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arXiv:1010.2729v2 [cond-mat.mes-hall] 27 Nov 2010

Spin-polarized currents in double and triple quantum dots driven by ac magnetic fields

Maria Busl and Gloria Platero

Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain

We analyze transport through both a double quantum dot and a triple quantum dot with inho-

mogeneous Zeeman splittings in the presence of crossed dc and ac magnetic fields. We find that

strongly spin-polarized current can be achieved by tuning the relative energies of the Zeeman-split

levels of the dots, by means of electric gate voltages: depending on the energy level detuning, the

double quantum dot works either as spin-up or spin-down filter. We show that a triple quantum

dot in series under crossed dc and ac magnetic fields can act not only as spin-filter but also as

spin-inverter.

PACS numbers: 72.25.Dc, 73.21.La

I.INTRODUCTION

A key aim in spintronics is the realization of spin-based

quantum information devices, where coherent electron

spin manipulation is a fundamental issue.1,2In semicon-

ductor quantum dots, coherent electron spin manipula-

tion can be realized by electron spin resonance (ESR),

where an oscillating magnetic field is applied to the sam-

ple in order to rotate the electron spin.3–8Together with

ESR, electron dipole spin resonance techniques — which

combine ac electric fields with spin-orbit interaction9or

with a dc magnetic field gradient10— have been imple-

mented in order to measure coherent rotations of one

single electron spin3,9in double quantum dots (DQDs).

Coherent spin rotations of one single spin have also been

proposed theoretically in triple quantum dots (TQDs)11

under crossed ac and dc magnetic fields.

In ESR experiments in quantum dot arrays, an impor-

tant issue is to individually address the electron spin in

each quantum dot. To this end, it has been proposed

to tune the Zeeman splitting, in order to manipulate the

electron spin independently in each dot.12The Zeeman

splitting in a quantum dot is determined by the inten-

sity of the applied dc magnetic field and the electron

g factor, ∆Z = gµBBdc. Hence different Zeeman split-

tings can occur in quantum dot arrays where the dots

have different g factors, or as well by applying different

magnetic fields to each quantum dot. Both alternatives

have been realized experimentally very recently: vertical

DQDs made out of different materials — e.g. GaAs and

InGaAs — show different g factors13and on the other

hand, in a sample with a spatially homogeneous g factor,

an additional micro-ferromagnet placed nearby creates a

different external magnetic field Bdcin each dot.10

The next logic step from DQDs to networks of quan-

tum dots is a TQD, in linear or triangular arrangement.

Both versions have been realized experimentally in the

last few years,14–16where tunneling spectroscopy and sta-

bility diagram measurements have been performed in or-

der to gain a deeper insight into the electronic configura-

tions in TQDs, which is necessary for potential three-spin

qubit applications. On the theoretical side, next to fun-

damental studies of their eigenenergy spectrum,17TQDs

have attracted interest mostly in a triangular arrange-

ment, where the system symmetry gives rise to funda-

mental coherence phenomena. In this context, so-called

“dark states”8,18,19and Aharonov-Bohm oscillations8,20

have been studied. TQDs have been used as a testing

ground for Kondo physics21and have been proposed as

current rectifiers22,23and spin-entanglers.24

In the present work, we are interested in single electron

manipulation, and therefore study theoretically transport

through both double and triple quantum dots. We calcu-

late the current and current spin polarization through a

DQD and a linear TQD array exposed to crossed dc and

ac magnetic fields. We consider an inhomogeneous dc

magnetic field that produces different Zeeman splittings

in the dots, while the g factor is the same in both dots.

For DQDs, a regime is considered where the system is

occupied either by zero or one electron. For TQDs, the

corresponding features are discussed for one or two elec-

trons in the system. With the single electron spin levels

resolved in each quantum dot, interdot tunneling is gov-

erned by definite spin selection rules, i.e. tunneling from

one dot to the other is only possible when two equal spin

levels are aligned. However, when an ac magnetic field is

applied, it rotates the spin and allows for spin-flip pro-

cesses along the tunneling that can lead to new features

in the current. This effect of an ac magnetic field has

been explored previously in a DQD,3where the authors

report single electron spin rotations by a combination of

an ac magnetic field and sharp electric pulses.

