Resonant and Kondo tunneling through molecular magnets

0.29 MB·English

by  Florian Elste

Resonant and Kondo tunneling through molecular magnets
Florian Elste1 and Carsten Timm2
1Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA
2Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany
(Dated: January 28, 2010)
Transport through molecular magnets is studied in the regime of strong coupling to the leads.
0
We consider a resonant-tunneling model where the electron spin in a quantum dot or molecule is
1
coupledtoanadditional local, anisotropic spinviaexchangeinteraction. Thetwooppositeregimes
0
dominated by resonant tunneling and by Kondo transport, respectively, are considered. In the
2
resonant-tunneling regime, the stationary state of the impurity spin is calculated for arbitrarily

n strong molecule-lead coupling using a master-equation approach, which treats the exchange inter-
a action perturbatively. We find that the characteristic fine structure in the differential conductance
J
persists even if the hybridization energy exceeds thermal energies. Transport in the Kondo regime

8 is studied within a diagrammatic approach. We show that magnetic anisotropy gives rise to the
2 appearance of two Kondopeaks at nonzero bias voltages.

] PACSnumbers: 73.23.Hk,75.20.Hr,73.63.-b,75.50.Xx
l
l
a
h
I. INTRODUCTION Recently, the Kondo effect in single-molecule magnets
-
s with easy-axis anisotropy has been studied by Romeike
e et al.10 Their model describes an anisotropic spin cou-
m Over the past few years the idea of integrating the
pled to metallic electrodes by an exchange interaction,
concepts of spintronics and molecular electronics has
t. developed into a new research field dubbed molecular in the absence of a bias voltage. This differs from the
a model studied here, in which the electronic spin in the
m spintronics.1,2 Progress has not only been stimulated by
relevant molecular orbital is coupled to an additional lo-
technologicalinterestsbuthasalsobeenaccompaniedby
- cal, anisotropic spin via an exchange interaction J, i.e.,
d the realization that magnetic single-molecule transistors
charge fluctuations are explicitly taken into account. In
n exhibit various fundamental quantum phenomena.3–9
o Amongmanypromisingideasdiscussedintheliterature, addition,weincludeanonzerobiasvoltage. Thepresence
c ofaKondoeffectforananisotropicspinisatfirstglance
particular attention has been paid to current-induced
[ surprisingsinceitrequirestwoapproximatelydegenerate
spin reading and writing, spin relaxation, entanglement,
low-energyspinstatesconnectedbyatermintheHamil-
2 quantum computation, and Kondo correlations.10–19
v tonian. ThesimplestHamiltonianforananisotropicspin
6 Anexperimentalrealizationofspintronicsdevicesmay S with S ≥1 exchange-coupled to an electronic spin s
be achieved by using single-molecule magnets in com-
5
9 bination with metallic (nonmagnetic or ferromagnetic) H =−K (Sz)2+Js·S (1)
2
1 leads. For molecular-memory applications, long spin-
. relaxation times are advantageous, which may be real-
9 does not provide such a term. Using a renormalization-
0 izedinmoleculeswithlargemagneticanisotropy,suchas groupapproach,Romeike et al.10 could show that quan-
9 molecules based on Mn12, Fe4, and Ni4.20–22 tum tunneling of the magnetic moment, which is de-
0 Controllinganddetectingthemolecularspinbymeans scribed by higher-order anisotropy terms not included
v: of electronic tunneling into source and drain electrodes in Eq. (1), may give rise to a Kondo peak in the linear
i poses a major challenge. While some approachesrely on conductance, centered at zero bias voltage. The Kondo
X
break junctions, others are based on a scanning tunnel- temperature is found to depend strongly on the ratio of
r ing microscope. In both cases, the coupling between the the applied magnetic field and the anisotropy barrier.
a
moleculeandtheleadscanvarybyseveralordersofmag- Further, Gonz´alez et al.34 have derived a Kondo Hamil-
nitude, thus giving rise to strikingly different transport tonian of the type studied in Ref. 10 from an electronic
regimes.3–6,23–32 model. Theyhaveshownthatatransversemagneticfield
In the regime of weak molecule-lead coupling, many can induce or quench the Kondo effect. This is due to
experimental features such as Coulomb blockade, spin Berry-phaseinterferencebetweendifferentquantumtun-
blockade, sequential tunneling, and cotunneling can be neling paths of the spin.
described within a master-equation or rate-equation ap- Koerting et al.32 consider the nonequilibrium Kondo
proachtreatingtheelectronictunnelingperturbatively.33 effect for a double quantum dot with four leads. By re-
However,forstrongcoupling,low-orderperturbationthe- movingtheleadscoupledtooneofthedotsonewouldob-
ory breaks down. It is then advantageous to treat the tain a model similar to ours. The main difference is that
electronic tunneling exactly, at the price of introducing we include charge fluctuations on the dot which is cou-
approximations elsewhere. pled to the leads, whereas Koerting et al.32 work in the
2
regime of weak tunneling and Coulomb blockade, where fieldinthatitgivesrisetoasplittingoftheKondopeak.
both quantum dots act as local spins. Furthermore,wefindasuppressionofthedifferentialcon-
In the present paper, we address the question of ductance with 1/ε6.
d
spin-dependent resonanttunneling andKondo tunneling The paper is organizedasfollows. InSec. II, we intro-
throughmolecularmagnets. Asnotedabove,weconsider duce our model. Section III considers transport within
a resonant-tunneling model where the electron spin on a master-equation approach, which allows us to study
the quantum dot or molecule is coupled to an additional magnetic nonequilibrium phenomena, whereas Sec. IV
local,anisotropicspinviaanexchangeinteractionJ. We considersadiagrammaticapproach,whichappliesto the
assume this interaction to be weak, which allows us to Kondo regime. In Sec. V we summarize and discuss our
employ perturbation theory for small J. The stationary results further. Some detailed calculations are relegated
current through the left and right leads is identical and to Appendices.
