Time-dependent single electron tunneling through a shuttling nanoisland

G. Cohen, V. Fleurov, and K. KikoinRaymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel

�Received 11 March 2009; revised manuscript received 5 May 2009; published 11 June 2009�

We offer a general approach to the calculation of single electron tunneling spectra and conductance of ashuttle oscillating between two half-metallic leads with fully spin-polarized carriers. In this case the spin-flipprocesses are completely suppressed and the problem may be solved by means of canonical transformation,where the adiabatic component of the tunnel transparency is found exactly, whereas the nonadiabatic correc-tions can be taken into account perturbatively. Time-dependent corrections to the tunnel conductance of mov-ing shuttle become noticeable at finite bias in the vicinity of the even/odd occupation boundary at the Coulombdiamond diagram.

DOI: 10.1103/PhysRevB.79.245307 PACS number�s�: 73.21.La, 73.23.Hk

I. INTRODUCTION

Single electron tunneling �SET� is a salient feature ofquantum transport in nanostructures. The SET phenomenonis observed in various systems, e.g., quantum dots �QDs� in atunnel contact with metallic electrodes,1,2 molecular bridgesbetween the edges of broken metallic wires,3–5 atoms andmolecules absorbed on metallic surfaces in a contact with thetip of tunnel microscope,6,7 etc. The study of electron tunnel-ing through the nano-object with time-dependent character-istics is one of the most challenging problems in this field.

There are several sources of time dependence, which maybe realized in practical devices. The simplest one is the time-dependent gate voltage vg�t� applied to the dot. It is wellknown8–10 that this time dependence may be converted intothe time dependence of tunnel matrix element. Another pos-sibility is the nanoelectromechanical shuttling �NEMS�,11

where the nanosize island suspended on a pivot12 or attachedto a string13 oscillates between the leads under the action ofan electromechanical force. In the case of molecular bridges,the vibration eigenmodes may be the source of the periodicaloscillations of tunneling parameters.5,14

Usually the tunneling between metallic leads and such anano-object is accompanied by many-particle Kondo screen-ing effect15,16 resulting in a specific type of zero-biasanomaly �ZBA� in the tunnel conductance. Modification ofKondo regime because of periodically modulated in timetunneling rate due to the center-of-mass oscillations wasstudied recently in several papers. If the oscillations are theeigenmodes of a nano-object �molecule�, then the Kondopeak �zero-bias anomaly in the tunnel conductance� maytransform into a dip due to the destructive interference withthe vibrational mode.14 In the case of adiabatic motion in-duced by electromechanical forces �NEMS� �Ref. 11�, theKondo temperature follows the periodical evolution of thedot position and increases eventually due to the effectivereduction in the average distance between the dot and theleads,17 which is determined by the mean-square displace-ment of the dot �in an analogy with the Debye-Waller effectin the scattering intensity�. A nonadiabatic enhancement ofKondo tunneling through such moving nano-object at a finitesource-drain bias has been studied recently.18

In the present paper we consider adiabatic and nonadia-batic time-dependent effects in conventional cotunneling be-

tween metallic leads due to a periodic modulation of thelead-dot tunneling rate. To suppress the many-particle Kondoscreening effects, one should consider the leads with mag-netically polarized electrons. Tunneling between the ferro-magnetically ordered leads was discussed in the context ofKondo effect.19–25 We are interested in the situation, wherethe spin-flip cotunneling is completely suppressed at smalllead-dot bias and low temperatures. Such tunneling regimemay be realized in half-metallic ferromagnets, where theFermi surface is formed only by the majority-spin electrons,whereas the spectrum of the minority-spin carriers is gapped.The electronic and magnetic properties of such metallic com-pounds are reviewed in Ref. 26. From the point of view ofexisting devices, where the leads are formed by two-dimensional �2D� electrons in degenerate semiconductors,the relevant material for our studies is a dilute magneticsemiconductor �Ga,Mn�As.27 The indirect magnetic ex-change between Mn impurity ions is responsible for thelong-range ferromagnetic order in this material. This ex-change is mediated by spin-polarized carriers near the top ofthe valence band. The tunnel current in the half-metallic re-gime arises due to the minority-spin hole cotunneling.

We will show that in the absence of Kondo effect theproblem of tunneling through a moving nanoisland �quantumdot� may be solved by means of time-dependent canonicaltransformation, which exactly takes into account both adia-batic and nonadiabatic lead-dot tunneling processes diagonalin the lead indices. The nondiagonal source-drain cotunnel-ing may be treated by the canonical transformation methodonly perturbatively, but the adiabatic and nonadiabatic con-tributions into tunnel conductance may be sorted out also inthis case. It will be shown that the time-dependent contribu-tion into current-voltage characteristics of moving nano-object is significant near the boundaries of Coulomb dia-monds.

II. MODEL

We choose for the realization of ac-driven tunnel conduc-tivity the simplest model of a nano-object, which is widelyused in the studies of SET. The nano-object is represented inthis model by the quantum well with a resonance level �d�see Fig. 1, upper panel�. The SET regime arises due to the

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Coulomb blockade effect: addition energy for the secondelectron in a singly occupied dot is �d+U�� j, where � j isthe tunneling rate to the left �j= l� and right �j=r� leads, andU is the capacitive energy of the dot. To separate the time-dependent SET from the Kondo ZBA, we consider tunnelingbetween spin-polarized leads, where spin-flip processes areinelastic, because the continuum of electron-hole pairs withthe opposite spins in the leads responsible for the Kondoeffect is gapped. Two types of spin-polarized �magneticallyordered� metallic leads are presented schematically in Fig. 1.In the middle panel the leads are formed by “half-metallicferromagnets”26 with gapped spectrum of minority-spin car-riers. In the lower panel characteristic for the p-type degen-erate dilute magnetic semiconductors,27 the carriers are theminority-spin holes.