In our work, as will be discussed in more detail below,

we will focus our attention on the polarizing effect of

an ac field, i.e. we will show that the combination of

inhomogeneous dc and ac magnetic fields in DQDs and

TQDs allows for the creation of spin-polarized currents

and thus for the design of spin-filters and spin-inverters.

The paper is organized as follows: In section II we

introduce the model and the technique used to calculate

transport through a DQD and TQD. Section III discusses

in detail the results of this paper. In section IIIA we first

briefly review the main result of a related experimental

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FIG. 1: (Color online) Schematic diagram of a DQD (above)

and TQD (below) exposed to crossed dc (Bdc) and ac (Bac)

magnetic fields. The electron spin is rotated once the ac fre-

quency matches the Zeeman splitting in one of the dots. In

the TQD, one electron is confined in the left dot (dot 1), such

that only an electron with opposite spin can enter the TQD.

The dots are coupled coherently by tunneling amplitudes tij

and incoherently to leads by rates ΓL and ΓR.

work that has recently been reported in the literature and

is important for further understanding. We then proceed

in the following paragraphs IIIB - IIID with a detailed

analysis of the main results of this paper, namely the

spin-polarized currents produced by a combination of dc

and ac magnetic fields at certain interdot level detun-

ings. The role of the system parameters involved in the

polarization mechanism — Zeeman splitting difference,

ac field amplitude and frequency and interdot tunneling

amplitude — is discussed. In section IIIE we present the

corresponding results obtained for a TQD. We end with

a summary of the main results in section IV.

II.MODEL AND TECHNIQUE

We consider a quantum dot array as shown schemati-

cally in Fig. 1. The dots are coupled to each other coher-

ently by a tunneling amplitude tij and are weakly con-

nected to source and drain contacts by rates ΓLand ΓR.

The total Hamiltonian of the system is:

H = H0

Dots+ H0

tij+ H0

B(t) + HT+ HLeads,

(1)

where the individual terms are

H0

Dots=

?

iσ

ξiσˆ c†

iσˆ ciσ+

?

i

Uiˆ ni↑ˆ ni↓+1

2

?

i?=j

V ˆ niˆ nj

H0

tij= −

?

ijσ

?

?

tij(ˆ c†

iσˆ cjσ+ ˆ c†

jσˆ ciσ)

HLeads=

l∈L,R,kσ

ǫlkσˆd†

lkσˆdlkσ

(2)

HT=

l∈L,R,kσ

γl(ˆd†

lkσˆ clσ+ ˆ c†

lσˆdlkσ).

The first term, H0

tum dots, with electrons coupled electrostatically. Here,

ξi stands for the single energy spectrum of an electron

located in dot i, and Uiand V are are the intra- and the

inter-dot Coulomb repulsion respectively. H0

the coherent tunneling between the dots, which in the

case of a DQD is given by t12and in a TQD by t12and

t23. The quantum dot array is coupled to leads which

are described by HLeads, and the coupling of the array to

the leads is given by HT. The magnetic field Hamilto-

nian consists of two parts, coming from a dc field Bdcin

z-direction, and an ac field Bacapplied in xy-direction:

Dots, describes an isolated array of quan-

tijdescribes

H0

B(t) =

?

i

[∆iSzi+ Bac(cos(ωt)Sxi+ sin(ωt)Syi)],

(3)

being Si=1

dot, and the sum running over index i = 1,2 for the DQD

and i = 1,2,3 for the TQD. Bdchas a different intensity

in each dot and thus produces different Zeeman splittings

∆i= gµBdci, while we consider the dots with equal g fac-

tor. Bacinduces spin rotations when its frequency fulfills

the resonance condition ω = ∆i. The time-dependent

Hamiltonian can be transformed by means of a unitary

transformation5,8U(t) = exp(−i[ωt?

tating reference frame. The resulting time-independent

Hamiltonian is then:

2

?

