PRL 95, 116806 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending9 SEPTEMBER 2005

Coulomb Promotion of Spin-Dependent Tunneling

L. Y. Gorelik,1 S. I. Kulinich,1,2 R. I. Shekhter,3 M. Jonson,3 and V. M. Vinokur4

1Department of Applied Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden2B. I. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkov, Ukraine

3Department of Physics, Goteborg University, SE-412 96 Goteborg, Sweden4Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 6043, USA

(Received 7 March 2005; published 9 September 2005)

0031-9007=

We study transport of spin-polarized electrons through a magnetic single-electron transistor (SET) inthe presence of an external magnetic field. Assuming the SET to have a nanometer size central island witha single-electron level we find that the interplay on the island between coherent spin-flip dynamics andCoulomb interactions can make the Coulomb correlations promote rather than suppress the currentthrough the device. We find the criteria for this new phenomenon—Coulomb promotion of spin-dependent tunneling—to occur.

DOI: 10.1103/PhysRevLett.95.116806 PACS numbers: 73.23.2b, 73.40.Gk

µ

µ

h

U

TL TR

L

R

FIG. 1. Sketch of the nanomagnetic SET device discussed inthe text: a quantum dot, modeled as a single spin-degenerateelectron level, is coupled to two leads with antiparallel magne-tization. TL;R are probability amplitudes for tunneling from thedot to the left and right leads, and U is the Coulomb interactionenergy of a doubly occupied dot. The potential difference �L ��R � eV between the leads is due to a bias voltage V. Anexternal magnetic field h induces flips between the spin-up andspin-down states (dotted arrows) on the dot.

Strong Coulomb correlations have important conse-quences for electronic transport on the nanometer lengthscale. Coulomb blockade (CB) of single-electron tunneling[1,2] is, e.g., the fundamental physical phenomenon behindsingle-electron transistor (SET) devices [3]. The electronspin comes into play if the source and drain electrodes inthe SET structure are made of magnetic material, allowingfor the electrons that carry the current to be spin polarized[4,5]. Almost fully spin-polarized materials are available[6]. Experiments have established that transport of spin-polarized electrons is sensitive to the relative orientation ofthe magnetization of the source and drain leads. This opensup the possibility for a spin-valve effect, where an externalmagnetic field controls the current. Switching the magne-tization in one of the leads is one way of achieving such acontrol [7]. Another approach [8]—to be pursued here—isto flip the spin of electrons carrying current from onemagnetic lead to another via a central nonmagnetic island(’’Coulomb dot’’), keeping the lead polarizations fixed.This approach brings into focus the role of the spin dy-namics in resonant electron tunneling, where electrons canbe trapped in a resonant state—e.g., on a Coulomb dot—for a long time. Since the electron level of the dot may bedoubly occupied by electrons with different spin, Coulombblockade of single-electron tunneling should have an im-portant influence on the spin-dependent resonant chargetransfer. Understanding how these two effects combine isof fundamental interest in the context of magneticnanostructures.

In this Letter we will consider electron transport throughthe simple magnetic SET device shown in Fig. 1. Here aCoulomb dot, subject to an external magnetic field, islocated between two spin-polarized leads. The externalmagnetic field is oriented perpendicular to the polariza-tions in the leads and is taken to be small enough not toaffect any lead magnetization. Assuming the dot to be ofnanometer size we consider only one electron energy levelon the dot. This level may, however, accommodate two

05=95(11)=116806(4)$23.00 11680

electrons of different spin and the Coulomb interactionbetween them significantly affects the charge transferthrough the device. The magnetic field in its turn, byinducing coherent spin-flip dynamics on the dot, actuallypromotes electron transport if the magnetization in theleads point in opposite directions. This effect is mostconspicuous when fully spin-polarized leads give rise toa ‘‘spin blockade’’ of tunneling. In this case an electron cantunnel from one lead to the other only if it spin flips while itresides on the dot. We have found that Coulomb correla-tions on the dot, if they are strong enough to prevent adouble occupancy of the resonant level, significantlystimulate such spin-flip processes and hence promotespin-dependent electronic tunneling.