Virtual tunneling results in a shift of level positions inQD. This renormalization �“Friedel shift”� is also spin de-pendent. As a result the spin polarization of QD adjusts tothat of the ferromagnetic lead �see calculations below�. AllKondo processes are quenched in this regime.

The Anderson Hamiltonian modeling SET has the form

H = Hd + Hb + Htun, �2.1�

where the terms

Hd = ��

E�������, Hb = �j=l,r

�k�

� jk�ajk�† ajk� �2.2�

describe the electron states in the isolated dot and two me-tallic �semiconductor� leads, respectively. We write Hd interms of its eigenstates ��� �Hubbard representation�. Thistrick allows one to take all intradot interactions into accountexactly even when the contact with the leads is switched on.The tunneling Hamiltonian

Htun = �jk�

�Vjkd�†ajk� + H.c.�

may be rewritten in the Hubbard representation by expand-ing the creation operator d�

† in terms of the configurationchange operators X��= ������ which connects the states inadjacent charge sectors with N and N−1 electrons in the dot.We confine ourselves with the simplest case, where onlythree charge sectors N=0,1 ,2 are involved in SET Hamil-tonian. Then Hamiltonian �2.1� acquires the following form:

H = �d��

X�� + E2X22 + �jk�

� jk�ajk�† ajk�

+ �jk�

�Vjkajk�† �X0� + �X�2� + H.c.� . �2.3�

Here the quantum numbers � ,�=0,� ,2 correspond toempty, singly, and doubly occupied states of QD, respec-tively, and E2=2�d+U is the energy of doubly occupied QD.The last term in this Hamiltonian is time dependent.

III. CANONICAL TRANSFORMATION OF ANDERSONHAMILTONIAN

Our program is to exclude the tunneling term from Hamil-tonian �2.3� by means of the canonical transformation

H = eSHe−S, �3.1�

then derive the tunnel current operator in the new basis andcalculate the tunnel conductance. It was shown in Ref. 28that this transformation may be performed exactly in the ab-sence of Coulomb correlation, provided the energy level �dfalls into the energy gap and remains there after renormal-ization �Friedel shift�. It will be shown below that the matrixS still may be found exactly in the presence of Coulombblockade under the same condition of discreteness of renor-malized d-level, provided the spin-flip processes arequenched. As may be perceived from Fig. 1, this condition isrealized for the majority-spin electrons in the case of half-metallic ferromagnet and for the minority-spin holes in thecase of p-type dilute magnetic semiconductor.

As usual, the canonical transformation is made by meansof the Baker-Hausdorff expansion

εd

V (t) V (t)l r

(1) (2)

left right

left right

εd+ U

(1)(2)

left right

dε σ

E E

EE

εd + U

FIG. 1. �Color online� Upper panel: model of a quantum dot ina time-dependent contact with leads. Middle and lower panels: den-sity of electron states in the leads and occupied and empty states inthe dot for N=1; mechanisms of spin-polarized electron cotunnel-ing �middle panel� and hole cotunneling �lower panel� are indicatedby arrows. Figures in parentheses point out the sequence of electrontunneling acts in the cotunneling processes.

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eSHe−S = �n=0

�1

n!†S,�S, . . . �S,H� . . .�‡ . �3.2�

In the noninteracting case, the second quantization operatorsin S and H possess Fermi-like commutation relations, theHamiltonian H is a quadratic form, and the tunnel operatorconserves spin, so the series in the right-hand side �r.h.s.� ofEq. �3.2� may be summed exactly.28 The Coulomb blockadeseparates the Hilbert space for the dot electron operators intothe charge sectors divided by the energy gaps. As a resultthese operators lose the simple Fermi statistics.

We are interested in the strong Coulomb blockade caseand start with the simplest case, when the ground state of thedot corresponds to N=1 in the limit of U� ��d−�F�, so thatthe doubly occupied states are completely suppressed. Thenonly the states �=� ,0 are retained in Hamiltonian �2.3�. Inparticular, the anticommutation relation for the Hubbard op-erators mixing the adjacent sectors N=0,1 has the form

�X�0,X0���+ = X��� + X00���, �3.3�

which follows from the obvious multiplication ruleX�1�1X�2�2 =�1�2

X�1�2. Even disregarding the spin-flip pro-cesses each commutation operation in expansion �3.2� gen-erates operators K�=X��+X00. In spite of this, the Baker-Hausdorff series still can be summed exactly, because theseoperators are idempotent, K�K�=K�, like the conventionalfermion occupation operators d�

†d�. This summation will beperformed in the following section.

The time-dependent problem is more complicated becausein that case the canonical transformation should be applied tothe operator

L = − i�

�t+ H . �3.4�

In Sec. III B we will show that the canonical transformationis operational in this case as well, at least in some importantspecial cases.

A. Time-independent transformation

In this section we generalize the canonical transformationproposed for an Anderson model applied to transition-metalimpurities in semiconductors.28,29 In those calculations theintraimpurity Coulomb interaction was taken into account inthe Hartree approximation. Here we take the Coulomb block-ade term exactly, when summing series �3.2�, at least in thecharge sectors N=0,1. The anti-Hermitian operator S islooked for in the form

S = �jk�

�ujk�ajk�† X0� − ujk�

� X�0ajk�� = ��

S�.

If the spin-flip processes are neglected, the canonical trans-formation is made for each spin projection separately. Onemay apply the transformation to the majority spin in the caseof electron tunneling and to the minority spin in the case ofhole tunneling in two models introduced above.

