Exchange effects on electron transport through single-electron spin-valve transistors

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Exchange effects on electron transport through single-electron spin-valve transistors

Wouter Wetzels and Gerrit E. W. Bauer

Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

Milena Grifoni

Institut für Theoretische Physik, Universität Regensburg, D-93035 Regensburg, Germany

共Received 9 August 2006; published 11 December 2006兲

We study electron transport through single-electron spin-valve transistors in the presence of nonlocal ex-change between the ferromagnetic leads and the central normal-metal island. The Coulomb interaction is described with the “orthodox model” for Coulomb blockade and we allow for noncollinear lead magnetization directions. Two distinct exchange mechanisms that have been discussed in the literature are shown to be of comparable strength and are taken into account on equal footing. We present results for the linear conductance as a function of gate voltage and magnetic configuration, and discuss the response of the system to applied magnetic fields.

DOI:10.1103/PhysRevB.74.224406 PACS number共s兲: 85.75.⫺d, 72.25.Mk, 73.23.Hk

I. INTRODUCTION

Downscaling magnetoelectronic devices to the nanometer regime implies that electron-electron interaction effects be-come prominent, as has been amply demonstrated by many experimental studies on the Coulomb blockade in double tunnel junctions with ferromagnetic elements. Measurements were done on systems with nonmagnetic islands contacted to ferromagnetic leads1–6 _{as well as for all-ferromagnetic} systems.7–9Much of the theoretical work focuses on F兩N兩F spin valves, in which the island is a normal metal共N兲 and the contacts are ferromagnets 共F兲 with variable magnetization directions. Initially, the interest was mainly focused on the giant magnetoresistance, i.e., the difference in the transport properties for parallel or antiparallel magnetizations.10–14 More recently, the interplay between spin and interaction ef-fects for noncollinear magnetization configurations has at-tracted quite some interest. Besides spin-valve systems with quantum-dot islands,15–21 _{also islands consisting of metallic} clusters,22 _{Luttinger liquids,}23,24 _{and nanomagnets}25 _were studied.

A single-electron spin-valve transistor 共SV-SET兲 is an

F兩N兩F spin valve with a sufficiently small normal-metal 共N兲

island that is coupled to the ferromagnetic leads by tunnel barriers. When the electrostatic charging energy of the island is larger than the thermal energy, charge transport can be controlled at the level of single-electron charges by varying bias and gate voltage, as is well known for nonmagnetic SET’s.26 _{With spin-dependent electron tunneling rates and} sufficiently long spin-decay lifetimes, a spin accumulation 共or nonequilibrium magnetization兲 that strongly affects elec-tron transport may build up in the nonmagnetic island.

In this paper, we discuss the transport characteristics of metallic SV-SETs in the Coulomb blockade regime, allowing for arbitrary, noncollinear magnetization directions. We ob-tain results to leading order in the transmission matrix ele-ments, and therefore higher-order cotunneling processes are not taken into account. In particular, we examine the influ-ence of exchange effects through F兩N tunnel contacts on the spin accumulation in the center island, presenting a more

complete discussion compared to that in Ref.22. We argue that two separate exchange effects have to be taken into ac-count. On one hand, there is the nonlocal interface exchange, let us call it “X1” in the following. In scattering theory for noninteracting systems it is described by the imaginary part of the spin-mixing conductance,27 _{while in the context of} current-induced magnetization dynamics X1 acts as an “ef-fective field.”28 _{Such an effective field has been found} ex-perimentally to strongly affect the transport dynamics in spin valves with MgO tunnel junctions.29_{This effect has recently} also been involved to explain magnetoresistance effects in carbon nanotube spin valves5 _{and called spdependent} in-terface phase shifts.30 _{The second exchange term} _{共“X2”兲 is} an interaction-dependent exchange effect due to virtual tun-neling processes that is absent in noninteracting systems. It has been considered for islands in the electric quantum limit, in which transport is carried by a single quantized level only.15 _{The X2 effect is potentially attractive for quantum} information processing, since it allows to switch on and off effective magnetic fields in arbitrary directions just by a gate electric potential. We compute here X2 for a metallic island in which size quantization is not important. We find that both exchange effects are of comparable magnitude and affect the transport properties in a characteristic way, but can be sepa-rated in principle by employing the gate dependence of X2. The paper is organized as follows. In Sec. II we introduce the model system for the SV-SET. In Sec. III the two relevant types of exchange processes are discussed. Charge and spin transfer rates are determined in Sec. IV. Finally, we present results for the transport characteristics as a function of mag-netic configuration, gate voltage, and applied magmag-netic field in Sec. V.