To understand this phenomenon qualitatively let us firstconsider a simplified setup, where a Coulomb dot with asingle level of energy � is linked only to the left metalliclead with chemical potential �L. If �L � �� �L, where

6-1 © 2005 The American Physical Society

PRL 95, 116806 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending9 SEPTEMBER 2005

�L is the width of the dot level, and if the lead is fullypolarized (spin-up), then the spin-up state on the dot will befully occupied if there is no external field present; the spin-down state will be completely empty. If a perpendicularmagnetic field h� �L=� (� is the Bohr magneton) isswitched on, spin-flip processes will be induced. The char-acteristic time �sf for these to populate the spin-down stateon the dot is an important quantity. It turns out that �sf

strongly depends on whether or not Coulomb interactionsprevent a second electron from tunneling onto the dotduring the spin-flip process.

To see this, let us first consider the CB regime, where theenergy difference �L � � is smaller than the Coulombinteraction energy U=2 between two dot electrons. Thetunneling of a second electron onto the dot is then blockedand the population of the spin-down state is simply con-trolled by the coherent spin dynamics of the one electronalready there. The probability amplitude Asf for a spin-fliptransition increases linearly with time, Asf � t=�h (here�h � @=�h), and the probability �# to find the electron inthe spin-down state is

�#�t� � jAsfj2 � �t=�h�2: (1)

The spin-flip time �sf may be estimated from the condition�#��sf� � 1. So, in the CB regime �sf � �CB

sf ’ �h.If, on the other hand,�L � � > U, the CB is lifted and a

second (spin-up) electron can tunnel onto the dot if there isa finite probability for its spin-up state to be unoccupied.This process couples the electronic state on the dot to alarge number of states in the lead and breaks the coherenceof any ongoing evolution of the dot spin state after a time�t ’ @=�L. The probability for a spin flip to occur duringthis time is Psf��t� � jAsf�t � �t�j2 � ��t=�h�

2. For lon-ger times t the probabilities for a spin-flip to occur in t=�tcoherent time intervals add incoherently. Therefore, if theCB is lifted, the probability to find the dot electron in thespin-down state can be written as

�#�t� ’ �t=�t�Psf��t� � �t�t�=�2h; (2)

and in this ‘‘free’’ regime the spin-flip time �sf � �Fsf ’�CB

sf ��h=�t�. It follows that �CBsf � �Fsf in a weak magnetic

field �h� �L=��. Hence, if tunneling of a second electronis blocked, the probability for the spin of the electronalready on the dot to flip is strongly enhanced.

Next we extend our qualitative discussion to the tunnel-ing of electrons through the entire SET device by switchingon the coupling between the spin-down dot state and thestates in a fully polarized (spin-down) right lead held atchemical potential�R. If�R � �� �R, where �R=@ is thetunneling rate between dot and right lead, the spin-downdot electron can tunnel to an empty state in the right lead.The resulting current through the SET is given by theproduct of the tunneling rate and the probability to findthe dot electron in the spin-down state. If the spin-flip rateis much smaller than �R=@ the population of the spin-down

11680

state can be estimated as �#�t � @=�R�. Since electronexchange with the right lead also restricts the coherentspin-flip time one has to put �t � minf@=�L; @=�Rg inEq. (2). Therefore, in a strongly asymmetric situation,when the ratio �R=�L is very different from one, thecurrent through the SET is

I �e@

��h�2

�R

�1 CB regime�c�minf1;�R=�Lg free regime�f�:

(3)

This expression is valid only when the spin-flip process isthe bottleneck for charge transfer, i.e., when �h� �R�L�.One concludes that, when �R=�L � 1, the current in theCB regime is larger by a factor �L=�R than in the freeregime. We will refer to this as ‘‘Coulomb promotion’’ ofspin-dependent tunneling. From Eq. (3) it also follows thatthe I-V curve is strongly asymmetric in the CB regime; thecurrent changes by a factor �R=�L if the sign of the biasvoltage is reversed.

