Transport through a double-quantum-dot system with noncollinearly polarized leads

PHYSICAL REVIEW B 77, 245313 共2008兲

Transport through a double-quantum-dot system with noncollinearly polarized leads R. Hornberger, S. Koller, G. Begemann, A. Donarini, and M. Grifoni Institut für Theoretische Physik, Universität Regensburg, 93035 Regensburg, Germany 共Received 29 November 2007; revised manuscript received 21 February 2008; published 12 June 2008兲 We investigate linear and nonlinear transport in a double quantum dot system weakly coupled to spinpolarized leads. In the linear regime, the conductance as well as the nonequilibrium spin accumulation are evaluated in analytic form. The conductance as a function of the gate voltage exhibits four peaks of different heights with mirror symmetry with respect to the charge neutrality point. As the polarization angle is varied, due to exchange effects, the position and shape of the peaks change in a characteristic way, which preserves the electron-hole symmetry of the problem. In the nonlinear regime, various spin-blockade effects are observed. Moreover, negative differential conductance features occur for noncollinear magnetizations of the leads. In the considered sequential tunneling limit, the tunneling magnetoresistance 共TMR兲 is always positive with a characteristic gate voltage dependence for noncollinear magnetization. If a magnetic field is added to the system, the TMR can become negative. DOI: 10.1103/PhysRevB.77.245313

PACS number共s兲: 73.63.Kv, 72.25.⫺b, 73.23.Hk, 85.75.⫺d

I. INTRODUCTION

Spin-polarized transport through nanostructures is attracting increasing interest due to its potential application in spintronics1,2 as well as in quantum computing.3 Downscaling magnetoelectronic devices to the nanoscale implies that Coulomb interaction effects become increasingly important.4,5 In particular, the interplay between spinpolarization and Coulomb blockade can give rise to a complex transport behavior in which both the spin and the charge of the “information carrying” electron play a role. This has been widely demonstrated by many experimental studies on single-electron transistors 共SETs兲 with ferromagnetic leads, with central element being either a ferromagnetic particle,6–8 normal metal particles,9,10 a two-level artificial molecule,11 a C60 molecule,12 or a carbon nanotube,13 showing the increasing complexity and variety of the investigated systems. Initially, the theoretical work was mainly focused on the difference in the transport properties for parallel or antiparallel magnetizations in generic spin-valve SETs.14–24 More recently, the interplay between spin and interaction effects for noncollinear magnetization configurations attracted quite some interest both in systems with a continuous energy spectrum,25–28 as well as in single-level quantum dots,29–36 many-level nanomagnets,37 and in carbon nanotube quantum dots.38 In the noncollinear case, a much richer physics is expected than in the collinear one. For example, two separate exchange effects have to be taken into account. On the one hand, there is the nonlocal interface exchange, in scattering theory for noninteracting systems described by the imaginary part of the spin-mixing conductance,39 which in the context of current-induced magnetization dynamics acts as an effective field.40 Such an effective field has been experimentally found to strongly affect the transport dynamics in spin valves with MgO tunnel junctions.41 This effect has also been recently involved to explain negative tunneling magnetoresistance 共TMR兲 effects in carbon nanotube spin valves13 and called spin-dependent interface phase shifts.22,42 The second exchange term is an interaction-dependent exchange effect due to virtual tunneling processes that is absent in noninter1098-0121/2008/77共24兲/245313共18兲

acting systems.25,28,30,38 This latter exchange effect is potentially attractive for quantum information processing since it allows to switch on and off magnetic fields in arbitrary directions just by a gate electric potential. Recently, there has been increasing interest in doublequantum-dot systems 共that can be realized, e.g., in semiconducting structures43 or carbon nanotubes44兲 as tunable systems attractive for studying fundamental spin correlations. In fact, the exchange Coulomb interaction induces a singlettriplet splitting, which can be used to perform logic gates.45 Moreover, Coulomb interaction together with the Pauli principle can be used to induce spin-blockade when the two electrons have triplet correlations.46–49 The Pauli spin-blockade effect can be used to obtain a spin-polarized current even in the absence of spin-polarized leads; it requires a strong asymmetry between the two on-site energies of the left and right dots. So far, transport through a double-dot 共DD兲 system with spin-polarized leads has been addressed in few theoretical23,24,50 and experimental11 works, for the case of collinearly polarized leads only. While Ref. 23 addresses additional Pauli spin-blockade regimes when one lead is halfmetallic and one is nonmagnetic, Ref. 24 focuses on the effects of higher order processes in symmetric DD systems, which can, e.g., yield a zero bias anomaly or a negative tunneling magnetoresistance. In Ref. 11, Coulomb blockade spectroscopy is used to measure the energy difference between symmetric and antisymmetric molecular states and to determine the spin of the transferred electron. In this work, we investigate spin-dependent transport in the so-far unexplored case of a DD system connected to leads with arbitrary polarization direction. Specifically, we focus on the low transparency regime where a weak coupling between the DD and the leads is assumed. Our model takes into account interface reflections as well as exchange effects due to the interactions and relevant for noncollinear polarization. We focus on the case of a symmetric DD, so that rectification effects induced by Pauli spin-blockade are excluded. In the linear transport regime, the conductance is calculated in closed analytic form. This yields four distinct resonant tunneling regimes, but due to the electron-hole sym-

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PHYSICAL REVIEW B 77, 245313 共2008兲

HORNBERGER et al. gate gate

Θ

FIG. 1. 共Color online兲 Schematic picture of the model: a doublequantum-dot system attached to polarized leads. The significance of the on-site and intersite interactions U and V, respectively, is depicted. The source and drain contacts are polarized and the direction ជ ␣ is indicated by the arrows. of the magnetizations m

metry of the DD Hamiltonian, each possess a symmetric mirror with respect to the charge neutrality point. However, by applying an external magnetic field, this symmetry is broken, which can lead to negative tunneling magnetoresistance features. Finally, in the nonlinear regime, some excitation lines can be suppressed for specific polarization angles, and negative differential features also occur. The method developed in this work to investigate charge and spin transport is based on the Liouville equation for the reduced density matrix 共RDM兲 in lowest order in the reflection and tunneling Hamiltonians. The obtained equations of motion are fully equivalent to those that could be obtained by using the Green’s function method30,51 in the same weaktunneling limit. The advantage of our approach is that it is, in our opinion, easier to understand and to apply for newcomers, as it is based on standard perturbation theory and does not require knowledge of the nonequilibrium Green’s function formalism. The paper is organized as follows. In Sec. II, we introduce the model system for the ferromagnetic DD single-electron transistor. In Sec. III, the coupled equations of motion for the elements of the DD reduced density matrix are derived. Readers not interested in the derivation of the dynamical equations can directly go to Secs. IV and V, where results for charge and spin transfer in the linear and nonlinear regimes, respectively, are discussed. Finally, we present results for the transport characteristics in the presence of an external magnetic field in Sec. VI. Conclusions are drawn in Sec. VII.

FIG. 2. 共Color online兲 The spin quantization axis of the doubledot, z䉺, is chosen to be perpendicular to the plane spanned by the ជ s, m ជ d in the leads. The latter enclose an magnetization directions m angle ⌰.

ˆ ,H ˆ that model the the spin-up state. The Hamiltonians H s d source and drain contacts 共␣ = s , d兲 read as follows: ˆ = 兺 共␧ − ␮ 兲c† c H ␣ k␴␣ ␣ ␣k␴ ␣k␴␣ ,

ˆ +H ˆ +H ˆ +H ˆ , ˆ =H ˆ +H H 䉺 s d T R

ˆ = H T

共t␣d␣† ␴ c␣k␴ 兺 ␣k␴ ␣

␣

accounting for the DD Hamiltonian, the source 共s兲 and drain 共d兲 leads, and the tunneling 共T兲 and reflection 共R兲 Hamiltonians. The two contacts are considered to be magnetized along an arbitrary but fixed direction determined by the magជ ␣. The two magnetization axes enclose netization vectors m an angle ⌰ 苸 关0 ° , 180°兴 共see Fig. 1兲. The spin quantization ជ ␣ of the axis zជ␣ in lead ␣ is parallel to the magnetization m lead. The majority of electrons in each contact will then be in

␣

+ t␣ⴱ c␣† k␴ d␣␴␣兲. ␣

共3兲

ˆ includes reflecThe so-called reflection Hamiltonian H R tion events at the lead-molecule interface.28,38 For strongly shielded leads, the overall effect is the occurrence of a small energy shift ⌬R induced by the magnetic field in the contacts and built up during several cycles of reflections at the boundaries. It reads ˆ =−⌬ H R R

共1兲

共2兲

where c␣† k␴ and c␣k␴␣ are electronic lead operators. They ␣ create, respectively annihilate, electrons with momentum k and spin ␴␣ in lead ␣. The electrochemical potentials ␮␣ = ␮0␣ + eV␣ contain the bias voltages Vs and Vd at the left and right lead with Vs − Vd = Vbias. There is no voltage drop within the DD. We denote in the following ␧k␴␣ − ␮␣ ª ␧␣k␴␣. Tunneling processes into and out of the DD are described ˆ . We denote with d† , d by H T ␣␴␣ the creation and destruc␣␴␣ tion operators in the DD. We assume that tunneling only can happen between a contact and the closest dot, so that we can use the convention that the lead indices ␣ = s , d correspond to ␣ = 1 , 2 for the DD. With t␣ the tunneling amplitude, we find

II. MODEL

We consider a two-level DD, or a single molecule with two localized atomic orbitals, attached to ferromagnetic source and drain contacts and with a capacitive coupling to a lateral gate electrode. The system is described by the total Hamiltonian,

␣

k␴␣

兺

␣=s,d

共d␣† ↑ d␣↑␣ − d␣† ↓ d␣↓␣兲. ␣

␣

共4兲

Finally, the DD Hamiltonian needs to be specified. As the spin quantization axis of the DD, zជ䉺, we choose the direction perpendicular to the plane spanned by zជs and zជd 共Ref. 30兲 共see Fig. 2兲. The two remaining basis vectors xជ 䉺 and yជ 䉺 are along zជs + zជd, respectively along zជs − zជd. The matrices that mathematically describe the above transformations read M s↔䉺 =

冉

冊

1 + e+i⌰/4 + e−i⌰/4 ⴱ 冑2 − e+i⌰/4 + e−i⌰/4 = M d↔䉺 .