II. MODEL SYSTEM

An SV-SET关see Fig.1共a兲兴 is composed of a small metal-lic cluster in contact with two large ferromagnetic electron reservoirs in thermal equilibrium characterized by magneti-zation directions mជ₁ and mជ₂with mជ_␣=共sin␪_␣, 0 , cos␪_␣兲共for

␣= 1 , 2兲, where␪₁=␪/ 2 and␪₂= −␪/ 2.

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The F_␣兩N contacts are tunneling barriers with conduc-tances that depend on the electron spin, G_␣↑↑for the majority and G_␣↓↓for the minority spin in the ferromagnet. The total conductance for contact␣ is then given by G_␣⬅共G_␣↑↑+ G_␣↓↓兲 and the contact polarization is defined as P_␣⬅共G_␣↑↑ − G_␣↓↓兲/共G_␣↑↑+ G_␣↓↓兲. The resistances R_␣= 1 / G_␣are taken to be much larger than the resistance quantum RQ= h / e2, which, at low enough temperatures and bias voltages, allows us to study the blockade of transport by the Coulomb interaction. The electron tunneling rates are governed by the change of electrostatic energy of the whole circuit upon transfer of an electron. The capacitances of the junctions C_␣determine the charging energy of the island.

. 共4兲 Then, for each contact, a tunneling Hamiltonian

HT␣=

兺

kqs

Tkqs␣ a␣ks †

c_␣qs+ H.c. 共5兲

describes the coupling to the island. The tunneling coeffi-cients are assumed to not significantly depend on energy on the scale of the charging energy. We discuss the exchange contribution, represented by the Hamiltonian H_ex␣, in the next section.

III. EXCHANGE EFFECTS

Here we discuss two different exchange effects that affect the electrons in the normal-metal island attached to magnetic contacts. These two flavors arise when the transport proper-ties for an SV-SET are determined to lowest order in the tunneling probabilities.

A. Nonlocal interface exchange (X1)

The nonlocal exchange coupling between ferromagnetic films through a normal-metal spacer is an important effect that determines the ground state of magnetic multilayers共see Ref.34for a recent review兲. Electrons in a normal metal that

are reflected at a contact to a ferromagnet pick up a phase depending on the electron spin relative to the magnetization direction. In sufficiently clean and narrow F兩N兩F structures, quantum well states are formed in N whose energy depends on the magnetic configuration through the spin-dependent phase. By a rotation of the magnetization directions the en-ergy spectrum and Fermi enen-ergy varies, causing the ground-state energy to depend on the relative angle ␪. In metallic FIG. 1. 共a兲 The spin-valve single-electron transistor: A small

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multilayers with a suitable spacer thickness, this can lead to an antiparallel ground state, which displays the celebrated giant magnetoresistance when the magnetizations are forced into a parallel direction by a magnetic field. Even when the ground-state energies are not significantly affected by the exchange coupling, configuration-dependent quantized states can still be observed in transport. This has been shown for high-quality planar tunnel junctions35_{as well as spin valves} in which the node is formed by single carbon nanotubes with a quantized energy spectrum.5,36 _{In Ref.}₃₀_{the effect of} in-terfacial phase shifts on the magnetoresistance of ballistic quantum wires between ferromagnetic leads was calculated. The spin-dependent phase shifts give rise to a slightly differ-ent quantization condition, which can spin-split the energy levels. Since we are here interested in classical islands with a continuous electron spectrum, we calculate energy shifts for a semiclassical island using the Bohr-Sommerfeld quantiza-tion rule in Appendix A.

Here we consider the limit of tunnel junctions between a normal-metallic island and ferromagnetic electrodes. The torques on the ferromagnets are then very small. The ex-change coupling does not significantly disturb the ferromag-nets in this limit, but persists to affect transport. The present study focusses on the charge-transport properties in the limit of small tunneling matrix elements, thus from the outset ex-cluding resonant tunneling, co-tunneling, or Kondo-type physics. The states on the island may be size quantized, i.e., the energy-level spacing exceeds the thermal energy 共“quan-tum dot”兲, or, in the opposite limit, better described by a semicontinuous density of states共“classical dot”兲. Here we concentrate on the latter, i.e., semiclassical, diffuse, or cha-otic islands, for which it can be shown quite generally that equilibrium spin currents are suppressed.37 _{The state of the} island is then characterized by a semiclassical charge and spin distribution function that has to be determined self-consistently as a function of the junction parameters and the applied voltages. For noninteracting systems, the spin and charge currents through an F兩N interface are determined not only by the conventional conductances G_␣↑↑ and G_␣↓↓ intro-duced above, but also by the complex spin-mixing conduc-tances G_␣↑↓,27 _{which are discussed in Sec. IV. The real part} Re G_␣↑↓is the material parameter that is proportional to the spin-transfer torque.38,39 _{The imaginary part Im G}

