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Cotunneling signatures of Spin-Electric coupling in frustrated triangular

molecular magnets

J.F. Nossa1, 2 and C.M. Canali1

1Department of Physics and Electrical Engineering,Linnaeus University, SE-39182 Kalmar, Sweden

2Solid State Physics/The Nanometer Structure Consortium,Lund University, Box 118, SE-221 00 Lund Sweden

(Dated: January 27, 2018)

The ground state of frustrated (antiferromagnetic) triangular molecular magnets is characterizedby two total-spin S = 1/2 doublets with opposite chirality. According to a group theory analysis[M. Trif et al., Phys. Rev. Lett. 101, 217201 (2008)] an external electric field can efficientlycouple these two chiral spin states, even when the spin-orbit interaction (SOI) is absent. Thestrength of this coupling, d, is determined by an off-diagonal matrix element of the dipole operator,which can be calculated by ab-initio methods [M. F. Islam et al., Phys. Rev. B 82, 155446 (2010)].In this work we propose that Coulomb-blockade transport experiments in the cotunneling regimecan provide a direct way to determine the spin-electric coupling strength. Indeed, an electric fieldgenerates a d-dependent splitting of the ground state manifold, which can be detected in the inelasticcotunneling conductance. Our theoretical analysis is supported by master-equation calculations ofquantum transport in the cotunneling regime. We employ a Hubbard-model approach to elucidatethe relationship between the Hubbard parameters t and U , and the spin-electric coupling constantd. This allows us to predict the regime in which the coupling constant d can be extracted fromexperiment.

I. INTRODUCTION

Molecular magnets (MMs)1 represent a richplayground for exploring quantum mechanics atthe nanoscale, and are intensively investigatedboth in condensed matter physics and chemistry.MMs, rationally designed and realized by chemicalengineering,2 are promising building blocks of elec-tronic devices for molecular spintronics,3,4 and forclassical5 and quantum information processing.6–8

For applications in quantum computation, MMswith frustrated antiferromagnetic coupling be-tween spins are particularly promising, since at lowenergies they behave effectively as magnetic two-level systems with long spin coherent times, whichcan be used as qbits to encode and manipulatequantum information.2,8 One outstanding issue inquantum information processing is the need of re-alizing fast control and switching between quan-tum spin states. Standard spin-control techniquessuch as electron spin resonance (ESR), carried outby time dependent magnetic fields, have limita-tions, since in practice it is difficult to achieveswitching times of the order of nanoseconds forlarge enough fields. The need to achieve spatialresolutions of the order of 1 nm represents anotherserious challenge for spin manipulations via mag-netic fields. For these reasons, control via electricfields seems to be a much more promising alterna-tive, since strong electric fields can be switched onand off fast, and applied selectively to nanoscaleregions.9–11

Electric control and manipulation of magneticproperties is an important topic in solid statephysics, presently studied in multiferroic materi-

als, dilute magnetic semiconductors and topologi-cal insulators. The electric control of nanomagnetspresents both hard challenges and novel possibili-ties. Since electric fields do not couple directly tospins, electric control can typically occur only in-directly, e.g., via a manipulation of the spin-orbitinteraction (SOI). Indeed, interesting spin-electriceffects induced solely by SOI have been realizedin semiconductor quantum dots.12 The applicabil-ity of this procedure in MMs on the other handis much harder, since the relative strength of theSOI scales with the volume of the system, imply-ing that impractically large electric fields are re-quired for systems of the order of a few nanome-ters. Therefore alternative schemes for efficientspin-electric coupling in MMs have been proposed.One example relies on the electric manipulation ofthe spin exchange constant13,14 which can triggervarious level crossings between magnetic states ofa different total spin. Here we are interested in an-other type of spin-electric coupling, made it possi-ble in certain antiferromagnetic MMs by the lack ofinversion symmetry, as proposed by Trif et al..15

It turns out that in some of these antiferromag-netic molecules, such as the triangular {Cu3} and{V3} MMs,16,17 and other odd-spin rings, an elec-tric field can couple spin states through a combina-tion of exchange and chirality of the spin-manifoldground state (GS). For triangular MMs this cou-pling is nonzero even in the absence SOI.

The low-energy physics of a triangular MMcan be described by three identical 1/2-spin Cucations, located at rj , j = 1, 2, 3, interacting viaan antiferromagnetic (Heisenberg) exchange cou-pling (see Fig. 1). The ground state consists of two

2

FIG. 1. Schematic representation of a triangularmolecule.

FIG. 2. The three independent spin configurations as-sociated with total spin projection Sz = 1/2. The twochiral ground states |E′

±, Sz = 1/2〉 are linear combi-nations of these states.

total-spin S = 1/2 doublets, |E′±, Sz = ±1/2〉, of

opposite spin chirality, E′±, which are degenerate

in the absence of spin-orbit interaction. (Here E′

refers to the two-dimensional irreducible represen-tation (IR) of the D3h symmetry group of the tri-angular MM, spanned by the two states, |E′

+, Sz〉and |E′

−, Sz〉.) The states |E′±, Sz = 1/2〉 can be

written as linear combinations of the three frus-trated spin configurations shown in Fig. 2.According to an analysis based on group

theory,15,18 the matrix elements of the componentsof the operator R =

∑3j=1 rj in the triangular

molecule plane, X± = ±X + iY , between statesof opposite chirality, do not vanish

e〈E′+, Sz|X−|E′

−, Sz〉 = e〈E′−, Sz|X+|E′

+, Sz〉 = 2id .(1)

In Eq. (1) e is the electron charge, i =√−1, and

the real number d has the units of an electric dipolemoment. All the other matrix elements ofR in thesubspace spanned by {|E′

±, Sz = ±1/2〉} are zero.The nonzero value of d is in fact related to theexistence of a nonzero electric dipole moment ineach of the three frustrated spin configurations ofFig. 2 that compose the chiral ground states.19–22

An electric field εεε couples to the triangular MMvia eεεε · R. Then the non-zero matrix elementsin Eq. (1) ensure that the amplitude of the spin-electric coupling between chiral states is linear

in the field. Note that the electric-field–inducedtransitions conserve the total spin. However, inthe presence of an additional small dc magneticfield that mixes the spin states, this spin-electriccoupling can generate efficient electric transitionsfrom one spin state to another.

The relevance of this spin-electric mechanismfor qubit manipulation and qbits coupling clearlydepends on the value of the electric dipole mo-ment d. It has been proposed15 that an experi-mental estimate of d can in principle be providedby ESR measurements in static electric fields.Nuclear magnetic resonance, magnetization andspecific heat measurements have also been pro-posed to determine the strength of the couplingexperimentally.18 As far as we know these mea-surements have not yet been performed. Theoret-ically, a Hubbard model approach can provide un-derstanding and a rough estimate of d in terms ofa small number of Hubbard model parameters.18

In practice, a microscopic evaluation of d canonly be provided by first-principles calculations.In fact, in Ref. 21 we have carried out DensityFunctional Theory (DFT) studies of a {Cu3} MM,and shown that d is of the order of e10−4a, wherea is the Cu-Cu separation. At electric fields of theorder of 108 V/m, easily accessible in the vicinityof a scanning tunneling spectroscope (STM) tip,a d of this size would ensure transition times ofthe order of 1 ns. More recent DFT calculations23

have shown that the value of d in other triangularmolecules, such as {V3} {Cu3O3} and {V15}, canbe one or two orders of magnitude larger than in{Cu3}.In this paper we carry out a theoretical study

of quantum transport through an individual trian-gular antiferromagnetic MM displaying the spin-electric coupling, arranged in a single-electrontransistor (SET) geometry. The main motivationof this work is to investigate whether the coher-ent coupling of the two spin chiral states inducedand controlled by an electric field has detectableconsequences on the transport properties of theMM. Our conclusion is that, in the cotunnelingregime of Coulomb blockade transport, the GS en-ergy splitting induced by the electric field shouldbe easily accessible and should provide a directestimate of the strength of the electric dipole mo-ment parameter d. In this coherent regime, higherexcited states of the MM could add as additionalauxiliary states that can be exploited to performquantum gates.15 We also show that similar resultscould be obtained by performing inelastic electrontunneling spectroscopy through the MM adsorbedon surface by means of STM techniques. For themodeling of the MM we use the Hubbard modelapproach introduced in Ref. 18. This approach isquite convenient and transparent to address theeffect of an applied electric field on the molecular

3

orbitals of the molecule leading to the spin- elec-tric coupling. The parameters of the model areextracted from our previous first-principles calcu-lations on {Cu3}. Quantum transport is studiedby means of a quantum master equation includ-ing both sequential and cotunneling contributions.Transport studies on triangular systems using asimilar formalism have been done recently.24–29

But our motivation is different and an analysis ofthe spin-electric effect in this system has not beenconsidered so far.The paper is divided into the following. four sec-

tions. In Sec. II we introduce a Hubbard approachto model the effect of the electric field leading tothe spin-electric coupling in terms of a few free pa-rameters. In Sec. III we introduce the model andthe formalism to study quantum transport and cal-culate the conductance in the sequential tunnelingand cotunneling regime. In Sec. IV we presenttransport results. Finally, we summarize the con-clusions of our work in Sec. V