共5兲

We express the DD Hamiltonian in the localized basis such that, e.g., 兩+ , −典 describes a state with a spin-up electron on site 1 and a spin-down electron on site 2 共with spin directions expressed in the spin-coordinate system of the DD兲. Such state can be obtained by applying creation operators on † † d2↓兩0典. In general, the orderthe vacuum state, i.e., 兩+ , −典 = d1↑

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TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM… TABLE I. Eigenstates of the double-dot system and corresponding eigenvalues and parity. In the limit 兩b兩 → ⬁ where the interdot hopping is unhindered, R → 0 and ␣0 → ␤0. For 兩b兩 → 0, i.e., no interdot hopping takes place, we find that R → + ⬁ and ␣0 → 1, ␤0 → 0 if U ⬎ V; state 兩2典 then becomes degenerate to 兩2⬘共0兲典, forming a Heitler–London state. In turn, if U ⬍ V, then R → −⬁ and ␣0 → 0, ␤0 → 1. Abbreviation

State

Eigenvalue

Spin

兩0典

兩0 , 0典

0

0

兩1e␴典

1 冑2 共兩 ␴ , 0典 + 兩0 , ␴ 典兲

␰⬘ + b

1/2

兩1o␴典 兩2典

1 冑2 共兩 ␴ , 0典 − 兩0 , ␴ 典兲 ␣0 冑2 共兩+ , −典 + 兩− , +典兲

␰⬘ − b 2␰⬘ + 2 共U + V − ⌬兲

1/2 0

2␰⬘ + V

1

2␰⬘ + U

0 0

1

␤0

+ 冑2 共兩2 , 0典 + 兩0 , 2典兲 兩+ , +典 1 冑2 共兩+ , −典 − 兩− , +典兲 兩− , −典 1 冑2 共兩2 , 0典 − 兩0 , 2典兲

兩2⬘共1兲典 兩2⬘共0兲典 兩2⬘共−1兲典 兩2⬙典 兩2⵮典

␤0

冑2 共兩+ , −典 + 兩− , +典兲

2␰⬘ + 2 共U + V + ⌬兲 1

number have total spin S = 0 and are not degenerate. In the case of the two-particle ground state, the parameters ␣0 and ␤0 determine whether the electrons prefer to pair in the same dot or are delocalized over the DD structure. Since the eigenstates are normalized to one, the condition ␣20 + ␤20 = 1 holds. The energy difference between the S = 0 ground state and the triplet is given by the exchange energy, 1 J = 共⌬ − U + V兲 = 2兩b兩共R + 冑1 + R2兲, 2 where ⌬ = 4兩b兩冑1 + R2 and R = 共V − U兲 / 共4兩b兩兲. Besides the triplet, one observes the presence of higher two-particle excited states with total spin S = 0. ˆ contain operators of the ˆ and H Finally, we remark that H T R † DD, d␣␴ and d␣␴␣, in the spin-coordinate systems of the ␣ ˆ is already expressed in terms of DD operaleads, while H 䉺 † tors d␣␴ and d␣␴䉺 with spin expressed in the coordinate 䉺 system of the DD. III. DYNAMICAL EQUATIONS FOR THE REDUCED DENSITY MATRIX

␣0

− 冑2 共兩2 , 0典 + 兩0 , 2典兲 1 3␰⬘ + U + 2V + b 冑2 共兩2 , ␴ 典 + 兩 ␴ , 2典兲 1 3␰⬘ + U + 2V − b 共兩2 , ␴ 典 − 兩 ␴ , 2典兲 冑2 兩2 , 2典 4␰⬘ + 2U + 4V in terms of R = 共U − V兲 / 共4兩b兩兲: 1 1 ⌬ = 4兩b兩冑1 + R2 , ␣0 = 冑2 冑 2 冑 2

兩3o␴典 兩3e␴典 兩4典

1/2 1/2 0

1+R −R 1+R

† † † † ing of the creation operators is defined as d1↑ d2↑d1↓d2↓兩0典. The DD Hamiltonian then reads

ˆ = H 䉺

␧␣d␣† ␴ 兺 ␣␴ 䉺

冉

䉺

d␣␴䉺 + b 兺 共d1†␴ d2␴䉺 + d2†␴ d1␴䉺兲

U + ␰− −V 2

␴䉺

冊兺 兺

䉺

2

2

␣=1 ␴䉺

䉺

d␣† ␴ d␣␴䉺 䉺

+ V共n1↑䉺 + n1↓䉺兲共n2↑䉺 + n2↓䉺兲,

+ U 兺 n ␣↑䉺n ␣↓䉺 ␣=1

共6兲

where the spin index 䉺 indicates that the operators are expressed in the spin-coordinate system of the DD. The tunneling coupling between the two sites is b, while U and V are on-site and intersite Coulomb interactions. In the following, we consider a symmetric DD with equal on-site energies ␧1 = ␧2. Thus, we can incorporate the on-site energies in the parameter ␰ proportional to the applied gate voltage Vgate. To understand transport properties of the two-site system in the weak-tunneling regime, we have to analyze the eigenstates of the isolated interacting system. These states, which are expressed in terms of the localized states, and the corresponding eigenvalues are listed in Table I.52 Table I also indicates the eigenvalues of the total spin operator. The ground states of the DD with odd particle number are spin degenerate. In contrast, the ground states with even particle

In this section, we shortly outline how to derive the equation of motion for the RDM to lowest nonvanishing order in the tunneling and reflection Hamiltonians. The method is based on the well known Liouville equation for the total density matrix in lowest order in the tunneling and reflection Hamiltonians. Equations of motion for the reduced density matrix are obtained upon performing the trace over the lead degrees of freedom,53 yielding, after standard approximations, Eqs. 共13兲 and 共14兲 below. In the case of spin-polarized leads, however, it is convenient to express the equations of motion for the RDM in the basis that diagonalizes the isolated system’s Hamiltonian and in the system’s spin quantization axis. After rotation from the leads’ quantization axis to the DD one we obtain 关Eq. 共21兲兴, which forms the basis of all the subsequent analysis. Let us start from the Liouville equation for the total density matrix ␳ˆ I共t兲 in the interaction picture, iប

d␳ˆ I共t兲 ˆ I 共t兲 + H ˆ I 共t兲, ␳ˆ I共t兲兴, = 关H T R dt

共7兲

ˆ transformed into the interaction picture by ˆ and H with H T R ˆ +H ˆ +H ˆ 兲共t−t 兲 I i/ប共H ˆ e−i/ប共Hˆ䉺+Hˆs+Hˆd兲共t−t0兲, where t ˆ 䉺 s d 0 H HT/R共t兲 = e T/R 0 indicates the time at which the perturbation is switched on. Integrating Eq. 共7兲 over time and inserting the obtained expression in the right-hand side of Eq. 共7兲 one equivalently finds 1 i ˆI i ˆI ˆ I共t0兲兴 − 关H ˆ I共t0兲兴 − 2 ␳˙ˆ I共t兲 = − 关H R共t兲, ␳ T共t兲, ␳ ប ប ប

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ˆ I 共t兲,关H ˆ I 共t⬘兲 + H ˆ I 共t⬘兲, ␳ˆ I共t⬘兲兴兴. +H R T R

冕

t

ˆ I 共t兲 dt⬘关H T

t0

共8兲

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HORNBERGER et al.

The time evolution of the RDM,

i I ˆ I 共t兲, ␳ˆ I 共t兲␳ˆ ␳ˆ 兴 ␳˙ˆ 䉺 共t兲 = − Trleads关H s d 䉺 R ប

I ␳ˆ 䉺 共t兲 ª Trleads关␳ˆ I共t兲兴,

共9兲

−

is now formally obtained from Eq. 共8兲 by tracing out the lead degrees of freedom. To proceed, we make the following standard approximations. 共i兲 The leads are considered as reservoirs of noninteracting electrons that stay in thermal equilibrium at all times. In fact, we only consider weak tunneling and, therefore, the influence of the DD on the leads is marginal. Hence, we can approximatively factorize the density matrix of the total system as I ␳ˆ I共t兲 ⬇ ␳ˆ 䉺 共t兲␳ˆ s␳ˆ d ,

冕

⬁

0

␣k␴␣

再兺 ilm

0

ˆ I 共t兲,关H ˆ I 共t − t⬙兲, ␳ˆ I 共t兲␳ˆ ␳ˆ 兴兴兲. dt⬙ Trleads共关H s d 䉺 T T

A. Contribution from the tunneling Hamiltonian

In the following, we derive the explicit expression for the GME in the basis of the isolated DD. For simplicity, we omit the contribution of the reflection Hamiltonian in a first instance. When we shall have obtained the final form of the GME due to the tunneling term, we will see that it is easy to insert the contribution from the reflection Hamiltonian. Let us then start from the tunneling Hamiltonian in the interaction picture,

共10兲

dt⬙ 兺 兩t␣兩2

冕

⬁

共11兲

ˆ I 共t兲 = H T

where ␳ˆ s and ␳ˆ d are time independent and given by the usual ˆ −␤H s/d thermal equilibrium expression for the contacts ␳ˆ s/d = e Zs,d , where ␤ is the inverse temperature and Zs/d are the partition sums over all states of lead s / d. ˆ . 共ii兲 We consider the lowest nonvanishing order in H T/R 共iii兲 We apply the Markov approximation, i.e., in the inI I 共t⬘兲 with ␳ˆ 䉺 共t兲. In other tegral in Eq. 共8兲, we replace ␳ˆ 䉺 words, it is assumed that the system loses all memory of its past due to the interaction with the lead electrons. Furthermore, being interested in the long term behavior of the system only, we send t0 → −⬁. We finally obtain the generalized master equation 共GME兲 for the reduced density matrix,

1 I ␳ˆ˙ 䉺 共t兲 = − 2 ប

1 ប2

t␣c␣† k␴ 共d␣␴ 兲ij兩i典具j兩exp关i共␧i − ␧ j + ␧␣k␴ 兲t/ប兴 兺 兺 ␣k␴ i,j ␣

␣

␣

␣

共12兲

+ H.c., 共d␣† ␴ 兲ij = 具i兩d␣† ␴ 兩j典 ␣ ␣

are the where 共d␣␴␣兲ij = 具i兩d␣␴␣兩j典 and electron annihilation and creation operators in the spinquantization axis of lead ␣ expressed in the basis of the energy eigenstates of the quantum dot system. To simplify Eq. 共11兲, standard approximations are invoked. 共i兲 The first one is the secular approximation: fast oscillations in time average out in the stationary limit we are interested in and thus can be neglected. Together with the relation Trleads共␳ˆ s␳ˆ dc␣† k␴ c␣⬘k⬘␴␣兲 = ␦kk⬘␦␣␣⬘␦␴␴⬘ f ␣共␧␣k␴兲, ␣ where f ␣共␧␣k␴兲 is the Fermi function, and the cyclic properties of the trace we get

I f ␣共␧␣k␴␣兲共d␣␴␣兲il共d␣† ␴ 兲lm兩i典具m兩␳ˆ 䉺 共t兲exp关i共␧m − ␧l + ␧␣k␴␣兲t⬙/ប兴 ␣

I 共t兲exp关− i共␧l − ␧m + ␧␣k␴␣兲t⬙/ប兴 + 兺 共1 − f ␣共␧␣k␴␣兲兲共d␣† ␴ 兲il共d␣␴␣兲lm兩i典具m兩␳ˆ 䉺 ␣

ilm

I 共t兲共d␣␴␣兲il共d␣† ␴ 兲lm兩i典具m兩exp关− i共␧i − ␧l + ␧␣k␴␣兲t⬙/ប兴 + 兺 f ␣共␧␣k␴␣兲␳ˆ 䉺 ␣

ilm

I 共t兲共d␣† ␴ 兲il共d␣␴␣兲lm兩i典具m兩exp关+ i共␧l − ␧i + ␧␣k␴␣兲t⬙/ប兴 + 兺 共1 − f ␣共␧␣k␴␣兲兲␳ˆ 䉺 ␣

ilm

I 共t兲 jl共d␣† ␴ 兲lm兩i典具m兩exp关+ i共␧m − ␧l + ␧␣k␴␣兲t⬙/ប兴 − 兺 共1 − f ␣共␧␣k␴␣兲兲共d␣␴␣兲ij␳ˆ 䉺 ␣

iljm

I 共t兲 jl共d␣␴␣兲lm兩i典具m兩exp关− i共␧l − ␧m + ␧␣k␴␣兲t⬙/ប兴 − 兺 f ␣共␧␣k␴␣兲共d␣† ␴ 兲ij␳ˆ 䉺 iljm