is related to the local electronic spectral function A (ω)
σ
on the molecule by the Meir-Wingreen formula37
II. MODEL
e Γ Γ
L R
hI i= dω [f (ω)−f (ω)] A (ω),
L 2π~ Γ +Γ L R σ Weconsideramagneticmoleculecoupledtotwometal-
σ Z L R
X lic leads. Electronic tunneling through the junction is
(2)
assumed to involve a single orbital with energy ε and
where σ is the spin, Γ is the broadening of the molecu- d
α spin s that is coupled to a local spin S via exchange in-
lar level due to the hybridization with lead α=L,R, to
teraction. The model is described by the Hamiltonian
be defined below, and f denotes the Fermi distribution
α
function of lead α. The spectral function is determined
H =H +H +H , (3)
by the imaginary part of the retarded Green’s function, 0 J mag
A (ω) = −2ImGret(ω). If transport is dominated by a
σ σσ where
single molecular level of energy ε , the coupling to the
d
lteioand,sAgi0v(eωs)ri=seΓto/[a(ωLo−reεnt)z2ia+nΓfo2r/m4]owfitthheΓsp=ecΓtral+fuΓnc-, H0 = εd d†σdσ+ ǫαka†αkσaαkσ
σ d L R σ αkσ
which manifests itself as a peak in the differential con- X X
ductance. For single-molecule devices, the excitation of + tαa†αkσdσ+t∗αd†σaαkσ (4)
additional degrees of freedom due to the electronic tun- αXkσ(cid:16) (cid:17)
neling is expected to translate into additional character-
is the resonant-tunneling Hamiltonian,
istic features in the current.
Weconsidertwocomplementarysituations. Thefirstis H =Js·S (5)
J
thecaseofarbitrarygateandbiasvoltagesbutexcluding
the region where the Kondo contribution to the current withs≡ σσ′d†σ(σσσ′/2)dσ′ istheexchangeinteraction
islarge. Withinamaster-equationapproachtreatingthe between the electrons in the molecular orbital and the
local exchange interaction perturbatively to second or- local spinPS, and
der, we calculate the transition rates between local-spin
states, showing that the spin can be driven out of equi- H =−K (Sz)2 (6)
mag 2
libriumevenforstrongmolecule-leadhybridization. Sig-
natures of inelastic tunneling such as the fine-structure describes the easy-axis magnetic anisotropy of the local
splitting of the differential-conductance peaks persist in spin. We choose the z axis as the easy axis. Here, d†
σ
the regime where the hybridization energy exceeds the creates an electron with spin σ and energy εd on the
thermal energy. molecule,whilea†αkσ createsanelectronwithenergyǫαk,
The secondcase concerns the regime of a large Kondo wavevectork,andspinσinleadα,whichisconsidereda
contribution to the differential conductance, which only noninteracting electron gas. The vector σ ≡(σx,σy,σz)
occurs for small bias voltages on the order of |eV| ∼ denotes the Pauli matrices. In break junctions produced
K2(2S −1), as we shall see. Here, transport is studied by electromigration, the onsite energy εd can be tuned
using a diagrammatic approach. We consider the case by applying a gate voltage.3–6
that the molecular orbital is far from resonance so that
the resonant-tunneling contributions are negligible. In
addition, this allows us to obtain analytical expressions. III. MASTER EQUATION FOR THE SPIN
We find that the magnetic anisotropy gives rise to the
appearance of two Kondo peaks in the differential con- The presence of strong coupling between the molecule
ductance at finite bias voltages ±V . This intrinsically and the leads prevents us from treating the hybridiza-
c
nonequilibrium Kondo effect is quite different from the tion term in Eq. (4) perturbatively. However, since the
zero-bias peak studied by Romeike et al.,10 which relies Hamiltonian becomes bilinear in the limit of vanishing
on higher-order anisotropies absent from our model. In exchange coupling, J = 0, our strategy is to diagonalize
our case, the magnetic anisotropy acts like a magnetic H +H exactly while treating H as a perturbation
0 mag J
3
up to second order. This approach allows us to study Here, operators O(t) with an explicit time argument,
the nonequilibrium dynamics of the molecular spin at including ρ (t), are in the interaction picture, O(t) =
J
finite bias voltages for arbitrary molecule-lead coupling ei(H0+Hmag)t/~Oe−i(H0+Hmag)t/~. Note that second-or-
strengths, provided that Kondo correlations do not lead der perturbation theory in the exchange coupling gives
to a diverging contribution from higher-order terms in the first non-vanishing correction to the conductance,
the expansion. since the expectation value hSi and thus all first-order
We start by rewriting H in terms of new opera- terms vanish exactly due to symmetry.
0
tors,35,36 We areinterestedinthestationarystate. The station-
ary density matrix ρ has to be diagonal in the basis of
H0 = ǫαkc†αkσcαkσ, (7) eigenstates |mi of SJz, since the full Hamiltonian H is
αXkσ invariant under rotation about the z-axis in spin space.
where Inserting Eq. (12) into Eq. (14) we thus obtain a Pauli
aαkσ = ηαα′kk′cα′k′σ, (8) master equation, also called rate equations, of the form
αX′k′ P˙m = Pm+1Rm+1→m+Pm−1Rm−1→m
dσ = ναkcαkσ, (9) −Pm(Rm→m+1+Rm→m−1) = 0 (15)
αk
X
for the occupation probabilities P of spin states |mi in
and m
the stationary state. The transition rates read
ηαα′kk′ = δαα′δkk′ − ǫαk−tαǫναα′k′k′′+iδ, (10) R =|hm±1|S±|mi|2 J2/4
tα m→m±1 2π~
ναk = ǫαk−εd− α′k′ ǫαk−ǫtα2α′′k′−iδ. (11) × dω|ν˜α(ω)|2|ν˜α′(ω−[±2m+1]K2)|2
For simplicity we assume reaPl tunneling amplitudes tα. Xαα′Z
In terms of the new operators, the exchange interaction ×[1−fα(ω)]fα′(ω−[±2m+1]K2). (16)
assumes the form
σ The spectral functions are given by
HJ =J να∗kνα′k′c†αkσ σ2σ′cα′k′σ′ ·S. (12) Γ
αα′Xkk′σσ′ |ν˜α(ω)|2 = (ω−ε )2α+Γ2/4 (17)
The time evolution of the density matrix ρ of the full d
system is described by the von Neumann equation, ρ˙ =
with Γ ≡ Γ +Γ and Γ ≡ 2πt2D . The densities of
−(i/~)[H,ρ]. The degrees of freedom of the local spin L R α α α
states for the leads, D , are taken as constants. Com-
α
are described by the reduced density matrix
pared to Eq. (11) we have approximated the self-energy
ρJ =Trelρ, (13) part of ναk by a constant and absorbed a factor 2πDα.