Above we used the stochastic picture for the exchange ofelectrons between the leads and the central dot [the scat-tering and master equation approaches give identical [9] ].The two-level spin system on the dot is, however, treatedquantum mechanically.

To provide a quantitative description of the Coulombpromotion phenomenon discussed above we consider aSET structure described by the Hamiltonian

H � H l � H d � H T;

H l �X�;�;�

"�;�;�ay�;�;�a�;�;�;

H d �X�

�ay�a� �U2ay" a

y# a"a#

��Xi;�;�0

hiay���;�

0

i a�0

H T �X�;�;�

T��ay�;�;�a� � H:c:�:

(4)

The first term describes noninteracting electrons in theleads and ay�;�;� �a�;�;�� is the creation (annihilation) op-erator for electrons in lead � with energy "�;�;� and spinprojection � � �"; #�. The electron density of states ��� ineach lead is assumed to be independent of energy butstrongly dependent on spin direction. The chemical poten-tials in he leads are �L;R � EF eV=2, where e is theelectronic charge, V > 0 is the bias voltage, and EF is theFermi energy of the ferromagnetic metal. The second termdescribes electronic states in the dot, their coupling to theexternal magnetic field ~h � �hx; 0; hz� and intradot elec-tron correlations characterized by the Coulomb energy U;the operator ay� �a�� creates (destroys) an electron withenergy � and spin �; ��;�

0

i�x;y;z are Pauli matrices. The lastterm represents spin-conserving tunneling of electrons be-tween dot and leads.

6-2

FIG. 2 (color online). Drop of the magnetic-field promotedcurrent �J, due to the lifting of the Coulomb blockade at V �U, as a function of the asymmetry parameter � for different leadpolarizations � (see text). The inset shows a domain in the �;�plane, where a negative differential conductance (NDC) may beobserved (dark). The drop in the total current associated with aNDC is of order Ik and peaks at � � 1, � ’ �1.

PRL 95, 116806 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending9 SEPTEMBER 2005

We focus on the Coulomb promotion phenomenondiscussed qualitatively above and assume the Coulombenergy U to be much larger than both the thermal en-ergy kT and the width and Zeeman splitting of the dotlevel. We furthermore let � � EF, and consider the currentfor bias voltages jVj � ��j ~hj;��� ; kT� and jV �Uj ���j ~hj;��� ; kT� (here ��� � 2���T2

� is the spin-dependentlevel width associated with tunneling to the lead �). If theexternal magnetic field is oriented along the polarizationaxis, charge transfer between the leads conserve spin pro-jection and may be described within the ‘‘orthodox’’ CBtheory [1]. A perpendicular magnetic field, on the otherhand, generates coherent spin dynamics and makes the spindegree of freedom relevant for the electron transport prob-lem. Intrinsic spin relaxation processes are slow on thescale of �h, which is about 0:1 �s in a field of 10 Oe, andwill be neglected [10].

The coupled processes of charge transfer and coherentspin dynamics are governed by a quantum master equationfor a reduced density operator ��t�, which can be derivedfrom the Liouville–von Neumann equation for the totalsystem by projecting out the degrees of freedom of theleads [13–16]. The operator ��t� acts on the Fock space ofthe quantum dot, which is spanned by the four basis vectorsj0i; j "i � ay" j0i, j #i � ay# j0i, and j2i � ay" a

y# j0i. In this

basis ��t� is a 4 4 matrix. The diagonal elements �0 �h0j��t�j0i and �2 � h2j��t�j2i represent the probabilitiesfor the dot to be unoccupied and doubly occupied, respec-tively. The singly occupied dot is described by the 2 2matrix block �1 � ����;�0 � h�j�j�

0i.For bias voltages satisfying the given conditions, the

time evolution of the probabilities �0, �2 and of the densitymatrix �1 is determined by the system of equations