␣

I 共t兲 jl共d␣† ␴ 兲lm兩i典具m兩exp关− i共␧i − ␧ j + ␧␣k␴␣兲t⬙/ប兴 − 兺 共1 − f ␣共␧␣k␴␣兲兲共d␣␴␣兲ij␳ˆ 䉺 ␣

iljm

冎

I 共t兲 jl共d␣␴␣兲lm兩i典具m兩exp关+ i共␧ j − ␧i + ␧␣k␴␣兲t⬙/ប兴 . − 兺 f ␣共␧␣k␴␣兲共d␣† ␴ 兲ij␳ˆ 䉺 iljm

␣

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TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

共ii兲 For the second approximation, we notice that we wish I 兩m典 of the RDM in the to evaluate single components 具n兩␳ˆ 䉺 system’s energy eigenbasis. Therefore, we assume that the DD is in a pure charge state with a certain number of electrons N and energy EN. In fact, in the weak-tunneling limit, the time between two tunneling events is longer than the

ENN ␳˙ nm 共t兲 = −

␲ 兺兵 ប ␣␴␣ l,l

冋 冋 冋 冋

兺

⬘苸兩N−1典

,

兺

j苸兩ENN典

兺

,

h,h⬘苸兩N+1典

共a兲 + f ␣共␧h − ␧ j兲D␣␴␣共␧h − ␧ j兲 +

i ␲

冕

共b兲 + 共1 − f ␣共␧ j − ␧l兲兲D␣␴␣共␧ j − ␧l兲 −

共c兲 + f ␣共␧h − ␧ j兲D␣␴␣共␧h − ␧ j兲 −

i ␲

冕

共d兲 + 共1 − f ␣共␧ j − ␧l兲兲D␣␴␣共␧ j − ␧l兲 +

time where relaxation processes happen. That is, we can neglect matrix elements between states with different numI ber of electrons and only regard elements of ␳ˆ 䉺 , which connect states with same electron number N and same energy I into submatrices labeled with N EN. So, we can divide ␳ˆ 䉺 and EN and find

其兩t␣兩2

d␧k i ␲

i ␲

f ␣共␧k兲D␣␴␣共␧k兲 ␧k − ␧h + ␧ j

冕

d␧k

再 d␧k

␣

␧k − ␧ j + ␧l

␧k − ␧h + ␧ j d␧k

共d␣␴␣兲nh共d␣† ␴ 兲hj␳EjmNN共t兲

共1 − f ␣共␧k兲兲D␣␴␣共␧k兲

f ␣共␧k兲D␣␴␣共␧k兲

冕

册 册

册

共d␣† ␴ 兲nl共d␣␴␣兲lj␳EjmNN共t兲 ␣

␳EnjNN共t兲共d␣␴␣兲 jh共d␣† ␴␣兲hm

共1 − f ␣共␧k兲兲D␣␴␣共␧k兲 ␧k − ␧ j + ␧l

册

␳EnjNN共t兲共d␣† ␴␣兲 jl共d␣␴␣兲lm

E N+1

共e兲 − 2共1 − f ␣共␧h − ␧ j兲兲D␣␴␣共␧h − ␧ j兲共d␣␴␣兲nh⬘共d␣† ␴ 兲hm␳h hh 共t兲 ␣

E N−1

共f兲 − 2f ␣共␧ j − ␧l兲D␣␴␣共␧ j − ␧l兲共d␣† ␴ 兲nl⬘共d␣␴␣兲lm␳l ll ⬘ ␣

ENN I,ENN In Eq. 共14兲, we used the notation ␳nm ª 具n兩␳ˆ 䉺 兩m典. By convention, 兵兺l,l⬘ , 兺 j , 兺h,h⬘ 其 means that in each line 关共a兲–共f兲兴, we sum over the indices occurring in this line only. Notice that the sum over j is restricted to states of energy E j = EN = En = Em. For the states with N ⫾ 1 electrons, we have to sum over all energies; therefore, we indexed the density matrix with Eh = Eh⬘ in line 共e兲 and El = El⬘ in line 共f兲. Further, we replaced the sum over k by an integral, 兺k →兰d␧␣k␴␣D␣␴␣共␧␣k␴␣兲, where D␣␴␣共␧␣k␴␣兲 denotes the density of states in lead ␣ for the spin direction ␴␣, and applied the following useful formula:

冕

冕 ⬙ 冕⬘ t

d␧␣k␴␣G共␧␣k␴␣兲

= ␲បG共E兲 ⫾ iប

dt e⫾共i/ប兲共␧␣k␴␣−E兲t⬙

⬘

再冎

共t兲 .

共14兲

B. Transformation into the spin-coordinate system of the double dot

In Sec. II, we introduced the transformation rules for changing from the lead spin coordinates ␴␣ into the DD spin coordinates ␴䉺. These rules give

冉 冊 冉 d␣† ↑

d␣† ↓

␣

=

␣

1

冑2

+ e−i⌰␣/2 + e+i⌰␣/2 − e−i⌰␣/2 + e+i⌰␣/2

冊冉 冊 d␣† ↑

䉺

d␣† ↓

䉺

,

共16兲

with ⌰s ª − ⌰2 , ⌰d ª + ⌰2 . Thus, Eq. 共14兲 can be easily expressed in the DD spin quantization axis. For example, it holds

0

d␧␣k␴␣

G共␧␣k␴␣兲 共␧␣k␴␣ − E兲

,

共15兲

where the prime at the integral denotes Cauchy’s principal-part integration. In our case, G共␧␣k␴␣兲 = D␣␴␣共␧␣k␴␣兲f ␣⫾共␧␣k␴␣兲 with f ␣+ = f ␣ and f ␣− = 1 − f ␣. In order to simplify the notations, we replaced ␧␣k␴␣ by ␧k in Eq. 共14兲.

D␣␴ d␴† d␴ 兺 ␴ ␣

␣

␣

␣

1 = 共D␣↑␣ + D␣↓␣兲 兺 ⌽␣␴䉺␴䉺d␣† ␴ d␣␴䉺 䉺 2 ␴䉺 1 + 共D␣↑␣ − D␣↓␣兲 兺 ⌽␣ⴱ ␴ −␴ d␣† ␴ d␣−␴䉺 , 䉺 䉺 䉺 2 ␴䉺 共17兲

where we introduced

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PHYSICAL REVIEW B 77, 245313 共2008兲

HORNBERGER et al.

⌽ ␣␴䉺␴⬘ ª 䉺

冦

⬘ ␴䉺 = ␴䉺

1 e e

i⌰␣

−i⌰␣

␴䉺 = ↑

⬘ =↓ ␴䉺

␴䉺 = ↓

⬘ = ↑. ␴䉺

冧

ˆI =−⌬ 兺 H R R ␣

再

⫾ ⫾ ⬘ 1 D␣+␣ f ␣ 共E兲 + D␣−␣ f ␣ 共E兲 ␴䉺 = ␴䉺 ª ⫾ ⫾ 2 D␣+ f ␣ 共E兲 − D␣− f ␣ 共E兲 ␴䉺 ⫽ ␴䉺 ⬘, ␣ ␣

ª

冕

⬘

⫾ d␧F␣␴ ␴⬘ 共␧兲共␧ 䉺 䉺

冎

− E兲−1 .

⬘ 䉺

兺

,

兺

ⴱ

␴䉺⫽␴䉺 ⬘

⌽␣␴ ␴⬘ 关d␴† d␴⬘ 兩j典 䉺jl 䉺lj 䉺 䉺

I ␳ˆ 䉺 共t兲d␴† d␴⬘ 兩j典具j兩兴. 䉺jl 䉺lj

R ␣␴䉺␴⬘ = 䉺

共19兲

1 ⌬R共␦␴䉺↑␦␴⬘ ↓ + ␦␴䉺↓␦␴⬘ ↑兲. 䉺 䉺 兩t␣兩2

共20兲

Now, we can add R␣␴䉺␴⬘ in Eq. 共14兲 in lines 共b兲 and 共d兲 to 䉺 find the final form of the complete master equation in the DD spin-coordinate system. It reads

兺

,

j苸兩ENN典

−

兺

In order to include this commutator in the master equation 关Eq. 共14兲兴, let us introduce the following abbreviation:

In order to give the full expression for the GME in the system’s eigenbasis, we need to compute the contribution from the reflection Hamiltonian in Eq. 共11兲. In analogy to what we did to evaluate the contribution from the tunneling ˆ into the interaction Hamiltonian, we must first transform H R picture and then perform the secular approximation to get rid ˆ in the DD of the time dependence. To start, we express H R spin quantization basis,

䉺

i 兺 ⌬R 兺 兺 ប ␣ j苸兩N典 l苸兩N−1典

I ⫻具j兩␳ˆ 䉺 共t兲

⬘苸兩N−1典

␴䉺⫽␴䉺 ⬘

i ˆ I , ␳ˆ I 共t兲␳ ␳ 兴 − Trleads关H s d R 䉺 ប =−

␲ 兺 兩t␣兩2 兺 兵 ប ␣=s,d l,l ␴ ,␴

ⴱ

⌽␣␴ ␴⬘ d␴† d␴⬘ 兩j典具j兩. 䉺jl 䉺lj 䉺 䉺

The commutator is easily evaluated to be

C. Contribution from the reflection Hamiltonian

ENN ␳˙ nm 共t兲 = −

兺

共18兲

and its related principal-part integral, ⫾ P␣␴ ␴⬘ 共E兲 䉺 䉺

j苸兩N典 l苸兩N−1典

For later convenience, we also define ⫾ F ␣␴ ␴⬘ 䉺 䉺

兺

h,h⬘苸兩N+1典

冋 冋 冋 冋

其

再

册

i + + † 共a兲 + ⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␧h − ␧ j兲 + P␣␴ ␴⬘ 共␧h − ␧ j兲 共d␣␴䉺兲nh共d␣␴⬘ 兲hj␳EjmNN共t兲 䉺 䉺 䉺 䉺 䉺 䉺 ␲

册

i − ⴱ − 共b兲 + ⌽␣␴ ␴⬘ F␣␴ ␴⬘ 共␧ j − ␧l兲 − 关P␣␴ ␴⬘ 共␧ j − ␧l兲 + R␣␴䉺␴⬘ 兴 共d␣† ␴ 兲nl共d␣␴⬘ 兲lj␳EjmNN共t兲 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺 ␲

册

i + + † 共c兲 + ⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␧h − ␧ j兲 − P␣␴ ␴⬘ 共␧h − ␧ j兲 ␳EnjNN共t兲共d␣␴䉺兲 jh共d␣␴⬘ 兲hm 䉺 䉺 䉺 䉺 䉺 䉺 ␲

册

i − ⴱ − 共d兲 + ⌽␣␴ ␴⬘ F␣␴ ␴⬘ 共␧ j − ␧l兲 + 关P␣␴ ␴⬘ 共␧ j − ␧l兲 + R␣␴䉺␴⬘ 兴 ␳EnjNN共t兲共d␣† ␴ 兲 jl共d␣␴⬘ 兲lm 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺 ␲ −

E N+1

†

共e兲 − 2⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␧h − ␧ j兲共d␣␴䉺兲nh⬘␳h hh 共t兲共d␣␴⬘ 兲hm ⬘ 䉺 䉺 䉺 䉺 ⴱ

+

E N−1

共f兲 − 2⌽␣␴ ␴⬘ F␣␴ ␴⬘ 共␧ j − ␧l兲共d␣† ␴ 兲nl⬘␳l ll ⬘ 䉺 䉺 䉺 䉺 䉺

共t兲共d␣␴⬘ 兲lm .