At zero temperature, the integrals can be evaluated
which is obtained by tracing out all electronic degreesof
analytically. In the limit of large bias voltages,the rates
freedom. Assuming that the large electronic subsystem,
approachthe constant value
which acts as a spin reservoir, is weakly perturbed by
theexchangecoupling,wereplacethefulldensitymatrix πJ2Γ Γ
L R
R =
by the direct product ρ ≃ ρJ ⊗ρel. We need a further m→m±1 2π~Γ[Γ2+(±2m+1)2K2]
approximation for the electronic density matrix ρ . We 2
el ×|hm±1|S±|mi|2. (18)
assume that different chemical potentials µ (µ ) are
L R
imposed for the left (right) lead far from the junction.
On the other hand, at zero bias only the rates involving
It would thus be natural to assume Fermi distributions
the absorptionof energy are finite, whereas the emission
f (ω) = f(ω−µ ) for the physical electrons created by
α α rates vanish.
a† . However, we need to make a reasonable assump-
αkσ Solving Eq. (15) allows us to compute the differential
tiononthetransformedc fermionsappearinginEqs.(7)
conductance of the molecular junction. The current op-
and(12). Sincec†Lkσ (c†Rkσ)createsanelectroninastate erator of lead α reads
with vanishing probability density far from the junction
e
in the right (left) lead, we assume the occupation num- Iα =−i~ tα a†αkσdσ−d†σaαkσ
bers of these states to be described by fα(ω). Xkσ (cid:16) (cid:17)
Making use of the Markov approximation that ρ e
changesslowly on the time scale of electronic relaxationJ, =i~ tανα∗′k′ηαα′k′k′′c†α′k′σcα′′k′′σ −h.c. .
we obtain Xkσα′kX′α′′k′′(cid:16) (cid:17)
(19)
1 t
ρ˙ (t)=− dt′Tr [H (t),[H (t′),ρ (t)⊗ρ ]].
J ~2 el J J J el In order to compute the spin-dependent contribution to
Z−∞
(14) the expectation value hI i ≡ TrI ρ = TrI (t)ρ(t) of
α α α
4
the totalcurrent,we usethe iterativesolutionofthe von (a)
Neumann equation,
1 ΓΓ = = K K/1/200
ρ(t)=−~12 t dt′ t′ dt′′[HJ(t′),[HJ(t′′),ρ(t′′)]]. 0.5 ΓΓΓΓ == == KK KK222/2222/100
Z−∞ Z−∞
(20)
>
A term containing ρ(−∞) has dropped out here since it
L 0
I
is linear in H and thus vanishes upon taking the trace. <
J
Making use of the Markov approximation we find
-0.5 1.1
1 t t′
hI i(2) = − dt′ dt′′
α ~2 -1 1
Z−∞ Z−∞
×Tr [[Iα(t),HJ(t′)],HJ(t′′)]ρ(t) (21) -20 -10 0 10 20
V (K /e)
for the second-order term. Carrying out the time inte- 2
(b)
grals and evaluating the spin sums as explained in Ap-
pendix A, we obtain
1
hILi(2) = 2πe~ J42 ΓL−δLα[ΓL+ΓR] 0.8 RR21→→10
αXα′α′′(cid:16) (cid:17) P2,P-2
P,P
× Pm 21|hm−1|S−|mi|2Iαα′α′′([−2m+1]K2) 0.6 P10 -1
m (cid:26)
X1 0.4
+ |hm+1|S+|mi|2Iαα′α′′([2m+1]K2)
2
0.2
+|hm|Sz|mi|2Iαα′α′′(0) (22)
(cid:27) 0
-20 -10 0 10 20
with V (K /e)
2
Iαα′α′′(E)≡ dω|ν˜α′′(ω)|2 FIG.1: (Color online) (a) Current-voltagecharacteristics for
Z
different hybridization energies, Γ=K /20, Γ=K /10, and
× |ν˜α(ω)|2|ν˜α′(ω−E)|2 [1−fα(ω)]fα′(ω−E) Γ = K . The inset shows a closeup2of the fine2structure
2
n−|ν˜α(ω)|2|ν˜α′(ω+E)|2fα(ω)[1−fα′(ω+E)] . aatndpooscictiuvpeabtiioans.p(rbo)baMbialigtnieesticPmtraanssiftuionnctiroantessoRf 2b→ia1s, VR1→fo0r
(2o3) Γ = K2/20. We assume symmetric couplings to the leads,
ΓL = ΓR, and symmetric capacitances, µL = eV/2, µR =
Equations (22) and (23) give the first non-vanishing cor- −eV/2, a local molecular spin of length S = 2, εd = 4K2,
rection to the zero-order current hI i(0), which is ob- andzerotemperature. Further,wesetJ =Γ/5. Currentsare
L given in units of (2e/~)ΓLΓR/Γ. Rates are given in units of
tained from the Meir-Wingreen formula [Eq. (2)] by in-
theirmaximum values, cf. Eq. (18).
serting the spectral function of the unperturbed system,
A0(ω) = Γ/[(ε −ω)2 +Γ2/4]. Note that Eq. (2) with
σ d
A = A0 is recovered by inserting the equilibrium den-
σ σ
sity matrix ρ0 and the current operator from Eq. (19)
into hI i0 =TrI ρ0.