␣

↑↓ _reflects the spin-dependent interface phase shifts and affects the magnetization and spin accumulation dynamics as an effec-tive exchange magnetic field parallel to the magnetization direction.28,38,40 _{Im G}

␣

↑↓ _{is relatively small for intermetallic} interfaces,41but is in general comparable in magnitude to the other conductance parameters.40,45_{The nonlocal interface} ex-change has been discussed in similar terms for spin valves consisting of Luttinger liquids with ferromagnetic contacts.23 The blocking of transport by the Coulomb charging is usually described by Fermi’s Golden Rule 共see below兲, which employs a probability共squared matrix elements兲 and energy conservation. As long as the charging energy is much smaller than atomic energy scales 共like the Fermi energy兲, the junction parameters such as the interface transparency and spin-mixing conductance are unaffected and the Cou-lomb blockade is governed by the energy conservation crite-rion only. This implies that the exchange effect can be de-scribed by the Im G_␣↑↓of the bare junction.

It remains to parametrize the exchange in the limit of the tunneling Hamiltonian, i.e., to lowest order in the interface transmission. We show below that this is achieved by adding the following exchange term H_ex␣to the Hamiltonian for the two leads:

H_ex␣=

兺

ks

⌬⑀␣ksc␣ks† c␣ks. 共6兲 The energy shifts⌬⑀_␣ks, see Eq.共A2兲, are proportional to the

inverse density of states, but they remain relevant for small level splitting because the dwell time is inversely propor-tional to the average energy-level separation or inverse den-sity of states ␦=␳N

−1

. This Hamiltonian is an effective Zee-man splitting caused by an exchange magnetic field in the direction of the magnetization, see Sec. IV B.

B. Virtual tunneling processes (X2)

The interface exchange term X1 is a property of the sepa-rate interfaces and they contribute independently. The second type of exchange 共X2兲 felt by the spins on the island is a property of the entire device. It originates from virtual tun-neling processes, corresponding to single-electron transfer from and to the cluster. In the tunneling regime, this process can be treated and understood in terms of perturbation theory. In the absence of tunneling, the number of electrons on the island is a good quantum number. The perturbation by the contact to the electrodes allows mixing in of states in which the number of electrons on the island is changed by unity, at the cost of the charging energy. In second-order perturbation theory this results in an energy gain represented by a sum over共virtually兲 excited states in which the Cou-lomb energy appears in the denominator and the tunneling probability in the numerator. When the leads are nonmag-netic, these virtual processes correspond to a quantum cor-rection to the average charge on the central electrode.42,43 This effect depends strongly on the applied gate voltage. When the unperturbed N + 1共N−1兲 particle ground state is tuned in energy just above the N particle state, the quantum correction will be large and positive共negative兲. At the de-generacy point, perturbation theory breaks down, but the en-suing divergence can be controlled by taking into account finite temperatures.

When the tunneling probabilities to the ferromagnetic contacts are spin dependent, the deviations from the exact quantized charge on the island become spin dependent, and therefore lead to a net excess of spins in the ground state that depends on the configuration of the contact magnetizations. For a symmetric spin valve it is easy to see that the island ground-state magnetization due to these virtual processes X2 is maximal for parallel magnetizations and vanishes for an-tiparallel ones.

The additional exchange affects nonequilibrium electron transport, in contrast to higher-order so-called co-tunneling processes, to the same order as the in- and out-tunneling processes. For a quantum-dot island with a single quantized level, König and Martinek15_{showed that in the case of}

non-collinear magnetizations the nonequilibrium spins on an

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precessing around the effective exchange field. This effect was also discussed for few-level quantum dots.20 _{Since X1} discussed in Sec. III A is a material constant, the gate voltage dependence of X2 provides a handle for an experimental discrimination of the two effects. We derive an expression for the effective X2 exchange field for a classical SV-SET in Sec. IV B.

IV. CHARGE AND SPIN TRANSPORT

We compute the transport characteristics of the SV-SET in lowest-order perturbation theory44 _{for a diffusive or chaotic} island in the sequential tunneling regime. The rate equations lead to a probability distribution for the excess number of charges nN. The excess spin accumulation sជcontains a large number of spins and we are interested in its average value in the steady state that is found from the condition具dsជ/ dt典=0.