共21兲

䉺

E N NN+1 E N+1 NN ENN ␳˙ nm 共t兲 = − 兺 Rnmjj⬘␳ jjN 共t兲 + 兺 Rnmhh⬘␳hhh 共t兲 ⬘ ⬘

D. Current formula

We now observe that Eq. 共21兲 can be recast in the following Bloch–Redfield form:

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jj⬘

hh⬘

+ 兺 Rnmll⬘ ␳ll l

NN−1 E N−1

ll⬘

⬘

共t兲,

共22兲

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

where the sums in Eq. 共22兲 run over states with fixed particle number: j , j⬘ 苸 兵兩ENN典其, h , h⬘ 苸 兵兩N + 1典其, l , l⬘ 苸 兵兩N − 1典其. The Redfield tensors are given by 共␣ = s , d兲 共Ref. 38兲 Rnmjj⬘ = 兺 NN

␣

兺 l共or h兲

共−兲NN+1

共+兲NN−1 关␦mj⬘共⌫␣共+兲NN+1 ,nhhj + ⌫␣,nllj 兲 + ␦nj共⌫␣,j⬘hhm

共−兲NN−1

+ ⌫␣,j⬘llm 兲兴,

共23兲

Rnmkk⬘ = 兺 共⌫␣,k⬘mnk + ⌫␣,k⬘mnk 兲, 共+兲NN⫿1

NN⫾1

共−兲NN⫿1

共24兲

␣

where the quantities ⌫共⫾兲NN⫾1 can be easily read out from ␣,njjk Eq. 共21兲. They are 共⫾兲NN+1

⌫␣,nhh⬘k =

兺

␴䉺␴䉺 ⬘

⫾ 共⫾兲NN−1

⌫␣,nll⬘k =

再

冋 册

␲ + ⌽␣␴䉺␴⬘ 兩t␣兩2 F␣␴ ␴⬘ 共␧h − ␧k兲 䉺 䉺 䉺 ប

冎

␴䉺␴䉺 ⬘

⫿

再

冋

␲ ⴱ − 兩t␣兩2 F␣␴ ␴⬘ 共␧k − ␧l兲 ⌽ 䉺 䉺 ប ␣␴䉺␴䉺⬘

i − 共P 共␧k − ␧l兲 + R␣␴䉺␴⬘ 兲 䉺 ␲ ␣␴䉺␴䉺⬘

⫻共d␣† ␴ 兲nl共d␣␴⬘ 兲l⬘k 䉺 䉺

冎

册

I = 2␣e Re兺

兺

N n,n⬘,j

−

N = 2 ↔ N = 3,

N = 3 ↔ N = 4.

共27兲

N = 0. In the case of an empty system, we have only one density matrix element in the corresponding block with fixed particle number N = 0, i.e.,

.

共+兲NN−1 E N ⌫␣,njjn⬘ 兲␳n nn,st . ⬘

N = 1 ↔ N = 2,

B. Elements of the reduced density matrix

I With the stationary density matrix ␳ˆ 䉺st being known, the current 共through lead ␣ = s / d = ⫾兲 follows from 共+兲NN+1 共⌫␣,njjn⬘

N = 0 ↔ N = 1,

In all of the four cases, we get a system of five coupled equations involving diagonal and off-diagonal elements of the RDM. The matrix elements of the dot operators between the involved states entering these equations are given in Appendix A. Before going into the details of these equations, it is instructive to analyze the structure and the physical significance of the involved RDM elements.

i + † P 共␧h − ␧k兲 共d␣␴䉺兲nh共d␣␴⬘ 兲h⬘k , 䉺 ␲ ␣␴䉺␴䉺⬘

兺

共N兲 in the equation for ␳˙ˆ 䉺 and over l , l⬘ in the equation for ˙␳ˆ 共N+1兲 to energy-ground states E共0兲 and E共0兲 because all the 䉺 N N+1 other transitions are exponentially suppressed by the Fermi function. Notice, however, that these two approximations are not appropriate for the principal-part terms since they are not 共N兲 共N+1兲 and ␳ˆ 䉺 energy conserving. The resulting equations for ␳ˆ 䉺 关Eqs. 共B1兲 and 共B2兲, respectively兴 can be found in Appendix B. In the following, we shall apply those equations to derive an analytical expression for the conductance in the four different resonant charge state regimes possible in a DD system, i.e.,

共25兲

We numerically solve Eq. 共22兲 and use the result to evaluate the current flowing through the DD, as will be Secs. V–VII. At low-bias voltages, however, we can make some further approximations to arrive at an analytical formula for the static dc.

共0兲 ␳00 共t兲 = :W0 ,

共28兲

describing the probability to find an empty double-dot system. N = 1. In this case, we have four eigenstates for the system, where the two even ones build the degenerate ground state and the two odd ones are excited states 共see Table I兲. In the low-bias regime, we only need to take into account transitions between ground states. Therefore, we have to deal with the 2 ⫻ 2 matrix,

冉

IV. LOW-BIAS REGIME

共1兲 共1兲 ␳1e↑1e↑ ␳1e↑1e↓ 共1兲 ␳1e↓1e↑

共1兲 ␳1e↓1e↓

冊 冉 =:

冊

W1↑ w1ei␣1 . w1e−i␣1 W1↓

共29兲

The total occupation probability for one electron is A. General considerations

A low-bias voltage ensures that merely one channel is involved with respect to transport properties. Here, we focus on gate voltages that align charge states N and N + 1. Moreover, we can focus on density matrix elements that involve 共0兲 only. In the following, the energy ground states EN共0兲 and EN+1 we shall use the following compact notations: 共0兲 I,EN N

␳ˆ 䉺

共N兲 ª ␳ˆ 䉺 ,

共N兲 共N兲 具n兩␳ˆ 䉺 兩m典 = ␳nm .

共26兲 共N兲 ␳ˆ 䉺

Evaluation of the current requires the knowledge of 共N+1兲 and ␳ˆ 䉺 , i.e., a solution of the set of coupled equations that are obtained from Eq. 共21兲 or, equivalently, from Eq. 共22兲. In the low-bias regime, this task is simplified since 共i兲 terms which try to couple states with particle numbers unlike N and N + 1 can be neglected; 共ii兲 we can reduce the sums over h , h⬘

W1 ª W1↑ + W1↓ .

共30兲

The meaning of the off-diagonal elements, the so-called coherences, becomes clear if we regard the average spin in the system, 1 Pauli 共1兲 ␳ˆ 䉺 共t兲兴, S共1兲 i = Tr关␴i 2

共31兲

where i = x , y , z and ␴Pauli are the Pauli spin matrices. This i yields

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S共1兲 x = w1 cos ␣1,

S共1兲 y = − w1 sin ␣1 ,

1 Sz共1兲 = 共W1↑ − W1↓兲. 2

共32兲 共33兲

PHYSICAL REVIEW B 77, 245313 共2008兲

HORNBERGER et al.

再

␲ ˙ Sជ 共1兲 = − 兺 兩t␣兩2k+2 2F␣+ ↑↑共␮2兲Sជ 共1兲 − 关F␣+ ↑↓共␮2兲W1 ប ␣=s,d

N = 2. For the case N = 2, we actually have six different eigenstates, but only one of them, 兩2典, is a ground state 共with spin S = 0兲, see Table I. Only this ground state must be considered in the low-bias regime, yielding 共2兲 ␳22 共t兲

共3兲 共3兲 ␳3o↑3o↑ ␳3o↑3o↓ 共3兲 共3兲 ␳3o↓3o↑ ␳3o↓3o↓

冊 冉 =:

冎

共34兲

= :W2 .

This element describes the probability to find a dot with two electrons. N = 3. In this case, we have again four eigenstates for the system, whereas the two odd ones build the degenerate ground state and the two even ones are excited. In the lowbias regime, we only need to deal with the 2 ⫻ 2 matrix involving the three-particle ground states,

冉

ជ␣+ − 2F␣− ↑↓共␮2兲W2兴m

W3↑

w 3e i␣3

w3e−i␣3

W3↓

冊

.

共35兲

As for the case N = 1, the off-diagonal elements yield infor1 Pauli 共3兲 ␳ˆ 䉺 共t兲兴 in the sysmation on the average spin S共3兲 i = 2 Tr关␴i tem through the following relations:

1 Sz共3兲 = 共W3↑ − W3↓兲. 2

共36兲 共37兲

N = 4. Finally, if the double quantum dot is completely filled with four electrons, we only have one nondegenerate state. Correspondingly, there is only one relevant RDM matrix element, 共4兲 ␳44 共t兲

= :W4 ,

共38兲

describing the probability to find four electrons in the system. The total spin is S = 0. Hence, we see that in all of the four cases 关Eq. 共27兲兴, we get a system of five equations with the five independent 共i兲 共i兲 physical quantities WN , WN+1 and S共i兲 x , S y , Sz with i = 1 or 3. C. Conductance formula

We shall exemplarily present results for the resonant transition N = 1 ↔ N = 2. For the other transitions, similar considerations apply. The quantities of interest are 共1兲 共1兲 W1 , W2 , S共1兲 x , S y , Sz , which are related through Eqs. 共32兲 共1兲 . From Eqs. and 共36兲 to the density matrix elements of ␳ˆ 䉺 共B1兲 and 共B2兲 and Table III, and with W1 = 1 − W2 we finally obtain the following:

共40兲

2 ª 共␣0 ⫾ ␤0兲2. All of the We have introduced the notation 4k⫾ nonvanishing principle-value factors P␣⫾↑↓ and the reflection parameter R␣↑↓ have been merged to the following compact form:

1 − 2 + P␣共␮1,兵E2其 − E共0兲 1 兲 ª − 关P␣↓↑共␮1兲 + R␣↓↑兴 − k+ P␣↑↓共␮2兲 2 1 1 + P␣+ ↑↓共␧2⬘ − ␧1e兲 − P␣+ ↑↓共␧2⬙ − ␧1e兲 4 4 共41兲

共0兲 where we introduced the chemical potential ␮N+1 = EN+1 共0兲 − EN and 兵E2其 denotes the four different two-particle energies. We notice that the set of coupled equation 关Eq. 共39兲 and 共40兲兴 for the evolution of the populations and of the spin accumulation has a similar structure to that reported in Refs. 28, 30, and 38 for a single-level quantum dot, a metallic island, and a single-walled carbon nanotube, respectively. Some prefactors and the argument of the principal-part terms, however, are DD specific. In particular, as in Refs. 28 and 30, we clearly identify a spin precession term originating from the combined action of the reflection at the interface and the interaction. The associated effective exchange splitting is ␥B1, where ␥ = −g␮B is the gyromagnetic ratio and

ជ ª 2 兺 兩t 兩2P m ជ B 1 ␥ ␣ ␣ ␣ ␣

共42兲

is the corresponding effective exchange field. We now focus on the stationary limit. In the absence of the precession term, the spin accumulation has only a S共1兲 y component since, due to our particular choice of the spin quantization axis, S共1兲 x = 0 holds. The exchange field tilts the accumulated spin out of the magnetizations’ plane and gives rise to a nonzero Sz共1兲 component proportional to B1 and S共1兲 y . To get further insight in the spin dynamics, we observe that since we are looking at the low-bias regime, we can linearize the Fermi function f ␣ in the bias voltage, i.e., TABLE II. Matrix elements for the N = 0 ↔ N = 1 transition induced by operators d␣↑ and d␣↓, ␣ = 1 , 2.