One straightforwardly derives the commutation relations

�S�,X�0� = �jk

ujk�ajk�† K� C�

†K�,

�S�,ajk�† � = − ujk�

� X�0.

To shorten the notations, we introduce the quantities C�†

=� jkujk�ajk�† and ��

2 =� jkujk�ujk�� . Besides, we omit the lead

index j and specify the band state by a single index k char-acterizing both the lead and the electron wave number. Usingthese definitions and the above-mentioned idempotency ofthe operator K�, we obtain the following expressions for thetransformed operators:

X�0 = eS�X�0e−S� = X�0 cos �� + C�†K���

−1 sin ��,

�3.5�

ak�† = eS�ak�

† e−S� = ak�† + uk�

� C�†K���

−2�cos �� − 1�

− uk�� X�0��

−1 sin ��, �3.6�

and

eSC�†e−S = C�

† + C�†K��cos �� − 1� − X�0�� sin ��.

�3.7�

The tunneling term in the transformed Hamiltonian iseliminated, provided

uk� =��Vk

��k� − Ed��tan ��

gk�Vk �3.8�

with

tan2 �� = − dL����d�

�=Ed�

, �3.9�

L���� = �k

�Vk�2

� − �k�

. �3.10�

Then the transformed Hamiltonian takes the form

H� = Ed�X�� + �kk�

�kk�� ak�

† ak��, �3.11�

where the renormalized level position is given by the equa-tion

Ed� =�d + Z� tan2 �� − �T� + T�

��tan ��

1 + tan2 ��

, �3.12�

with Z�=��−2�k��k�uk�

� uk� and T�=��−1�k�Vkuk�

� �. Substitu-tion of Eqs. �3.8�–�3.10� into Eq. �3.12� transforms it into theconventional form28,29

Ed� = �d + L��Ed�� , �3.13�

where the self-energy L��E� has only real part �Friedel shift�,provided the level Ed� remains within the gap, which is thecase for �=↑ in the configuration presented in the middlepanel of Fig. 1.

The canonical transformation for the second spin compo-nent �=↓ should be done more carefully, because the barelevel �d falls into continuum of spin-down states. One should

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be cautious with turning to the thermodynamic limit, wherethe sum in the right-hand side of Eq. �3.10� transforms intothe integral and acquires the imaginary part, thus making theHamiltonian non-Hermitian. The problem can be avoided.The recipe is to keep the spectrum of electrons in the leadsdiscreet when doing the canonical transformation. Then Eq.�3.13� has N+1 solutions �i, where N is the number of statesin the valence band �k↓. Using Eqs. �3.9� and �3.10�, thecorresponding coefficients �i may be found. In accordancewith Eq. �3.5�, the factor cos �i determines the weight of thed component in the hybridized wave function of the dot elec-tron in the state i. One may identify the state i=m producingthe maximum value of cos �i with the center of future Frie-del resonance, which arises in the thermodynamic limit N→�. In this sense the state Ed↓�m↓ is formally definedfrom the equation �m↓=�d+L↓��m↓�. This level is shown bythe dashed line in the middle panel of Fig. 1. It correspondsto the bunch of excited states of QD coupled to the leadswith N=1 and spin oriented antiparallel to that of the mag-netized leads.

The spectrum of continuous part of Hamiltonian �3.11� isdetermined by the expression

�kk�� = �k�kk� + Wkk�

� �3.14�

containing the “scattering” matrix element, which eventuallypredetermines the tunnel current. The general form of thismatrix element is

Wkk�� =

uk�uk���

��2 K���2�d − �k� − �k����1 − cos ���

−T� + T�

�

2

2 cos3 �� − 3 cos2 �� + 1

cos �� sin ���

+ K��Vkuk��� + uk�Vk�

� �sin ��

��

. �3.15�

Using Eqs. �3.8� and �3.12� we get

T� = T�� = −

1

tan ��

L��Ed��

and after some algebraic manipulations the scattering matrixelement is eventually transformed into a quite compact ex-pression,

Wkk�� = VkVk�

� K�� 1

�k�

+1

�k���R���� +

L��Ed���k��k��

R2����� ,

�3.16�

where

R���� =�1 + tan2 �� − 1

tan2 ��

�3.17�

and �k�=Ed�−�k�. Equation �3.16� holds only in the staticcase or, as we will see below, for adiabatically slow timevariations of the tunneling amplitudes. If we want to studynonadiabatic corrections a more general formula �3.15�should be used.

Equation �3.16� generalizes the familiar second-order ex-pression for the single electron tunneling amplitude throughthe QD, which takes into account both the renormalization ofthe energy level of dot electron �3.13� and the reconstructionof the band continuum �3.6�. Far from the resonance tunnel-ing regime �in the center of the Coulomb diamond diagram;see, e.g., the middle panel of Fig. 1�, one may neglect thesecond term in the square bracket of Eq. �3.16�, and thetunneling matrix element acquires the simple form Wlk,rk�

�

=Jlk,rk�K�R����, where the first factor is the off-diagonal ma-

trix element of indirect exchange between the leads and thedot due to electron cotunneling and the second and the thirdfactors regulate the occupation of the dot level and the nor-malization of electron wave functions, respectively. Thetransparency of QD is, of course, exponentially weak

Now we turn to the resonance regime illustrated by Fig. 2,where the level �d

�2�=�d+U driven by the gate voltage vgapproaches the level �F from above. The level �d

�2� may beoccupied only by a spin-down electron. Since it falls into thegap of spin-down density of states in the leads, the canonicaltransformation introduced above may be performed in asimilar way, provided the addition energy for the first elec-tron ��1���d falls deep enough below �F and excitations ofelectrons in this level are suppressed. In this regime � ,�=� ,2, and one should use the commutation relations

�X�2,X2���+ = X��� + X22��� �3.18�

instead of Eq. �3.3�. Correspondingly, one should insert K�

=X��+X22 in equations

X2� = �X2� cos �� + C�†K���

−1 sin ��, �3.19�

a�† = ak�

† + uk�� C�

†K���−2�cos �� − 1� − uk�

� �X2���−1 sin �

�3.20�

for the transformed creation operators.Then the transformed Hamiltonian for spin-down elec-

trons has the form

source dε(1)

drain

N=1 N=2

E E

dε(2) eW

FIG. 2. �Color online� Cotunneling mechanism in the resonanceregime for second electron in QD on the boundary of Coulombwindow.