II. HUBBARD MODEL APPROACH TOTHE SPIN-ELECTRIC COUPLING

In this section we introduce the Hubbard modelapproach developed in Ref. 18 to analyze the spin-electric coupling. This approach is very useful forthree reasons. Firstly, it describes the effect ofthe applied electric field on the orbital degrees offreedom of the molecular magnet (MM), and there-fore it elucidates the emergence of the spin-electriccoupling at the microscopic level. Secondly, it per-mits the description of the spin-electric coupling interms of a few parameters that can be evaluatedby first-principles methods. Last but not least, itprovides the natural framework to study later onquantum transport.Before we introduce the Hubbard model, it is

convenient to summarize the main results of thespin-electric coupling using the language of a spinHamiltonian,15 in part already anticipated in theintroduction, which will then emerge again fromthe Hubbard model.The ground-state manifold of a three-site spin

s = 1/2 Heisenberg antiferromagnet, withisotropic exchange constant J , is given by the twodoubly-degenerate chiral doublets

|E′±, Sz =

1

2〉 = 1√

3

(| ↓↑↑〉+ ǫ±| ↑↓↑〉+ ǫ∓| ↑↑↓〉

),

(2)where ǫ± = exp (±2πi/3). These states are eigen-states of the total spin S2 with eigenvalue S = 1/2,and of the total z component Sz, with eigenvalue1/2. The three spin configurations in Eq. (2) areshown in Fig. 2). Similar linear combinations canbe written for the 2 eigenstates of Sz with eigen-

value 1/2. These states are also eigenstates of thez-component of the chiral spin operator

Cz =4√3s1 · s2 × s3 , (3)

with eigenvalue ±1. (The ± in E′± refers to this

quantum number.)The lowest excited state, separated from the

ground state (GS) by an energy of order J , is thefourfold degenerate eigenstate of S2, with eigen-value 3/2. The element of this quartet that isan eigenstate of Sz with eigenvalue 1/2, is writ-ten in terms of the same three spin configurationsof Fig. 2 as

|A′1, Sz =

1

2〉 = 1√

3

(| ↓↑↑〉+ | ↑↓↑〉+ | ↑↑↓〉

). (4)

The four states |A′1, Sz〉 form four A′

1 one-dimensional IR of the symmetry group D3h. Notethat the expectation value of Cz for the states|A′

1, Sz〉 vanishes.The SOI-induced Dzyaloshinskii-Moriya (DM)

interaction splits the chiral GS manifold into twotwo-dimensional subspaces. As we discussed in theintroduction, an electric field couples states of op-posite chirality. These two interactions can be rep-resented by the following low-energy effective spinHamiltonian15

Hspineff = ∆SOICz Sz + dεεε ·C‖ (5)

where C‖ ≡ (Cx, Cy, 0) is the component of thechiral operator in the xy plane. In Eq. (5) theenergy ∆SOI is proportional to the SOI couplingstrength, and turns out to be equivalent to theDM coupling constant D. The parameter d is theelectric dipole moment introduced in Eq. (1). Wewill now see how this effective spin Hamiltonianemerges from the Hubbard model approach.18

The second quantized one-band HubbardHamiltonian reads

HU = −∑

i,j

∑

α

{

tijc†iαcjα+h.c.

}

+1

2U∑

i

ni↑ ni↓ ,

(6)

where c†iα (ciα) creates (destroys) an electron with

spin α at site i, niα = c†iαciα is the particle num-ber operator and tij is a spin-independent hop-ping parameter. More precisely, the index i la-bels a Wannier function localized at site i. Thefirst term represents the kinetic energy describingelectrons hopping between nearest-neighbor sitesi and j. For D3h symmetry this term is charac-terized by a hopping parameter tij = t. The sec-ond term is an on-site repulsion energy of strengthU , which describe the energy cost associated withhaving two electrons of opposite spin on the samesite. In this model the interaction energy between

4

electrons which are not on the same site is com-pletely neglected. The Hubbard model is the sim-plest model describing the fundamental competi-tion between the kinetic energy and the interactionenergy of electrons on a lattice.The spin-orbit interaction in the Hubbard

model is described by adding the following spin-dependent hopping term18,30–32

HSOI =∑

i,j

∑

α,β

{

c†iα

(

iPij

2·σσσαβ

)

cjβ+h.c.}

, (7)

where σσσ = σxx + σy y + σz z is the vector of thethree Pauli matrices. A commonly used notationfor the Pauli matrices is to write the vector indexi in the superscript, and the matrix indices as sub-scripts, so that the element in row α and columnβ of the ith Pauli matrix is σi

αβ , with i = x, y, z.Here the vector Pij is proportional to the matrixelement of ∇∇∇V × p between the orbital parts ofthe Wannier functions at sites i and j; V is theone-electron potential and p is the momentum op-erator. Clearly the spin-orbit term has the formof a spin-dependent hopping, which is added tothe usual spin-independent hopping proportionalto t. In Eq. (7), spin-orbit coupling induces aspin precession about Pij when an electron hopsfrom site i to site j. This form of the spin-orbitinteraction is a special case of Moriya’s hoppingterms33 in the limit that all but one orbital en-ergy is taken to infinity,31 and it is consistent withour choice of a one-band Hubbard model. The xand y components of Pij describe processes withdifferent spin, and because of the αv symmetry,Pij = pez. Therefore, because of the symmetryof the molecule, the free Hubbard parameters arereduced to three, namely, t, U and p.The final expression of the Hamiltonian describ-

ing the electrons in a triangular molecule, includ-ing the spin-orbit interaction, is

HU+SOI =∑

i,α

{

c†iα(− t+ iλSOIα

)ci+1α + h.c.

}

+∑

i,α

(

ǫ0niα +1

2Uniα niα

)

, (8)

where λSOI ≡ p/2 = Pij/2 · ez is the spin-orbitparameter, ǫ0 is the on-site orbital energy, and α =−α.We want to treat the two hopping terms pertur-

batively on the same footing, by doing an expan-sion around the atomic limit t/U , λSOI/U → 0.In many molecular magnets t ≫ λSOI. This turnsout to be the case also for {Cu3}.34 In othermolecules the two hopping parameters are of thesame order of magnitude.We are interested in the half-filled regime. From

second-order perturbation theory in t/U , an an-tiferromagnetic isotropic exchange term emerges

and it splits the spin degeneracy of the low-energysector of the Hubbard model, which is defined bythe singly-occupied states.The perturbative method requires the definition

of the unperturbed states being the one-electronstates

|φαi 〉 = c†iα |0〉 , (9)

singly-occupied three-electron states

|ψαi 〉 =

3∏

j=1

c†jαj|000〉 =

3∏

j=1

∣∣φ

αj

j

⟩, (10)

with αj = α for j 6= i and αj = α, for j = i.Finally the doubly-occupied three-electron states

∣∣ψα

ij

⟩= c†i↑c

†i↓c

†jα |000〉 , (11)

with i = 1, 2, 3 and j 6= i. Note that the statesin Eqs. (9)-(11) are eigenstates of the Hamilto-nian, Eq. (8), only in the absence of the hop-ping and spin-orbit parameter and with energiesǫ0, 3ǫ0 and 3ǫ0 +U , respectively. These states arenot yet symmetry adapted states of the D3h pointgroup. Symmetry adapted states can be found us-ing the the projector operator formalism.18,35 One-electron symmetry adapted states can be writtenas a linear combinations of one-electron states, Eq.(9),

∣∣∣Φα

A′1

⟩

=1√3

3∑

i=1

|φαi 〉 , (12)

and

∣∣∣Φα

E′±

⟩

=1√3

3∑

i=1

ǫi−11,2 |φαi 〉 , (13)

where A′1 and E′

± are one-dimensional and two-dimensional IR in the D3h point group, respec-

tively, and ǫk = exp((2πi/3)k

)1,2is a phase fac-

tor. The three-electron symmetry adapted statesfor singly-occupied magnetic centers can be writ-ten as

∣∣∣ψ1α

A′1

⟩

=1√3

3∑

i=1

|ψαi 〉 , (14)

and

∣∣∣ψ1α

E′±

⟩

=1√3

3∑

i=1

ǫi−11,2 |ψα

i 〉 , (15)

The states |ψ1αE′

+

〉 and |ψ1αE′

−

〉 have total spin

S = 1/2 and z-spin projection Sz = ±1/2. Thesestates are formally identical to the chiral statesgiven in the Eq. (2), and are eigenstates of theHubbard Hamiltonian when t = λSOI = 0. The

5

tunneling and spin orbit interaction (SOI) mix thesingly-occupied and doubly-occupied states. Sym-metry properties of the D3h point group dictatethat the tunneling and SOI terms in the HubbardHamiltonian transform as the irreducible (IR) A′

1.Therefore, only states transforming according tothe same IR could be mixed. The first-order cor-rection in t/U and λSOI/U is obtained by mixingin doubly-occupied states18

|Φ1αE′

±〉 ≡ |ψ1α

E′±〉+ (ǫ12 − 1)(t± αλSOI)√

2U|ψ2α

E′1±

〉

+3ǫ11(t± αλSOI)√

2U|ψ2α

E′2±

〉 , (16)

where

|ψ2αE

′1±

〉 = 1√6

3∑

i=1

ǫi−11,2 (|ψα

i1〉+ |ψαi2〉) , (17)

and

|ψ2αE

′2±

〉 = 1√6

3∑

i=1

ǫi−11,2 (|ψα

i1〉 − |ψαi2〉) , (18)

are three-electron symmetry adapted states fordoubly-occupied magnetic centers.In the small t/U , λSOI/U limit, we can re-

sort to a spin-only description of the low-energyphysics of the system. The ground state mani-fold (corresponding to the states in Eq. (16)) isgiven by the two chiral spin states of Eq. (2). Inthis low-energy regime, the orbital states corre-spond to the singly-occupied localized atomic or-bitals. The lowest energy states have total spinS = 1/2 and chirality Cz = ±1. Using the sameperturbative procedure, we can construct approx-imate Hubbard model states corresponding to theS = 3/2 excited-state quartet of Eq. (4). To firstorder in t/U and λSOI/U one obtains

|Φ1αA′

1〉 = |ψ1α

A′1〉 (19)

The energy of the S = 3/2 quartet is 3J/2 higherin energy than the energy of the chiral GS dou-blets, with J ≈ 4t2/U .We now introduce the effect of the external elec-

tric field. An external electric field εεε can couple tothe molecule via two mechanisms. The first mech-anism that we will study is by the modification ofthe on-site energies ǫ0 via the Hamiltonian