˙ = − ␲ 兺 兩t 兩2k2 关2F+ 共␮ 兲W − 4F− 共␮ 兲W W 1 ␣ + 1 2 ␣↓↓ 2 ␣↓↓ 2 ប ␣=s,d

ជ ␣兴, − 4F␣+ ↑↓共␮2兲Sជ 共1兲 · m

ជ␣ P␣共␮1,兵E2其 − E共0兲 1 兲m

− k−2 P␣+ ↑↓共␧2⵮ − ␧1e兲,

W3 ª W3↑ + W3↓ .

S共3兲 y = − w3 sin ␣3 ,

␲k+2

⫻ Sជ 共1兲 .

The total occupation probability for three electrons is

S共3兲 x = w3 cos ␣3,

2

共39兲 245313-8

兩1e↑典 d1↑ : 具0兩 d1↓ : 具0兩 d2↑ : 具0兩 d2↓ : 具0兩

1 冑2

0 1 冑2

0

0↔1 兩1e↓典 0 1 冑2

0 1 冑2

兩1o↑典

兩1o↓典

1 冑2

0

0 1

− 冑2 0

1 冑2

0 1

− 冑2

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

f ␣共␰兲 = 共1 + e␤共␰+eV␣兲兲−1 ⬇ f共␰兲„1 − f共− ␰兲e␤V␣…. 共43兲 By introducing the polarization of the contacts, p ␣共 ␰ 兲 ª

D ␣↑␣共 ␰ 兲 − D ␣↓␣共 ␰ 兲 D ␣↑␣共 ␰ 兲 + D ␣↓␣共 ␰ 兲

ric case where both leads have the same properties, which, in particular, means that tunneling elements, polarizations, density of states, and reflection amplitude are equal, t1 = t2 = :t,

,

D1 = D2 = :D,

⫾

we can express the F␣␴ ␴⬘ factors as 䉺 䉺 F␣⫾↑↑共␰兲

=

G12共⌰兲 =

p␣共␰兲F␣⫾↑↑共␰兲,

where D␣ = D␣↑␣ + D␣↓␣. It is also sufficient for our calculations to regard the density of states as a constant quantity, D␣共␰兲 = D␣. Consequently, the polarization is also constant, p␣共␰兲 = p␣. Finally, we focus in the following on the symmet-

⌫ 2 2 f共␮2兲f共− ␮2兲 e ␤k+ 2 f共− ␮2兲 + 1

再

⫻ 1−

p2 sin2共 ⌰2 兲

1 + 关B1/f共␮2兲2⌫k+2兴2 cos2共 ⌰2 兲

冎

再

where 兩具i + 1兩d†兩i典兩 is a shortcut notation for the nonvanishing 共0兲 i + 1兩d␣† 䉺兩E共0兲 matrix elements 兩具Ei+1 i i典兩 calculated in Tables † II–V. It holds 兩具1兩d␣䉺兩0典兩 = 兩具4兩d␣† 䉺兩3典兩 = 1 / 冑2 and 兩具2兩d␣† 䉺兩1典兩 = 兩具3兩d␣† 䉺兩2典兩 = k+. Moreover, we gathered together the principal-part contributions and the ones coming from the reflection Hamiltonian in the effective magnetic fields,

ជ = Bជ , B 2 1 ជ = Bជ ª 2 兺 兩t 兩2P⬘ 共␮ ,E共0兲 − 兵E 其兲m B 3 4 2 ជ ␣. ␥ ␣ ␣ ␣ 4 3 The latter are defined in terms of the following function: 1 + 2 − P␣⬘ 共␮4,E共0兲 3 − 兵E2其兲 ª − 关P␣↓↑共␮4兲 + R␣↓↑兴 − k+ P␣↑↓共␮3兲 2 1 1 + P␣− ↑↓共␧3o − ␧2⬘兲 − P␣− ↑↓共␧3o − ␧2⬙兲 4 4 −

. 共45兲

共46兲

is present 共i.e., odd filling: one or three electrons兲 the effective magnetic field can have an influence. That is why correspondingly the renormalized effective exchange splitting vanishes for even fillings, namely, below 0 ↔ 1 and 2 ↔ 3, respectively, above the 1 ↔ 2 and 3 ↔ 4 resonances. This can nicely be seen from Fig. 3 共remember that Vgate ⬀ −␰, so “below” means larger, “above” means smaller ␰兲, where the four different factors 共␥Bi兲2 / 兵f共共−1兲i␮i兲⌫其2 are plotted. The curves belonging to the resonances involving the half-filling do not immediately go to zero but show a more complex behavior with some small intermediate peaks due to the influence of the various excited states present for a twoelectron population of the dot. As we expect, G01共␰兲 and G34共␰兲 关G12共␰兲 and G23共␰兲, respectively兴 are mirror symmetric with respect to each other when the gate voltage is varied. This in turn reflects the electron-hole symmetry of the DD Hamiltonian. The parameters of the figures are chosen to be as follows 共b ⬍ 0兲: kBT = 4 ⫻ 10−2兩b兩,

− ␧2⵮兲.

Moreover, a closer look to Eq. 共46兲 shows that its angular dependence is strongly coupled to the square of the ratio 共␥Bi兲 / 兵ប⌫f关共−1兲i␮i兴其, which is the effective exchange splitting rescaled by the coupling and the Fermi function. The ratio occurs in the denominators, and its value depends on the gate voltage. As the change of Bi under the variation in the gate voltage is comparatively small, the factor dominating the gate voltage evolution is the Fermi function. This accounts for the population of the dot: only if a nonzero spin

冎

Similarly, we find for an arbitrary resonance 共i = 0 , 1 , 2 , 3兲,

p2 sin2共 ⌰2 兲 ⌫ 2 † 2 f共␮i+1兲f共− ␮i+1兲 Gii+1共⌰兲 = e ␤兩具i + 1兩d 兩i典兩 1− , 2 1 + f„共− 1兲i␮i+1… 1 + 关Bi+1/f共共− 1兲i+1␮i+1兲2⌫兩具i + 1兩d†兩i典兩2兴2 cos2共 ⌰2 兲

k−2 P␣− ↑↓共␧3o

共44兲

R1␴䉺−␴䉺 = R2␴䉺−␴䉺 = :R.

Upon introducing the linewidth ⌫ = 2ប␲ D兩t兩2, the conductance G12 = I12 / Vbias for the resonant regime N = 1 ↔ N = 2 reads

1 ⬇ D␣共␰兲f共⫾ ␰兲共1 ⫿ f共⫿ ␰兲e␤V␣兲, 2 F␣⫾↑↓共␰兲

p1 = p2 = :p,

U = 6兩b兩,

ប⌫ = 4 ⫻ 10−3兩b兩, V = 1.6兩b兩,

共47兲

and p = 0.8, R = 0.05D. As expected, the peaks are mirror symmetric with respect to the half-filling gate voltage. Notice also the occurrence of different peak heights, both in the parallel and in the antiparallel case. For both polarizations, the principal-part terms entering Eq. 共46兲 vanish, the spin accumulation is entirely in the magnetization plane, and the peak ratio is solely determined by the ratio of the ground state overlaps 2 / k+2. For polarization angles ⌰ ⫽ 0 , ␲, the ra-

245313-9

PHYSICAL REVIEW B 77, 245313 共2008兲

HORNBERGER et al.

TABLE III. Matrix elements for the N = 1 ↔ N = 2 transition induced by d†␣↑ and d†␣↓, ␣ = 1 , 2. The notation 兩2⬘共sz兲典, with sz = 0 , ⫾ 1, specifies which one of the triplet elements is addressed. 1↔2 兩1e↓典

兩1e↑典 d†1↑:

d†1↓:

d†2↑:

d†2↓:

具2g兩

0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

␣ 0+ ␤ 0 2

1 冑2

0

具2g兩

− ␣ 0− ␤ 0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

0

具2g兩

1

0 0 0 0

2

0 1

2

− ␣ 0+ ␤ 0

2

0

2

0 0

1

2

1 冑2

0 1 −2 ␣ 0− ␤ 0 2

0 0 ␣ 0+ ␤ 0 2

0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

1 − 冑2

0

0 0 0 0

−2 0 1 −2

具2g兩

− ␣ 0− ␤ 0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

0 1 −2 0

1

− ␣ 0+ ␤ 0

2

0

2

0 0 1

− 冑2 0 0

1

2 ␣ 0− ␤ 0 2

tio is also determined by the nontrivial angular and voltage dependence of the effective exchange fields. Finally, as expected from the conductance formula Eq. 共46兲, the conductance is suppressed in the antiparallel compared to the parallel case. The four conductance peaks are plotted as a function of the gate voltage in Fig. 4 for the polarization angles ⌰ = 0 and ⌰ = ␲, top and bottom figures, respectively. These features of the conductance are nicely captured by the color plot of Fig. 5, where numerical results for the conductance plotted as a function of gate voltage and polarization angle are shown. The conductance suppression nearby ⌰ = ␲ is clearly seen. In the following, we analyze in detail the single resonance transitions. Due to the mirror symmetry, it is convenient to investigate together the N = 0 ↔ N = 1, N = 3 ↔ N = 4 and N = 1 ↔ N = 2, N = 2 ↔ N = 3 resonances. We use the convention that for a fixed resonance, the parameter ␰ = 0 when ␮N+1 = 0.