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H↓ = Ed↓X↓↓ + �

kk�

�kk�� ak↓

† ak�↓, �3.21�

where Ed↓ and �kk�� are given by the same equations �3.13�

and �3.14� as in the previous case, but with the energy �d�2�

substituting for �d and the correlation function K� taken fromEq. �3.18�. It should be stressed that the tunneling throughthe QD is impossible at zero source-drain bias because of thespin blockade in spin-polarized electrodes. Only spin-up car-riers exist around Fermi level, and these electrons may beinjected into the QD only when accompanied by the spin-flipexcitations given by the operators X↓↑ in the intermediatestate with N=1 of cotunneling process. These processes areinelastic and exponentially weak ��V2 in transparency and�V4 in conductance�. A more detailed discussion of the spin-dependent tunneling is postponed until Sec. IV A.

B. Time-dependent transformation

As was discussed above experimental conditions may becreated when the tunneling matrix element Vk�t� in Hamil-tonian �2.3� becomes time dependent. The canonical trans-formation �3.1�, as described in the previous section, cannotbe straightforwardly applied. Its generalization is in order.

We start with the time-dependent Schrödinger equation

L = 0 �3.22�

�see Eq. �3.4�� and look for the time-dependent transforma-

tion matrix S�t�, which transforms Eq. �3.22� into

L = 0 �3.23�

with transformed operator

L�t� = eS�t�He−S�t� − ieS�t� �

�te−S�t�

for the new wave function =eS�t� . The Hamiltonian H sat-isfying Eq. �3.23� can be written as

H = eSHe−S + i�0

1

d� e�SSe−�S �3.24�

�see Appendix A�.Hamiltonian �3.24� contains now two terms, of which the

first one is just a modification of Hamiltonian �3.1� of thetime-independent case. It means that all the equations in Sec.III A hold except for Eq. �3.8�, which defines the coefficientsuk� of the canonical transformation. These coefficients mustbe now found anew. There is also the second term in theright-hand side of Eq. �3.24�, which is responsible for nona-diabatic effects.

In order to find the canonical transformation parametersuk� we write explicitly the condition

uk���−1���d − Z��sin �� cos �� − ��k� − Z��sin ��

+ T� cos2 �� − T�� sin2 � − T� cos ���

+ Vk cos �� = − iuk�

sin ��

��

− iuk�

��k��uk��uk��

� 1

2��3 �sin �� cos �� + �� − 2 sin ���

+ uk��� uk��

1

2��2 1 −

sin �� cos ��

���� = 0 �3.25�

of elimination of the QD-lead tunneling in the transformedHamiltonian. Here both the tunneling amplitude Vk� andtransformation parameter uk� are functions of time. Condi-tion �3.25� contains a number of terms with the time deriva-tives uk. Neglecting these time derivatives would correspondto the adiabatic approximation where the variation in thetunneling amplitude is slow enough and the whole electronsystem always has enough time to readjust to the varyingtunneling amplitude without additional level mixing. Thenone can check straightforwardly that Eq. �3.8� with the time-dependent Vk�t� solves Eq. �3.25�.

Now we carry out a more general analysis going beyondthe adiabatic approximation. For this sake we multiply Eq.�3.25� by uk�

� and sum over k, which leads to the equation

��d − Z��tan �� + T� − T�� tan2 ��

= − id

dttan �� − i

tan ��

2��2 �

k�

�uk��uk��� − uk��

� uk��� .

�3.26�

Substituting Eq. �3.26� into Eq. �3.25� we get the equation

uk����k� − Z��tan �� + T�� − �Vk�

= − itan ��

��2 uk��

k�

uk��� uk�� + iuk� tan ��.

�3.27�

Neglecting the time derivatives of transformation parameteruk�t� in the r.h.s of Eq. �3.27�, which corresponds to theadiabatic approximation, yields Eq. �3.8� with the time-dependent tunneling amplitude Vk�t�. All the other equationsobtained in Sec. III A also hold. Accounting for the r.h.s. ofEq. �3.27� allows one to obtain nonadiabatic corrections.

Having in mind that the quantities �d, ��, and Z� areexplicitly real, we separate real and imaginary parts in Eq.�3.26� and thus get two equations

T� − T�� = − 2i�� �3.28�

and

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��d − Z��tan �� +T� + T�

�

2�1 − tan2 ���

= − itan ��

2��2 �

k�

�uk��uk��� − uk��

� uk��� . �3.29�

These equations may be instrumental in looking for solutionsuk��t� for specific problems. It follows from Eq. �3.28� thatthe quantity T�, which was real for the time-independentcase, remains real also in the adiabatic approximation for thetime-dependent case.