H0d−ε =

∑

α

3∑

i=1

(−eri · εεε) c†iαciα, (20)

where ri is the coordinate vector of the ith mag-netic center. From Fig. 3, the on-site electricHamiltonian can be written as

H0d−ε = −ea

∑

α

[εy√3c†1αc1α − 1

2

(

εx +εy√3

)

c†2αc2α

+1

2

(

εx − εy√3

)

c†3αc3α

]

, (21)

FIG. 3. Coordinates of magnetic centers in a triangularmolecule. ri is the coordinate of the ith electron.

where εx,y are the in-plane coordinates of the elec-tric field, e the electron charge and a the distancebetween magnetic centers.The second mechanism is given by the modifi-

cation of the hopping parameters tii+1 and it canbe written as

H1d−ε =

∑

α

3∑

i=1

tεεεii+1,αc†iαci+1α +H.c., (22)

where tεεεii+1,α =⟨φαi | − er · εεε|φαi+1

⟩are the modi-

fied hopping parameters due to the external elec-tric field εεε, φαi are the Wannier states localizedon the ith magnetic center with spin α. Theseinduced hopping parameters can be written astεεεii+1,α =

∑

q qαii+1εq, with q

αii+1 = −e

⟨φαi |q|φαi+1

⟩

and q = x, y, z. D3h point group symmetry prop-erties, given by the dipole selection rules, reducethe number of free parameters induced by the elec-tric field. Finding these free parameters is notan easy task when the basis set is composed oflocalized Wannier orbitals. In order to investi-gate the effect of the electric field on the trian-gular molecule, we switch from the localized Wan-nier basis set to the symmetry adapted basis setΓ = A′

1, E′±. Then we apply the transition dipole

selection rules to the new induced hopping pa-rameters. In the symmetry adapted states, thehopping-Hamiltonian, Eq. (22), reads

H1d−ε =

∑

α

∑

ΓΓ′

tεεεΓ,Γ′,αc†ΓαcΓ′α +H.c., (23)

where Γ,Γ′ = A′1, E

′+, E

′−, t

εεεΓ,Γ′,α =

∑

q qαΓΓ′Eq,

with q = x, y, z and qαΓΓ′ = −e 〈φαΓ |q|φαΓ′〉. Here

c†Γα(cΓα) creates (destroys) an electron in theadapted state Γ with spin α. Note that in Eq. (23)all the possible transitions are included, even thosebetween states of the same symmetry adapted ba-sis set. Dipole transition rules then will select theallowed transitions and the corresponding states.Although symmetry properties control the dipoletransition rules, they do not allow us to calculate

6

the strength of the transitions. Detailed exper-imental measurements and/or accurate ab-initiocalculations have to be carried out to determinethem. In the D3h point group, the (x, y) and z-coordinates span as the E′ and the A′

1 IR, respec-tively. We have grouped x and y because theyform a degenerate pair within the E′ represen-tation. From character tables of the D3h pointgroup, the only allowed transitions correspond to

⟨

φαE′+

∣∣∣ x∣∣∣φαE′

−

⟩

= −i⟨

φαE′+

∣∣∣ y∣∣∣φαE′

−

⟩

≡ −dEE

e⟨

φαA′1

∣∣∣ x∣∣∣φαE′

+

⟩

= −i⟨

φαA′1

∣∣∣ y∣∣∣φαE′

+

⟩

≡ −dAE

e(24)

⟨

φαA′1

∣∣∣ x∣∣∣φαE′

−

⟩

= i⟨

φαA′1

∣∣∣ y∣∣∣φαE′

−

⟩

≡ −dAE

e

where dEE and dAE are the only two free param-eters to be determined. Here we have used thesymmetry rule that the product f1⊗ f2⊗ f3 6= 0 ifit spans the A1 representation. All the other pos-sible transitions are not allowed within the D3h

symmetry group. Inserting these allowed transi-tions into the Hamiltonian, Eq. (23), we have18

H1d−ε =

∑

α

[

dAE

(

Ec†A′1αcE′

−α + Ec†A′

1αcE′

+α

)

+ dEE Ec†E′−αcE′

+α

]

+H.c., (25)

where E = εx+iεy and E = εx−iεy. Note that theparameters dAE and dEE tell us about the possibledipole-electric transitions between states that spanthe A′

1-E′± and E′

+-E′− IR, respectively. From Eq.

(15) we can see that the chiral states also span theE± IR.To take even more advantage of the symme-

try of the triangular molecule, we now write therelationship between the second quantized opera-

tors c†iα, ciα and the symmetry adapted operators

c†Γα, cΓα. From Eqs. (9),(12) and (13), we have

c†A′1α

c†E′+α

c†E′−α

=

1 1 11 ǫ ǫ2

1 ǫ2 ǫ

c†1αc†2αc†3α

, (26)

where we have used ǫ4 = ǫ. From the last equationwe can write the localized second quantized opera-tors as a linear combination of symmetry adaptedoperators

c†1αc†2αc†3α

=

1 1 11 ǫ2 ǫ1 ǫ ǫ2

c†A′1α

c†E′+α

c†E′−α

. (27)

Now we can write the rest of the perturbedHamiltonian, namely the H0

d−ε on-site electricfield Hamiltonian (Eq. (21)) and HSOI spin-orbit

Hamiltonian (Eq. (7)), in terms of the symmetryadapted operators

H0d−ε = − iae

2√3

∑

α

[

Ec†E′+αcA′

1α − Ec†E′

−αcA′

1α

+Ec†E′−αcE′

+α

]

+ H.c., (28)

and

HSOI =√3λSOI

∑

α

α(

c†E′−αcE′

−α − c†E′

+αcE′

+α

)

.

(29)We conclude this section with the following im-

portant considerations1. With the use of the symmetry properties ofthe triangular molecule, the Hubbard model in thepresence of SOI (Eq. (29)) and an external electricfield (Eqs. (25) and (28)), can be parametrized byfive free parameters: t, U , λSOI, dEE and dAE . Fora realistic molecular magnet, t, U , λSOI can be ex-tracted from first-principles calculations, as for ex-ample done in Ref. 34 for {Cu3}. An analogous de-termination of the single-particle parameters dEE

and dAE has not been attempted so far. For lo-calized orbitals, one expects ea >> dEE , dAE , andthis the assumption that we will make in the pa-per.2. Eqs. (25) and (28) and Eq. (29) are completelyconsistent with the effective spin Hamiltonian re-sult of Eq. (5), in that they imply a splitting ofthe chiral GS by the SOI, and a linear coupling ofthe same states by an electric field. Note also thatthe SOI does not mix states of different chiralityand/or spin.3. Clearly Eqs. (25) and (28) and Eq. (29) aresingle-particle Hamiltonian. In order to extractthe electric-dipole moment d and the DM splitting∆SOI appearing in Eq. (5), one has to take ma-trix elements of these Hamiltonians between many-body states |Φ1α

E′±

〉 defined in Eq. (16). For the

matrix elements of the electric field Hamiltonianone finds18

∣∣∣

⟨

Φ1αE′

−

∣∣∣H0

d−ε

∣∣∣Φ1α

E′+

⟩∣∣∣ ≃

∣∣∣∣

t3

U3Eea

∣∣∣∣, (30)

∣∣∣

⟨

Φ1αE′

−

∣∣∣H1

d−ε

∣∣∣Φ1α

E′+

⟩∣∣∣ ≃

∣∣∣∣

4t

UEdEE

∣∣∣∣. (31)

It follows that the electric-dipole moment d of thespin electric coupling is given by a combination of∣∣∣t3

U3 ea∣∣∣ and

∣∣ 4tU dEE

∣∣.

4. In the presence of an electric field, the degen-erate GS chiral manifold {|Φ1α

E′±

〉} is replaced by

the coherent linear superpositions

∣∣χα

±(εεε)⟩==

1√2

(

|Φ1αE′

+〉+±|d · εεε|

d · εεε |Φ1αE′

−〉)

(32)

7

FIG. 4. (Color online) Electric-field–induced splitting∆E(ε) of the chiral ground state energy for a triangularmolecular magnet at half-filling (N = 3), as a functionof field strength ε and t/U . At these small/moderatevalues of the field, ∆E(ε) depends linearly on ε.

with energies

E±(ε) = E±(0)± d ε/√2 (33)

Note that spin degeneracy is preserved, even whenSOI is included. The electric-field-induced split-ting of the chiral GS, ∆E(ε) ≡ E+(ε) − E−(ε), isproportional to ε, at least in this approximation,in agreement with the effective spin Hamiltonianapproach. We will refer to the states

∣∣χα

±(εεε)⟩as

mixed chiral states. They will play a crucial rolein transport.

5. Eqs. (25) and (28) show that an electric field, infact, can couple {|Φ1α

E′±

〉} with |Φ1αA′

2

〉. However thiscoupling, which in principle could affect Eq. (32) isnot important, since these states are separated byan energy of order J . We will therefore disregardit.

In Figs. 4 and 5 we plot the computed energysplitting of the chiral GS, ∆E(ε), induced by anelectric field of strength ε, as a function of ε andt/U . The splitting is, as expected, linear in ε atsmall fields. This is the landmark of the spin-electric coupling. However, at larger field, we findalso a quadratic dependence. It seems that, de-spite the large value of U , the system has a sizablepolarizability, leading to an rather strong inducedelectric dipole moment in the presence of a field.This is responsible for the quadratic contributionin ∆E(ε).

All the calculations on the model presented inthe next section are obtained by exact diagonaliza-tion of the Hubbard model for N = 2, 3, 4 fillingor charge states. It turns out, however, that forthe values of the parameters relevant for {Cu3},the perturbative results in t/U are typically quiteclose to the exact results.