σσ′ ˆ c†

iσσσσ′ˆ ciσ′ the spin operator of the ith

iSzi]) into the ro-

H0

B=

?

i

[(∆i− ?ω)Szi+ BacSxi].

(4)

The dynamics of the system is given by the time evolu-

tion of the reduced density matrix elements ρmn, whose

equations of motion read, within the Born-Markov-

approximation:

˙ ρmn(t) = − i?m|[H0

+

Dots+ H0

(Γnkρkk− Γknρnn)δmn

tij+ H0

B,ρ]|n?

(5)

?

k?=n

− Λmnρmn(1 − δmn)

The commutator accounts for the coherent dynamics in

the quantum dot array, tunneling to and from the leads

is governed by transition rates Γmn from state |n? to

state |m?, and decoherence due to interaction with the

reservoir is considered in the term Λmn=1

2

?

k(Γkm+

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Γkn). The transition rates are calculated using Fermi’s

golden rule:

Γmn=

?

l=L,R

Γl[f(Em− En− µl)δNm,Nn+1+

(1 − f(Em− En− µl))δNm,Nn−1],

(6)

where Em−Enis the energy difference between states |m?

and |n? of the isolated quantum dot array, and ΓL,R =

2πDL,R|γL,R|2are the tunneling rates for each lead. The

density of states DL,Rand the tunneling couplings γL,R

are assumed to be energy independent. We set ΓL =

ΓR= Γ.

We consider strong Coulomb repulsion, such that the

DQD can be occupied with at most one extra electron.

It is then described by a basis of 5 states, namely: |0,0?,

|↑,0?, |↓,0?, |0,↑?, |0,↓?. With a bias applied from left to

right, current I flows whenever dot 2 is occupied:

IDQD= Γ(ρ|0,↑?+ ρ|0,↓?)

(7)

In the TQD, one electron is confined in the left dot (dot 1,

see Fig. 1, lower panel), and the chemical potential of the

left lead is such that only an electron with the opposite

spin can enter the TQD. Considering here as well strong

Coulomb repulsion, we allow only for one additional elec-

tron to enter the TQD. The full two-electron basis for the

TQD contains fifteen two-electron states, and one zero-

and six one-electron states. For the scope of this paper,

it is sufficient to look at transport around the triple point

(2,0,0) ↔ (1,1,0) ↔ (1,0,1). The number of relevant

basis states is then reduced to eleven, which are

• 1-electron states: |↑,0,0?, |↓,0,0?

• 2-electron states: |↑,↑,0?, |↑,↓,0?, |↓,↑,0?, |↓,↓,0?

|↑,0,↑?, |↑,0,↓?, |↓,0,↑?, |↓,0,↓?

|↑↓,0,0?.

The current from left to right through the TQD is calcu-

lated summing over all states that include an electron in

the right dot (dot 3):

ITQD= Γ?ρ|↑,0,↑?+ ρ|↑,0,↓?+ ρ|↓,0,↑?+ ρ|↓,0,↓?

The spin-resolved currents hence are

?

(8)

I↑= Γ?ρ|↑,0,↑?+ ρ|↓,0,↑?

I↓= Γ?ρ|↑,0,↓?+ ρ|↓,0,↓?

?

?.

(9)

The spin polarization of the current is defined as

P =I↑− I↓

I↑+ I↓,

(10)

where I↑(I↓) is the ↑(↓)-current.

FIG. 2: (Color online) Current I versus detuning ǫ in an un-

driven DQD with different Zeeman splittings. Here maximal

current flows, when ǫ = 0, and this central current decreases

for increasing δ, since then parallel spin-levels are more sepa-

rate. Parameters (e = ? = 1, in meV): t12 = 0.005, Γ = 0.001,

∆1 = 0.025 (Bdc ≈ 1T), and the current I is normalized in

units of the hopping Γ to the leads.