Returning back to the transformed Hamiltonian �3.24�, wenote that the first term eSHe−S is now time dependent due tothe time dependence of S. Carrying out the transformation inthe same fashion, as in the previous section, we get the time-dependent energy level

Ed� = Ed��a� + Ed�

�b�, �3.30�

where

Ed��a� =

�d + Z� tan2 �� − �T� + T���tan ��

1 + tan2 ��

�3.31�

is the same energy level �3.12� as before but with the coef-ficients Z� ,T� ,T�

� ,�� depending parametrically on time t.Thus, in the adiabatic approximation the time dependence ofthe resonance level position Ed�

�a��t� is determined by Eq.�3.13� with the time-dependent self-energy part

L���,t� = �k

�Vk�t��2

� − �k�

. �3.32�

There is also the nonadiabatic correction

Ed��b��t� = −

1

2i

sin2 �

�2 �k

�uk�uk�� − uk�

� uk�� . �3.33�

To calculate the coefficients, one has to specify the form oftunneling amplitudes Vk�t�. An example of time-dependenttunneling will be considered in the next section.

IV. TUNNEL CONDUCTANCE

In this section we study the tunnel conductance basing on

the transformed Hamiltonian H. The tunnel current may becalculated, e.g., by means of the Keldysh technique,30 wherethe bias eV is included in the zero-order Hamiltonian and thescattering �Wlk,rk� is considered as a perturbation. First, wecalculate the spin-polarized current through an immovablequantum dot and then discuss the modulation of this currentdue to the oscillatory motion of the dot.

A. Tunneling through static dot

Far from the boundary between the two adjacent chargesectors with N=1 and N=2, where both levels �d↑

�1� and �d↓�2�

are far from the chemical potential �, the Keldysh methodapplied to Hamiltonian �3.21� in a single loop approximationgives the conventional golden rule equation

I = e2�

�kk�

��

�Wlk,rk�� �2��lk� + e� − �rk���

��f��lk���1 − f��rk���� − f��rk����1 − f��lk���� ,

�4.1�

where e� is the source-drain bias, f��k�� is the Fermi distri-bution function for spin-polarized electrons, and the scatter-ing amplitude Wlk,rk� is defined in Eq. �3.16�.

The standard Coulomb diamond diagram for tunnelingconductance G�vg ,e�� is distorted in the region of gate volt-ages vg corresponding to the change in QD occupation �N=1�→ �N=2�. In the case of completely spin-polarized dotsthe Coulomb step corresponding to the resonance E�N=1,vg�=E�N=2,vg�−� in the current-voltage characteristicsis absent at zero bias and zero temperature due to the spinblockade. The chemical potential � is pinned to the Fermilevel of spin-up electrons �F↑, but the resonance level Ed↓=E�N=2�−E�N=1� belongs to the down-spin electron �seeFig. 2�. Thus the tunneling at zero bias e�=0 is suppressedby the spin blockade. This blockade may be surmounted bymeans of a finite source-drain bias compensating the energygap. However, the conditions for the onset of spin-up andspin-down tunnel currents are different.

The boundary of the Coulomb diamond with N=1 forspin-down electrons follows the evolution of the resonancelevel Ed↓ until this level approaches from above the bottomof conduction band �c↓. When it crosses the band edge, theresonance tunneling is no more possible at eV��cv, so theCoulomb resonance line for G�vg ,e�� deviates from the lin-ear behavior, as it is sketched in Fig. 3 �solid line�. Negative-bias part of the Coulomb diamond corresponds to the holetunneling through the occupied resonance level in the sectorN=2. It is distorted in the same way in the region of vgwhere this level matches the top of the valence band �secondbranch of the solid line�. When the down-spin level is deepenough in the conduction or valence band, the linear behav-ior is restored again.

The blockade for spin-up electrons near the boundary�N=1� / �N=2� is lifted at e� compensating the spin-flip ex-

gv

−∆ e∆ cv

N=2

N=1

cvυ

FIG. 3. Coulomb diamonds diagram for tunneling conductanceG�vg ,e�� near the border �N=1� / �N=2� for the down-spin and up-spin electrons �solid and dashed lines, respectively�. Three bars onthe vg axis mark the values of the gate voltage corresponding to thelevel Ed↓ in resonance with �c, �F, and �v �bottom up�.

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citation in the dot. As was mentioned above, the spin-upelectron tunneling is allowed provided the dot is excited tothe state Ed↓ given by the solution of Eq. �3.13� for the dou-bly occupied dot, and the splitting energy may be estimatedas �↑↓��L�Ed↑�−L�Ed↓�� �see Ref. 25 for experimental de-termination of such splitting in the Kondo tunneling regime�.The line of resonance tunneling for spin-up electrons isdrawn in Fig. 3 by the dashed curve.

Besides, the tunneling transparency is especially sensitiveto the position of the dot level in the near vicinity of bandedges. The intensity of standard resonance tunneling lines isenhanced for the spin-down electrons in the region of �e��+�cv ,�d−vg��c� and �e��−�cv ,�d−vg��v�. In order toinvestigate this enhancement, we find the explicit equationfor the tunnel conductance from Eqs. �4.1� and �3.16�. Trans-forming summation over kk� by integration over ��� in theusual way and performing standard calculations, one gets theequation

G↓�vg,�cv� =e2

h

�l�r

�2�2�R2��↓�

�Ed↓ − �c�2�1 +

R��↓�L�Ed↓��cv

�2

�4.2�

for the threshold value of e�→�cv+o in case of 2D electrongas in the planar leads. Here �i=2�Vi

2Si is the tunneling ratefor the lead i obtained in the approximation of Vik=Vi andconstant electron density of states Si���=Si, which is validfor 2D electrons. It should be taken into account that thedefinition of the position of resonance level Ed↓ falling intothe conduction-band continuum implies the procedure de-scribed below Eq. �3.13�. The function L��� has a logarith-mic singularity at the band edge, Re L��→�c��−ln���−�c� /�cv�. As a result, Eq. �3.13� has either one or two so-lutions depending on a position of the level �d↓