A. Charge transfer

The operators for the excess number of electrons on the island and on the two leads are nN=兺kscks†cks and nF␣ =兺ksa_␣ks† a␣ks, respectively. The unpaired spin angular mo-mentum on the cluster is written as sជ=共ប/2兲兺kss⬘cks†␴ជss⬘cks⬘, where␴ជ=共␴x,␴y,␴z兲 is the vector of Pauli spin matrices. It is convenient to introduce a vector chemical potential⌬ជ␮in the island, with size 兩⌬␮ជ兩=2兩具sជ典兩/共␳Nប兲共see also Ref. 13兲, where␳Nis the density of states at the Fermi energy. We can take into account Stoner enhancement intra-island exchange effects in terms of the static susceptibility␹s, and we may also write⌬␮= 2␮B2兩具sជ典兩/共␹sប兲. We denote the unit vector in the direction of the spin accumulation by sˆ.

The charge current is equal to the expectation values for the rate of change of nN. In terms of the tunneling Hamil-tonian HT= HT1+ HT2the time evolution is given by

冔

_o,

where 具¯典o denotes an expectation value with respect to Hamiltonian H

⬘

. The electrochemical potentials of the two reservoirs are␮cF1= eV / 2 and␮cF2= −eV / 2. It is convenient to introduce grand canonical Hamiltonians including the chemical potentials as23,44

KN= HN−ប−1⌬␮ជ· sជ, 共8兲

KF_␣= HF_␣−␮cF_␣nF_␣. 共9兲 The time dependence cks共t兲=e共i/ប兲KNtckse−共i/ប兲KNt can be for-mulated in terms of the projection operators

uˆ↑共sˆ兲 =1

2共I + sˆ ·␴ជ兲, 共10兲

uˆ↓共sˆ兲 =1

2共I − sˆ ·␴ជ兲, 共11兲 where I is the unit matrix, by making use of the equality

e共i/ប2兲⌬␮ជ·sជtcps⬘e−共i/ប

2_{兲⌬␮ជ·sជt}

=

兺

s⬙

关e−共i/ប兲共⌬␮/2兲t_uˆ↑_共sˆ兲 + e共i/ប兲共⌬␮/2兲tuˆ↓共sˆ兲兴s⬘s⬙cps⬙.

共12兲 The leads and the island are supposed to be in thermal equilibrium, so that 具c_ks†_⬘ck⬘s⬙典o= f共⑀ks⬘兲␦kk⬘␦s⬘s⬙, with Fermi-Dirac distribution f共⑀兲⬅共1+e␤⑀_兲−1_{, where} _␤ _{is the inverse} temperature. Using the expression for the matrix elements

关U共␪␣兲us⬙共sˆ兲U共␪␣兲†兴s⬘s⬘= 1

2共1 + s

⬘

s

⬙

⬙

⌬␮ 2

冊

册

, 共14兲 where F共⑀兲⬅⑀共1−e−␤⑀_兲−1 _{and E}

m⬅e2共m−CGVG/ e兲2/ 2C. The relation between the up and down spin conductances共

G_␣↑↑ and G_␣↓↓兲 and the tunneling coefficients is G_␣ss =共␲e2_/_ប兲_␳

N␳F␣s兩Ts␣兩2, where兩Ts␣兩2is the value of兩Tkqs␣ 兩2at the Fermi energy averaged over all the modes.␳F␣sis the spin-dependent density of states in ferromagnet␣.

In the low-bias regime considered here we can linearize Eq. 共14兲 in ⌬␮ and ␮cF␣. The resulting expression for the rate for electron tunneling through contact␣, increasing the excess number of electrons nNfrom “0” to “1,” is denoted by ⌫_␣0→1_{. The analogous rate for removing one electron when n}

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Ⰶe2_{/ 2C, and restrict ourselves to a gate voltage range for} which the excess number of electrons nN alternates between “0” and “1” 共0⬍CGVG⬍e兲, knowing that the results will periodically repeat with this period. The center of the Cou-lomb oscillation for transitions between nN= “ 0” and “1” electrons is at CGVG= e / 2.