FIG. 5. (Color online) The same as in Fig. 4, but forlarger values of the electric field, showing a quadraticdependence of ∆E(ε) due to an induced electric dipolemoment.

FIG. 6. Schematic representation of the transport ge-ometry with a triangular molecular magnet. Picturemodified from the original work by Fuechsle et.al..Reprinted by permission from Macmillan PublishersLtd: Nature Nanotechnology 7, 242246, copyright(2012)

III. TRANSPORT MODEL AND MASTEREQUATION APPROACH

A. Transport setup

We are interested in studying quantum trans-port through a triangular molecular magnet(MM), weakly coupled to conducting leads, gated,and with the possibility of an extra external elec-tric field for control of the spin-electric coupling.The transport regime that we have in mind ispredominately controlled by Coulomb blockadephysics. Later in this section we will also com-ment on the possibility of employing inelastic elec-tric tunneling spectroscopy without the presenceof charging effects.

A possible transport geometry is schematicallyshown in Fig. 6. The MM is placed on a surface(semiconducting or insulating.). Two conductingcoplanar leads acting as source (S) and drain (D)are constructed on the surface, for example usingtechniques recently to realize a single-atom tran-

8

sistor.36 The molecule is weakly coupled to S andD leads via ligands. Two in-plane gates (G1 andG2) are also patterned on either side of the trans-port channel. The orientation of the MM on thesurface is such that the electric field from the gateis orthogonal to the plane of the MM, and it issimply used as a capacitative coupling to controlthe chemical potential of MM. Alternatively, Sand D nanoleads and gate electrodes can be con-structed by nano-lithography by depositing metalatoms (e.g., Au) on an insulating surface. Finally,a STM tip is positioned in the vicinity of the MM(see the blown-up region of the device close to theMM). This electrode is supposed to provide an-other strong and localized electric field to manip-ulate the MM states via the spin-electric couplingdiscussed in the previous section.The construction of the device described here is

very challenging. But we rely on recent progressin STM nano-lithography, and especially in fun-cionalizing MMs on surfaces.A second possibility is to study transport in

a single-electron transistor (SET) built in moretraditional molecular electronic device. MMs arepresently being successfully investigated with thistechniques.14,37? ? ,38 Here the challenge is to pro-vide an independent extra gate electrode (besidesthe ordinary back gate) to reliably generate an in-plane electric field triggering the spin-electric cou-pling.In the following we will assume that the fol-

lowing three features are present in our system:(i) source and drain leads weakly coupled to themolecule, providing a bias voltage Vb for electrictransport; (ii) a gate voltage generating a variablepotential Vg on the molecule able to manipulate itscharge state; (iii) a third independent local elec-tric field εεε, of strengths typically attainable in thevicinity of a STM, with a component in the planeof the MM.

B. Hamiltonian of the transport device

The Hamiltonian of the system, schematicallyrepresented in Fig. 6, is the sum of three terms

H = HL/R +Hmol +HTL/R , (34)

where

HL/R =∑

kα

εL/Rk a†L/RkαaL/Rkα (35)

describes free (i.e., noninteracting) electrons in theleft/right conducting lead (source/drain). Here,

the operator a†L/Rkα (aL/Rkα) creates (destroys)

one electron with wave vector k and spin α in

the left/right lead, respectively with energy εL/Rk .

The tunnel junctions representing the coupling be-tween leads and MM are described by the tunnel-ing Hamiltonian

HTL/R =

∑

kmα

(

TL/Rkmαa

†L/Rkαcmα +H.c.

)

, (36)

where TL/Rkmα is the tunneling amplitude, c†mα (cmα)

creates (destroys) an electron in a single particlestate with quantum numbers m and α inside theMM. The tunneling Hamiltonian HT

L/R is treated

as a perturbation to Hmol and HL/R.The general form of the MM Hamiltonian is

given by

Hmol = H0 +HU +Ht +HSOI +HEF, (37)

where

H0 =∑

j

∑

α

(ǫj − e Vg) c†jαcjα, (38)

with Vg the gate voltage. HU = U∑

j nj↑nj↓

with U the on-site Coulomb repulsion parameter

and njα = c†jαcjα the number operator. Ht =

t∑

j

∑

α c†jαcj+1α +H.c. the hopping Hamiltonian

with t the hopping parameter. HEF = H1d−ε +

H0d−ε the electric field Hamiltonian defined in Eqs.

(25) and (28) andHSOI the spin-orbit Hamiltoniandefined in Eq. (29).We assume the Coulomb interaction between

electrons in the MM and those in the environment,to be determined by a single and constant capac-itance C = CL + CR + Cg, where CL/R and Cg

are the capacitances of the right/left lead and thegate electrode, respectively. Another assumptionis that the single-particle spectrum is independentof these interactions.Quantum transport, e.g. the calculation of the

tunneling conductance as a function of bias andgate voltages, can now be studied by means ofa quantum master equation. General derivationsof these equations have recently appeared in theliterature,39–41 together with several approximatesolutions applied to SETs with quantum dots42

and molecules,40,41,43 including MMs.39,44–47 Thesimplest strategy is to solve these equations per-turbatively in the tunneling Hamiltonian.48

C. Coulomb blockade Regime, SequentialTunneling

In the regime of weak coupling between leadsand molecule, transport occurs via the so-calledsequential tunneling.48 We review here the maincharacteristics of this regime an the steps leadingto the calculation of the current.48 In this regimethe conductance of the tunnel junctions should

9

be much smaller than the quantum of conduc-tance GQ = 2e2/h. The electron tunneling ratesΓ should be much smaller than the charging en-ergy Ec of the molecule and the the temperature:hΓ ≪ kBT ≪ Ec. The time between two tunnelingevents ∆t is the longest time scale in the regime. Inparticular ∆t ≫ τφ, where τφ is the electron phasecoherence. This guarantees that once the electrontunnels in, it has the time to loose its phase coher-ence before it tunnels out. Therefore the chargestate can be treated classically and superpositionof different charge states is not allowed. Only one-electron transitions between leads and moleculeoccur in the system. These transitions are char-acterized by rates Γij , where i, j are the initialand final system states of the system involved inthe electron transfer. The system is described bystationary non-equilibrium populations Pi of thestate i. These occupation probabilities can be ob-tained from the master equation

d

dtPi =

∑

j(j 6=i)

(ΓijPj − ΓjiPi) . (39)

The first RHS term represents events where theelectron tunnels into the state i from the statej, while the second RHS term represents eventswhere the electron tunnels out from the state iinto the state j. These probabilities obey the nor-malization condition

∑

i

Pi = 1 . (40)

In the steady state, the probabilities are time-independent dPi/dt = 0. Therefore, Eq. (39) canbe written as

0 =d

dtPi =

∑

j(j 6=i)

(ΓijPj − ΓjiPi) . (41)

In the regime of sequential tunneling the tran-sition amplitudes are computed by first-order per-turbation theory in the tunneling HamiltonianHT ,Eq. (36). Therefore the transition rates from statei to state j, through the left/right lead, are givenby Fermi’s golden Rule

ΓL/Ri→j =

2π

h

∑

i,j

∣∣∣

⟨

j∣∣∣HT

L/R

∣∣∣ i⟩∣∣∣

2

Wiδ(Ej − Ei) ,

(42)where Wi is a thermal distribution function andEj −Ei gives the energy conservation. The states|i〉 and |j〉 are the unperturbed system states andare defined as a product of the molecule and leadstates |i〉 = |imol〉 ⊗ |il〉 ⊗ |ir〉. Transition ratesdepend on whether an electron is leaving or en-tering the molecule through the left or right lead.Inserting the tunneling Hamiltonian Eq. (36) into

the Fermi’s golden Rule, Eq. (42), the transitionrates become48,49

ΓL/R,−i→j = γ

L/R,−ji

[1− fL/R(E)

], (43)

ΓL/R,+i→j = γ

L/R,+ji

[fL/R(E)

], (44)

where

γL/R,−ji = ΓL/R

∑

m,α

|〈j |cm,α| i〉|2 (45)

and

γL/R,+ji = ΓL/R

∑

m,α

∣∣⟨j∣∣ c†m,α

∣∣ i⟩∣∣2

(46)

are the transition matrix elements between thestates j and i of the molecule (we have nowdropped the label ”mol”); E = Ej − Ei is theenergy difference between molecule many-electron

states, and fL/R(E) =[e(E−µL/R)/kBT + 1

]−1is

the Fermi function. Here the combination betweenthe tunneling amplitudes T

L/Rm,α and the left/right

lead density of states DL/R(iL/R) is assumed to be

constant: ΓL/R = (2π/h)∣∣∣T

L/Rm,α

∣∣∣

2

DL/R(iL/R) =

(2π/h)∣∣TL/R

∣∣2DL/R(iL/R). The full transition

matrix in the master equation, Eq. (39) is the sumof all contributions of electrons tunneling out orinto the molecule, Eqs. (43) and (44):

Γij = ΓL,+ij + ΓR,+

ij + ΓL,−ij + ΓR,−

ij . (47)

The stationary rate equation, Eq. (41), is a sys-tem of linear equations and has to be solved nu-merically for a system of n many-electron statesthat are taking into account. We can rewrite it asa matrix equation

0 =n∑

j

ΛijPj , (48)

where

Λij = Γij − δij

n∑

k=1

Γkj . (49)

There must exist a physical solution to Eq. (48).Therefore we replace the first line of of this equa-tion by the normalization condition, Eq. (40), fix-ing Λ1j = 1. Thus we can write

δ1i =

n∑

j

ΛijPj (50)

instead Eq. (48). Because Coulomb block-ade is typically studied at low temperaturessome transitions rates might become exponentiallysmall. This leads to numerical problems in solving

10

Eq. (50). Then some of the states do not con-tribute and one has to develop a convenient trun-cation method.45

Finally, the current flowing through left leadcoming into the molecule must be equal to the cur-rent flowing through right lead coming out fromthe molecule. Knowing the occupation probabili-ties, Eq. (41), the current through the system isdefined as42

I ≡ IL/R = (−/+)e∑

i,j(j 6=i)

Pj

(

ΓL/R,−ij − Γ

L/R,+ij

)

(51)This expression contains implicitly the bias and

gate voltages. Therefore IV curves can be obtainedfor finite values of these voltages. The bias deriva-tive of the current gives the differential conduc-tance G. When plotted as a function of the biasVb, the current has steps in correspondence of val-ues of Vb at which new transitions involving twocontiguous charge states are energetically allowed.At low voltages – smaller than the charging energy– this is not possible and the current is blocked.In correspondence of these transitions, the con-ductance as function of Vb displays peaks. Whenplotted simultaneously as a function of both Vband Vg, the conductance displays a characteristicdiamond pattern, the so-called stability diagram:inside each diamond a given charge state is stableand the current is blocked.