兩1o↑典 0 1

− 冑2 0 0 0 0 ␣ 0− ␤ 0 2

0 1 −2 0 1 −2 ␣ 0+ ␤ 0 2

2

0 1

−2 0 1

2

− ␣ 0− ␤ 0

2

0 0 0 1

− 冑2 0 0

0

␣ 0− ␤ 0 2

1 − 冑2

0

0 0 0 0

−2 0

− ␣ 0+ ␤ 0

2

0 1 −2 0 1

−2

− ␣ 0− ␤ 0

2

1

1

2 ␣ 0+ ␤ 0 2

0 0 0 1

− 冑2 0 0

cal expressions 关Eq. 共46兲兴 共continuous lines兲 perfectly match the results obtained from a numerical integration of the master equation 关Eq. 共21兲兴 with the current formula 关Eq. 共25兲兴. We also can see that the maxima of the conductance decrease with ⌰ growing up to ␲. It can be shown that the peaks for ⌰ = 0 and ⌰ = ␲ lie at the same value of ␰ because the effective fields Bi exactly vanish due to trigonometrical prefactors. In other words, virtual processes captured in the effective fields Bi do not play a role in the collinear cases. For noncollinear configurations, however, the peak maxima are shifted toward the gate voltages where an odd population of the dot dominates because there the effective exchange field can act on the accumulating spin and make it precess, which eases tunneling out. These findings are in agreement with results obtained for a single-level quantum dot,30 a metallic island,28 and carbon nanotubes.38 To quantify the relative magnitude of the current for a given polarization angle ⌰ with respect to the case ⌰ = 0, we introduce the angledependent TMR as

1. Resonant regimes N = 0 ^ N = 1 and N = 3 ^ N = 4

The expected mirror symmetry of G01 and G34 is shown in Fig. 6, where the conductance peaks are plotted for different polarization angles ⌰ of the contacts. Notice that the analyti-

兩1o↓典 − ␣ 0+ ␤ 0

TMRNN+1共⌰, ␰兲 = 1 − For the 0 ↔ 1 transition, it reads

245313-10

GN,N+1共⌰, ␰兲 . GN,N+1共0, ␰兲

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

TABLE IV. Matrix elements for the N = 2 ↔ N = 3 transition governed by d␣↑ and d␣↓, ␣ = 1 , 2. 2↔3 兩3o↓典

兩3o↑典 ␣ 0+ ␤ 0 2

具2g兩 具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

0 1 −2 0

2

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

1 冑2

0

0 0 0 0

−2 0

具2g兩

− ␣ 0− ␤ 0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

0 1 −2 0

d2↑:

p2 sin2共 ⌰2 兲 1+

␮1兲⌫

2

1 2 ␣ 0− ␤ 0 2

共 兲

兴cos2 ⌰2

TABLE V. Matrix elements for the N = 3 ↔ N = 4 transition induced by the operators d†␣↑ and d†␣↓, ␣ = 1 , 2.

兩3o↑典 d†1↑ : 具2 , 2兩 d†1↓ : 具2 , 2兩 d†2↑ : 具2 , 2兩 d†2↓ : 具2 , 2兩

0 1 冑2

0 1

− 冑2

3↔4 兩3o↓典

兩3e↑典

兩3e↓典

1 冑2

0

1 − 冑2

0

− 冑2 0

1 − 冑2

0

1

1

− 冑2

0 1

− 冑2 0

1

− 冑2 0 0 − ␣ 0+ ␤ 0

2

1 − 冑2

0 1

0 0 0 0

2 − ␣ 0+ ␤ 0 2

at ⌰ = ␲. For the remaining polarization angles, ⌰ ⫽ 0 and ⌰ ⫽ ␲, the TMR is gate voltage dependent and positive. The behavior of the TMR as a function of the gate voltage is shown in Fig. 7. To understand the gate voltage dependence

0 0

0

1

TMR01共␲, ␰兲 = p2 ,

2

0

1 −2 ␣ 0+ ␤ 0 2

␣ 0+ ␤ 0 2

Hence, the TMR vanishes for ⌰ = 0 and takes the constant value,

− ␣ 0− ␤ 0

2

0 0

共48兲

1

1

1

.

−2 0 1 −2

0

−2 0

2

0 0 0 0

1 冑2

0 0 0 0

− ␣ 0+ ␤ 0

0

0

0

0 0

1 冑2

0 0

1 冑2

1 冑2

␣ 0− ␤ 0 2

0

0

具2⬘共+1兲兩 具2⬘共0兲兩 具2⬘共−1兲兩 具2⬙兩 具2⵮兩

关B21/f 2共−

1

1 −2 ␣ 0− ␤ 0 2

0 0

0

2

2

具2g兩

d2↓:

0

2 ␣ 0+ ␤ 0 2

− ␣ 0− ␤ 0

0

␣ 0− ␤ 0 2

1

0 0

−2

− ␣ 0+ ␤ 0

兩3e↓典

0 1 −2 0

1 冑2

1

具2g兩

d1↓:

TMR01 =

0 0

2

0 1

2 − ␣ 0− ␤ 0 2

of the TMR at noncollinear angles, we have to remember that the dot is depleted with increasing ␰. For the 0 ↔ 1 transition, this means that at positive ␰, the dot is predominantly empty, so that an electron that enters the dot also leaves it fast. In this situation, the TMR is finite and its value depends in a complicated way on the amplitude of the exchange field. At negative ␰, the DD is predominantly occupied with an 10 (γBi )/(¯hΓf ((−1)i µi ))

d1↑:

0

兩3e↑典

Resonance 0↔ 1 1↔ 2 2↔ 3 3↔ 4

8 6 4 2 0 -20

-15

-10

-5

0

5

10

15

20

ξ [|b|]

FIG. 3. 共Color online兲 Gate voltage dependence of the renormalized effective exchange splitting entering the conductance formula 关Eq. 共46兲兴. Notice the mirror symmetry of the 0 ↔ 1 with the 3 ↔ 4 curve and of the 1 ↔ 2 with the 2 ↔ 3 one. 245313-11

PHYSICAL REVIEW B 77, 245313 共2008兲

resonance regime

4↔3

3↔2

2 ↔⇐ 1

Conductance G01 [e2/h]

HORNBERGER et al. 1↔0

Conductance G[e2/h]

0.08 0.06 0.04

0.08

0.04 0.02

0.02

-2

-6

-4

-2

0 ξ[|b|]

2

4

6

8

0.08

0.03

Conductance G[e2/h]

0.04 0.02

0.01

-2

-6

-4

-2

0 ξ[|b|]

2

4

6

8

FIG. 4. 共Color online兲 Conductance at low bias for the parallel case ⌰ = 0 共top兲 and antiparallel case ⌰ = ␲ 共bottom兲. Notice the different peak heights and the mirror symmetry with respect to the half-filling value ␰ = 0. The conductance in the antiparallel configuration is always smaller than that in the parallel one.

electron that can now interact with the exchange field, which makes the spin precess and thus eases tunneling out of the dot. Consequently, GNN+1共⌰ , ␰兲 ⬇ GNN+1共0 , ␰兲 and the TMR vanishes. Finally, Fig. 8 illustrates the angular dependence of the normalized conductance for three different values of the gate voltage. We detect a common absolute minimum for the conductance at ⌰ = ␲, i.e., transport is weakened in the antiparallel case. The width of the curves is dependent on the renormalized effective exchange 共␥Bi兲 / 共⌫f兲. The larger its value, the narrower the curves because the spin precession can equilibrate the accumulated spin for all angles but ⌰ = ␲. Notice again the equivalence of the curves belonging to ␰ = ⫾ 2兩b兩 for the 1 ↔ 2 resonance to the curves with ␰ = ⫿ 2兩b兩 for the 3 ↔ 4 resonance.

V. NONLINEAR TRANSPORT

In this section, we present the numerical results deduced from the general master equation 关Eq. 共21兲兴 combined with the current formula 关Eq. 共25兲兴. We show the differential condI ductance dV 共␰ , Vbias兲 for the three distinct angles ⌰ = 0, ⌰

For the resonant 1 ↔ 2 and 2 ↔ 3 transitions, qualitatively analogous results as for the 0 ↔ 1 and 3 ↔ 4 transitions are found. Thus, exemplarily, we only show the angular depenG[e2/h]

T M R34

0.08 0.06 0.04 0.02 0 π 2

π

3 π 2

2π

Θ[rad]

FIG. 5. 共Color online兲 Conductance as a function of the polarization angle and of the gate voltage. The minimal conductance peaks occur as expected at ⌰ = ␲.

2

dence of the normalized conductance in Fig. 9, showing the expected absolute conductance minimum at ⌰ = 0.

2. Resonant regimes N = 1 ^ N = 2 and N = 2 ^ N = 3

6 4 2 0 -2 -4 -6

0 ξ[|b|]

FIG. 6. 共Color online兲 The conductance G01共␰兲 共upper figure兲 关G34共␰兲 共lower figure兲兴 vs gate voltage for different polarization angles. The mirror symmetry of the conductance peaks for the 0 ↔ 1 and 3 ↔ 4 transitions is clearly observed. Notice the excellent agreement between the prediction of the analytical formula 关Eq. 共46兲兴 共continuous lines兲 and the results of a numerical integration of Eq. 共21兲 with Eq. 共25兲 共symbols兲.

T M R01

-8

ξ[|b|]

2

Θ = 0.0π Θ = 0.5π Θ = 0.7π Θ = 0.9π Θ = 1.0π

0.06

0.02

0

0 ξ[|b|]

Conductance G34 [e2/h]

-8

Θ = 0.0π Θ = 0.5π Θ = 0.7π Θ = 0.9π Θ = 1.0π

0.06

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2

Θ = 0.0π Θ = 0.5π Θ = 0.7π Θ = 0.9π Θ = 1.0π

0 ξ[|b|]

2

Θ = 0.0π Θ = 0.5π Θ = 0.7π Θ = 0.9π Θ = 1.0π

0 ξ[|b|]

2

FIG. 7. 共Color online兲 TMR for the 0 ↔ 1 共upper figure兲 and 3 ↔ 4 共lower figure兲 transitions vs gate voltage. The TMR is always positive and for collinear lead magnetizations, ⌰ = 0 and ⌰ = ␲, independent of gate voltage.

245313-12

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

12 8 4 0 -4 -8 -12

0.8

eVbias [|b|]

G01(Θ)/G01(Θ = 0)

1

ξ = −2

0.6

ξ=0 ξ = +2

0.4 0

π 2

π Angle Θ[rad]

3π 2

2π

12 8 4 0 -4 -8 -12

0.8 ξ = −2 ξ=0 ξ = +2

0.4 0

π 2

π Angle Θ[rad]

3π 2

= ␲2 , and ⌰ = ␲, see Fig. 10, top, middle, and bottom, respectively. The results confirm the electron-hole symmetry and the symmetry upon bias voltage inversion I共␰ , Vbias兲 = −I共␰ , −Vbias兲. In all of the three cases, we can nicely see the expected three closed and the two half-open diamonds, where the current is blocked and the electronic number of the double-dot system stays constant. At higher bias voltages, the contribution of excited states is manifested in the appearance of several excitation lines. One clearly sees that transiG12(Θ)/G12(Θ = 0)

1 0.8 0.6

ξ = −2

0.4

ξ = +2

ξ=0 π 2

π Angle Θ[rad]

3π 2

2π

G23(Θ)/G23(Θ = 0)

1 0.8 0.6

ξ = −2

0.4

ξ = +2

ξ=0

0

π 2

π Angle Θ[rad]

3π 2

0.06 0.04 0.02 0

2π

12 8 4 0 -4 -8 -12

2π

FIG. 9. 共Color online兲 G12共⌰兲 / G12共0兲 共upper figure兲 关G23共⌰兲 / G23共0兲 共lower figure兲兴 vs polarization angle. Notice the overall agreement of the analytical predictions 关Eqs. 共45兲 and 共46兲兴 共continuous lines兲 with the data 共symbols兲 coming from numerical solutions of the equations for the reduced density matrix.

eVbias [|b|]

FIG. 8. 共Color online兲 G01共⌰兲 / G01共0兲 共upper figure兲 关G34共⌰兲 / G34共0兲 共lower figure兲兴 vs polarization angle. For all the three chosen values of the gate voltage, the curve displays an absolute minimum at ⌰ = ␲. Notice the overall agreement of the analytical predictions 关Eq. 共46兲兴 given by the continuous curves with outcomes of a numerical solution of the master equation 关Eq. 共21兲兴 together with Eq. 共25兲 共symbols兲.