The steady state on the island is characterized by a con-stant spin accumulation 共to be determined below兲 and the probabilities p0and p1that there are “0” or “1” excess elec-trons. We have p0+ p1= 1. The rate equation for the probabili-ties is

dpn/dt = − pn共⌫n→n+1+⌫n→n−1兲 + pn+1⌫n+1→n+ pn−1⌫n−1→n. 共17兲 From the condition of detailed balance, p₀⌫0→1_{= p}

1⌫1→0, we find p₀共V,VG,⌬ជ␮兲 = f共E0− E1兲 + ␤f共E₀− E₁兲f共E₁− E₀兲 G₁+ G₂ ⫻

兺

␣

冉

G_␣␮cF␣− P␣G␣ ⌬␮ 2 sˆ · mជ␣

冊

. 共18兲 The expression for the conductance of the SV-SET as a func-tion of the spin accumulafunc-tion can now be calculated and reads G共V,VG,⌬ជ␮兲 = − ep0⌫1 0→1_{+ ep} 1⌫1 1→0 = G1G2 G1+ G2 ␤共E0− E1兲 2 sinh␤共E0− E1兲 ⫻

冉

1 − ⌬␮ 2eVsˆ ·共P1mជ1− P2mជ2兲

冊

. 共19兲 B. Spin accumulation

dsជ dt=

i

ប关HT,sជ兴, 共21兲

with an expectation value that to second order in HTreads

冓

dsជ共t兲 dt

冔

= i ប

冕

−⬁ t dt

⬘

冓

冋

dsជ共t兲 dt ,HT共t

⬘

兲

册

冔

.

The spin current共rate of change of the spin angular momen-tum兲 due to tunneling when m excess electrons are on the island reads关cf. Eq. 共14兲兴

冓

dsជ dt

冔

n_N=m = ប 4e2

兺

␣s_⬙ 共G␣s

⬙

sˆ + P␣G␣mជ␣兲

冋

− F

冉

− Em−1+ Em−␮cF␣+ s

⬙

⌬␮ 2

冊

+ F

冉

− Em+1+ Em+␮cF␣− s

⬙

⌬␮ 2

冊

册

+ ប 4␲e2

兺

␣s_⬙ P_␣G_␣s

⬙

共mជ_␣⫻ sˆ兲

冉

冕

d⑀1

冕

⬘

d⑀2 f共⑀₁兲关1 − f共⑀₂兲兴关⑀2−⑀1+ Em−1− Em+␮cF␣− s

⬙

⌬␮/2兴 −

冕

d⑀1

冕

m

共⌬⑀␣m↑−⌬⑀␣m↓兲具sជ典 ⫻ mជ␣, 共25兲 where M is the number of transport channels in the normal metal and the energy shifts⌬⑀are found in Eq.共A2兲.

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G_␣ss⬘⬅ e 2 h

兺

nm 关␦nm− rs␣ nm_共r s⬘␣ nm_兲*_{兴, 共s,s}

_⬘

_{苸 ↑,↓兲. 共26兲} Here n and m denote the transport channels in the normal metal and r_↑nm_␣ and r_↓nm_␣ are the corresponding spin-dependent reflection coefficients. The contact conductances for spin-up and spin-down electrons are G_␣↑↑ and G_␣↓↓ and the mixing conductance G_␣↑↓governs the transverse spin currents that are absorbed and reflected by the ferromagnet ␣. The current polarized normal to the magnetization but in the plane of sជ and mជ_␣ is proportional to Re G_␣↑↓ and describes the spin transfer to the magnet, thereby dissipating the spin accumu-lation. In the case of tunnel junctions Re G_␣↑↓→G_␣/ 2. The out-of-the sជ, mជ_␣ plane spin current is caused by reflection processes that make spins precess around mជ_␣and is propor-tional to Im G↑↓. This mixing conductance has been evalu-ated from first principles for various contact materials and is small for intermetallic interfaces because positive and nega-tive contributions in the space spanned by the transport chan-nels average out.41 _{However, there is no general reason that} Im G↑↓should be smaller than G or Re G↑↓. It is known to be quite large for the Fe兩InAs interface45 _{and found to be very} significant for the magnetization dynamics of MgO magnetic tunnel junctions.29 _{For a simple model barrier discussed in} Appendix B, we find the value Im G↑↓/ G = −0.26. Using the relation between the reflection phases and the energy shifts as derived in Eq.共A2兲, we can rewrite the contribution given

in Eq. 共25兲 in terms of the imaginary part of the mixing

conductance as共cf. Ref.22兲

冓

dsជ dt

= ប 2e2 ␤共E0− E1兲 2 sinh␤共E0− E1兲

再

G1G2 G1+ G2 eV共P1mជ1 − P2mជ2兲 − 共G1+ G2兲 ⌬␮ 2 关sˆ + 共sˆ · bជ兲bជ兴

冎

+g␮B ប Bជeff⫻ 具sជ典 − 具sជ典 ␶sf = 0, 共30兲 where bជ⬅ P1G1 G1+ G2 mជ1+ P2G2 G1+ G2 mជ2. 共31兲