�2�−vg relativeto the band edge.31

The resonance factor �Ed↓−�c�2 in the denominator andthe singular factor L�Ed↓� in the numerator of Eq. �4.2� resultin a noticeable enhancement of G at the boundary of Cou-lomb diamond. This enhancement is characterized by theevolution of the ratio

��vg� =R2��↓�

�Ed↓ − �c�2�4.3�

from its value in the middle of Coulomb diamond to that inthe vicinity the point �d�vg�=�c. Here and below we omit thesuperscript �2� in the notation of �d. Far from the Coulombresonance the difference between �Ed↓� and �d is small andtan2 �↓�V2 / ��d−�c�2�1. The function R��↓� tends to 1/2 inthis limit, so that the factor � may be estimated as ��1 /4��d−�c�2. Near the band edge �c the factor R��↓��cot �↓=��Ed↓−�c� /�r �see Eqs. �3.9� and �3.17��, so that����Ed↓−�c��r�−1. Numerical estimates of this enhancementare presented at Fig. 4. In these calculations the density ofstates was assumed to be constant in the lower part of 2Dconduction band, so that the self-energy in the r.h.s. of Eq.�3.13� may be approximated as L�E���r ln�E /D�, where Dis the effective width of conduction band and the argument Eis complex. Then the solutions E=Ed↓ of Eq. �3.13� are ex-pressed via the Lambert W function W�n ,x�,

Ed↓ = − �rW�n,− �r−1e−�d/�r� �4.4�

�here the reference point is �c=0, all energy parameters aremeasured in units D, and index n enumerates branches of theW function�. There are two solutions for Ed↓ near the bandedge.31 The principal branch n=0 gives the discrete level inthe gap, and the branch n=1 corresponds to the resonance inthe band, which is of interest for us here.

Thus the magnitude of conductance peak on the resonancelines of Coulomb diamond diagram for spin-down electrons�Fig. 3� as a function of the gate voltage vg should follow thecurves plotted in Fig. 4. The amplification reaches its maxi-mum when the difference �Ed↓−�c� comes up to �r; therefore,the smaller the �r, the stronger the enhancement factor �.The factor R��↓�L�Ed↓� /�cv in the square brackets in Eq.�4.2� behaves as �1/2 ln � in the vicinity �= �Ed↓−�c� of theband edge and thus gives additional contribution to this en-hancement. Both analytical and numerical estimates confirmthis statement. Experimentally, this effect should be observedas an increase in the tunneling conductance on the boundaryof Coulomb diamond in the vicinity of the threshold value ofe�th=�cv. A similar effect should arise on the hole side of theCoulomb diamond diagram �e�th=−�cv�, when the occupiedlevel Ed↓ crosses the top of the down-spin valence band�solid lines in Fig. 3�.

To conclude the dependence of tunneling conductance one� and vg near the occupation border �N=1� / �N=2� radicallydiffers from that for the standard Coulomb diamond diagram.�i� The Coulomb step G�vg� is absent at �e��→0. �ii� Thediamond boundaries are split due to the spin polarization ofcarriers. The spin-up boundary is symmetric relative to thechange in the bias sign, and the threshold value for G↑�e�� atvg corresponding to a Coulomb resonance for the spin-upelectrons is given by the induced molecular field for dotelectrons e�th=Ed↓−Ed↓ �3.13�. The threshold value forG↓�e�� is e�th=�cv and the spin-down resonance lines are

P r i m e d o t l e v e l

K 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4

Amplificationfactor

0

1 0

2 0

3 0

4 0

FIG. 4. �Color online� Amplification factor ���d�2�� for two val-

ues of tunneling rate � /D=0.01 �lower curve� and � /D=0.02 �up-per curve� as a function of the prime dot level position �d�vg�. Theevolution of the resonance level Ed↓ is given by Eq. �4.4� with n=1. This resonance approaches the conduction-band edge with �d

→+0 and the amplification factor �4.3� achieves its maximal valueunder these conditions.

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shifted downward relative to G↑ for the left-right tunneling�positive bias� and upward for right-left tunneling �negativebias�. �iii� The amplitude of the Coulomb resonance lineG↓�vg ,e�� in the vicinity of e�th should follow the profileshown in Fig. 4.

B. Tunneling through moving dot

As is shown above, the canonical transformation allowsone to distinguish between the adiabatic and nonadiabaticcontributions to the tunneling amplitude. Let us first discussthe adiabatic corrections to the inelastic current given by Eq.�4.1� and illustrated by Fig. 4. In the vicinity of the point�N=1� / �N=2� of the Coulomb diamond diagram, the adia-batic position of the down-spin level Ed↓

�a� given by the solu-tion of the equation

Ed↓�a��t� = �

k

�Vjk�t��2

Ed↓�a��t� − �k

�4.5�

rocks around �cr or �vl �see Fig. 2�. This solution dependsparametrically on time via the oscillating tunnel coupling�Vjk�t��2. The above analysis of the static case prompts thattime-dependent corrections become significant at a finite biasclose to the threshold value e�th= ��cv. The adiabatic evo-lution of the level Ed↓

�a� in time near �c is given by the sameequation �4.4�, where the tunneling rate �r parametricallydepends on t. If the time-dependent perturbation is weak incomparison with the static value of tunneling rate, the tem-poral component of Ed↓

�a� may be found perturbatively. Rep-resenting the tunneling rate as �r=�r0+��t� and expandingEq. �4.4� around the time-independent value marked by in-dex “0,” we get

Ed↓�a��t� = Ed↓

�0� −W0

1 + W0 �d

�r0+ W0���t� . �4.6�

When deriving Eq. �4.6�, the equality W�x�=x�1+W�x��dW /dx is used.