0

[e2/h] 0.08

eVbias [|b|]

G34(Θ)/G34(Θ = 0)

1

0.6

dI dV

-8

-6

-4

-2

0 ξ[|b|]

2

4

6

8 dI

FIG. 10. 共Color online兲 Differential conductance dV for the parallel ⌰ = 0 共top兲, perpendicular ⌰ = ␲ / 2 共middle兲, and antiparallel ⌰ = ␲ 共bottom兲 configurations. The two half diamond and three diamond regions correspond to bias and gate voltage values where transport is Coulomb blocked. The excitation lines, where excited states start to contribute to resonant transport, are clearly visible in all of the three cases. However, a negative differential conductance is observed in the perpendicular case, while some excitation lines are absent in the antiparallel configuration.

tion lines present in the parallel case are absent in the antiparallel case. Moreover, in the case of noncollinear polarization, ⌰ = ␲ / 2, negative differential conductance 共NDC兲 is observed. In the following, we want not only to explain the origin of these two features, but alongside also give another example for spin-blockade effects, which play a decisive role in the DD physics. As a starting point, we plot in Fig. 11 共top兲 the current through the system for the three different angles ⌰ = 兵0 , ␲2 , ␲其 at a fixed gate voltage ␰ = 4兩b兩 and positive bias voltages. We recognize that for eVBias ⬍ 2.4兩b兩, the current is Coulomb blocked in all of the three cases. In this configuration, exactly one electron stays in the double-dot. From about eVbias ⱖ 2.4兩b兩, the channel where the ground state energies ␮1 and ␮2 are degenerate opens 共兩1e␴典 ↔ 兩2典 transition兲 and the current begins to flow. With increasing bias, more and more transport channels become energetically favorable. In particular, for all the polarization angles ⌰, we observe two consecutive steps corresponding to 兩0典 ↔ 兩1e␴典 and 兩1e␴典 ↔ 兩2⬘典 transitions. The latter, occurring at about eVbias = 4兩b兩, involves the excited two-particle triplet states 兩2⬘共Sz兲典. The next excitation step, indicated with a circle in Fig. 11 共top兲, belongs to the 兩1o␴典 ↔ 兩2⬙典 transition. The associated

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PHYSICAL REVIEW B 77, 245313 共2008兲

HORNBERGER et al. 1.2

Θ=0 Θ = π2 Θ=π

Current I[eΓ]

1 0.8



0.6 0.4 0.2 0

Occ. probability triplet states



 

0

1

2

4

6 8 10 eVbias [|b|]

12

14

16

4

6 8 10 eVbias [|b|]

12

14

16

Θ=0 Θ = π2 Θ=π

0.8 0.6 0.4 0.2 0

0

2

FIG. 11. 共Color online兲 Current 共top兲 and triplet occupation 共bottom兲 for the two collinear 共⌰ = 0 , ⌰ = ␲兲 cases and the perpen␲ dicular case 共⌰ = 2 兲 at a fixed gate voltage ␰ = 4兩b兩. Notice the occurrence of a pronounced negative differential conductance feature for perpendicular polarization ⌰ = ␲ / 2.

line is missing for the antiparallel configuration, as well as the lines corresponding to 兩1o␴典 ↔ 兩2⵮典, 兩1e␴典 ↔ 兩2⬙典, 兩2典 ↔ 兩3o␴典, and 兩1e␴典 ↔ 兩2⵮典. Crucially, in all of these transitions, a two-particle state with total spin zero is involved. In order to explain the absence of these lines, let us, e.g., focus on the first missing step corresponding to the 兩1o␴典 ↔ 兩2⬙典 resonance. In the parallel case 共say, both contacts polarized spin-up兲, there is always an open channel corresponding to the situation in which the spin in the DD is antiparallel to that in the leads 共i.e., 兩1o−典兲. In the antiparallel case 共say, source polarized spin-up and drain polarized spindown兲, originally a spin-down might be present in the dot. An electron that enters the DD from the source must then be spin-up 共in order to form the 兩2⬙典 state兲, but as the drain is down-polarized, it will be the spin-down electron that leaves the DD, which corresponds to a spin flip. Now, the presence of a spin-up electron in the DD prevents a majority 共another spin-up兲 electron from the source to enter the DD, such that we end up in a blocking state. The transition is hence forbidden. A similar yet different spin-blockade effect determines the occupation probabilities for the triplet state 共Fig. 11, bottom兲. Naturally, for all angles, the probability to be in the triplet state increases above the resonance at eVbias = 4兩b兩, but interestingly, such probability is largest in the antiparallel case. This is due to the fact that a majority spin in the parallel configuration 共spin-up兲 can be easily transmitted through the DD via the triplet states 兩2⬘共1兲典 or 兩2⬘共0兲典. In the antiparallel case, however, a blocking state establishes 共say, again source polarized spin-up and drain polarized spin-down兲. Let initially a spin-down electron be present on the DD. From the source electrode, most likely a majority electron 共polarized spin-up兲 will enter the dot. Now, just as in the previous case, the consecutive tunneling event will cause a spin flip in the DD because the spin-down electron 共majority electron of the

drain兲 will leave the dot. So, the DD is finally in a spin-up state, and once the next majority spin-up electron from the source enters, the DD ends up in the triplet state 兩2⬘共+1兲典 and will remain there for a long time due to the fact that the majority spins in the drain are down polarized. Hence, the triplet state 兩2⬘共+1兲典 acts as a trapping state. Notice that the two distinct spin-blockade effects are different from the Pauli spin-blockade discussed in the DD literature.46–49 Moreover, the second effect, relying on the existence of degenerate triplet states, is also different from the spin-blockade found in Ref. 30 for a single-level quantum dot. Finally, let us turn to the negative differential conductance, which occurs for noncollinearly polarized leads 共see the dashed blue lines in Fig. 11兲 and which we find to become more evident for higher polarizations 共not shown兲. By neglecting the exchange field, we would just expect the magnitude of the current for the noncollinear polarizations to lie somewhere in between the values for the parallel and the antiparallel current because the noncollinear polarization could, in principle, be rewritten as a linear combination of the parallel and the antiparallel configuration. Now, the effect of the exchange is to cause precession and therewith equilibration of the accumulating spin, which corresponds to shifting the balance in favor of the parallel configuration, i.e., enhancing the current. The decisive point is that the exchange field is not only gate dependent but also bias voltage dependent and reaches a minimum around eVbias ⬇ 8兩b兩. This explains the decrease of the current up to this point. Afterward, the influence of the spin precession regains weight. The same consideration applies for the other NDC regions observed in Fig. 10, e.g., in the gate voltage region ␰ ⬇ 2兩b兩 involving the N = 0 ↔ N = 1 transition, as described in Ref. 30.

VI. EFFECTS OF AN EXTERNAL MAGNETIC FIELD

In this section, we wish to discuss the qualitative changes brought by an external magnetic field applied to the DD. Specifically, the magnetic field is assumed to be parallel to the magnetization direction of the drain. For simplicity, we focus on the experimental standard case of parallel and antiparallel lead polarizations and of low-bias voltages. Then, the magnetic field causes an energy shift ⫿EZeeman depending on whether the electron spin is parallel or antiparallel, respectively, to it. For collinear polarization angles, the principal-part contributions vanish, and the equations for the RDM are easily obtained. We exemplarily report results for 0 ↔ 1 and 1 ↔ 2 transitions. Let us then consider the parameter regime nearby the 0 ↔ 1 resonance and setup a system of three equations with three unknown variables, W0, W1↑, and W1↓. The first equation corresponds to the normalization condition W1↑ + W1↓ + W0 = 1. The remaining equations are the equations of motion for W1↑/↓, which can be written as

245313-14

˙ = − ␲ 兺 兩t 兩2关F− 共␮ 兲W − F+ 共␮ 兲W 兴, 共49兲 W 1↑ ␣ 1↑ 0 ␣↑ 1↑ ␣↑ 1↑ ប ␣=s,d

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…

Conductance G [e2/h]

˙ = − ␲ 兺 兩t␣兩2关F− 共␮ 兲W − F+ 共␮ 兲W 兴, 共50兲 W 1↓ 1↓ 0 ␣↓ 1↓ ␣↓ 1↓ ប ␣=s,d where

␮1↑/↓ = ␮1 ⫿ EZeeman ,

共51兲

F␣⫾␴ 共E兲 = D␣␴䉺 f ⫾共E兲. 䉺

and D␣↑/↓ = D␣⫾␣. On the other hand, Ds↑/↓ = Ds⫿s and Dd↑/↓ = Dd⫾d in the antiparallel case. Upon considering symmetric contacts 共t1 = t2 = t, D1 = D2 = D兲, we find the following in the parallel case:

=

⌫e2 f共− ␮1↑兲f共− ␮1↓兲 8␤

⫻

p关f共␮1↑兲 − f共␮1↓兲兴 + f共␮1↑兲 + f共␮1↓兲 . f共− ␮1↑兲f共␮1↓兲 + f共− ␮1↑兲f共− ␮1↓兲 + f共␮1↑兲f共− ␮1↓兲 共52兲

For the antiparallel case, we obtain G01共⌰ = ␲兲 = G01共⌰ = 0兲

1 − p2关f共␮1↑兲 + f共␮1↓兲兴 . p关f共␮1↑兲 − f共␮1↓兲兴 + f共␮1↑兲 + f共␮1↓兲 共53兲

Analogously, we find the following for the 1 ↔ 2 transition: G12共⌰ = 0兲 =

⌫e2k+2 f共␮2↑兲f共␮2↓兲 2␤

⫻

p关f共− ␮2↓兲 − f共− ␮2↑兲兴 + f共− ␮2↑兲 + f共− ␮2↓兲 , f共− ␮2↑兲f共␮2↓兲 + f共− ␮2↑兲f共− ␮2↓兲 + f共␮2↑兲f共− ␮2↓兲 共54兲

G12共⌰ = ␲兲 = G12共⌰ = 0兲 ⫻

1 − p2关f共− ␮2↑兲 + f共− ␮2↓兲兴 . p关f共− ␮2↓兲 − f共− ␮2↑兲兴 + f共− ␮2↑兲 + f共− ␮2↓兲 共55兲

The remaining resonances are analogously calculated. Figure 12 shows the four conductance resonances for the parallel and antiparallel configurations. Strikingly, the applied magnetic field breaks the symmetry between tunneling regimes 0 ↔ 1 and 3 ↔ 4, as well as between 1 ↔ 2 and 2 ↔ 3 resonances in case of parallel contact polarizations. The reason for this behavior is the following: in the low-bias regime, transitions between ground states dominate transport. In particular, the magnetic field removes the spin degeneracy of 兩1e␴典 and 兩3o␴典 states, such that states with spin aligned to the external magnetic field are energetically favored. Therefore, the transport electron in tunneling regime 0 ↔ 1 is a majority spin carrier. For the case 3 ↔ 4, however, two of the three electrons of the ground state 兩3o↑典 have spin-up, such that the fourth electron that can be added to the DD has to be