III. RESULTS

A.Undriven case: Bac = 0

Let us now start to describe transport through a DQD

(see Fig. 1, upper panel).

produce within our theoretical framework the results re-

cently reported by Huang et al..13The authors have

shown that in transport through DQDs with different

Zeeman splittings a so-called spin bottleneck situation

can occur: When either ↑- or ↓-levels are aligned, trans-

port is suppressed, whereas the current is largest in the

configuration where the interdot level detuning ǫ is set to

half the Zeeman energy difference.

Applying a dc magnetic field in z-direction produces a

Zeeman splitting ∆z, which we consider inhomogeneous:

∆1 ?= ∆2, and δ = ∆2− ∆1.

onto the ↑(↓)-level in dot 1 that is far from resonance

from the corresponding spin-level in dot 2, a spin block-

ade or bottleneck situation arises: Spin is conserved at

tunneling, so the electron remains in dot 1 without being

able to tunnel to dot 2. This blockade is only relieved

by a finite level broadening and coupling to the leads.

The maximal current occurs then for the most symmet-

ric level arrangement, that is when neither ↑- nor ↓-levels

are in resonance, but when they are symmetrically placed

around each other (see Fig. 2). Increasing the Zeeman

splitting difference δ maintains the bottleneck situation,

but the central current decreases, since it is a consequence

of the level hybridization of the same spin-levels due to

tunneling. Hence, the further separated they are, the less

current flows. Notice that the current only depends on

the Zeeman splitting difference δ and not on the absolute

values.

Interdot tunneling conserves spin and the current

through the sample is completely unpolarized. In ac mag-

netic fields however, the electron spin undergoes rotations

and the spin selection rules thus do not apply any more.

In this section we will re-

If an electron tunnels

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For certain detunings, this will lead to spin-polarized cur-

rents, as we will see in the next section.

B. Resonance condition: ω = ∆1

With a circularly polarized ac magnetic field Bac ap-

plied to the DQD, the transformed Hamiltonian H0

reads:

H0=

−∆1

2+ω

Bac

2

−t12

0

2

Bac

2

2−ω

0

−t12

−t12

0

2+ω

Bac

0

∆1

2

−t12

Bac

2

−∆2

2− ǫ

2

∆2

2−ω

2− ǫ

,

(11)

where ǫ is the detuning between dot 1 and dot 2.

For the ease of its analysis, Hamiltonian (11) can be

seen as a a pair of two-level systems coupled by t12. In

a two-level system, the important physical quantities are

the energy difference (“detuning”) of the two levels and

the coupling between them. In the present case, note

that t12 couples only levels with the same spin, which

are detuned by ±δ/2+ǫ, where δ = ∆2−∆1. Moreover,

within each dot the different spin-levels are coupled by

Bac/2 and “detuned” by ω − ∆1,2(see diagonal elements

in (11)). Therefore, depending on the ac frequency ω, the

energy levels in either left or right dot are renormalized

to the same energy. In the other dot however, since there

ω ≷ ∆i, the renormalized splitting between the spin-

levels becomes smaller when ω < ∆ior bigger for ω > ∆i.

We will focus first on the resonance condition ω = ∆1,

as it is the most relevant here.

In order to understand the effect of Bacon the system,

let us look at the eigenstates of the isolated dots 1 and

2. In dot 1, since ω = ∆1, the eigenstates are |ψ1?±=

1

√2(|↑1? ± |↓1?) and their eigenenergies differ by Bac. In

dot 2 however, since it is out of resonance, the eigenstates

depend both on δ and Bac:

|ψ2?±=

1

N±(−δ ±?B2

ac+ δ2

Bac

| ↑2? + | ↓2?)

(12)

Here N±=√2/Bac

??