D. Cotunneling Regime

When the coupling to the leads becomesstronger the description of transport based on in-coherent sequential tunneling is no longer enough.In particular higher-order tunneling processes inwhich the electron tunnels coherently through clas-

sically forbidden charge states. As a result, forvalues of the voltages where sequential tunnel-ing predicts a blocking of the current, a smallleakage current is in fact possible though theseprocesses.48 The simplest example of these pro-cesses is second order in the tunneling Hamilto-nian, and it is known as cooperative tunneling orcotunneling. Typically for the cotunneling regimekBT < hΓ ≪ Ec.Cotunneling can be either elastic or inelastic.

In the former case the energies of the initial andfinal state are the same, while in the latter theenergies are different. Signatures for these pro-cesses have also been observed in single-moleculejunctions.14,37,38 Beyond the sequential tunnelingregime, the tunneling Hamiltonian must be re-placed by the T -matrix, which is given by48

T = HT +HT 1

Ej −H0 + iηT , (52)

where Ej is the energy of the initial state |j〉 |n〉,where |j〉 refers to the equilibrium state on the leftand right lead and |n〉 is the initial molecular state,η = 0+ is a positive infinitesimal andH0 = Hmol+HL/R. To second order, the transition rates fromstate |j〉 |n〉 to |j′〉 |n′〉 with an electron tunnelingfrom lead α to the lead α′ are given by

Γnj;n′j′

αα′ =2π

h

∣∣∣∣〈j′| 〈n′| HT 1

Ejn −H0 + iηHT |n〉 |j〉

∣∣∣∣

2

×δ(Ej′n′ − Ejn) , (53)

where Ej′n′ and Ejn are the energies of the finaland initial states, respectively. Here |j′〉 |n′〉 =

a†α′k′σ′aαkσ |j〉 |n′〉. Inserting the tunneling Hamil-tonian, Eq. (36), in last equation and after somealgebra (see Appendix A) one can get the expres-sion for the transition rates for processes from leadα till lead α′ and from molecular state |n〉 to thestate |n′〉:

Γn;n′

αα′ =∑

σσ′

γσαγσ′

α′

∫

dεf (ε− µα) (1− f (ε+ εn − εn′ − µα′))

×∣∣∣∣∣

∑

n′′

{

Aσ∗n′′n′Aσ′

n′′n

ε− εn′ + εn′′ + iη+

Aσ′

n′n′′Aσ∗nn′′

ε+ εn − εn′′ + iη

}∣∣∣∣∣

2

, (54)

where σ is the electron spin, f(ε) is the Fermi dis-tribution function, µα is the chemical potential ofthe lead α, µL − µR = −eV/2, |n′′〉 is a virtual

state, Aσ′

ij = 〈i| cσ′ |j〉 and Aσ∗ij = 〈j| c†σ |i〉. Here

γσα is the tunneling amplitude. Note that |n〉 and|n′〉 are states with the same number of particles.We have not taken into account processes changingthe electron number by ±2 units.43,50

The transition rates in Eq. (54) cannot be eval-uated directly because of the second-order poles inthe energy denominators. A regularization schemehas been carried out to fix these divergences andobtain the cotunneling rates.51,52 Here it is im-portant to mention that these divergences are, infact, an artifact of the T -matrix approach ratherthan a real physical problem. The fourth-order

11

Bloch-Redfield quantum master equation (BR)and the real-time diagrammatic technique (RT)approaches to quantum transport have been devel-oped to avoid any divergences and therefore no ad

hoc regularization to cotunneling is required.40,41

Nevertheless, the T -matrix approach agrees with

these two approaches and gives good reasonableresults deep inside the Coulomb blockade region.46

We expect to catch all the relevant physics for oursystem with the T -matrix approach. After the reg-ularization scheme is implemented, we get the tun-neling rates defined as (see Appendix B)

Γn;n′

αα′ =∑

σσ′

γσαγσ′

α′

[∑

k

(A2J(E1, E2, εak) +B2J(E1, E2, εbk)

)+ 2

∑

q

∑

k 6=q

AkAqI(E1, E2, εak, εaq)

+2∑

q

∑

k 6=q

BkBqI(E1, E2, εbk, εbq) + 2∑

q

∑

k

AkBqI(E1, E2, εak, εbq)

]

(55)

where Ak = Aσ∗kn′Aσ′

kn, Bk = Aσ′

n′kAσ∗nk, εak = εn′ −

εk, εbk = εk−εn, E1 = µα and E2 = µα′+εn′−εn.Here I and J are integrals that come out from theregularization scheme, and are defined in Eqs. (B1)and (B2), respectively.The complete master equation, including both

sequential and cotunneling contributions, finallyreads

d

dtPi =

∑

j(j 6=i)

(ΓijPj − ΓjiPi)

+∑

αα′j

(

Γjiαα′Pj − Γij

αα′Pi

)

,(56)

and the current through the system is now givenby

I ≡ IL/R = (−/+)e∑

i,j(j 6=i)

Pj

(

ΓL/R,−ij − Γ

L/R,+ij

)

+(−/+)e∑

i,j(j 6=i)

Pj

(

ΓjiLR/RL − Γij

RL/LR

)

(57)

As mentioned above, cotunneling gives rise to asmall current inside a Coulomb-blockade diamondregion of a given charge state. At small values ofthe bias voltage, smaller than any excitation ener-gies for the given charge state, we are in the regimeof elastic cotunneling and the current is propor-tional to the bias voltage. At voltages correspond-ing to the transition energy to the first excitedstate of the same charge state, a new cotunnelingtransport channel becomes available and the slopeof the linear dependency of the current increases.This signals the first occurrence of inelastic co-

tunneling. Upon further increasing the bias, otherupward changes of the slope of the current occurin correspondence to energies at which higher ex-cited states become available. It follows that thedifferential conductance displays steps that resem-ble the IV curve in the sequential tunneling regime.Note however, that the nature of the two curves isvery different: at low bias the conductance is fi-nite (elastic cotunneling). Furthermore the width

of the steps in the cotunneling conductance givesthe energy difference between states of the same

charge state, fixed by the specific Coulomb dia-mond of the stability diagram. Therefore, cotun-neling is an excellent tool to investigate directly

the excitation energies of a given charge state. In-deed cotunneling spectroscopy has been used to in-vestigate electronic, vibrational and magnetic ex-citations in nanostructures such as a-few-electronsemiconductor quantum dots,53 carbon nanotubequantum dots,54,55 metallic carbon nanotubes,56

and single-molecule junctions.57–59

At this point, before analyzing the transport re-sults of our model, it is useful to make a connec-tion with inelastic electron tunneling spectroscopy(IETS), studied for example by electron tunnel-ing from a scanning tunneling spectroscope (STM)tip through a molecule adsorbed on a surface60,61.The reader familiar with IETS easily recognizesthat the differential conductance versus appliedvoltage for this case is very similar to the cotun-neling conductance of Coulomb blockade. Thissimilarity is not accidental: the physics is essen-tially the same in both cases, since it involves thecoherent electron tunneling through a finite sys-tem, whose internal degrees of freedom (e.g., vi-brational, magnetic and electronic) can be excitedby the process. The mathematical formulationof the problem is very similar in the two cases.There is one noticeable difference. In IETS bySTM the coupling between the molecule and the(conducting) substrate is much stronger that thecoupling between the STM tip and the molecule.Therefore typical IETS setups can be viewed asstrongly asymmetric Coulomb-blockade systems,when these are studied in the cotunneling regime.

These considerations suggest an alternative wayto investigate the spin-electric coupling in trian-gular MMs via quantum transport. In the setupof Fig. 6 we can imagine that transport throughthe MM occurs between the STM and the sub-strate. on which the MM is placed. Now the gates

12

IS = 1 �2, E-

' MIS = 1 �2, E+

' M

HS = 0L HS = 1L

2 3 4-14

-12

-10

-8

-6

-4

-2

0

N

Ene

rgyH

eVL

FIG. 7. (Color online) Low-energy spectrum of the tri-angular molecular magnet described by the Hubbardmodel, Eq. (8), for different charge states or electronfilling, N = 2, 3, 4. Here the Hubbard model parame-ters, t = −0.051, U = 9.06, λSOI = 0.0004 (all in eV),are taken from first-principles calculations34 for the{Cu3} molecular magnet. A gate voltage Vg = U/2has been added to rigidly shift the spectrum of thesystem for a given N . The total spin of the groundstate (GS) for the different charge states is indicatedin parenthesis. The GS for the N = 3-particle sys-tem corresponds to the chiral states, E′

±, defined inEq. (16).

and leads constructed on the surface could providethe external electric field responsible for the spin-electric tunneling. For this purpose the plane ofthe triangular MM should be parallel to the sur-face of the substrate. In this case the detectionand coherent manipulation of the low-energy chiralstates of the MM would occur by means of IETS.