This time dependence turns into the corresponding adia-batic time dependence of tunnel conductance, mainly via theenhancement factor ��Ed↓� �4.3�. One may expect that theslow adiabatic variations in G�t� will be especially distinct atvg corresponding to steep slopes of ��t� �Fig. 4� at e�� ��cv near the boundary �N=1� / �N=2� of the Coulombdiamond diagram �Fig. 3�.

C. Weak time-dependent perturbation

Calculation accounting for nonadiabatic corrections totunnel conductance is generally an extremely complicatednonlinear problem. To make it tractable, we assume that thetime-dependent part of Vjk� is only a small periodic pertur-bation with respect to the time-independent part Vjk�

�0� andconsider only the linear response given by the first harmon-ics. The nonlinear effects will be discussed separately. Thisapproach allows one to pick up first nonadiabatic correctionsto tunneling amplitude Wkk�, which turns out to be smalleverywhere except in the vicinity of band edges.

Let us assume that the tunneling integral has the form

Vk�t� = Vk�0� + Vk

�1� cos �t �4.7�

�here and below indices j� are omitted for the sake of brev-ity�. Here Vk

�0��Vk�1�. The solution of Eq. �3.25� is looked for

in the form

uk = uk�0� + uk

�1� cos �t + ivk�1� sin �t , �4.8�

where the time-dependent corrections to uk�0� are also small.

Then we vary all the coefficients in Eq. �3.28� with respect touk

�0�=gkVk�0� �see Eq. �3.8�� and Vk

�0� and collect separately allthe terms containing cos �t and sin �t, respectively. Substi-tuting then uk in Eq. �3.15�, we reduce the scattering ampli-tude Wkk� to the following form:

Wkk� = Wkk� + Wkk��1� cos �t + �Wkk�

�2� sin �t , �4.9�

where the coefficients of the cosine and sine terms can beexplicitly calculated. The cosine coefficient

Wkk��1� = �

q�Wkk�

VqVq

�1� +Wkk�

Vq� Vq

�1��� �4.10�

is obtained by varying Eq. �3.16� over the hybridization po-tential Vq, whereas the sine coefficient

Wkk��2� = i�

q

gq2�Wkk�

uqVq

�1� −Wkk�

uq� Vq

�1��� + Wkk��2�,nonad

�4.11�

is obtained by varying the more general equation �3.15� overthe transformation parameter uq. After the variation, uk in theform Eq. �3.8� can be substituted. Then

Wkk��2�,nonad = − i�Vk�

�1��Vk�0� − Vk

�1�Vk��0��

�k��k��

R����

+Vk

�0�Vk��0��

2�k��k��

R2�����k�

Vk��1�Vk�

�0�� − Vk��0��Vk�

�0�

�k���k���

with

R� =R�

�1 + tan2 ��

.

The most divergent terms in Eqs. �4.10� and �4.11� appearwhen we vary only explicitly written Vk

�0� in Eq. �3.16� anduk� in Eq. �3.15�. As a result keeping only these most diverg-ing terms we have

Wkk��1� = �Vk

�0�Vk��1�� + Vk�

�0��Vk�1��K� 1

�k�

+1

�k���R����

�4.12�

and

Wkk��2� = − i�Vk

�0�Vk��1�� − Vk�

�0��Vk�1��K� 1

�k�2 +

1

�k��2 �R����

��

tan ��

.

�4.13�

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The corresponding corrections to the tunneling transpar-ency near the conductance band edge behave as

G↓�1��vg,�cv� �

e2

h

�l�r�1�

�2�2�R2��↓�

�Ed↓ − �c�2cos �t �4.14�

and

G↓�2��vg,�cv� � ���

e2

h

�l�r�2�

�2�2�R2��↓�

�Ed↓ − �c�5/2�

2��r

sin �t ,

�4.15�

where

�r�m� = 2�Sr�i�m−1�Vr

�1�Vr�0�� + �− i��m−1�Vr

�0�Vr�1���

is the adiabatic correction to the tunneling rate for m=1 andthe correction due to weak nonadiabatic effect for m=2.Equation �4.15� uses the fact that

�↓ →�

2, tan �↓ =�−

dL

d��� �r

�Ed↓ − �c�

for Ed↓→�c when �Ed↓−�c���r. We use here the approxi-mation Vk�

�m��Vr�m� similarly to the one used in Eq. �4.2�.

The term with the sine in Eq. �4.9� causes a phase shiftbetween the oscillations of the hybridization parameter andthe resulting current through the dot. If we neglect the kdependence of coefficients �4.10� and �4.11�, this phase shiftcan be readily found:

� = − arctan G↓�2��vg,�cv�

G↓�1��vg,�cv�

� .

It is expected to be generally rather small due to the factorgk�, which is usually very small unless the dot level ap-proaches the band edge �c. Close to the edge this factordiverges and results in an increasing phase shift,

� � − arctan��r

�2�

�r�1���r�Ed↓ − �c�

� − arctan��r

�2�

�r�r�1� .

As a result the phase shift � may become essential at Ed↓→�c.

The nonadiabatic corrections to tunneling conductance ac-quire the simple form of phase shift in oscillating cosinefunction only as long as the parameter fk=gk��� /� issmall and the perturbative approach is valid. With increasingfk one may expect appearance of higher harmonics n� inoscillating conductance. With an increase in the amplitudeV�1� the language of quasienergy levels32 is more appropriate.We plan to discuss this approach elsewhere.