The total effective magnetic field Bជeffconsists of the external magnetic field and contributions from the exchange effects X1 and X2, and reads

Bជeff共VG兲 = Bជ+ BជX1+ BជX2共VG兲, 共32兲 with BជX1= ប ␳Ng␮Be2

兺

␣ Im G_␣↑↓mជ_␣, 共33兲 Bជ_X2共VG兲 = − ប 2␳Ng␮Be2 共G1+ G2兲bជ

冋

1 ␲f共E0− E1兲 ⫻

冕

d⑀f

⬘

共⑀兲␩

冉

⑀+ E0− E−1, e2 C

冊

+ 1 ␲f共E1− E0兲 ⫻

冕

d⑀f

⬘

共⑀兲␩

冉

⑀+ E1− E0, e2 C

冊

册

. 共34兲 Here we introduced15 ␩共␧,U兲 ⬅

冕

⬘

d␼

冉

1 − f共␼兲 ␼ − ␧ + f共␼兲 ␼ − ␧ − U

冊

= − Re

冋

⌿

冉

1 2+ i␤␧ 2␲

冊

−⌿

冉

1 2+ i␤共␧ + U兲 2␲

冊

册

, 共35兲 where ⌿共z兲 is the Digamma function. In Appendix C we discuss the derivation of the expression for Bជ_X2in more de-tail and comment on the differences compared to the case of a single-level quantum dot.

V. RESULTS AND DISCUSSION

Figure2shows the magnitude of the total effective mag-netic field Bជeffas a function of gate voltage共solid line兲 for a FIG. 2. The effective magnetic-field strength兩Bជ_eff兩 as a function of gate voltage共solid line兲 for a spin valve in the parallel configu-ration. The parameters are G₁= G₂, P₁= P₂= 0.7, Im G₁↑↓= Im G₂↑↓ = G1/ 4, and e2/共2C兲=10kBT. The imaginary part of the mixing

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symmetric spin valve with parallel magnetizations 共it van-ishes for the antiparallel configuration兲 and a polarization

P1= 0.7. The X1 term is a constant that does not depend on gate voltage共dotted line兲. BជX2vanishes when CGVGequals 0,

e / 2, and e. At these points, contributions from incoming and

outgoing electrons cancel each other共see Appendix C兲. The curve repeats as a function of gate voltage with period e / CG. The spin accumulation on the island found from Eq. 共30兲

tends to suppress the current through the system. Spin-flip and exchange effects that dissipate or dephase the spin accu-mulation therefore increase the conductance. As a reference we list here the conductance G共␪兲 for a spin valve without

interaction, with equal conductance parameters for the left

and the right tunneling barrier G1= G2, P1= P2:

G共␪兲 =G1 2

冉

1 − P₁2G1共G1+ 2Gsf兲sin2␪/2 G₁+ 2Gsf+关2 cos共␪/2兲Im G1↑↓兴2

冊

. 共36兲 The final result for the symmetric spin valve with interaction can be obtained simply from this expression by the substitu-tions G₁→ ␤共E0− E1兲 2e sinh␤共E0− E1兲 G₁, 共37兲 Im G₁↑↓→e 2 ប ␳Ng␮BBeff 2 cos共␪/2兲. 共38兲

For nonmagnetic contacts共P1= 0兲 this result reduces to the known expression for normal-metal single-electron transistors.46

As shown in Fig.3共a兲, changing the relative strengths of X1 and X2, or, since the X2 contribution is proportional to the polarization of the leads, Im G_␣↑↓/ P_␣G_␣, qualitatively modifies the current profile of the Coulomb oscillations. The constant offset given by BX1skews the exchange field around

CGVG= e / 2, causing asymmetric conductance curves. When the offset starts to dominate the symmetry gets restored. The X2 contribution vanishes when the Coulomb blockade is lifted共CGVG= e / 2兲, so the angular dependence of the con-ductance for different values of Im G₁↑↓/ G₁ in Fig. 3共b兲 re-flects only the X1 effect. The curve is a simple cosine for Im G₁↑↓= 0, but is sharpened for larger Im G₁↑↓because of the dephasing of the spin accumulation occurring for noncol-linear angles. In Fig. 3共c兲 Im G₁↑↓/ G₁ is fixed to 0.25 and curves are plotted for different values of the gate voltage. It can be seen that the angular dependence differs because the X2 depends on VG in an asymmetric way around

CGVG= e / 2.