IV. RESULTS AND DISCUSSION

We now discuss quantum transport for the setupof Fig. 6 We first construct the relevant low-energymany-body states for the charge states containingN = 2, 3, 4 electrons. For this purpose we use heHubbard model introduced in Sec. II. The parame-ters of the model are taken from the first-principlesstudies on the {Cu3} triangular molecular magnet(MM) by Ref. 34. We have t = −51 meV, U = 9.06eV, λSOI = 0.4 meV. The model is solved exactlyfor N = 2, 3, 4. We label the many-body stateswith their electron number N (the charge state),total spin S and z-component of the total spin Sz

62

In case of additional degeneracy, we will use addi-tional quantum numbers to specify the states, e.g.,for the the chiral degeneracy for the N = 3 groundstate (GS), we will add E′

±.The low-energy levels for the three contiguous

charge states are shown in Fig. 7. To the ener-gies calculated with the Hubbard model, we haveadded a gate voltage term −eVgN = −U/2N ,which shifts rigidly the spectra of the differentcharge states with respect to each other. This

FIG. 8. (Color online). Schematic energy diagram ofa triangular molecular magnet in the presence of anexternal electric field ε. Only the ground state (GS)of the N = 2, 3, 4-particle system and the lowest ex-cited states of the N = 3 system are included. Thenumbers in parenthesis corresponds to the total spinS. The electric field lifts the N = 3 GS degeneracy,and mixes the chiral states defined in Eq. (16). The“mixed chiral states”, are now labeled by χα

±, with χα−

being the GS. The GS splitting ∆E is linear in ε atlow fields. Here we have used the same parameters ofFig. 7, plus eaε = 0.487eV, and dEEε = 0.1eaε. Theelectric field is applied in the plane of the triangle,perpendicularly to line joining vertexes 1 and 2 of thetriangle. Also shown in the figure with dashed-coloredlines are allowed inelastic cotunneling transitions, oc-curring via N = 2, 4 virtual ground states, 20 and 40,respectively. Red, black and green dashed lines corre-spond to transitions: χ− ↔ χ+ (∆E), χ+ ↔ S = 3/2and χ− ↔ S = 3/2, respectively.

choice makes the spectra of the N = 2 and N = 4charge states more symmetric with respect to theN = 3 states. We will also use this value of thegate voltage below, in the study of cotunnelingtransport, to make sure that the system is stablein the middle of the N = 3 Coulomb diamond.

For the present choice of the Hubbard parame-ters, these states are well described by the pertur-bative analysis of Sec. II. As discussed there, theGS for the N = 3 charge state (lowest middle line)is four-fold degenerate, and it corresponds to thestates defined in Eq. (16). In Fig. 7 the same linedenotes the position of the S = 3/2 excited state,whose separation from the GS is not visible on thisenergy scale.

We now consider the presence of a strong andlocalized electric field, generated, for example, bya scanning tunneling spectroscope (STM) tip posi-tioned nearby the MM. We will consider values of εup to a maximum equal 0.1V/A, which can be eas-ily attained with a STM.63,64 For a {Cu3}MM, thedistance between magnetic ions is a = 4.87A. For aspin-electric coupling strength d = ea, which is themaximum value estimated in Ref. 15, the energyscale eaε is equal to 0.487 eV when ε = 0.1 V/A.As discussed in Sec. II, we model the effect of theelectric field in the Hubbard approach via the pa-

13

FIG. 9. (Color online) Differential conductance as afunction of the bias and gate voltages in the sequen-tial tunneling regime (stability diagram), showing theCoulomb diamonds for three contiguous charge statesN = 2, 3, 4. Only the corners of the diamonds areshown. The arrows indicate the electron transitionsresponsible for peaks in the conductance. States arelabeled following the notation of Fig. 8. The calcula-tions are done for a symmetric device at temperatureT ∼ 10−2K (kBT ∼ 0.001meV). The parameters forthe Hubbard model are the same of those in Fig. 8, Alocal electric field ε = 0.1 V/A, is also included, caus-ing a spin-electric coupling of the N = 3 chiral statesand a ground state splitting ∆E.

rameters a, dEE , dAE entering the single-particleHamiltonians in Eqs. (25) and (28). Here we takedEE = 0.1ea and dAE = 0. The effect of the fieldon the low-energy spectrum of the MM is shownin Fig. 8, with the expected splitting and mixingof the GS chiral states for the N = 3 charge state.In the absence of spin orbit interaction (SOI) the“mixed chiral states” |χα

−(ε)〉 and |χα+(ε)〉 (with

|χα−(ε)〉 being the GS) are still spin (α = ±1/2)

degenerate. As we saw, their splitting ∆E(ε) isproportional to ε. It is interesting to note that,the (small) spin-orbit coupling given in Eq. (29),mixes a little bit |χα

−(ε)〉 and |χα+(ε)〉. However,

since the effect is the same for α = ±1/2, the dou-ble degeneracy of the GS and the first excited stateis preserved, and the splitting remains of the orderof ∆E(ε).Shown on the same figure are also the four-fold

degenerate (N= 3, S = 3/2) excited state and theN = 2 and N = 4 GS, having spin S = 0 andS = 1 respectively. The N = 2(4) GS has totalspin S = 0(1) and spin projection Sz = 0(0). Therest of the energy spectrum is not shown in Fig. 8.In Fig. 9 we plot the Coulomb blockade stabil-

ity diagram, that is, the differential conductancein the sequential tunneling regime as a functionof bias and gate voltages. The calculations aredone for a symmetric device, where the capaci-tances and tunneling resistances for the two junc-tions are the same. The temperature is taken tobe T ∼ 10−2K (kBT ∼ 0.001meV). The calcula-tions are done for the parameters of Fig. 8, andan electric field ε = 0.1 V /A is included, generat-ing a GS splitting ∆E for the N = 3 charge state.

The picture displays familiar Coulomb diamondsfor the three contiguous charge states N = 2, 3, 4,inside which the current is zero. The lines delim-iting these diamonds represent the onset of tun-neling current, where the conductance has peaks.They correspond to real transitions between statesof two contiguous charge states N → N ± 1. Thefirst lines where this happens involve the transitionbetween the corresponding GSs. Other lines, par-allel to these, involve transitions between excitedstates, which become occupied out of equilibrium.We do not include any energy or spin relaxationmechanism in these calculations.

We now consider transport in the cotunnelingregime. In Fig. 10 we plot the differential con-ductance as a function of the bias voltage Vb, forVg = U/2, which locates the system in the middleof N = 3 Coulomb diamond, that is, deep insidethe Coulomb blockade regime. Here the sequentialtunneling current is suppressed, and transport isentirely due to cotunneling. The conductance isnonzero even at zero bias, due to elastic cotunnel-ing. At Vb ≈ 1.1 meV, the conductance has a firststep, indicated by the red dashed line. The stepsignals the onset of inelastic cotunneling, whichtakes place when the bias voltage provides enoughenergy for the final occupation of the lowest ex-cited state of the N = 3 charge state (N = 3, χα

+),via the virtual transition from the (N = 3, χα

−)GS to the (N = 2, S = 0), (N = 4, S = 1) GSs.Therefore, the width of this first step provides adirect estimate of the energy splitting between themixed chiral states, (N = 3, χα

+) and (N = 3, χα−),

caused by the spin-electric coupling. Increasingfurther the bias, other two cotunneling channelsopen up, causing the appearance of two other stepsin the conductance. The first one, quite small, in-dicated by the black dashed line, is related withthe first occupation of the (N = 3, S = 3/2) ex-cited state, which occurs via the virtual transi-tion from the (N = 3, χα

+) excited state to the(N = 2, S = 0), (N = 4, S = 1) GSs. Notethat the state (N = 3, χα

+) is already occupiedbecause of the first inelastic cotunneling transi-tion. The second (higher) step, indicated by agreen dashed line, is again due to the occupation ofthe (N = 3, S = 3/2) as a final state, but thoughthe virtual transition from the (N = 3, χα

−) GS tothe (N = 2, S = 0), (N = 4, S = 1) GSs.

The cotunneling conductance pattern dependson the external electric field ε. In Fig. 11 we plotthe conductance as function of the external elec-tric field, ε and bias voltage, Vb. As expected, thevalue of the voltage where the first inelastic stepoccurs increases with the field. Variations of theposition of the other two inelastic steps in the con-ductance as a function of ε are also visible: at lowfields, where the splitting of the chiral GS van-ishes, the other two inelastic steps involving the

14

dI/d

V (

eΓ/V

)

Vb(V)

1 1.5 2 2.5x 10

−3

0

1

2

3

4

x 10−6

∆E

χ+α↔S=3/2

χ−α↔S=3/2

FIG. 10. (Color online) Cotunneling differential con-ductance as a function of the bias voltage for param-eters as in Fig. 7. The states involved are labeled asin Fig. 8. At low voltage, transport is through elasticcotunneling. The red-dashed line corresponds to thefirst onset of inelastic cotunneling, due to the occupa-tion of the lowest excited state (N = 3, χα

+), througha virtual transition (N = 3, χα

−) ground state (GS)→ (N = 2, S = 0), (N = 4, S = 1) GSs. The black-dashed line and green-dashed line indicate inelastic co-tunneling steps caused by the final occupation of the(N = 3, S = 3/2) excited state via the virtual transi-tions from (N = 3, χα

±) to the (N = 2, S = 0), (N =4, S = 1) GSs.

FIG. 11. Cotunneling differential conductance as afunction of the bias voltage and the local electric fieldtriggering the spin-electric coupling.