2↔3

0.06 0.04

3↔4

1↔2

0.02 -6

-4

-2

0

ξ[|b|]

2

4

6

FIG. 12. 共Color online兲 Conductance vs gate voltage for parallel 共continuous line兲 and antiparallel 共dashed lines兲 contact configurations and Zeeman splitting EZeeman = 0.05兩b兩. The magnetic field breaks the mirror symmetry with respect to the gate voltage in the parallel configuration.

a minority spin carrier. Therefore, the conductance gets diminished with respect to the 0 ↔ 1 transition, and the mirror symmetry present in the zero field case is broken. Analogously, the broken symmetry in the case of the 1 ↔ 2 and 2 ↔ 3 transitions can be understood. Correspondingly, the TMR can become negative for values of the gate voltages around the 2 ↔ 3 and 3 ↔ 4 resonances. We observe that a negative TMR has recently been predicted in Ref. 42 for the case of a single impurity Anderson model with orbital and spin degeneracies 共see Fig. 13兲. In that work, a negative TMR arises due to the assumption that multiple reflections at the interface cause spin-dependent energy shifts. In our approach, however, where the contribution from the reflection Hamiltonian is treated to the lowest order, see Eq. 共19兲, such spin-dependent energies originate from the magnetic-fieldinduced Zeeman splitting. VII. CONCLUSIONS

In summary, we have evaluated linear and nonlinear transport through a double-quantum-dot 共DD兲 coupled to polarized leads with arbitrary polarization directions. Due to strong Coulomb interactions, the DD operates as a singleelectron transistor, a F-SET, at low enough temperatures. A detailed analysis of the current-voltage characteristics of the DD and comparison with results of previous studies on other 6

0.4 0.2

TMR

G01共⌰ = 0兲

0↔1

0.08

-

0.0

-0.2 -0.4 -8

-4

0

4

8

ξ[|b|] FIG. 13. 共Color online兲 TMR vs gate voltage in the presence of an external magnetic field. In contrast to the zero field case, the TMR can become negative in the vicinity of the 2 ↔ 3 and 3 ↔ 4 resonances.

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HORNBERGER et al.

F-SET systems with noncollinear polarization 共a single-level quantum dot,30 a metallic island,28 and a carbon nanotube38兲 brings us to the identification of universal behaviors of a F-SET, i.e., a behavior shared by any of those F-SETs, independent of the specific kind of conductor that is considered as the central system, as well as system-specific features. The presence of an interfacial exchange field together with an interaction-induced one is universal. These exchange fields can act only for noncollinear polarizations and cause a precession of the accumulated spin on the dot and therewith ease the tunneling. This effect has various implications. It determines, e.g., the gate and angular dependences, as well as the height of single conductance peaks and can yield negative differential conductance features. Another universal feature is the occurrence of a negative tunneling magnetoresistance—even in the weak-tunneling limit—if a Zeeman-splitting exists. The following features are specific to the DD system: in the low-bias regime, the problem can be analytically solved, and for tunneling regimes 0 ↔ 1 and 1 ↔ 2 共2 ↔ 3 and 3 ↔ 4, respectively兲, the system behaves equivalent to a single-level quantum dot, where the Coulomb blockade peaks are found to be mirror symmetric with respect to the charge neutrality point. This mirror symmetry reflects the electron-hole symmetries of a system and is therefore typical for the DD, as well as the ratio of the peak heights. An external magnetic field lifts this symmetry and can cause a negative tunneling magnetoresistance. In the nonlinear bias regime, the presence of various excited states gives rise to interesting DD specific features. For example, a suppression of several excitation lines for an antiparallel lead configuration originates from a spin-blockade effect. It occurs because a trapping state is formed whenever a transition involves a two-electron state with total spin zero. A second spin-blockade effect we described involves the two-electron triplet state. The common mechanism of these two spin-blockades is the following: in both cases, a tunneling event can only occur if initially the dot is populated with an unpaired electron possessing the majority spin of the drain. The second step is that a majority electron of the source will enter, forming a spin-zero state.

共N兲 ␳˙ nm 共t兲 = −

␲ 兺 兩t␣兩2 兺 ប ␣=s,d ␴ ,␴ 䉺

+ ⌽ ␣␴䉺␴⬘

䉺

⬘ 䉺

再兺

l苸兩N−1典

,

兺

共0兲 ,N典 j苸兩EN

,

兺

Then, the first electron can leave the dot, causing a spin flip. If the triplet state is involved, the dot will be left in a trapping state once a second majority electron from the source enters. Otherwise, we are directly in a blocking state. Finally, for noncollinear lead polarizations, negative differential conductance can be observed. All in all, due to their universality and the multiplicity of their properties, F-SETs based on DD systems seem good candidates for future magnetoelectronic devices. ACKNOWLEDGMENTS

Financial support under the DFG Program Nos. SFB689 and SPP1243 is acknowledged. APPENDIX A: MATRIX ELEMENTS OF THE DOT OPERATORS

Tables II–V show all the possible matrix elements 具N − 1兩d␣␴䉺兩N典 and 具N兩d␣† ␴ 兩N − 1典 with ␣ = 兵1 , 2其 and ␴䉺 䉺 = 兵↑ , ↓其, which occur in the master equations 关Eqs. 共B1兲 and 共B2兲兴. Notice that we only need to illustrate either the matrix elements 具N − 1兩d␣␴䉺兩N典 or 具N兩d␣† ␴ 兩N − 1典 because they are 䉺 complex conjugated to each other. APPENDIX B: MASTER EQUATION FOR THE REDUCED DENSITY MATRIX IN THE LINEAR REGIME

We explicitly report here the coupled equations of motion 共N兲 共N+1兲 共t兲 and ␳˙ nm 共t兲 of the RDM to be solved in for elements ␳˙ nm the low-bias regime. They are obtained from the generalized master equation 关Eq. 共21兲兴 upon observing that 共i兲 in the linear regime, terms that couple states with particle numbers unlike N and N + 1 can be neglected; 共ii兲 we can reduce the sum over h and h⬘ and over l , l⬘ only to energy ground states. In the remaining energy nonconserving terms, the sum has to 共0兲 − EN共0兲 being the go also over excited states. With ␮N+1 ª EN+1 chemical potential, we finally arrive at the following two master equations:

,

共0兲 h,h⬘苸兩EN+1 ,N+1典

兺 ˆ

h苸兩N+1典

冎再

⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␮N+1兲共d␣␴䉺兲nh共d␣␴⬘ 兲hj␳共N兲 jm 共t兲 䉺 䉺 䉺 䉺 +

†

i + i − † ⴱ P␣␴ ␴⬘ 共␧hˆ − ␧ j兲共d␣␴䉺兲nhˆ共d␣␴⬘ 兲hˆ j␳共N兲 关P␣␴ ␴⬘ 共␧ j − ␧l兲 + R␣␴䉺␴⬘ 兴共d␣† ␴ 兲nl共d␣␴⬘ 兲lj␳共N兲 jm 共t兲 − ⌽␣␴ ␴⬘ jm 共t兲 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺 䉺␲ 䉺 ␲

i + + † † ˆ ˆ P 共␧hˆ − ␧ j兲␳共N兲 + ⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␮N+1兲␳共N兲 nj 共t兲共d␣␴䉺兲 jh共d␣␴⬘ 兲hm − ⌽␣␴䉺␴䉺 nj 共t兲共d␣␴䉺兲 jh共d␣␴⬘ 兲hm ⬘ 䉺 䉺 䉺 䉺 䉺 ␲ ␣␴䉺␴䉺⬘

冎

i − ⴱ − † 共N+1兲 + ⌽␣␴ ␴⬘ 关P␣␴ ␴⬘ 共␧ j − ␧l兲 + R␣␴䉺␴⬘ 兴␳共N兲 共t兲共d␣† ␴ 兲 jl共d␣␴⬘ 兲lm − 2⌽␣␴䉺␴⬘ F␣␴ ␴⬘ 共␮N+1兲共d␣␴䉺兲nh⬘␳h⬘h 共t兲共d␣␴⬘ 兲hm , 䉺 䉺 nj 䉺 䉺 䉺 䉺 䉺 䉺␲ 䉺 䉺 䉺 共B1兲 245313-16

PHYSICAL REVIEW B 77, 245313 共2008兲

TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM… 共N+1兲 ␳˙ nm 共t兲 = −

␲ 兺 兩t␣兩2 兺 ប ␣=s,d ␴ ,␴ 䉺

⬘ 䉺

再

兺

,

ˆl苸兩E ,N典 N

兺

,

共0兲 l,l⬘苸兩EN ,N典

兺

共0兲 j苸兩EN+1 ,N+1典

兺

,

h苸兩EN+2,N+2典

冎再

i + P ⬘ 共␧h − ␧ j兲 䉺 ␲ ␣␴䉺␴䉺

⌽ ␣␴䉺␴⬘

i − † ⴱ − ⴱ † 共N+1兲 ˆ ⫻共d␣␴䉺兲nh共d␣␴⬘ 兲hj␳共N+1兲 jm 共t兲 + ⌽␣␴䉺␴⬘ F␣␴䉺␴⬘ 共␮N+1兲共d␣␴䉺兲nl共d␣␴䉺 ⬘ 兲lj␳ jm 共t兲 − ⌽␣␴䉺␴䉺 ⬘兴 ⬘ ␲ 关P␣␴䉺␴䉺 ⬘ 共␧ j − ␧l兲 + R␣␴䉺␴䉺 䉺 䉺 䉺 ⫻共d␣† ␴ 兲nlˆ共d␣␴⬘ 兲ˆl j␳共N+1兲 jm 共t兲 − ⌽␣␴䉺␴⬘ 䉺

䉺

䉺

i + † ⴱ − P 共␧h − ␧ j兲␳共N+1兲 共t兲共d␣␴䉺兲 jh共d␣␴⬘ 兲hm + ⌽␣␴ ␴⬘ 关F␣␴ ␴⬘ 共␮N+1兲兴␳共N+1兲 共t兲 nj nj 䉺 䉺 䉺 䉺 䉺 ␲ ␣␴䉺␴䉺⬘

i − ⴱ ⴱ + 共t兲共d␣† ␴ 兲 jlˆ共d␣␴⬘ 兲ˆlm − 2⌽␣␴ ␴⬘ F␣␴ ␴⬘ 共␮N+1兲 ⫻共d␣† ␴ 兲 jlˆ共d␣␴⬘ 兲ˆlm + ⌽␣␴ ␴⬘ 关P␣␴ ␴⬘ 共␧ j − ␧lˆ兲 + R␣␴䉺␴⬘ 兴␳共N+1兲 䉺 䉺 䉺 䉺 nj 䉺 䉺 䉺␲ 䉺 䉺 䉺 䉺 䉺 䉺 共N兲

冎

⫻共d␣† ␴ 兲nl⬘␳l⬘l 共t兲共d␣␴⬘ 兲lm . 䉺 䉺

共B2兲

Notice that we kept the sums over excited states l , hˆ in Eq. 共B1兲 and ˆl , h in Eq. 共B2兲, which are responsible for the virtual transitions.

1 S.

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