B2

ac+ δ2± δ?B2

?B2

ac+ δ2

?

are the

normalization factors. The eigenenergies associated to

these states are separated by

forward to show that for Bac≪ δ, the eigenstates in dot 2

are almost pure ↑(↓)-states, i.e. the spin-mixing is weak.

Regarding the detuning ǫ, we distinguish three different

level arrangements, see Fig. 3, upper panel: In case I, the

↑- and ↓-levels in dot 1 are aligned with the ↑-level in dot

2, case II is the symmetric situation, and in case III the

levels in dot 1 are in resonance with the ↓-level in dot 2.

In Fig. 3, lower panels, we plot the current I through

the driven DQD and the polarization P as a function of

the level detuning ǫ. It shows two peaks at ǫ ≈ ±δ/2. At

these lateral peaks, corresponding to case I and III, the

current is strongly spin-polarized: an electron in dot 1 is

ac+ δ2. It is straight-

FIG. 3: (Color online) Upper panel: Energy level distribution

for different detunings ǫ in a DQD driven by Bac. When ω =

∆1, the levels in dot 1 renormalize to the same energy (their

eigenenergies are split by Bac, see text), and the levels in dot

2 get closer or farther apart than in the undriven case. Middle

panel: Spin-resolved currents I↑ and I↓ vs. detuning ǫ. At

ǫ ≈ ±δ/2, the current is strongly ↑(↓)-polarized, compared to

the undriven current I0. Lower panel: Polarization P versus

the detuning ǫ. Note the strong polarization (P ≈ ±1) around

ǫ ≈ ±δ/2. Parameters in meV (e = ? = 1): Γ = 0.001,

t12 = 0.005, Bac = 0.005 (≈ 0.2T), ∆1 = 0.025 (Bz1 ≈ 1T),

∆2 = 0.1.

rotated by the ac field which breaks the spin bottleneck

and the electron can thus tunnel to dot 2, where the

spin-levels are almost pure, or — speaking in terms of

the rotating field — the ac frequency in dot 2 is far off

resonance and cannot rotate the electron there. We thus

arrive at one of the main result of this paper: under

the condition ω = ∆1, dot 2 acts as a spin-filter, and it

depends on ǫ, whether it filters ↑- or ↓-electrons. Notice

that the current I only depends on δ and not on the

absolute values ∆1,2.

For the purpose of a spin-filter, one has to answer the

question as to how reliable the mechanism is, and how it

depends on the different system parameters. Both strong

polarization and measurable currents are desirable. Here,

we discuss the sensibility of the spin filtering mechanism

towards the interplay between tunneling t12, ac field in-

tensity Bacand Zeeman splitting difference δ.

In order to get more insight into the problem, we obtain

the current I analytically for certain limits: At symmet-

ric detuning ǫ = 0 (case II), the current is unpolarized

and reads

I0

Γ=

4t2

12) + (Γ2+ δ2)(Γ2+ 10t2

12(4B2

ac+ Γ2+ δ2)

4B2

ac(Γ2+ 10t2

12+ δ2).

(13)

I0decreases for large δ and increases with growing Bac.

In the limit of very large t12, the total current I saturates

to I/Γ(t12→ ∞) = 2/5. For the limiting cases of Bacwe

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FIG. 4: (Color online) Polarization P versus t12 , Bac and δ

in ac-driven DQD at detuning ǫ = ±δ/2 for ω = ∆1: Left

and middle panel: For both small t12 and Bac, spin-polarized

current flows. |P| becomes smaller as Bac and t12grow. Right

panel: P is zero at δ = 0 and increases with δ. Parameters

see Fig. 3.

get

lim

Bac→0

I↑,↓

Γ

I↑,↓

Γ

=

2t2

12+ 4ǫ2+ δ2

2t2

12

Γ2+ 10t2

12

Γ2+ 10t2

(14)

lim

Bac→∞

=

12+ 4ǫ2.