L L If the magnetic anisotropy is large compared to the
A simple interpretation of Eq. (22) is possible for the
hybridizationenergy,K ≫Γ,the generalexpressionfor
special case of a local spin of length S = 1/2, for which 2
the current in Eq. (22) simplifies to
the magnetic anisotropy K is irrelevant and can be set
2
to zero. For this case we obtain
hILi(2) = 2πe~ J2S(S4 +1) ΓLΓΓR hILi(2) = 2πe~ J42 ΓL−δLα[ΓL+ΓR] Iαα′α′′(0)
Γ 3 αXα′α′′(cid:16) (cid:17)
× dω (ω−ε )2+Γ2/4 [fL(ω)−fR(ω)]. (24) × Pm|hm|Sz|mi|2, (25)
Z (cid:20) d (cid:21) m
X
Here, the third power of the spectral function ap-
pears, since the current operator and the two exchange-
interaction operators in Eq. (21) are each bilinear in since the integrals Iαα′α′′([±2m + 1]K2) are negligible
fermionic operators. compared to Iαα′α′′(0). Assuming symmetric capaci-
5
(a) characteristic fine structure of the current step at the
Coulomb-blockade threshold is due to the second-order
1 ΓΓ == KK2//2100 contribution, hILi(2), whereas the main step is mostly
2 coming from hI i(0). The fine structure persists as long
Γ = K L
0.5 2 asthehybridizationenergyΓremainssmallcomparedto
the magnetic anisotropy K . Note that the broadening
2
>
of the steps is due to Γ > 0, and not to the temper-
L 0
I
< ature, for which we assume T ≪ Γ. For bias voltages
below |eV| = 2ε , the current and all magnetic excita-
d
-0.5 1.1 tions are thermally suppressed. However, as soon as the
chemical potential of one lead aligns with the resonance
-1 1 of the molecule, the current increases to its maximum
value. The current-inducedmagnetic transitionsbecome
-20 -10 0 10 20
V (K /e) energetically accessible at the same time, as shown in
2
Fig.1(b),resultinginnonequilibriumprobabilitiesP of
m
(b)
thedifferentspinstates. Inthelimitoflargebiasvoltages
all spin states are equally occupied, P =1/(2S+1).
1 R m
2→1
R Interestingly,thepresenceofmagneticanisotropyleads
1→0
0.8 P,P to negative differential conductance in the vicinity of
2 -2
P,P |eV| = 2ε . The underlying mechanism shall be ex-
1 -1 d
P plained briefly. According to Eq. (25), the magnetic
0.6 0
states with maximum quantum numbers m=±S domi-
nate the current since the current is proportional to the
0.4
average P |hm|Sz|mi|2. Each decrease in P thus
m m ±S
causes a decrease in the current. Therefore, the spin-
0.2 P
dependent contribution to the current is large at low
bias voltages, where P = 1/2 and P m2 = 4,
0 ±2 m m
-20 -10 0 10 20 and small at high bias voltages, where P±2 = 1/5 and
V (K /e) P m2 =2. P
2 m m
We next consider the situation of strongly asymmet-
P
FIG. 2: (Color online) (a) Current-voltagecharacteristics for ric molecule-leadcouplings andcapacitances,i.e., |t |≪
L
different hybridization energies, Γ=K2/20, Γ=K2/10, and |tR|, ΓL ≪ ΓR, and µL ≃ eV, µR ≃ 0. Note that
Γ = K2. The inset shows a closeup of the fine structure this regime is naturally realized in many experimental
at positive bias. (b) Magnetic transition rates R2→1, R1→0 setups,whereasperfectlysymmetriccouplingsareingen-
and occupation probabilities Pm as functions of bias V for eralmoredifficulttoachieve. Thecurrent-voltagecurves
Γ = K /20. We assume strongly asymmetric couplings to
2 andthebiasdependenceofthemagneticexcitationrates
the leads, ΓL ≪ ΓR, and strongly asymmetric capacitances,
are shown in Fig. 2. Due to the asymmetric coupling
µL = eV, µR = 0, a local molecular spin of length S = 2,
εd = 4K2, and zero temperature. Further, we set J = Γ/5. and εd > 0, the current is suppressed for negative bias
Currentsaregiveninunitsof(2e/~)ΓLΓR/Γ. Ratesaregiven voltages. However, the characteristic steps correspond-
in unitsof their maximum values, cf. Eq. (18). ing to excitations of the molecular spin reappear at pos-
itive bias. Only their abscissas are reduced by a factor
of 2, since the chemical potential of the left lead is now
µ = eV instead of µ =eV/2. The device thus acts as
L L
tances, one finds, in the limit of large bias voltages, a rectifier. Note the small ohmic contribution with con-
stant slope for large coupling Γ in Fig. 2(a). We return
2e Γ Γ
hILi(0) → ~ LΓR, (26) to this point below.
