Spin Tunneling and Inversion Symmetry
ENRIQUE DEL BARCO
www.physics.ucf.edu/~delbarco
Department of Physics – UCF
QCPS II 2009 - Vancouver
Orlando
Spin Tunneling and Inversion Symmetry
ENRIQUE DEL BARCO, CHRISTOPHER RAMSEY (UCF)
Nature Physics 4, 277-281 (2008)
STEPHEN HILL
SONALI J.
(NHMFL and Physics Department, FSU – Tallahassee)
SHAH, CHRISTOPHER C. BEEDLE AND DAVID N.
(Chemistry Department, UCSD – La Jolla-San Diego)
PHILIP C.E. STAMP AND IGOR TUPITSYN
(PITP-Physics, UBC, Vancouver)
HENDRICKSON
THE MOLECULE
5/2
2
2
5/2
2
S=7
5/2
5/2
2
5/2
2
5/2
2
[Mn12(Adea)8(CH3COO)14]·7CH3CN
Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004).
MAGNETIZATION - QTM
1.0
Tc ~0.3K
 = 0.2 T/min
-6
-7
M/Ms
0.5
0.0
T = 0.90 K
T = 0.80 K
T = 0.70 K
T = 0.65 K
T = 0.60 K
T = 0.50 K
T = 0.41 K
T = 0.27 K
-0.5
HL
+1
TB
~0.9K
-1.0
-0.6
-0.3
0.0
0.3
0.6
+2
+3
+4
+5
S= 7
D = 0.4K
+6
mS = +7
T = 0.9K
H (T)
S = 7, D = 0.4 K
1.0
MAGNETIZATION - QTM
 = 0.2 T/min
HL
-0.6
-0.3
5
0
0.00
0.0
H (T)
0.29
H (T)
HT
0.3
Energy (K)
T = 0.90 K
T = 0.80 K
T = 0.70 K
T = 0.65 K
T = 0.60 K
T = 0.50 K
T = 0.41 K
T = 0.27 K
-15
Ms =5
HMR =6
 k  D / gB
-20
-25
0.00
s
Ms =7
0.29
0.58
H (T)
0.6
k=2
10
-1.0
?
exc.
dM/dH (a.u.)
-0.5
k = 1(S)
15
exc.
0.0
k = 1(A)
-10
k=0
M/Ms
0.5
0.58
H
  DS z2  g B S z H z
S = 7, D = 0.4 K
THE MOLECULE
5/2
2
2
5/2
2
S=7
5/2
5/2
2
5/2
2
5/2
2
[Mn12(Adea)8(CH3COO)14]·7CH3CN
Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004).
d
d*
d
d
d
d
d
d
d
d*
d
d
[Mn12(Adea)8(CH3COO)14]·7CH3CN
Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004).
davg~3.17Å
J ~2-5 cm-1
d*~3.49Å
J* <<J
Foguet-Albiol, D. et al., Angew. Chem. Int. Edn 44, 897–901 (2005)
THE MOLECULE
THE MOLECULE
d*
7/2
7/2
d*
[Mn12(Adea)8(CH3COO)14]·7CH3CN
Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004).
EXCHANGE-COUPLED SPINS
J S S  J S S  S S 
H  H  H  JS
1


2
Hi  D7 / 2 Si2,z  E7 / 2 Si2,x  Si2,y  BSi  gˆ  H Si  7 / 2
z z
z 1 1 22

x
1
x
2
D7 / 2  0.865K (~ 2D)
y
1
y
2
E7 / 2  0.156K (~ 2E ) J z  J   0.39K
QUANTUM TUNNELING BTW. STATES OF DIFFERENT SPIN LENGH
QUANTUM INTERFERENCE
HARD
HL