(15)

For Bac → 0, i.e. in the undriven case, the current

is unpolarized and maximal at ǫ = 0 and decreases for

growing δ, see Eq.(14). Notice that in the opposite limit,

i.e. for large Bac(Eq.(15)), the current is the same as in

the undriven case for δ = 0. In this case, the difference

of the eigenenergies in each isolated dot becomes Bacin

both dots and the spins are mixed almost equally strongly.

The polarized side-peaks therefore disappear in favor of

the unpolarized central current peak, see also Eq.(13).

Numerical analysis for intermediate field and tunnel-

ing amplitude yields that when t12 and Bac become of

the order of δ, the current is practically unpolarized. We

find that at Bac/t12≈ 1.5, the polarization is strongest,

when δ/t12 is at least one order of magnitude bigger

than Bac/t12. It can be shown numerically that for

t12,Bac≪ δ the position ǫ of the side-peaks is ǫ ≈ ±δ/2.

The larger δ, the further separated the peaks correspond-

ing to I↑ and I↓. As a consequence, also the polariza-

tion is stronger for large δ, since the overlap of the spin-

resolved currents tends to zero.

In order to illustrate the effect of tunneling t12, ac field

intensity Bac and Zeeman splitting difference δ on the

polarization P, we calculate P at ǫ = ±δ/2 (Fig. 4). In

the left and middle panel, one can appreciate that for

both small t12and Bac, P ≈ ±1, and it becomes smaller

as t12and Bacincrease (for constant δ). The right panel

in Fig. 4 shows the polarization for increasing δ: the

larger δ, the stronger P .

C.Resonance condition: ω = ∆2

When the ac field instead fulfills the resonance con-

dition ω = ∆2, the energy renormalization due to ω is

FIG. 5: (Color online) Density plots of the current I ver-

sus detuning ǫ and Zeeman splitting difference δ. Left side:

ω = ∆1: For growing δ and ǫ, the current I splits off in two

branches (light-colored regions), which are spin-polarized in

opposite direction (cf. previous section). Right side: ω = ∆2:

Current flows only around δ = ǫ = 0 (light-colored region);

P = 0. Parameters see Fig. 3.

reversed in the two dots as compared to ω = ∆1, and

now the energy levels in dot 2 become degenerate. The

analytical limits described for ω = ∆1hold here as well:

For large t12and Bac, the current becomes unpolarized,

and at ǫ = 0, it follows Eq.(13). However, out of these

limits, transport behavior here is very different from the

case ω = ∆1: At detunings ǫ ≈ ±δ/2, spin bottleneck

occurs similar as was shown in the undriven case: Since

dot 1 is out of resonance, the ac field can not rotate the

electron there, hence tunneling to dot 2 is strongly sup-

pressed. The maximal (unpolarized) current then flows

for ǫ = 0 and no side-peaks appear.

In summary, at ω ≈ ∆1, dot 2 can always act as a

spin-filter. The mixing of ↑- and ↓-states due to the ac

field is always stronger in dot 1 than in dot 2, no matter if

∆1≷ ∆2. The ac field mixes ↑- and ↓-states in dot 1 such

that at ǫ ≈ ±δ/2, the electron tunnels onto the almost

pure ↑- or ↓-levels in dot 2, which thus filters the spin and

gives rise to spin-polarized currents. This is opposed to

the case ω = ∆2: Here, due to spin bottleneck, tunneling

to dot 2 is only possible around ǫ = 0, where the current

is totally unpolarized. This behavior is shown in Fig. 5 in

two density plots of the current I versus detuning ǫ and

δ = ∆2−∆1for the two cases ω = ∆1(left) and ω = ∆2

(right). In the left plot, one can clearly see the formation

of the two spin-polarized current branches, which move

far apart as δ and ǫ grow. In contrast to that, the right

plot shows that current only flows for both ǫ = 0 and

δ = 0, and no spin-polarized side-peaks arise.