We finally turn to the full bias and gate-voltage de-
2e Γ Γ 3S(S+1) J2
hI i(2) → L R , (27) pendence of the current. In both the symmetric and
L ~ Γ 4 Γ2
the asymmetric case, selection rules for the spin require
forthe zero-orderandsecond-ordercontribution,respec- changes in the magnetic quantum number by ∆m = 0
tively. Note that the ferromagnetic or antiferromagnetic or ∆m = ±1, where ∆m = 0 corresponds to elastic and
sign of J does not affect the results in the present ap- ∆m=±1to inelasticscatteringevents,cf.Fig.3(a). In-
proximation. elastic scattering processes appear as additional steps in
We first consider the situation of symmetric molecule- the current and give rise to the magnetic fine structure
lead couplings and capacitances, i.e., |t | = |t |, Γ = in the two dimensional density plots of the second-order
L R L
Γ , and µ = eV/2, µ = −eV/2. Figure 1(a) shows contributiontothecurrentasafunctionofbiasandgate
R L R
the current-voltage characteristics up to second order in voltages shown in Figs. 3(b), (c). We can now under-
J for the case of a local spin of length S = 2. The stand the origin of the weak ohmic conduction seen in
6
(a) sary to resum the divergences to all orders in J.40–44
Thetotalcurrentthroughthemoleculeisrelatedtothe
E local electronic spectral function A (ω) = −2ImGret(ω)
σ σσ
bytheMeir-Wingreenformula,Eq.(2),whereGret (ω)=
σσ′
d(t−t′)eiω(t−t′)Gret (t−t′) denotes the Fourier trans-
σσ′
form of the retarded single-particle Green’s function
R
Grσeσt′(t,t′)≡−iθ(t−t′) dσ(t),d†σ′(t′) . (28)
Makinguseofthetransformat(cid:10)i(cid:8)ondefinedinE(cid:9)q(cid:11)s.(8)–(11)
m requires to compute the finite-temperature time-ordered
-2 -1 0 +1 +2
Green’s function
(b) (c)
12 12 Gαα′kk′σσ′(τ,τ′)≡− Tτcαkσ(τ)c†α′k′σ′(τ′) . (29)
e)6 e)6 Our strategy is to expand(cid:10)Gαα′kk′σσ′ in powers(cid:11)of J.
K /2 K /2 forImnuolrad,erwteonoebetdainthteheimcuargriennatryfropmartthoefMtheier-Weleicntgrroeneinc
(0 (0
V V Green’s function,
-6 -6
Im Grσeσt(ω)=Im ναkνα∗′k′Gαreαt′kk′σσ(ω), (30)
-12 -12 Xσ ααX′kk′σ
-6 -3 0 3 6 -6 -3 0 3 6
d (K 2 ) d (K 2 ) where Gαreαt′kk′σσ(ω) denotes the retarded Green’s func-
tion. All non-vanishing diagrams of the local Green’s
FIG.3: (Coloronline)(a)Levelschemeshowingallspintran-
sitions to order J2. (b),(c) Two-dimensional density plots of function up to third order in J are shown in Fig. 4, fol-
the absolute value of the second-order current contribution lowing the notation of Ref. 38. We again consider the
hILi(2) as a function of bias voltage V and gate potential εd case of strongly asymmetric couplings, i.e., |tL| ≪ |tR|,
for Γ = K2/20. εd is controlled by the gate voltage. We ΓL ≪ ΓR, and µL ≃ eV, µR ≃ 0. As we shall see,
choosethesameparametersasinFig.1. Bright(dark)colors Eq. (30) is then dominated by the contribution from the
correspond to high (low) currents. In (b) we assume sym- right electrode, α = α′ = R. This allows us to describe
metric couplings, µL = eV/2, µR = −eV/2, ΓL = ΓR, while the molecular degrees of freedom by a thermal equilib-
in (c) we assume asymmetric couplings, µL = eV, µR = 0, riumdistributionfunctionthat is independentofthe ap-
ΓR ≫ΓL. pliedbiasvoltageandtoobtainananalyticalexpression.
S S
Fig. 2(a) for Γ = K . What we are seeing is the tail of
2
the current step at ǫ = 0 in Fig. 3(c), which is consid- = +
d
erably broadened for Γ = K . Note that we here have
2
ǫ =4K = 4Γ, i.e., we are only 4Γ away from the step.
d 2
Since this distance does not depend on the bias voltage,
S S S
the conductivityisessentiallyconstant,leadingto ohmic
behavior. +
IV. KONDO TRANSPORT
FIG.4: Non-vanishingdiagramsoftheimpurityGreen’sfunc-
Second-order perturbation theory in the exchange in- tion up to third order in J, following Ref. 38. Diagrams in-
teraction J fails, even for small J, if the prefactors of cludingfermionloopsvanishexactlyandarenotshown. Spin
higher-ordertermsdiverge. ThisisthecaseintheKondo averages are denoted by dotted lines.
regime. Logarithmicdivergencesoftheconductancefirst
appear in terms of third orderin J.40–44 (In this section,
weassumeantiferromagneticexchange,J >0.) Studying The evaluation of Eq. (30) is shown in Appendix B.
the emergence of Kondo correlations thus requires to go We obtain
beyond the second-order master-equation approach dis-
2
Γ/2 ε
cussedinSec.III. ForsufficientlysmallJ andsufficiently Im Gret(ω)≃− + d ImΣret(ω)
large thermal energies, the conductance is dominated by σ σσ ε2d+Γ2/4 (cid:18)ε2d+Γ2/4(cid:19)
X
the third-order contribution, which we calculate in this (31)
section. At lower temperatures, it would become neces- with
7
π 1−f(ω+E −E)
ImΣret(ω) = − J2ν (ε ) P m l δ |hm|Si|ni|2
0 d m nl
2 1−f(ω)
(
mnl i
X X
x x
−iJν (ε ) ǫ hm|Si|nihn|Sj|lihl|Sk|mi ln +ln ,
0 d ijk
ijk (cid:20) (cid:12) (ω+Em−En)2+T2(cid:12) (cid:12) (ω+En−El)2+T2(cid:12)(cid:21))
X (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)p (cid:12) (cid:12)p (cid:12) (32)
(cid:12) (cid:12) (cid:12) (cid:12)
where
2
0.6 G (e /h)
Γ/2π
ν (ε )= . (33)
0 d ε2+Γ2/4
d
0.5 T = 0.1 K2
In the derivation we have assumed the molecular level
to be far from resonance, i.e., |ε | to be large compared
d T = 0.5 K2
to K2S, T, and Γ, but still small compared to the band 0.4
width x of the leads. Details are discussed in Appendix T = K 2
B. We have also assumed |ω| ≪ |ε |, the significance
d
of which will become clear in the following step. Under 0.3
these conditions, the resonant-tunneling contribution to T = 5 K2
the differential conductance, which we have studied in
Sec. III, is negligible compared to the Kondo contribu-
-10 -5 0 5 10
tion.