_�0 � �Tr f�Lg�0� Tr f�R�1g

_�2 � �Tr f�R� �U� V��L�g�2� �V �U�Tr f~�L�1�g

_�1 � i��hi�i; �1� �1

2f�R; �1g �

1

2�V �U�f~�L; �1g

� �L�0� �~�R� �U� V�~�L��2; (5)

where �����"���#��I=2���"���#���z=2, ~�� � �x���x,

�i are the Pauli matrixes, and we have set e � @ � 1.The stationary solution of Eq. (5), and hence the sta-

tionary current, depends on the relation between the biasvoltage V and the CB energy U. In particular, the thirdequation tells us that the CB significantly decreases therelaxation of the nondiagonal elements of �1. As thesedescribe the coherent spin-flip dynamics, the spin-flip pro-cess turns out to be faster in the CB regime than in the freeregime without Coulomb blockade.

Within our approximations the current I through theSET can be calculated from the formula

I � Tr f�R��1 � �2I�g: (6)

11680

It is convenient to think of the current as the sum of abackground current I�0� � I�h � 0� and a magnetic-fieldpromoted (MFP) current J�h� � I � I�0�. Substituting thestationary solution of Eq. (5) into Eq. (6), one finds

Il � I�0�l � Jl�h� � I�0�l � Jlh2x=�h

2l � h

2x�; (7)

where the label l � c; f indicates whether we are in the CBregime �V <U� or in the free regime �V > U�. The quan-tities I�0�l , Jl, and hl do not depend on the transversemagnetic field and are given by

I�0�c ��L�R

�L��� Jc; Jc �

�L��#L�"R��"L�#R���

"R��#R�

��L�����"�# ��#L�"L�;

h2c ��

�2 �R��"�# ��#L�"L�

�L��

�1

4���hz�2

�2R

�;

I�0�f ��L�R

�� Jf; Jf �

��"L�#R��#L�"R�2

��"�#;

h2f ��

�2�"�#�

1

4���hz�2

�2

�: (8)

Here �� � ����� , �� � ����� , and � � ��;���� .Our results Eqs. (7) and (8) for the current are valid for

any values of ~h and ��� that meet the conditions formulatedabove. In this Letter we are particularly interested insituations where the Coulomb promotion phenomenoncan be observed. Therefore, we assume in the analysis tofollow that the leads are made from the same magneticmaterial and are polarized in opposite directions [17]. Thisimplies that �#L=�"L � �"R=�#R � �1� ��=�1� ��, where� �0; 1� is a polarization parameter. The limiting values� � 1 and � � 0 denote fully polarized and unpolarizedleads, respectively. Furthermore, we take the magneticfield to be directed perpendicular to the magnetization inthe leads.

6-3

FIG. 3. Current I through an asymmetric �� � 0:9� magneticSET structure as a function of transverse magnetic field h andbias voltage V. The magnetization of source and drain is anti-parallel; the polarization parameter � � 0:98. For fixed mag-netic field the current changes abruptly at the CB threshold,where V � U. The background current at h � 0 and the large-field current h� �=� increase as expected when the CB islifted. In contrast, in some interval of intermediate magneticfields (controlled by the polarization parameter �), the currentdrops when the CB is lifted. This is one signature of theCoulomb promotion phenomenon described in the text.

PRL 95, 116806 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending9 SEPTEMBER 2005

From Eq. (8) it is then clear that Jc=h2c > Jf=h2

f andhf > hc. Invoking in addition Eq. (7) we conclude that theMFP current is larger in the CB regime than in the ‘‘free’’regime if h < hc, where hc is of order 100 Oe for a 1 Gohmdevice and larger for less resistive devices. It follows thatfor h < hc the MFP current drops when the bias voltagelifts the CB. The magnitude of the drop �J�h� � Jc�h� �Jf�h� depends on the degree of spin polarization as well ason the magnetic field. It also depends on the asymmetryof the SET structure, characterized by the parameter� � ��L � �R�=�. The magnitude of �J � max�J�h�—plotted in Fig. 2, where Ik � �L�R=�, and accessible toexperiments—is substantial for a wide range of parameters� and �.