(N = 3, S = 3/2) excited state occur at the samebias. Surprisingly, the height of the inelastic stepsis not strongly affected by the electric field. Theonly exception is the second step, whose height be-comes very small at the maximum value of ε, asalso shown in Fig. 10.In Fig. 12 we plot ∆E, extracted from the po-

sition of first inelastic step, as a function of ε. Apolynomial fitting of ∆E vs. ε finds, besides aquadratic contribution due to an induced electricdipole moment, a linear term, which dominatesat low fields, and it is the landmark of the (lin-ear) spin-electric coupling. Interestingly, the ex-

2 4 6 8 10x 10

8

0

0.2

0.4

0.6

0.8

1

1.2x 10−3

∆ E

(eV

)

ε (V/m)

∆EFit

Fitting curve:E

0−p.ε−αε2

p = 5.76x10−33 C.m

FIG. 12. Energy splitting of the N = 3 chiral groundstate, ∆E, caused by the spin-electric coupling, as afunction of the external electric field. The values of∆E correspond to the position of the first conductancestep in Fig. 11. The fitting curve contains a linear termproportional to a dipole moment p = 5.76 10−33 C m,in agreement with the first-principles calculations on{Cu3} molecular magnet of Ref. 21.

0 0.2 0.4 0.6 0.8 1x 10

−3

1

2

3

x 10−6

dIdV

(eΓ

/V)

Vb(V)

eaε=0.0eaε=0.487

FIG. 13. (Color online) Cotunneling differential con-ductance versus bias voltage with (dashed red line) andwithout (blue solid line) external electric field, causingthe spin-electric coupling. Here we have used the sameparameters of Fig. 8.

tracted value of the proportionality coefficient ofthe linear term, i.e. the “electric dipole moment”p = d/

√2, is equal to 5.76 10−33 C m, which

is consistent with the value found previously byab-initio methods for {Cu3} molecular magnet.21

This indicates that our choice of the spin-electricparameter dEE = 0.1ea (see Eqs. (24) and (25) ) isin the right ballpark. In principle, the curve plot-ted in Fig. 12 can be directly extracted from exper-imental measurements of the conductance in thecotunneling regime. From this curve, the strengthof electric dipole moment d can be estimated.

The cotunneling conductance for both ε = 0(blue line) and ε = 0.1 V/A (red dashed line) is

15

plotted in Fig. 13. At zero field, the splitting of theN = 3 GS, controlling the onset of inelastic cotun-neling, is brought about only by the SOI-inducedDzyaloshinskii-Moriya interaction, which splits thechiral states without mixing them. This splitting ispredicted to be very small, both experimentally15

(∆SOI = 0.04 meV) and theoretically (∆SOI =0.02 meV)34. The value extracted from the cotun-neling conductance of Fig. (13) is consistent withthis estimate. A measurement of this splittingfrom cotunneling experiments is also in principlepossible but probably very challenging. The valueof the elastic cotunneling conductance is slightlylarger when the ε-field is absent than in the pres-ence of the field. However value of the inelasticconductance is the same with and without field.The fact that inelastic cotunneling sets in at verydifferent thresholds with and without field suggeststhe possibility of using this system as a switchingdevice, which can be controlled electrically, possi-bly by a time-dependent field.

V. CONCLUSIONS

In summary, we have carried out a theoreticalstudy of quantum transport through an antiferro-magnetic triangular molecular magnet (MM), in asingle-electron transistor setup. The interplay ofspin frustration and lack of inversion symmetry inthis MM is responsible for the existence of an ef-ficient spin-electric coupling, which can affect thenon-linear transport regime. When a strong local-ized electric field is applied to the molecule, thespin-electric coupling causes a splitting betweenthe two doubly-degenerate spin chiral states thatcompose the ground state of the MM. We haveshown that this energy splitting and, consequentlythe strength of the spin-coupling, should be di-rectly accessible through experiments by measur-ing the inelastic cotunneling conductance in theCoulomb blockade regime. Both single-electrontransistors (SETs) used in molecular spintron-ics and inelastic electron tunneling spectroscopy(IETS) of molecules on surfaces addressed with aSTM could be employed to study this effect.Our theoretical approach was based on a Hub-

bard model,15,18 where the spin-electric couplingcan be described in terms of a few microscopicparameters derivable from first-principles calcu-lations. We have shown that the value of thestrength of spin-electric coupling estimated fromtunneling transport is consistent with the valuecalculated by first-principles methods.21

Antiferromagnetic molecules, like the one con-sidered here, characterized by ground states com-posed of chiral pairs of spin-1/2 doublets, could beused to create pairs of quasi-degenerate qbits. Thepossibility of coherently coupling these two qbitselectrically and detecting their quantum superpo-sition state in electronic transport is an interestingtopic that should further investigated.The effect of an external magnetic field, not

considered in this paper, can be used for gain-ing full control of the ground-state manifold. Fur-thermore, higher excited states of the system canplay a role as auxiliary states employed to per-form quantum gates. As we have shown in ourstudy of the cotunneling conductance (see Fig. 11),these higher states can also be manipulated electri-cally and brought closer to or further apart fromthe ground-state manifold. One important issuethat we have not discussed in this work is the ef-fect of spin relaxation on transport. This certainlyplays a crucial role in determining the robustnessof the coherent superposition induced by the elec-tric field.

ACKNOWLEDGMENT

We would like to thank D. Loss and D. Stepa-nenko for several important discussions and clar-ifications on the spin-electric coupling in molecu-lar magnets, and M. Islam for an ongoing collab-oration on the same subject. We would like tothank Dr. Magnus Paulsson for his help in de-veloping the codes used in this work. This workwas supported by the School of Computer Science,Physics and Mathematics at Linnaeus University,the Swedish Research Council under Grants No:621-2007-5019 and 621-2010-3761, and the Nord-Forsk research network 080134 “Nanospintronics:theory and simulations”.

Appendix A: Explicit derivation of Eq. (54)

Here we demonstrate the Eq. (54). We study the transition rates up to four order. The transitionrate from state |j〉 |n〉 to |j′〉 |n′〉 with one electron tunneling from lead α to the lead α′ is given by

Γnj;n′j′

αα′ =2π

h

∣∣∣∣〈j′| 〈n′|HT 1

Ejn −H0 + iηH

T |n〉 |j〉∣∣∣∣

2

δ(Ej′n′ − Ejn) ,

where Ej′n′ and Ejn are the energies of the final and initial states, respectively. HT =∑

α=L,R

tα∑

kσ

(

a†αkσcσ + c†σaαkσ

)

is the tunneling Hamiltonian Eq. (36) with TL/Rkmα = tα. H0 =

16

Hmol + Hleads and η is a positive infinitesimal number. Here |j′〉 |n′〉 = a†α′k′σ′aαkσ |j〉 |n′〉. |j〉 (|n〉)refers to the equilibrium state of the left and right Fermi sea (molecule). The total cotunneling rates fortransitions that involve virtual transitions between two n, n′-occupied molecule states are then given by

Γnj;n′j′

αα′ =2π

h

∑

kk′σσ′

∣∣∣∣∣〈j| 〈n′| a†αkσaα′k′σ′

∑

α′′′

t∗α′′′

∑

k′′′σ′′′

(

a†α′′′k′′′σ′′′cσ′′′ + c†σ′′′aα′′′k′′′σ′′′

)

× 1

Ejn −H0 + iη

∑

α′′

tα′′

∑

k′′σ′′

(

a†α′′k′′σ′′cσ′′ + c†σ′′aα′′k′′σ′′

)

|n〉 |j〉∣∣∣∣∣

2

δ(Ej′n′ − Ejn)

=2π

h

∑

kk′σσ′

∣∣∣∣∣〈j| 〈n′| a†αkσaα′k′σ′

∑

α′′′k′′′σ′′′

∑

α′′k′′σ′′

t∗α′′′tα′′

×

a†α′′′k′′′σ′′′cσ′′′

1

Ejn −H0 + iηa†α′′k′′σ′′cσ′′

︸ ︷︷ ︸

= 0, n-2 states

+ a†α′′′k′′′σ′′′cσ′′′

1

Ejn −H0 + iηc†σ′′aα′′k′′σ′′

+c†σ′′′aα′′′k′′′σ′′′

1

Ejn −H0 + iηa†α′′k′′σ′′cσ′′ + c†σ′′′aα′′′k′′′σ′′′

1

Ejn −H0 + iηc†σ′′aα′′k′′σ′′

︸ ︷︷ ︸

= 0, n+2 states

× |n〉 |j〉|2 δ(Ej′n′ − Ejn)

Γnj;n′j′

αα′ =2π

h

∑

kk′σσ′

∣∣∣∣∣〈j| 〈n′| a†αkσaα′k′σ′

∑

α′′′k′′′σ′′′

∑

α′′k′′σ′′

t∗α′′′tα′′

{

c†σ′′′aα′′′k′′′σ′′′

1

Ejn −H0 + iηa†α′′k′′σ′′cσ′′

+a†α′′′k′′′σ′′′cσ′′′

1

Ejn −H0 + iηc†σ′′aα′′k′′σ′′

}

|n〉 |j〉∣∣∣∣

2

δ(Ej′n′ − Ejn)

=2π

h

∑

kk′σσ′

∣∣∣∣∣

∑

α′′′k′′′σ′′′

∑

α′′k′′σ′′

t∗α′′′tα′′

{

〈j| 〈n′| a†αkσaα′k′σ′c†σ′′′aα′′′k′′′σ′′′

1

Ejn −H0 + iηa†α′′k′′σ′′cσ′′ |n〉 |j〉

+ 〈j| 〈n′| a†αkσaα′k′σ′a†α′′′k′′′σ′′′cσ′′′

1

Ejn −H0 + iηc†σ′′aα′′k′′σ′′ |n〉 |j〉

}∣∣∣∣

2

δ(Ej′n′ − Ejn) (A1)

Here n and n′ are states with the same number of particles. Now we take a look at the numeratorterms

〈j| a†αkσaα′k′σ′aα′′′k′′′σ′′′a†α′′k′′σ′′ |j〉 = −〈j| a†αkσaα′′′k′′′σ′′′aα′k′σ′a†α′′k′′σ′′ |j〉= −f (ε− µα) δαα′′′δkk′′′δσσ′′′

× (1− f (ε+ εn − εn′ − µα′)) δα′α′′δk′k′′δσ′σ′′

and

〈j| a†αkσaα′k′σ′a†α′′′k′′′σ′′′aα′′k′′σ′′ |j〉 = 〈j| a†αkσaα′k′σ′

✘✘✘✘✘✘✘✘✘✿

0

δα′′′α′′δk′′′k′′δσ′′′σ′′ − aα′′k′′σ′′a†α′′′k′′′σ′′′

|j〉

= −〈j| a†αkσaα′k′σ′aα′′k′′σ′′a†α′′′k′′′σ′′′ |j〉= 〈j| a†αkσaα′′k′′σ′′ |j〉 〈j| aα′k′σ′a†α′′′k′′′σ′′′ |j〉= f (ε− µα) δαα′′δkk′′δσσ′′

(1− f (ε+ εn − εn′ − µα′)) δα′α′′′δk′k′′′δσ′σ′′′

Here we have used a Taylor series expansion on the operator 1/(Ejn−H0) = (1/Ejn)∑∞

l=0(H0/Ejn)l.