V (K 2 /e)
Inthe low-temperaturelimitT ≪Γ,derivativesofthe
FIG. 5: (Color online) Differential conductance Gin unitsof
Fermi functions with respect to the bias voltage become e2/hfordifferentthermalenergiesT (inunitsofK )obtained
2
deltafunctionsandthedifferentialconductancesimplifies
fromEq.(34)asafunctionofbiasvoltageV inunitsofK /e.
2
to Here we assume a local molecular spin of length S = 2 and
e2 Γ Γ Γ choose Jν0(εd) = 1, x = 100K2, and ΓR = 100ΓL. Note
G≃ L R that the parameters Γ, εd, and J leave the curves for G (in
2π~ Γ (ε2d+Γ2/4 arbitrary units) invariant except for changing the constant
offset.
2
ε
−2 d ImΣret(eV) . (34)
ε2+Γ2/4
(cid:18) d (cid:19) )
Note that the argument ω of Σret(ω) is eV. The as- In our case, the splitting of the Kondo peak as a con-
sumption|ω|≪|ε |madeabovethuscorrespondsto|ε | sequence of magnetic anisotropy is more similar to the
d d
also being large compared to the bias, |eV|. The spec- situation of a quantum dot in an external magnetic field
tral function has logarithmic divergences for T → 0 at with Zeeman energy B, where a Zeeman splitting of the
the transition energies of the molecule, E −E , cor- energy levels leads to the occurrence of two conductance
m n
responding to virtual transitions between two magnetic peaks at eV ≃ ±B in the Kondo regime.45 At higher
states|miand|ni. Onerecoverstheprefactor3πJ2D0/8, temperatures, T ≫K2,the twoKondopeaks mergeinto
see Ref. 38, for the case of spin S = 1/2 and the (then a single peak centered at zero bias due to the thermal
irrelevant) anisotropy set to K =0. excitation of spin states with higher energy.
2
Numerical results for nonzero temperatures are shown We now turn to the Kondo temperature T . Poor
K
in Fig. 5. The differential conductance G diverges loga- man’s scaling for the equilibrium case results in T =
K
rithmically for T → 0 at critical bias voltages V = ±V 0 because the matrix elements of S± between the two
c
witheV =E −E =K (2S−1)sincetheemergence degenerate ground states of the local spin vanish for our
c S−1 S 2
of Kondo correlations requires the bias voltage to ex- model if S > 1/2.10 Since a Kondo effect evidently does
ceedthe energyof the transitionfrom the groundstates, occurat nonzerobias,this resultis clearlynotsufficient.
m = ±S, to the first excited states, m = ±(S − 1). A rough estimate of the Kondo temperature T can be
K
Note that G(V) is symmetric for positive and negative obtained as the temperature for which the second-order
bias, in spite of the highly asymmetric coupling since it andthird-ordertermsbecomeequalinEq.(32). We find
isprobingthe electronicspectralfunction. The situation T ∼ exp[−1/αν (ε )J], where α is a number of the
K 0 d
is quite different from the case considered by Romeike orderofunity. Inthe limitK →0,wherethe twopeaks
2
et al.,10 which concerns a zero-bias peak resulting from inFig.5 wouldmerge,werecoverthe resultα=2foran
quantumtunnelingbetweenthetwostates|Siand|−Si. isotropic spin.
8
Sincewefocusonthecaseofstronglyasymmetriccou- First, it would be interesting to include a local Coulomb
plings, where the molecular degrees of freedom are in interaction U between the electrons on the molecule.
equilibrium with one of the two leads, the logarithmic However, due to the large hybridization there are no
divergences are cut off by temperature or the applied states with large probability on the dot and the effect of
bias voltage, respectively, in our perturbative approach, U is expected to be relatively weak. We expect that for
see Eq. (32). The divergence for T → 0 is unphysi- very large U an equilibrium Kondo resonance could oc-
cal and would likely be removed by a resummation of cur as a zero-biaspeak in the differential conductance in
higher-orderterms. ByanalogytoRef.45,weconjecture additiontothenonequilibriumKondoeffectdescribedin
that the divergence is ultimately cut off by a voltage- thispaper. Second,thepresenceofanexternalmagnetic
dependent spin-relaxation rate. fieldmightleadtoaninterestinginterplaywiththesplit-
While we have so far discussed the dependence on the ting ofthe Kondopeaks due to the magnetic anisotropy.
bias voltage,see Fig. 5, we now turn to the gate voltage. Finally, it would be desirable to combine the two cases
The gate voltage shifts the on-site energy ε and thus studied here and to analyze the Kondo effect in mag-
d
enters the expression for the current through the square netic molecules in the resonant-tunneling regime, where
of the spectral function Γ/[ε2 + Γ2/4] and the square resonant-tunneling contributions to the conductance are
d
of the factor ε /[ε2 +Γ2/4]. In particular, we obtain a not negligible and the spin is driven out of equilibrium
d d
suppression of G∝1/ε6 in the limit of strong detuning, by the current.
d
ε ≫Γ.
d
Acknowledgments
V. CONCLUSIONS
We would like to thank A. Donabidowicz-Kolkowska,
We have studied the spin-dependent electronic trans- D. R. Reichman, and A. J. Millis for useful discussions.
port through magnetic molecules for strong coupling to Financial support by the Deutsche Forschungsgemein-
theleads. Ourdiscussionhasfocusedontwocomplemen- schaft is gratefully acknowledged.
tary regimes.
