Transcript Bose-Einstein Condensation of Spins in NiCl2

Bose-Einstein Condensation and
Magnetostriction in NiCl2-4SC(NH2)2
Vivien Zapf
National High Magnetic Field Laboratory
Los Alamos National Lab
National High Magnetic
Field Laboratory
Pulsed magnets to 75 T
M(H), r(H), magnetostriction, ESR, optics, etc
300 T single-turn (ms pulse)
Coming soon: 60 T and 100 T
long pulse magnets (2 s)
300 T single-turn nondestructive magnet
Chuck Mielke, Ross McDonald, et al
10 mm long.
10mm ID.
Capacitor bank pulses a short (ms) mega-amp current pulse
to achieve ultra high magnetic fields.
Low inductance
capacitor bank.
L = 18 nH, C = 144 mF,
V = 60 kV, E = 259kJ.
1st megagauss shot
Single turn magnet coil,
L = 7 nH.
(February 8th 2005)
Peak current 4 MA.
 2 ms rise time,
 dB/dt ≈ 108 Ts-1
3 orders of magnitude faster
than standard short pulse
magnets at the NHMFL.
NiCl2-4SC(NH2)2 (DTN)
No Haldane gap
Ni
Cl
Ni S = 1
Other BEC compounds
All Cu spin-1/2 dimers
Cl
•TlCuCl3 (wrong symmetry)
•BaCuSi2O6
(3D -> 2D crossover)
Jchain/kB = 1.72 K
• CsCuCl4
c
a
a
Jplane/kB = 0.17 K
NiCl2-4SC(NH2)2
 
H  D S j
z 2
 
 gmB H z  S j  S J v Si  S j
z
z
vij 
j
j
Spin-orbit
coupling
(DTN)
c
H
Zeeman term
a
Antiferromagnetic
exchange
1
Ni2+ S = 1: Triplet split by spin-orbit coupling
0.8
TetragonalElattice kz=0 (FM)
Sz = -1
Sz = +1
Sz = -1
Sz = +1
S=1
D~8K
D~8K
=0
Sz = 0
Hc1 = 2.1 T
0.4
0.2-1 -0.5 0
0.5
- -/2 0 -/2
1
kz
kz= (AFM)
Sz
0.6
M. Kenzelmann et al
H
Hc2 = 12.6 T
14
c
H
a
Mz (x103 emu/mol)
12
BEC/AFM
10
8
6
H || c
4
2
0
0
Hc1
5
H (T)
10
Ni spins
A. Paduan-Filho et al, Phys. Rev. B 69, 020405(R) (2004)
Hc2
15
Note: Approximate theory neglects Sz = -1 state.
z
z
i
Complete
theory:
  a S  0  be S  1
spin language
H.-T. Wang and Y. Wang, Phys. Rev. B 71, 104429 (2005)
K.-K. Ng and T.-K.iLee, cond-mat/0507663
  a 0  be 1
boson language
Example: treat upper state as an energetically unfavorable double
occupancy
Emptystate
site
Occupied boson
Sz = -1
E
or T(K)
H ~ Mz ~ N (# of bosons)
Sz = 1
D
AFM
0
FM
XY AFM order
b or N
Boson filling fraction/Number of bosons
H || c
1
Spin language Hamiltonian
 
Η  D S
i
Spin-orbit
coupling
z 2
i
 
 gmB H  S   J Si S j
z
i
i
 i , j 
Zeeman term
AFM exchange
S+ -> b+ (boson creation operator)
Boson language Hamiltonian (neglecting Sz = -1 term)
H









J
b
b
b
b

b
b

b
b

h
D
b
  i i j j i j j i eff  i bi
i , j 
Repulsion
(2nd order in N)
hopping
(Hardcore Constraint: One boson per site)
i
number operator
Boson mapping
Spins
Bosons
|Sz = 1> state
occupied boson |1>
|Sz= 0> state
unoccupied boson |0>
Order parameter:
Staggered magnetization Mx
Order parameter:
Boson creation operator b† = S+
Magnetic field
Chemical potential (H ~ N ~ m)
This also works for S=1/2: |↑> = occupied, |↓> = unoccupied
Spins are prevented from obeying fermion statistics since no realspace overlap between states allowed
Why do the bosons obey number
conservation?
|Sz = 1> = occupied boson
|Sz = 0> = unoccupied boson
b†
= boson creation operator
b
Tetragonal symmetry of crystal:
a
U(1) symmetry of spins.
Symmetric under rotation by  in a-b plane
• Hamiltonian must be independent of 
• Rotation by :
b† → b† ei
and b → be-i
• H can only contains b†b or bb† terms ( = number operator)
•
(b† ei be-i

b† b = N
• Hamiltonian commutes with boson number operator
• Boson number is conserved.
H
Symmetry in plane
Energy landscape of spins
Ni
XH
E()
S