As can be seen in Fig.4, the shape of the Coulomb oscil-lation can develop minima when the polarization is high and the magnetizations are nearly antiparallel. At the values of gate voltage where the X1 and X2 exchange effects cancel, the spin accumulation is not dephased and the conductance is suppressed.

Figure5 shows results for the conductance and spin ac-cumulation as a function of applied magnetic field in the x 共solid line兲, y 共dashed兲, and z 共dotted兲 directions. The spin

valve is again symmetric with P1= 0.7 and Im G1↑↓= G1/ 4. The angle ␪ is fixed to ␲/ 2 and CGVG= e / 2. The conduc-tance then depends only on the x component of the spin accumulation关see Eq. 共19兲兴. Without applied magnetic field,

the spin accumulation has components in the x and y direc-tions, while Bជeff is in the y direction. The results can be understood in terms of the dephasing of the spin accumula-FIG. 3. 共a兲 Coulomb oscillations at fixed angle ␪=␲/2 for a symmetric SV-SET with ratio Im G₁↑↓/ G₁= 0共solid兲, 0.25 共dashed兲, and 1共dotted兲 in units of G1. The polarization P is 0.7 and Gsf= 0.

共b兲 Conductance as a function of the angle for the same parameters as in共a兲, with CGVGfixed to 0.5.共c兲 Conductance as a function of

␪ with Im G1↑↓/ G1= 0.25. Results are shown for CGVGequal to 0.45

共dashed兲, 0.5 共solid兲, and 0.55 共dotted兲.

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tion by the magnetic-field induced precession that, for suffi-ciently large and noncollinear magnetic fields, quenches the spin accumulation. This “Hanle” effect is responsible for the conductance minimum at negative applied magnetic field in the y direction. In Fig.5共c兲only two curves are visible be-cause the curves for magnetic fields in the x and y directions overlap.

VI. SUMMARY

We studied the transport properties of single-electron spin valve transistors as a function of the magnetization configu-rations in the orthodox model of the Coulomb blockade. Two types of exchange effects between the spin accumulation on the island and the lead magnetizations play a role: a nonlocal interface exchange effect 共X1兲 and exchange due to virtual tunneling processes共X2兲. For metallic dots these two effects are found to be of comparable magnitude. We predict that a line-shape analysis of the Coulomb oscillation peaks should help to experimentally disentangle the two contributions. Ad-ditional information can be obtained by the Hanle effect.

ACKNOWLEDGMENTS

We would like to thank J. König, J. Martinek, M. Braun, and Y. V. Nazarov for useful discussions. This research was supported by the NWO and by the DFG via the SFB 689, and supported in part by the National Science Foundation under Grant No. PHY99-07949.

APPENDIX A: ENERGY SHIFTS

Let us consider a normal-metal island in contact to a fer-romagnet by a tunnel barrier 共see Fig. 6兲 without Coulomb

interaction. We wish to calculate the spin-dependent shifts of the energy levels due to the presence of the F兩N contact. In Ref. 30 an analogous calculation was done for a ballistic one-dimensional quantum wire. Here we consider an island in the quasiclassical regime, i.e., the de Broglie wavelength is much smaller than the size of the island.

The Bohr-Sommerfeld quantization rule47 1 ប

冖

p m_{共x兲dx +}_␾ 0 m +␾s m = 2␲

冉

n +1 2

冊

共A1兲

can be used to find the energy shifts, where pm共x兲 is the classical momentum for an electron in mode m, and n is an integer. The integral is over a whole period of the classical motion in the quasiclassical region. The total phase shift due to the reflections at the turning points is␾₀m+␾s

m

, where␾₀mis the spin-independent phase shift picked up during the reflec-tions from the boundaries for an isolated island without con-tact to the ferromagnet. The small spin-dependent phase shift

␾s

m_{Ⰶ1 arises from the weak coupling to the ferromagnet.} The phase shifts have to be computed quantum mechanically via the spin-dependent reflection coefficients rs

mm

for mode m at an interface that is assumed to be specular共see also Ap-pendix B兲.