For the first regime, we have presented a description
oftransportintermsofamasterequationthatkeepsthe Appendix A: Calculation of the current
electronic tunneling exactly, holds for arbitrary bias and
gatevoltages,andtreatsthelocalexchangeinteractionJ In this appendix we give details on the derivation of
perturbativelyatsecondorder. Thisapproachisthusap- Eqs. (22) and (23). We start from Eq. (21),
plicableforsmallJ. Wehavederivedthebias-dependent
magnetictransitionratesshowingthatthetunnelingcur- 1 t t′
rent can be used to drive the molecular spin out of equi- hIαi(2) = − ~2 dt′ dt′′
librium. Further, we have shown that the characteristic Z−∞ Z−∞
×Tr [[I (t),H (t′)],H (t′′)]ρ(t). (A1)
fine structure of the differential-conductance peaks per- α J J
sists for strong molecule-leadcoupling, where the broad-
Inserting the expressions for the current operator I ,
ening of the peaks is determined by the hybridization α
Eq. (19), and for the exchange interaction H , Eq. (12),
energies. J
we find
The perturbative expansion in J fails if Kondo corre-
lations contribute significantly to the transport. In this et J2 t t′
case, prefactors of the third- and higher-order terms in hI i(2) =i L dt′ dt′′Tr δ δ
L ~ 4~2 σσ1 σ1σ2
J diverge for T → 0. The Kondo correlations can be- Z−∞ Z−∞ kσ 123456
X X
come important for small bias voltages on the order of
|eV| ∼ K2(2S −1). Here, transport is described by the × η1Lk∗ν2ν3∗ν4ν5∗ν6−ν1∗η2Lkν3∗ν4ν5∗ν6
Meir-Wingreen formula in combination with a diagram- (cid:18) (cid:19)
maticcalculationofthe localelectronicspectralfunction × c†(t)c (t),c†(t′)c (t′)σ ·S(t′) ,
1 2 3 4 σ3σ4
ofthe molecule. We haveassumedthe molecularlevelto
(cid:20)h i
befarfromresonance,whichontheonehandmakessure
c†(t′′)c (t′′)σ ·S(t′′) ρ(t), (A2)
thatthe resonant-tunnelingcontributionsto the conduc- 5 6 σ5σ6
(cid:21)
tancearesmallandwhichontheotherallowsustoobtain
analyticalresults. We have shownthatKondo peaks ap- where we have assumed t to be real. Here, the short-
L
pearatfinitebiasvoltagesproportionaltotheanisotropy hand notation j =1,2,3,4,5,6stands for (α ,k ,σ ).
j j j
energy of the molecular spin. Introducing τ =t−t′ and τ′ =t′−t′′ and assuming a
Our results leave several avenues for future research. product state gives
9
et J2 ∞ ∞
hI i(2) =i L dτ dτ′ ηLk∗ν ν∗ν ν∗ν −ν∗ηLkν∗ν ν∗ν
L ~ 4~2 1 2 3 4 5 6 1 2 3 4 5 6
Z0 Z0 k 123456
X X (cid:0) (cid:1)
× δ δ δ f f (1−f )ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ f (1−f )(1−f )ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~
23 45 16 2 4 6 14 25 36 2 4 6
n(cid:2)−δ δ δ f f (1−f )ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~+δ δ δ (1−f )f (1−f )ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~
14 25 36 2 4 6 23 45 16 2 4 6
×Tr 2S(0)·S(−τ′)ρ
J J (cid:3)
− δ δ δ f (1−f )f ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ (1−f )(1−f )f ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~
23 45 16 2 4 6 14 25 36 2 4 6
(cid:2)−δ14δ25δ36(1−f2)f4f6ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~+δ23δ45δ16(1−f2)(1−f4)f6ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~
×Tr 2S(0)·S(τ′)ρ . (cid:3)(A3)
J J
o
This result can be rewritten as
et J2 ∞ ∞
hI i(2) =i L dτ dτ′ ηLk∗ν ν∗ν ν∗ν −ν∗ηLkν∗ν ν∗ν
L ~ 4~2 1 2 3 4 5 6 1 2 3 4 5 6
Z0 Z0 k 123456
X X (cid:0) (cid:1)
× δ δ δ f (1−f )ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ f (1−f )ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~ Tr 2S(0)·S(−τ′)ρ
23 45 16 4 6 14 25 36 2 6 J J
−nδ(cid:2) δ δ (1−f )f ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ (1−f )f ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~ T(cid:3)r 2S(0)·S(τ′)ρ .
23 45 16 4 6 14 25 36 2 6 J J
(cid:2) (cid:3) (Ao4)
The sums over spin indices are simplified by making use Noting that Eqs. (10) and (11) imply
of the identities
ηLk =δ +iπD t ν (ǫ ) (A7)
σσσ′ ·S1 σσ′σ·S2 = 2S1·S2, k 1 Lα1 L L α1 1
σσ′ X
X
σσσ′ ·S1 σσ′,−σ·S2 = 0. (A5) and
σσ′
X
Γ
ν (ǫ)−ν (ǫ)∗ =i |ν (ǫ)|2, (A8)
In the coefficients ναk in Eq. (11), we approximate the α α t α
α
self-energy part by a constant, as we did in Sec. III,
we arrive at the following expression for the tunneling
t
α
ναk =να(ǫαk)≃ . (A6) current:
ǫαk−εd−iΓ/2
et J2 ∞ ∞
hI i(2) =i L dτ dτ′ (δ −iπD t ν∗)ν ν∗ν ν∗ν −ν∗(δ +iπD t ν )ν∗ν ν∗ν
L ~ 4~2 Lα1 L L 1 2 3 4 5 6 1 Lα2 L L 2 3 4 5 6
Z0 Z0 123456
X (cid:2) (cid:3)
× δ δ δ f (1−f )ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ f (1−f )ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~ Tr 2S(0)·S(−τ′)ρ
23 45 16 4 6 14 25 36 2 6 J J
−nδ(cid:2) δ δ (1−f )f ei(ǫ6−ǫ2)τ/~ei(ǫ6−ǫ4)τ′/~−δ δ δ (1−f )f ei(ǫ4−ǫ2)τ/~ei(ǫ6−ǫ2)τ′/~ T(cid:3)r 2S(0)·S(τ′)ρ .