The inset of Fig. 2 shows that the effect of Coulombpromotion on the total current is most prominent forstrongly asymmetric SET structures with highly spin-polarized leads, i.e., when � ’ 1 and � ’ 1 as in Fig. 3.This is because with fully polarized leads, � � 1, thebackground current vanishes so that the MFP current givesthe total current. The Coulomb promotion then manifestsitself as a negative differential conductance at bias voltageV � U. It is interesting to note that if � � 1, � ’ �1, andh=hc � 1 (the case qualitatively discussed above) thenEqs. (7) and (8) reproduce Eq. (3) to leading order in smallparameters. If the leads are not completely spin polarizedthe CB of the background current competes with theCoulomb promotion of the MFP current. As a result thenegative differential conductance appears only in a smallregion where the parameter �> 0:9.

11680

In conclusion, we have studied resonant tunneling ofspin-polarized electrons through a magnetic SET devicewith a central island subject to an external magnetic fielddirected perpendicular to the magnetization in the leads.The combined effects of spin-dependent tunneling betweena source and a drain with antiparallel magnetizations andof Coulomb correlations were considered. We find that aCoulomb blockade preventing the single-electron level onthe central ‘‘Coulomb dot’’ to be doubly occupied maysignificantly stimulate the transport of electrons throughthe device. This effect gives rise to a new phenomenon—Coulomb promotion of spin-dependent tunneling.

This work was supported in part by the EuropeanCommission, Project No. FP6-IST-003673 CANEL. Theviews expressed by the authors are not necessarily those ofthe Commission. Financial support from the Swedish SSF,the Swedish VR, and from the U.S. DOE Office of Sciencethrough Contract No. W-31-109-ENG-38 is also gratefullyacknowledged.

6-4

[1] R. I. Shekhter, Zh. Eksp. Teor. Fiz. 63, 1400 (1972); I. O.Kulik and R. I. Shekhter, Zh. Eksp. Teor. Fiz. 68, 623(1975).

[2] C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991); D. V.Averin, A. N. Korotkov, and K. K. Likharev, Phys. Rev. B44, 6199 (1991).

[3] D. V. Averin and K. K. Likharev, in MesoscopicPhenomena in Solids, edited by B. L. Altshuler, P. A.Lee, and R. A. Webb (Elsevier, New York, 1991), p. 173.

[4] I. Zutic et al., Rev. Mod. Phys. 76, 323 (2004).[5] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.

Halperin, cond-mat/0409242 [Rev. Mod. Phys. (to bepublished)].

[6] Collosal Magnetoresistive Oxides, edited by Y. Tokura(Gordon and Breach, New York, 2000).

[7] J. Z. Sun et al., Appl. Phys. Lett. 69, 3266 (1996).[8] L. Y. Gorelik et al., Phys. Rev. B 71, 035327 (2005).[9] D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. B 67,

085316 (2003).[10] The electron spin relaxation time in C�60 exceeds 1 �s

[11]; in a semiconductor quantum dot it exceeds 50 �s[12].

[11] V. N. Popok et al., Solid State Commun. 133, 499 (2005).[12] R. Hanson et al., Phys. Rev. Lett. 91, 196802 (2003).[13] S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932

(1996).[14] Yu. V. Nazarov, Physica B (Amsterdam) 189, 57 (1993).[15] Bing Dong, H. L. Cui, and X. L. Lei, Phys. Rev. B 69,

035324 (2004).[16] S. Braig and P. W. Brouwer, Phys. Rev. B 71, 195324

(2005).[17] For identical leads with parallel (or no) polarization, spin

flips do not affect currents since J�h� � 0. This contradictsRef. [15], where the current due to spin-independenttunneling was found to depend on the spin-flip rate.