17

Taking into account last delta rules, we have

〈n′| c†σ′′′cσ′′ |n〉 =∑

n′′

〈n′| c†σ |n′′〉 〈n′′| cσ′ |n〉 =∑

n′′

(〈n′′| cσ |n′〉)† 〈n′′| cσ′ |n〉 =∑

n′′

Aσ∗n′′n′Aσ′

n′′n

and

〈n′| cσ′c†σ |n〉 =∑

n′′

〈n′| cσ′ |n′′〉 〈n′′| c†σ |n〉 =∑

n′′

〈n′| cσ′ |n′′〉 (〈n| cσ |n′′〉)† =∑

n′′

Aσ′

n′n′′Aσ∗nn′′

where Aσ′

n′n′′ = 〈n′| cσ′ |n′′〉 and Aσ∗nn′′ = 〈n′′| c†σ |n〉. Here n′′ represents a intermediate state.

Thus Eq. (A1) becomes

Γnj;n′j′

αα′ =2π

h

∑

kk′σσ′

∣∣∣∣∣

∑

α′′′k′′′σ′′′

∑

α′′k′′σ′′

t∗α′′′tα′′

{

−〈j| 〈n′| a†αkσaα′k′σ′aα′′′k′′′σ′′′c†σ′′′

1

εn′ − εn′′ − ε+ iηcσ′′ |n〉 a†α′′k′′σ′′ |j〉

+ 〈j| 〈n′| a†αkσaα′k′σ′a†α′′′k′′′σ′′′cσ′′′

1

εn − εn′′ + ε+ iηc†σ′′ |n〉 aα′′k′′σ′′ |j〉

}∣∣∣∣

2

δ(Ej′n′ − Ejn)

Γn;n′

αα′ = 2 |tα|2 |tα′ |2∑

σσ′

να(σ)να′ (σ′)

∫

dεf (ε− µα) (1− f (ε+ εn − εn′ − µα′))

×∣∣∣∣∣

∑

n′′

{

Aσ∗n′′n′Aσ′

n′′n

ε− εn′ + εn′′ + iη+

Aσ′

n′n′′Aσ∗nn′′

ε+ εn − εn′′ + iη

}∣∣∣∣∣

2

=∑

σσ′

γσαγσ′

α′

∫

dεf (ε− µα) (1− f (ε+ εn − εn′ − µα′))

×∣∣∣∣∣

∑

n′′

{

Aσ∗n′′n′Aσ′

n′′n

ε− εn′ + εn′′ + iη+

Aσ′

n′n′′Aσ∗nn′′

ε+ εn − εn′′ + iη

}∣∣∣∣∣

2

︸ ︷︷ ︸

Q

(A2)

Appendix B: Explicit derivation of Eq. (55)

The absolute value in Eq. (A2) can be written as

Q =

∣∣∣∣∣

∑

n′′

{

Aσ∗n′′n′Aσ′

n′′n

ε− εn′ + εn′′ + iη+

Aσ′

n′n′′Aσ∗nn′′

ε+ εn − εn′′ + iη

}∣∣∣∣∣

2

=

(

Aσ′∗1n A

σ1n′

ε− εn′ + ε1 − iη+

Aσn1A

σ′∗n′1

ε+ εn − ε1 − iη+

Aσ′∗2n A

σ2n′

ε− εn′ + ε2 − iη+

Aσn2A

σ′∗n′2

ε+ εn − ε2 − iη

+Aσ′∗

3n Aσ3n′

ε− εn′ + εn′′ − iη+

Aσn3A

σ′∗n′3

ε+ εn − ε3 − iη

)

×(

Aσ∗1n′Aσ′

1n

ε− εn′ + ε1 + iη+

Aσ′

n′1Aσ∗n1

ε+ εn − ε1 + iη+

Aσ∗2n′Aσ′

2n

ε− εn′ + ε2 + iη+

Aσ′

n′2Aσ∗n2

ε+ εn − ε2 + iη

+Aσ∗

3n′Aσ′

3n

ε− εn′ + εn′′ + iη+

Aσ′

n′3Aσ∗n3

ε+ εn − ε3 + iη

)

18

Q =∑

k

(

(Aσ∗kn′Aσ′

kn)2

(ε− εn′ + εk)2 + η2+

(Aσ′

n′kAσ∗nk)

2

(ε+ εn − εk)2 + η2

)

+2Re∑

q

∑

k<q

(

Aσ∗qn′Aσ′

qn

ε− εn′ + εq + iη

Aσ∗kn′Aσ′

kn

ε− εn′ + εk − iη+

Aσ′

n′qAσ∗nq

ε+ εn − εq + iη

Aσ′

n′kAσ∗nk

ε+ εn − εk − iη

)

+2Re∑

q

∑

k

(

Aσ∗kn′Aσ′

kn

ε− εn′ + εq − iη

Aσ′

n′kAσ∗nk

ε+ εn − εk − iη

)

Thus Eq. (54) becomes

Γn;n′

αα′ =∑

σσ′

γσαγσ′

α′

∫

dεf (ε− µα) (1− f (ε+ εn − εn′ − µα′))

×∣∣∣∣∣

∑

n′′

{

Aσ∗n′′n′Aσ′

n′′n

ε− εn′ + εn′′ + iη+

Aσ′

n′n′′Aσ∗nn′′

ε+ εn − εn′′ + iη

}∣∣∣∣∣

2

=∑

σσ′

γσαγσ′

α′

∫

dεf (ε− µα) (1− f (ε+ εn − εn′ − µα′))

×[∑

k

(

(Aσ∗kn′Aσ′

kn)2

(ε− εn′ + εk)2 + η2+

(Aσ′

n′kAσ∗nk)

2

(ε+ εn − εk)2 + η2

)

+2Re∑

q

∑

k<q

(

Aσ∗qn′Aσ′

qn

ε− εn′ + εq + iη

Aσ∗kn′Aσ′

kn

ε− εn′ + εk − iη+

Aσ′

n′qAσ∗nq

ε+ εn − εq + iη

Aσ′

n′kAσ∗nk

ε+ εn − εk − iη

)

+ 2Re∑

q

∑

k

(

Aσ∗kn′Aσ′

kn

ε− εn′ + εq − iη

Aσ′

n′kAσ∗nk

ε+ εn − εk − iη

)]

Γn;n′

αα′ =∑

σσ′

γσαγσ′

α′

∫

dεf (ε− E1) (1− f (ε− E2))

×[∑

k

A2

(ε− εak)2 + η2(Integral type J)

+∑

k

B2

(ε− εbk)2 + η2(Integral type J)

+2Re∑

q

∑

k<q

Ak

ε− εak + iη

Aq

ε− εaq − iη(Integral type I)

+2Re∑

q

∑

k<q

Bk

ε− εbk + iη

Bq

ε− εbq − iη(Integral type I)

+ 2Re∑

q

∑

k

Ak

ε− εak + iη

Bq

ε− εbq − iη

]

(Integral type I)

where Ak = Aσ∗kn′Aσ′

kn, Bk = Aσ′

n′kAσ∗nk, εak = εn′ − εk, εbk = εk − εn, E1 = µα and E2 = µα′ + εn′ − εn.

Integral type I

I(E1, E2, ε1, ε2) = Re

∫

dεf(ε− E1) [1− f(ε− E2)]1

ε− ε1 − iγ

1

ε− ε2 + iγ

=nB(E2 − E1)

ε1 − ε2Re

{

ψ

(1

2+iβ

2π[E2 − ε1]

)

− ψ

(1

2− iβ

2π[E2 − ε2]

)

−ψ(1

2+iβ

2π[E1 − ε1]

)

+ ψ

(1

2− iβ

2π[E1 − ε2]

)}

(B1)

19

Here ψ is the digamma function, nB is the Bose function and β = 1/kBT .

Integral type J

J(E1, E2, ε1) =

∫

dεf(ε− E1) [1− f(ε− E2)]1

(ε− ε1)2 + η2

=β

2πnB(E2 − E1)Im

{

ψ′

(1

2+iβ

2π[E2 − ε1]

)

− ψ′

(1

2+iβ

2π[E1 − ε1]

)}

(B2)

Thus Eq. (B1) becomes

Γn;n′

αα′ =∑

σσ′

γσαγσ′

α′

[∑

k

(A2J(E1, E2, εak) +B2J(E1, E2, εbk)

)

+2∑

q

∑

k 6=q

(AkAqI(E1, E2, εak, εaq) +BkBqI(E1, E2, εbk, εbq))

+ 2∑

q

∑

k

AkBqI(E1, E2, εak, εbq)

]

(B3)

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