23 45 16 4 6 14 25 36 2 6 J J
(cid:2) (cid:3) (Ao9)
SincewehaveassumedρJ tobediagonalinthestationary Appendix B: Calculation of the impurity Green’s
state, we finally obtain Eqs. (22) and (23). function
InordertousetheMeir-Wingreenformulaforthecon-
ductance, we have to compute the imaginary part of the
10
Green’s function in Eq. (30). We consider the situation into averages of pairs. We follow Ref. 38 in evaluating
of strongly asymmetric molecule-lead couplings and ca- the spin averages. Expanding the electronic Matsubara-
pacitances, i.e., |t | ≪ |t |, Γ ≪ Γ , and µ ≃ eV, Green’s function in powers of J and organizing the ex-
L R L R L
µ ≃0. pansion in terms of topologically distinct diagrams, one
R
SinceWick’stheoremdoesnotapplytospinoperators, obtains38
averages of products of spin operators do not factorize
∞ J n β β
Gαα′kk′σσ′(τ,τ′)= −~ dτ1··· dτn Tτ Si1(τ1)···Sin(τn)
nX=0(cid:18) (cid:19) Z0 Z0 i1X···inσ1···σXn,σ1′···σn′ D (cid:2) (cid:3)E0
σ σ
× Tτ Bσ†1(τ1) σ21σ1′ Bσ1′(τ1)···Bσ†n(τn) σ2nσn′ Bσn′ (τn)cαkσ(τ)c†α′k′σ′(τ′) , (B1)
(cid:28) h i(cid:29)0
where β ≡ 1/T denotes the inverse thermal energy. For where we have used that Trσiσj = 2δ . Here, E ≡
ij m
convenience, we have defined Bσ ≡ αkναkcαkσ. All −K2m2 denotes the magnetic anisotropy energy in the
non-vanishingdiagramsupto thirdorderinJ areshown spinstate |miwithoccupationprobabilityP . Thespin
m
in Fig. 4. The linear term vanishePs, since hSi = 0. averages in Eq. (B1) are to be evaluated for the unper-
Diagrams with fermion loops are zero for the following turbed Hamiltonian H ,38 leading to P ∝ e−βEm.
mag m
reasons:38 a loop with a single fermion line results in We restrict ourselves to the off-resonance situation, i.e.,
taking the trace of the Pauli matrix in the vertex,which the dark region in Fig. 3(c) with negligible resonant-
yields zero. A loop with two fermion lines appearing in tunneling differentialconductance,where the spin essen-
the third-order diagrams gives rise to a trace over two tially remains in equilibrium. This is certainly satisfied
Pauli matrices, Trσiσj = 2δ . The resulting spin aver- if |ε | is large compared to the energy scales relevant for
ij d
age hTτ[Si1(τ1)Si2(τ2)Si3(τ3)]i0, with at least two of i1, the Kondo contributions, K2S and T.
i , and i equal, vanishes. Equation (B5) contains a sum over leads, α = L,R,
2 3 1
Splitting off the zero-orderterm, the Green’s function andafactoroft2 underthesum. Sincewehaveassumed
α1
in Eq. (B1) can be written as38 strongly asymmetric couplings, |t | ≪ |t |, the sum is
L R
dominatedby the contributionfrom the rightlead, α =
Gαα′kk′σσ′(τ,τ′)=Gα0kσ(τ,τ′)δαα′δkk′δσσ′ R. Dropping the term with α = L, we note that 1the
1
β β Green’s function G0 (τ,τ′) in Eq. (B3) only contains
+ dτ1 dτ2Gα0kσ(τ,τ1)Σαα′kk′σσ′(τ1,τ2) the Fermi distributiRoknσfunction for the right lead, which
×Z0Gα0′k′σZ′0(τ2,τ′), (B2) IismfpRor(tǫaRnkt)ly=, thfe(ǫrReksu)lt=ing1/e(xepβrǫeRsksi+on1)is, isnindceepeµnRden=t o0f.
where the unperturbed Matsubara-Green’s function in the bias voltage.
the imaginary-time domain is given by Furthermore, we see that Eq. (B5) contains a factor
Gα0kσ(τ,τ′)=−[θ(τ −τ′)−f(ωαk)]e−ωαk(τ−τ′)/~ ttαhteαc′o.nFtrriobmutEioqn.f(r3o0m) lweaedosbαta,iαn′tihseprsoapmoertfiaocntaolrtosot2αtth2αa′t.
(B3) Since we have assumed |t | ≪ |t |, we can neglect all
L R
with ωαk ≡ǫαk−µα. In the frequency domain we have contributions except for α = α′ = R. We will keep only
1 these contributions from now on.
Gα0kσ(iωn)= iωn−ωαk, (B4) TakingtheFouriertransformofEq.(B2)andperform-
ing the analytic continuation, we obtain the retarded
whereiωn isafermionicMatsubarafrequency. Notethat Green’s function
weareonlyinterestedinthespintraceoftheself-energy,
σΣrαeαt′kk′σσ, which enters in the Meir-Wingreen for- GrReRtkk′σσ(ω)=GrReRt,k0k′σσ(ω)
mula.
1
PThe second-order term of the self-energy yields +(cid:20)P ω−ǫRk −iπδ(ω−ǫRk)(cid:21) ΣrReRtkk′σσ(ω)
Xσ Σ(α2α)′kk′σσ(τ1,τ2)= 2J~22 α1Xk1σ1Xm,nGα01k1σ1(τ1,τ2) ×(cid:20)P ω−1ǫRk′ −iπδ(ω−ǫRk′)(cid:21), (B6)
×να∗kνα′k′|να1k1|2 where P denotes the principal value. We assume |εd| to
× |hm|Si|ni|2e(Em−En)(τ1−τ2)/~Pm, (B5) be large not only compared to K2S and T but also to
Γ. One can then show that the delta-function terms are
i
X