E7 / 2 Si2,x  Si2,y

BERRY PHASE INTERFERENCE OF TWO COUPLED TUNNELING SPINS

HT
NEW TOPOLOGICAL EFFECT
SINGLE SPIN
INTERACTING SPINS
Classical spin precession
Classical coupled-spins precession
Sjoqvist, PRA (2000)
i.e. Wagh et al., PRL (1998)
Pancharatnam (1956)
(light interference)
Quantum Tunneling Spin
Berry (1984)
(quantal systems)
Coupled Tunneling Spins
THEORY
Aharanov and Anandan (1987)
(generalization Hilbert space)
THEORY
Loss et al., PRL (1992)
Von Delft et al., PRL (1992)
Garg, EPL (1993)
EXPERIMENT
Fe8: Wernsdorfer & Sessoli, Science (1999)
Mn12: del Barco et al., PRL (2003)
Mn12 -tBuAc: da Silva Neto et al., (2008)
.
.
.
(??)
EXPERIMENT
Mn12 wheel: Ramsey et al., Nature Physics (2008)
SYMMETRY RULES
H  H1  H 2  JS1  S2
ANTI-SYMMETRIC TERM NEEDED
Dzyaloshinskii–Moriya interaction
H DM
  
 D  S1  S 2
NOT ALLOWED ON A DIMER MODEL
with
INVERSION SYMMETRY
SYMMETRY RULES
Wernsdorfer, arXiv:0804.1246v1,v2,v3
a - Dimer model not valid
Rejected by NP: See our response in arXiv:0806.1922
7/2
Wernsdorfer, PRB (2008)
a - Dimer model identically used in a Mn6 wheel (CI)
b - DM interaction used to explain results
7/2
Wernsdorfer, PRL (2008)
a - Dimer model used in an “identical” Mn12 wheel
b – DM interaction used to explain results
SYMMETRY RULES
Wernsdorfer, arXiv:0804.1246v1,v2,v3
a - Dimer model not valid
Rejected by NP: See our response in arXiv:0806.1922
7/2
Wernsdorfer, PRB (2008)
a - Dimer model identically used in a Mn6 wheel (CI)
b - DM interaction used to explain results
7/2
Wernsdorfer, PRL (2008)
a - Dimer model used in an “identical” Mn12 wheel
b – DM interaction used to explain results
Wernsdorfer-justification:
1) Disorder
2) Local DM interactions are not forbidden
del Barco et al., PRL (2009)
1) Disorder
2) Local DM interactions are not forbidden
SYMMETRY RULES
2
5/2
7/2
5/2
2
D=0
d1
center of inversion
5/2
middle point
2
2
center of inversion
middle point
5/2
5/2
2
2
5/2
7/2
SYMMETRY RULES
D D0 tilted
0 parallel
(Wernsdorfer,
to z-axis PRL)
(Ramsey, Nature Physics)
The Hamiltonian of the coupled half-wheels:
Η  H1  H 2  H12
7/2
Each half-wheel:


H i   DS iz  E S  S   B Si  gˆ  H
center of inversion2
middle point

2
ix
2
iy

Exchange coupling:
Η12  Η12S  Η12A
7/2
Symmetric exchange:
Η12S  JS1  S 2
Antisymmetric exchange (DM interaction):

  
Η12A  D  S1  S 2

SYMMETRY RULES
z
 D
5/2
2

2
d1
5/2
*
D
2
center of inversion
*
2
5/2
middle point
5/2
y
5/2
2
2
5/2
x
SYMMETRY RULES
z
 D
5/2
2

2
d1
5/2
2
center of inversion
z
 D
2
5/2
y
middle point

5/2
y
5/2
x
2
2
5/2
x
SYMMETRY RULES
H
H
Center of Inversion
SYMMETRY RULES
z
z
’ D’
5/2
22
3/2
5/2

d1
center ofx inversion
z
z
’ D’
 D
middle point
y
’2
middle point

5/2
x
2
2
y
5/2
’
2
2
y
(d ,J)
 D
5/2
2
y
5/2
3/2
x
(d’<d ,J’>>J)
x
SYMMETRY RULES
The Hamiltonian of
z 4 coupled quarter-wheels:
z
’ D’
y
2
i


2
ix
Exchange coupling:
’ D’
y

x
Ηyij  Η ijS  Η ijA
2
10
-7
10
o
’Symmetric exchange:
 