D.Non-resonant driving

If the ac frequency does not match any of the Zee-

man splittings ∆1,2, the effective finite Zeeman split-

tings are ∆∗

1,2= ∆1,2− ω. It is easy to prove that for

ω = (∆1+ ∆2)/2 = ωs, there is a “symmetric” situa-

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tion, namely ∆∗

this case, the mixing of the spin-states within each dot

is equal in both dots, or in other words, both dots are

equally far from resonance with the ac field. Regarding

interdot tunneling, the levels are resonant at ǫ = 0, giv-

ing rise to one unpolarized current-peak. At all other

detunings ǫ, spin bottleneck avoids the formation of po-

larized side-peaks. In Fig. 6 we show the total current

I (upper left) and spin-resolved currents I↑(upper mid-

dle), I↓ (upper right) vs. detuning ǫ and frequency ω,

for ∆1< ∆2. In order to appreciate the different current

intensities, we plot in the lower panel the total current

versus the detuning ǫ for the three relevant frequencies

ω = ∆1,∆2,ωs. Note the regimes for ω, as discussed

in the previous sections: For ω = ∆2, spin bottleneck

only allows for a very weak and unpolarized current to

flow around ǫ = 0. When the frequency matches the

symmetric value ωs, at ǫ = 0 one sharp and unpolarized

current peak arises, as predicted. Further decreasing of

the frequency splits the current into two branches, which

are enhanced and broadened as ω ≈ ∆1. The sidearms

correspond to either ↑(middle panel)- or ↓(right panel)-

electrons.For any off-resonant frequency, the current

depends not only on δ as in the resonant case, but also

on the absolute values ∆1,2. Hence the position of the

side-peaks is not ǫ ≈ ±δ/2, but follows a different be-

havior. This explains the kink in Fig. 6 (upper panel)

around ω = ∆1.

We want to stress that, in the ac-driven DQD, spin-

polarized currents can be achieved both for ∆1 > ∆2

or ∆1 < ∆2, since by varying the frequency ω one can

always tune one Zeeman splitting to be smaller than the

other, as schematically indicated by the renormalization

of the energy levels due to ω (see Fig. 3, upper panel).

In contrast to that, a static magnetic field set-up — for

example, considering dc magnetic fields in x-direction25

— would only produce polarized currents for ∆1< ∆2.

1= (∆1− ∆2)/2, and ∆∗

2= −∆∗

1. In

E.A triple quantum dot as spin-inverter

Now we want to implement the spintronic functional-

ity of the spin-filter device towards a spin-inverter, and

to this end we consider a TQD. Our goal is to produce

spin-polarized incoming current Iinand oppositely spin-

polarized outgoing current Iout.

We consider the TQD in a regime where only 2 elec-

trons can be in the TQD at a time, and one electron is

confined electrostatically in the left dot (dot 1, cf. Fig. 1,

lower panel). This confinement is necessary to introduce

spin correlations in the dot, such that only an electron

with opposite spin can enter the TQD. The incoming

current is then either ↑- or ↓-polarized, depending on the

position of the energy levels in the adjacent dot. The ac

field frequency ω is in resonance with the central dot (dot

2), ω = ∆2, in order for the right dot (dot 3) to act as the

filter dot. The TQD is here operated at the triple point

(2,0,0) ↔ (1,1,0) ↔ (1,0,1). We restrict the discussion

FIG. 6: (Color online) Upper panel: Density plots of the to-

tal current (left) and spin-resolved currents I↑(middle) and I↓

(right) vs. detuning ǫ and ac frequency ω for ∆1 < ∆2. The

lighter the color, the higher the current. Note that only very

low current flows in the frequency range ω > ωs around ǫ = 0.

At ω = ωs and ǫ = 0, one sharp unpolarized peak arises.