V. CONCLUDING REMARKS

We have displayed in this paper an approach to the Ander-son model for a half-metallic electron liquid in a tunnel con-tact with a moving nanoshuttle under strong Coulomb block-ade. It is shown that in the situation where the spin-flipcotunneling processes are suppressed at low energies, theexact canonical transformation eliminating the tunneling

term in the Anderson Hamiltonian exists even in the presenceof strong Hubbard repulsion in the shuttle and time-dependent lead-shuttle tunneling. This canonical transforma-tion in principle allows one to sort out the slow adiabaticrenormalization of the energy levels and the tunnel transpar-ency and to consider nonadiabatic corrections at least pertur-batively. One may also include the inelastic spin-flip pro-cesses in the canonical transformation in the fourth order ofperturbation theory in Vi, but these weak corrections do notchange the above qualitative picture. We also have calculatedthe weak nonadiabatic corrections to tunneling transparencyin a specific model of periodic sinusoidal motion of theshuttle. Undoubtedly, strong nonadiabatic effects should betaken into account in a more refined scheme: in the case of aperiodic time-dependent perturbation one should appeal tothe Floquet theorem in the time domain and use the quasien-ergy language.32,33

Because of the energy gap for spin-flip processes in thehalf-metallic leads the Kondo cotunneling processes are sup-pressed, and the zero-bias anomaly in the tunnel conductanceis absent. Physical manifestations of the shuttling mechanismunder discussion arise in the form of time-dependent en-hancement of conductance at finite bias near the boundariesof Coulomb diamonds on the phase diagram G�vg ,e��. Theboundaries themselves are distorted due to the complete spinpolarization of carriers �see Fig. 3�, and even the Coulombblockade step corresponding to occupation change from oddto even number of electrons in the dot is absent at zero bias.One may expect appearance of quasienergy satellites in thispart of the phase diagram G�vg ,e�� when the shuttle motionis essentially nonadiabatic. This regime is a subject for futurestudies.

Experimental investigation of shuttling in a single elec-tron tunneling regime is a rapidly developing field. Recentprogress in revealing the features of Coulomb blockade inmechanical single electron transistors13,34 supports our hopethat the studies of adiabatic and nonadiabatic shuttling willobtain firm experimental substantiation in a near future.

APPENDIX A: TIME-DEPENDENT CANONICALTRANSFORMATION

To diagonalize the Schrödinger equation �3.23� for the

wave function =eS�t� , we conjecture the form

H = eSHe−S + S1

for the canonically transformed Hamiltonian. Then substitut-

ing H and into Eq. �3.23�, we obtain

i �

�teS� + ieS�

�t= �eSHe−S + S1�eS . �A1�

The time derivative of exponent is found by means of theoperator equation

d

dteS = �

0

1

e�SSe−��S−1�d� = eS��0

1

e−�SSe�Sd�� . �A2�

Then using Eqs. �A2� and �3.22� in Eq. �A1� yields

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i���=0

1

d� e�SSe−�S�eS + eSH = eSH + S1eS .

from where we straightforwardly get S1 and Eq. �3.24�.

APPENDIX B: TUNNELING AMPLITUDE FOR WEAKPERIODIC POTENTIAL

To derive the first nonvanishing time-dependent correc-tions to the tunneling procedure, we vary the terms in Eq.�3.28�. This variation procedure results in the two linearequations

uk�1��

Vk�0� − uk

�1�Vk�0��

− �gk�Vk�0��

vk�1� − vk

�1��

Vk�0��

= − gk�Vk�0��

Vk�1� − Vk

�0�Vk�1��

� ,

vk�1��

Vk�0� + Vk

�0��

vk�1� + �gk�Vk

�0��

uk�1� + uk

�1��

Vk�0�� = 0,

�B1�

where gk is explicitly real as defined in Eq. �3.8�. Spin index� is suppressed.

The first of these equations contains only explicitly imagi-nary terms whereas the second one contains only real terms.Therefore we have only two equation for four real variables�two complex variables�. It is readily verified that

uk�1� =

gk

1 − ��gk�2Vk�1�,

vk�1� = −

�gk2

1 − ��gk�2Vk�1� �B2�

is a particular solution of the set of equations �B1�. One canalso see that in the limit gk�→0 this solution reduces to theadiabatic approximation where we can take the results ob-tained in Sec. III A for the time-independent problem andsubstitute there the tunneling amplitude �4.7� slowly varyingin time.

The general solution of the corresponding homogenousset of equations reads

u� k�1� = −

1

2 �gk +

1

�gk��C1k + iC2k�

+ �gk −1

�gk� Vk

�0�

2Vk�0�� �C1k − iC2k� ,

v� k�1� = C1k + iC2k. �B3�

It should be added to the particular solution �B2�. The pa-rameters C1k and C2k are meanwhile arbitrary. These param-eters determined from the original cancellations�3.25�–�3.29� turn out to me small, C1 ,C2��gk��3, andmay result at most in nonadiabatic corrections ��gk��2.

In order to calculate the tunnel current we need the scat-tering matrix element

Wkk� = Wkk�ad + Wkk�

nonad, �B4�

where the first term Wkk�ad is obtained by substituting solutions

�4.8� and �B2� into Eq. �3.15� and keeping the terms linear ingk�. This contribution remains finite in the adiabatic limitgk�→0. However it contains also nonadiabatic correctionsdue to the third term in Eq. �4.8�. These corrections are de-termined by the equation

Wkk�nonad = i�uk�

� uk − ukuk�� �

1 − cos �

�2

+ iukuk�� �cos � − 1�2

2�4 �k�

�uk�uk�� − uk�

� uk�� ,

�B5�

which contains derivatives uk and, hence, is purely nonadia-batic, i.e., it disappears in the limit gk�→0. Keeping onlyterms linear in gk� in the scattering matrix element �B4�,we come to Eq. �4.9�.

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