XY
model
Ni
X H
Ising
model
Limitations
• Boson number only conserved in statistical average
• Spin fluctuations, lattice fluctuations limit super current
lifetimes
• Small corrections to XY model: DM interactions, dipoledipole, and spin-orbit coupling
Experimental Tests for BEC
14
12
E or
T(K)
Mz
10
6
H || c
4
Sz = 1
D
8
2
0
XY AFM order
0
5
H (T)
10
15
H || c
0
N
nk 
1
e  k  m  / kBT  1
1
N0  N    k  m  / kBT
1
k e
N0  M z  T
3/ 2
TC ~ N 2 / 3
where
k ~ k 2
N ~M ~H
TC ~ H 2 / 3
HC ~ T 3 / 2
Quantum Phase Transition:
universality class of a BEC
Thermal phase transition (XY AFM)
phase driven (d=3, z=1)
T
XY AFM
H
Hc2
Hc1
Quantum Phase Transition (BEC)
amplitude driven (d=3, z=2)
Mz  T

H c  H c1  T

3D BEC:  = 3/2
• BaCuSi2O6
(3D -> 2D crossover)
3D Ising:  = 2
• CsCuCl4
2D “BEC”:  = 1
• NiCl2-4SC(NH2)2
Measuring Specific Heat
Sapphire platform
C=Q/DT
Quasi-Adiabatic
C=t/k
Thermal Relaxation Time
Magnetocaloric Effect
10
0.44
H = 10 T
8
0.43
T (K)
C/T (mJ/mol K2)
Specific Heat
6
0.42
inflec. point
4
0.41
2
0.4
0
0
0.4
0.8
1.2
increasing H
decreasing H
0.39
10
1.6
11
12
13
14
B(T)
T (K)
1.2
1
0.8
0.6
Magnetocaloric effect
0.4
Specific heat
0.2
0
0
2
4
6
8
H (T)
10
12
14
V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison, C. D. Batista, M. Kenzelmann, C.
Niedermayer, A. Lacerda, and A. Paduan-Filho, Phys. Rev. Lett., 96, 077204 (2006)
1.2
1
0.8
0.6
Magnetocaloric effect
Specific heat
0.4
H – Hc  TN
0.2
0
0
2
4
6
8
H (T)
10
12
H – Hc  TN   2 (3D Ising magnet)
3/2 (3D BEC)
1 (2D “BEC”)
14
Hc  Hc1  T 
Mz  T 
2.5
Ising
magnet
2
1.5

3D BEC
1
H-Hc1 = aT
2D BEC
 = 1.47 ± 0.10
0.5
 = 1.5 BEC)
0
0
0.2
0.4
0.6
0.8
1
Tmax/1.2 K
Windowing Technique: see
V. S. Zapf, et al, Phys. Rev. Lett., 96, 077204 (2006)
S. Sebastian et al, Phys. Rev. B 72, 100404(R) (2005)
A. Paduan Filho, unpublished
Spin wave theory
1.2
Inelastic Neutron Diffraction
and magnetization :
1
D  8.12K
J av  0.69K
Tc (K)
0.8
0.6
Magnetocaloric effect
Specific heat
Predictions (3-level system):
0.4
Hc1  2.2T
0.2
Hc 2  10.85T
predicted Hc2
predicted Hc1
0
0
2
4
6
8
H (T)
10
12
14
C. D. Batista, M. Tsukamoto, N. Kawashima, in progress
V. S. Zapf et al, Phys. Rev. Lett. 96, 77204 (2006).
Magnetostriction
H
c
0.025
a
Capacitance
T = 25 mK
0.020
H || c
Hc2
0.015
CuBe
spring
DL/L (%)
0.010
Lc
Hc1
0.005
0.000
-0.005
Titanium Dilatometer
(design by G. Schmiedeshoff)
V. Correa, V.S. Zapf, T. Murphy, E. Palm,
S. Tozer, A. Lacerda, A. Paduan-Filho
-0.010
La
0
2
4
6
8
B (T)
10
12
14
Ni++
Hc2
Magnetostriction
0.025
0.020
0.015
DL/L (%)
J
JJ
=
JJ 1.7 K
0.010
Hc1
0.005
Lc
La
0.000
-0.005
-0.010
c
H
J = 0.17 K
a
0
2
4
6
8
B (T)
10
12
14
1.2
0.025
1
0.020
0.8
0.015
0.6
DL/L (%)
Tc (K)
Hc2
Magnetocaloric effect
Specific heat
0.4
0.010
Hc1
0.005
predicted Hc2
predicted Hc1
-0.005
-0.010
0
2
4
6
8
H (T)
10
12
Lc
La
0.000
0.2
0
Magnetostriction
14
0
2
4
6
8
B (T)
Summary
BEC confirmed experimentally via H-Hc1 ~ T and M ~ T
Magnetostriction effect distorts phase diagram with increasing
magnetic field
10
12
14
Future work:
Frustration-induced symmetry change?
?
Acknowledgements
Universidade de Sao Paulo, Brazil
Armando Paduan-Filho
(Crystal growth and Magnetization)
LANL
Cristian Batista (T-11 theory group)
Paul Scherrer Institute and
ETH, Zürich, Switzerland
B. R. Hansen
M. Kenzelmann (Neutron scattering)
NHMFL-LANL
Diego Zocco
Marcelo Jaime
Neil Harrison
Alex Lacerda
University of Tokyo
Mitsuaki Tsukamoto
Naoki Kawashima (Monte Carlo Simulations)
NHMFL-Tallahassee
Victor Correa (Magnetostriction)
Tim Murphy
Eric Palm
Stan Tozer
Occidental College
George Schmiedeshoff (dilatometer design)
NSF
NHMFL
DOE