Berry-Phase Induced Kondo Effect in Single-Molecule Magnets
Tulane University (November 2008)
Eduardo Mucciolo
University of Central Florida
Department of Physics
Collaborators: Michael Leuenberger (UCF) Gabriel Gonzalez (UCF)
Overview
- Single-molecule magnets:
- Single-molecule electronics
- Kondo effect in SMMs
- Berry-phase blockade in single molecules
Single-molecule electronics:
1 - 10 nm
IVbias
Vgateelectrode(source)
electrode(drain)
molecule
insulated metal plate(gate)
molecular transistor
- fast operation- large energy scales (eV)- quantum effects at high temperatures ?!
Potential advantages:
Fabrication of the nano gaps: Electromigration and breaking of nanowires
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40
2
4
6
8
10
Curre
nt (m
A)
Voltage (V)
4K
(Enrique del Barco, UCF)
Multiwire chip: Trial and error approach
6µm
70 nm
Single-electron transistors: What is actually measured?
μD < μN+1 < μS
μN+1
μN
D
μS
S
μD
μN-1
finite currentwith finite bias
Ec+ΔE
μN+1
μN-1
μN
D
μS
S
μD
μN < μS , μD < μN+1
(Coulomb blockade)
no current
N+1N
dI/dVI
μS = μN+1 = μD
D
μS
S
μDμN+1
μN
μN-1
finite current with(nearly) zero bias
Early experiments: Non magnetic molecules
Coulomb blockade and the Kondoeffect in single-atom transistors
Jiwoong Park*†‡, Abhay N. Pasupathy*‡, Jonas I. Goldsmith§,Connie Chang*, Yuval Yaish*, Jason R. Petta*, Marie Rinkoski*,James P. Sethna*, He´ctor D. Abrun˜a§, Paul L. McEuen* & Daniel C. Ralph*
* Laboratory of Atomic and Solid State Physics; and § Department of Chemistryand Chemical Biology, Cornell University, Ithaca, New York 14853, USA† Department of Physics, University of California, Berkeley, California 94720,USA
NATURE | VOL 417 | 13 JUNE 2002 |www.nature.com/nature
Several other groups around the world have also observed similar effects in molecular transistors
A recent but extensive literature on this technique already exists.
Alternative route to molecular transistors: STM of molecules
(Photo credit: Ben Utley)
I
VbiasAdvantages: fabrication, control
Drawbacks: no gating
How to increase the functionality of a molecular transistor ?
Combine electronics with magnetism:
Our Motivation:
(i) Use magnetic molecules
(ii) Use magnetic contacts
How do QTM and Berry phase interference
manifest themselves in electronic transport
through a single SMM?
- Spin-current blockade
- Kondo effect
-S S
N
N+1
polarized leads
QTM
unpolarized leads
no QTM
N+1
N
-S S
Berry-phase blockade:
fully polarized leads partially polarized leads
G. Gonzalez and M. Leuenberger [PRL 98, 256804 (2007)]
tunneling
tunneling
controlled by the Berry phase(transversal field)
G. Gonzalez, M. Leuenberger, ERM [PRB 78, 054445(2008)]
Dan Ralph (Cornell)Herre van der Zant (Delft)
Enrique del Barco (UCF)
Recent experiments (Mn12): Still not very conclusive...
No experiment has yet seen a unambiguous manifestation of QTM,much less Berry phaseinterference...
Kondo effect: The case of quantum dots and molecules attached to leads
!
(E)
" (E)
EF
KT > T
KT
E
E
"
Sharp resonance at the Fermi level appears at T >
E
At low temperatures, spin flip processes strongly renormalize the conductance
final statevirtual (intermediate) stateinitial state
no Coulomb blockade!
The Kondo effect in a non-magnetic single-electron transistorhas already been observed by several groups...
Another way of probing QTM: Kondo effect
... but not yet for SMMs.
Unconventional Kondo effect in SMMs: How it happens
(suppressed at zero bias)2
U
2
1
s = s =
S =m S =m!1
S =m!
t t
U
1
!( S = !1,
2 2zz
z zz
z
~ t
! s = +1)z
!1 +1
inelastic
( Em ≠ Em-1 )
M. Leuenberger and ERM [PRL 97, 126601 (2006)]
Jzelastic2
2
U
1
s = s =
S =m S =m
S =m!
t t
U
1
!( S = 0,
2
!1
2zz
z zz
z
~
! s = 0)z
!1
t no spinflipping
z
2
!E
S =!m
s =
S =m
t
2z
z
!1
s = +1)( S = !2m, !!
~
z z
S =m!z 2
1
1
S =!m+z1
2
t
2
+1s =z
1
UUE!