S
xΗ ij  J ij Si  S j
(d’<d ,J’>>J)
J12  J 34  J w
Center of inversion symmetry imposes:
o
12 = 0
34 = 1
3/2
-6
k=1(A) (K)
middle point
i , j (i  j )
ij
x
10
center ofx inversion
z
 D
y H



2
H i   DS  E S  Siy   B Si  gˆ  H
2
iz
-5
z
(d ,J)
Each quarter-wheel:
’
3/2
Η  HD
i 
o
34 = 5
o
34 = 10
o
34 = 30
o
34 = 90
o
34 = 120
J 23  J 41  J s ( J w ) = 150
o
34
o
 
*
* 
k = 1(A) is degenerate
-8
10
0.0
34 = 170
Antisymmetric exchange (DM interaction):
0.1

  
A
Η0.2
Si  S j
ij  Dij  0.3
HT (T)

o
34 = 180
0.4
SYMMETRY RULES

 (K)
10
-4
-5
m
ed
iu
m
10
ha
rd
k = 1(S)
k=0
y
10
-6
o
~30
10
x
-7
k = 1(A)
-6
10
-7
 (K)
10
12 = 0o
12 = 30o
12 = 60o
12 = 90o
-0.4
0.0
0.4
HT (T)
0.8
SYMMETRY RULES
In a centro-symmetric molecule
local DM-interactions MUST BE related by inversion symmetry and
DO NOT BREAK THE DEGENERACY
BETWEEN LEVELS OF OPPOSITIVE PARITY
independently of how complex the Hamiltonian is
because PARITY (good quantum number) MUST BE CONSERVED
SYMMETRY RULES
when inversion symmetry is not present
BOTH SYMMETRIC and ANTISYMMETRIC INTERACTIONS
CAN BREAK DEGENERACIES
DM-interactions are important in S = 1/2 systems
ONLY SOURCE OF DEGENERACY BREAKING
(Kagome lattice – weak ferromagnetism)
but never mix states of opposite parity in a system with inversion symmetry
E. del Barco, S. Hill and D. N. Hendrickson, Phys. Rev. Lett. in press (2009)
E. del Barco et al., In preparation
Dipolar fields? (Philip?)
z
 D
5/2
2

2
d1
5/2
2
center of inversion
z
 D
2
5/2
y
middle point

5/2
y
5/2
x
2
2
5/2

x
CONCLUSIONS
Quantum superposition of states with different spin length in a SMM
New topological effect: Quantum phase interference of two coupled tunneling spins
Local DM interactions in a centro-symmetric SMM do not break the degeneracy
between states of opposite parity
Del Barco Lab
Low temperature nanomagnetism
Single-molecule magnets
FM thin films and nanowires
Nanoparticles
Low temperature nanotransport
Molecular spintronics
Single-electron transistors
Low-dimensional systems
i.e. graphene, nanowires,
nanoparticles, molecules,…
Physics collaborations
Stephen Hill (NHMFL-FSU)
Masa Ishigami, Robert Peale, Lee Chow (UCF)
Agustin Camon, Fernando Luis (UZ-Spain)
Javier Tejada (UB-Spain)
Oliver Waldmann (U.Freiburg-Germany)
Andrew Kent (NYU)
XiXiang Zhang (KAUST)
Eduardo Mucciolo, Michael Leuenberger (UCF)
Philip Stamp, Igor Tupitsyn (UBC-Canada)
Chemistry collaborations
David Hendrickson (UCSD)
George Christou (UF)
Eugenio Coronado (UV-Spain)
Florenzio Hernandez (UCF)
Joel Miller (UU)
SISTER MOLECULES
[Mn12(Edea)8(CH3CH2COO)14]
[Mn12(Adea)8(CH3COO)14].7CH3CN
[Mn12(Edea)8(CH3COO)2(CH3CH2COO)12]
d
d
d*
d
d*
d*
d*
d
d*/davg = 1.093
J*/Javg
d*
d
<
>>
S=7
d*/davg = 1.100
J*/Javg
dM/dH (a.u.)
15
S = 7/2 + 7/2
10
5
0
0.00
0.29
H (T)
0.58
>
<<
d*
d
d*/davg = 1.091
J*/Javg
S=7