From Eq.共A1兲, we see that increasing the quantum

num-ber n by one corresponds to introducing an extra phase pe-riod that increases the kinetic energy by M /␳N, where ␳Nis the density of states of the island and M is the number of modes. The energy shift for an electron in mode m is there-fore, to linear order in␾s

m , ⌬⑀ms= − M ␳N ␾s m 2␲. 共A2兲

The effect of the interface on the island states can be taken into account by introducing an effective Hamiltonian as in FIG. 5. 共a兲 Conductance as a function of a magnetic field

ap-plied along the x共solid兲, y 共dashed兲, or z 共dotted兲 direction in units of G1. The SV-SET has symmetric junction parameters, with

polar-izations P1= 0.7, Im G1↑↓= G1/ 4, and Gsf= 0. The magnetizations are

fixed to mជ1/2=共±1,0,1兲/

冑

2, yielding an angle ␪=␲/2 and CGVG

= e / 2.共b兲–共d兲 The x, y, and z components of the spin accumulation for the same parameters. The curves in共c兲 for magnetic fields in the

x and y directions overlap.

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Eq.共6兲. In the case of a spin-independent tunneling barrier to

a ferromagnet, the spin splitting of the energy levels is small, of the same order as the transmission probability共see Appen-dix B兲.

APPENDIX B: RECTANGULAR BARRIERS Here we evaluate the spin-mixing conductance G↑↓for a model barrier, giving more details of the results of Ref.22. We consider a smooth rectangular barrier between a normal metal and a Stoner-model ferromagnet. The solution of the Schrödinger equation for spin s in the normal metal,

␺s

m_{共x,y,z兲 can be used to determine the reflection} coeffi-cients rs

mm

for each mode m. It reads

␺s m_{共x,y,z兲 =}␹ m_共x,y,z兲

冑

kN m 共e ik_Nmx_{+ r} s mm e−ikN m x_兲, where␹m_{共y,z兲 is the transverse wave function and k}

N m

is the longitudinal wave number for mode m in the normal metal. In terms of the wave numbers in the normal metal kN

m , barrier kB m , and ferromagnet kFs m

for a given energy, the reflection coefficient for mode m at the barrier reads

rs mm =␳共kN m ,kB m_{兲 + e}2iakB m ␶共kN m ,kB m_兲_␳_共k B m ,kFs m_兲_␶_共k B m ,kN m_兲, 共B1兲 where a is the barrier thickness and

␶共k1,k2兲 ⬅ 2

冑

k1k2 k1+ k2 , 共B2兲 ␳共k1,k2兲 ⬅ k1− k2 k1+ k2 . 共B3兲

For a tunneling barrier, kB m

is imaginary and the spin-dependent correction to the reflection coefficient is exponen-tially small in the barrier thickness.

For a numerical estimate we use a Fermi energy in the normal metal of 2.6 eV, a barrier height of 3 eV and barrier thickness of a = 1 nm. The Fermi momenta in the ferromag-net are taken to be kF↑= 1.09 Å−1and kF↓= 0.42 Å−1 共charac-teristic for Fe, see Ref.39兲.

For the spin-mixing conductance G↑↓, Eq. 共26兲, we find

that Im G↑↓/ G = −0.26 for this choice of parameters. The ef-fective field due the interface exchange effect is therefore not negligible compared to the conductance parameters. More realistic electronic structure calculations45 _{should be carried} out to obtain better estimates.

APPENDIX C: X2 EXCHANGE IN CLASSICAL DOTS Here we present more details concerning the derivation of Eq.共34兲 for the effective exchange field X2 in classical

_X2,n N=1 .

Using the spin currents from Eq.共22兲, we obtain, e.g.,

冓

dsជ dt

冔

_X2,n N=0 = ប 4␲e2

兺

_␣s_⬙P␣G␣s

⬙

共mជ␣⫻ sˆ兲

冉

冕

d⑀1

冕

⬘

d⑀₂ f共⑀1兲关1 − f共⑀2兲兴关⑀2−⑀1+ E−1− E0+␮cF␣− s

⬙

⌬␮/2兴 −

冕

d⑀₁

冕

⬘

d⑀₂ f共⑀2兲关1 − f共⑀1兲兴关⑀2−⑀1+ E0− E1+␮cF␣− s

⬙

⌬␮/2兴

冊

. 共C1兲

The first term in brackets describes virtual processes in which an electron tunnels out of the island, and the second term corresponds to incoming electrons. The expressions for the energy differences are given by

E−1− E0=共CGVG+ e/2兲e/C, 共C2兲

E0− E1=共CGVG− e/2兲e/C. 共C3兲 Because of the periodicity in the gate voltage

冓

dsជ dt

冔

_X2,n N=1 =

冓

dsជ dt

冔

_X2,n N=0 with VG→ VG− e/CG.

We can now rewrite Eq. 共C1兲 in terms of the function

␩共⑀, U兲, defined in Eq. 共35兲. The probabilities p0and p1 are taken from Eq. 共18兲. After linearization in V and ⌬␮, we arrive at the expression Eq.共34兲.

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