraboride family of compounds
q Help identify dominant interactions driving observed magnetic phases
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Magnetization plateaus in extended SSM (a) (b)
J2
J3
J1
Ø Effect of J3 explored in detail
Ø ZF: (π,π) AFM order for FM and weak AFM J3 in the Ising limit
Ø (π,0) AFM order for moderate to strong AFM J3 – observed in ErB4
Ø 1/3 plateau persists to finite J3
Ø ½ plateau appears for any J3 ≠ 0
Ø Finite exchange interactions induce superfluidity at boundary between plateaus
Columnar AFM order
Magnetic phases driven by J3
K. Wierschem and P.S., PRL 110, 207207 (2013)
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Spin supersolid in extended SSM
0
0.05
0.1
0.15
0.2
0.25
mc2
0
0.2
0.4
0.6
0.8
1
mz
L=12L=24
1 2 3 4hz
0
0.005
0.01
ms2
1 2 3 4hz
0
0.02
0.04
0.06
0.08
ρ s
0 0.05 0.1 0.15 0.21/L
0.040.060.080.10.12
0 0.05 0.1 0.15 0.21/L
0.02
0.03
0.04
0.05
hz=1.75
hz=2.75
hz=3.75
hz=1.75
(a) (b)
J2
J3
J1
Ø Spin supersolid GS observed at densities close to, but less that half-filling
Ø Longitudinal AFM order at (π,0) and (π,π)
Ø Transverse AFM order at (π,π)
Ø Simple mechanism based on delocalization of holes in the half-plateau by 1st order process
Ø Magnetization behaviour qualitatively similar to ErB4, but spin SSOL phase not yet reported
K. Wierschem and P.S., PRL 110, 207207 (2013)
Delocalization of a single flipped spin at the ½ plateau
Transverse and longitudinal structure factors in the spin SSOL phase
Simulation results showing the appearance of spin SSOL
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Magnetic phases in generalized SSM
J2
J3
J1
J4
(a) (b)
3J
4J
I II
III IV
(a) (b)
J2
J3
J1
-2 -1 0 1 2J3
-1
-0.5
0
0.5
1
J 4
0 - 1
0 - 1/2 - 1
0 - 1/3 - 1/2 - 1
0 - 1/2 - 1
0 - 1/3 - 1/2 - 5/9 - 1
0 - 1/3 - 5/9 - 1
0 - 1
/3 -
1/2
- 5/9
- 1
ErB4
TmB4
SS1
SS2
Ø Inclusion of J4 changes plateau sequence significantly
Ø AFM J3 and J4 destabilises the ½ plateau – emergence of the 5/9 plateau
Ø Different structures for same plateau realised for different parameters
Ø Spin-SSOL stablised over extended parameter ranges
Evolution of plateau sequence for finite J4 in the XXZ model
Plateau sequence in the Ising limit for finite J4
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Understanding plateaus in generalized SSM
Ø Explain spin textures at plateaus in terms of a plaquette unit cell
Ø Construct a “pinwheel” around a square with no diagonal bond
Ø Many different configurations possible (for Ising spins)
Ø Local dimer states determine nature of spin modulation
Ø Many possibilities for same plaquette magnetic moment
Ø Leads to multiple distinct plateaus with same magnetization
Ø Possible because of additional interactions
Ø Stable for finite exchange - Confirmed by QMC simulations in XXZ model
Columnar AFM
½ plateau
1/3 plateau
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Electronic transport in rare earth tetraborides
Ø Itinerant electrons in rare earth tetraborides couple to local moments
Ø Electronic transport affected significantly by underlying magnetic texture
Ø Study within the framework of Shastry Sutherland Kondo lattice model (SSKLM)
Ø Control electronic transport by applied magnetic field and magnetism by driving current
Ø Small change in applied magnetic field changes magnetic structure – interesting magneto-electric phenomena
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Conclusions
v Interplay between geometric frustration, strong interaction and high
magnetic field results in many novel quantum phases on the Shastry-Sutherland lattice
v Rare earth tetraborides present experimental realizations of many of these phases
v Metallicity makes these quantum magnets even more interesting
v Magnetic phases explored in great detail – transport promises new phases and phenomena
The best is yet to come !