2U
t J⊥elastic( Em = E-m )
spinflipping
local exchange term
HKondo =∑
k,α
(ξk +
gµBH∗xsx2
)ψ†kαψkα +Hex
Kondo effect in SMMs: detailed theory
HSMM =∑
m∗
Em∗ |m∗〉〈m∗|, Em∗ = E−m∗ [∆m∗,−m∗(H
∗
x) = 0]
Hex =1
2
∑
m
∑
k,k′
[
j(m)+ Σ
(m)+ ψ
†k↓ψk′↑ + j
(m)− Σ
(m)− ψ
†k↑ψk′↓ + j
(m)z Σ
(m)z
(
ψ†k↑ψk′↑ − ψ
†k↓ψk′↓
)]
project it onto the {|m〉} subspace
pseudo-spin flipping terms
Σ(m)z
=1
2(|m〉〈m|− |− m〉〈−m|)
Σ(m)± = | ± m〉〈∓m|
pseudo-spin operators
j(m)± = J±〈±m|S±|∓ m〉
anisotropic coupling constants
j(m)z = 2Jz〈m|Sz|m〉
H = HSMM + HKondoeffective Hamiltonian
Kondo effect in SMMs: microscopic derivation of coupling constants
jz = ± 2t2[
U + ∆0(U + ∆0)2 −∆21
+U + ∆0
(U −∆0)2 −∆21
]≈ ± 4t
2
U
j⊥ = 4t2[
∆1(U + ∆0)2 −∆21
+∆1
(U −∆0)2 −∆21
]≈ 8t
2 ∆1U
The sign depends on the intermediate spin state of the molecule!
|jz|! j⊥
1
0
E
E
E
E
E
(0)
(0)
(1)(!1)
(!1)
a,m
s,m
a,m
a,m
s,m
!1
q=!1 q=0 q=1
U
2
U
2
0
0
1
!1
!1
Es,m1(1)
anisotropic exchange interaction
energy levels relevantto the Kondo effect
S1 = S0 −12
S1 = S0 +12
AF
FM
Kondo effect in SMMs: poor man’s Renormalization Group
H =∑
m
[
Em
(
Σ(m)z
)2+ η(m)H∗xΣ
(m)x
]
+∑
k,α
ξkψ†kαψk,α + Hex
total Hamiltonian in projected subspace
η(m) = 1 −νj
(m)⊥
2
Knight shift molecule’s pseudo-spin couples to transversal field
1
2ν√
Carctanh
(
√
C
jz
)
= ln
(
D̃
TK
)
j2z− j2⊥ = C > 0
0 2 4 6 8 10
T / TK
-1
-0.5
0
0.5
1
1.5
! (
T /
TK)
H∗x −→ H∗
x/η(m)
D̃≈T
... and solutions
dη
dζ=
ν2
2(j+ + j−) jz
djz
dζ= −2ν j+j−
dj±
dζ= −2ν j±jz
Renormalization Group flow equations... ζ = ln(D̃/D)
!
ED
D
~
Kondo effect in SMMs: Conductance
G(T ) = G0
∫
dω
(
−df
dω
)
π2ν2
16
∑
m e−Em/kBT |A(m)(ω)|2∑
m e−Em/kBT
A(m)ω≈D̃
≈ j(m)⊥,ω≈D̃
=√
C
[
(ω/TK)2ν√
C
(ω/TK)4ν√
C − 1
]
Linear conductance (T dependence)scattering amplitude
1st order perturbation theory
singularity for T
Kondo effect in SMMs: Berry phase oscillations
The tunnel splitting is an oscillating function ofthe transverse magnetic field due to the Berryphase interference.
two-fold degeneracypoints (Kondo effect)
2
1
3
H
!
0
1 H2 H3< <
G
eV
H
1) The Kondo peak splitting is a non-monotonic function of the transverse magnetic field.
Consequences:
2) The period of Berry oscillations is renormalized by the Kondo effect (strongly temperature dependent, with a universal function form).
Kondo effect in SMMs: Ni4 , the best candidate
10-4
10-3
10-2
10-1
100
!E
S,-
S (
H ) [K]
T / TK = 2
T / TK = 3
T / TK = 10
0 1 2 3 4 5 6 7
H [T]
10-4
10-3
10-2
10-1
100
!E
S,-
S (
H ) [K]
" = 34#
" = 41#
Spin tunneling splitting Δ (numerical simulations)
[Ni4(ROH)4L4O12] (R=Me,Et) Sieber et al. (2005)
=
Some estimates (Ni4):
TK ≈ D exp[
−arctanh(√
C/jz)
/2ν√
C]
ν(Jz − J±) ≈ 0.15
∆E = D =∣
∣A[
S2− (S − 1)2
]∣
∣ ≈ 9.3 K
TK ≈ 1.2 K (m = ±4)
crucial requirements:i) large spin tunnel splitting
ii) large coupling to states in the leads
See also related work by the Aachen group (H. Schoeller).
issues under investigation: i) quantitative theory for transport (NRG, DMRG?)
ii) spin/angular momentum relaxation in isolated molecules
M. Leuenberger and ERM [PRL 97, 126601 (2006)]
Some Questions and Challenges for the STM group:
(1) How does the SMM bind to metallic surfaces?
(2) Where does the additional electron go in a SMM?chemistry/electronic structure
(3) Can the SMM be manipulated by the SMT tip? (move it, flip it, and extract or modify ligands) technical challenge
(4) Does the tip position change the electric response of the SMM?
(5) Can a SP-STM measure the magnetization curve of a SMM (quantum tunneling, coherent oscillations, decoherence)?
physics
SMM
SP!tip
nonmagnetic substrate
magnetic island
SMMs have intrinsically largemagnetization and stronganisotropy, so a magneticisland may not be necessary.
The End