Lowering ω further, the current splits into two arms and suc-

cessively grows, until around ω = ∆1, current is strongly en-

hanced and polarized, since the sidearms stem from either ↑-

or ↓-electrons, see middle and right upper panel. Lower panel:

Current versus ǫ for the three different situations ω = ∆1,

ω = ∆2 and ω = ωs. One can appreciate the big difference in

the current intensities: Only for ω = ∆1, polarized sidepeaks

arise. For ω = ∆2, current flows weakly around ǫ = 0 and for

ωs, only at ǫ = 0 a sharp current peak appears. Parameters

see Fig. 3.

for simplicity to the case where the Zeeman splittings are

∆1= ∆3> ∆2, although this condition is not necessary,

as long as ∆1,3?= ∆2.

From the previous sections we already know that de-

pending on the detuning, the dot connected to the drain

can act as ↑- or ↓-filter. In a TQD, there is one more

degree of freedom compared to the DQD regarding the

“detuning” between the dot levels. Without loss of gen-

erality, we can fix the energy level of dot 1, and move the

energy levels of dot 2 and 3 (which is experimentally real-

ized by applying gate voltages to the corresponding dots).

Under these conditions, there are then four relevant en-

ergy level configurations, which are shown in Fig. 7, lower

panel. In two of the configurations (I and II), the TQD

acts as a spin-polarizer, and in the other two (III and IV)

the electron spin is inverted. We hereby arrive at another

important result of our work: A TQD can be tuned as

both spin-polarizer and spin-inverter, by confining one

electron in the left dot and adjusting the gate voltages at

two of the three dots. Then electrons coming from the

left lead can only enter with a distinct spin-polarization,

which depends on the level position of the central dot.

As the magnetic field Bacis turned on with frequency

ω = ∆2, the electron spin coming from dot 1 is rotated

Page 7

7

FIG. 7: (Color online) Total current I and spin-resolved cur-

rents I↑ and I↓ vs. gate voltages Vg2 and Vg3 applied to the

central dot (dot 2) and the right dot (dot 3) in a TQD exposed

to crossed Bdc and Bac. Here ω = ∆2 and ∆1 = ∆3 > ∆2.

Four relevant level configurations can occur due to adjust-

ment of Vg2 and Vg3: in cases I and II, current through the

TQD is polarized in one spin-direction, and in cases III and

IV, the electron spin is inverted. In order not to overload the

figure, we indicate only the spins of the incoming and outgo-

ing electrons, but note that always one electron is confined in

an off-resonant state in the left dot (dot 1, cf. Fig. 1). Param-

eters in meV (e = ? = 1): Γ = 0.01, t12, 23 = 0.01, Bac = 0.01

(≈ 0.4T), ∆1 = ∆3 = 7∆2, ∆2 = 0.025 (Bdc≈ 1T), U = 1.0.

in dot 2, whereas dot 1 and dot 3 due to their different

Zeeman splittings are far off resonance from the ac field.

Dot 3 then acts as spin-filter and, depending on the rel-

ative position of its energy levels with respect to dot 2,

a ↑- or ↓-polarized current is produced, similar as in the

DQD described in the previous sections.

We plot the total Itotal and spin-resolved currents I↑

and I↓versus the two gate voltages applied to dot 2 and

dot 3 in Fig. 7, together with sketches of the correspond-

ing energy level distribution. In situations I and IV, dot

2 is energetically in resonance with the ↑-level in dot 1.

Therefore, only ↑-electrons coming from the left lead will

be able to tunnel to dot 2. Here they are inverted due

to ω = ∆2, where the renormalized energy levels have

been depicted schematically as we did for the DQD. It

depends then on the level position of dot 3, if the out-

going current is spin-up (case I) or spin-down (case IV)

polarized. An analogue situation occurs for cases II and

III: the energy level of dot 2 is such that only ↓-electrons

can tunnel from dot 1 to dot 2. Again, after rotation

due to the ac field in dot 2, in dot 3 the spin is filtered

without inversion (case II) or inverted (case III).

IV.CONCLUSIONS

In summary, we have analyzed spin current polariza-

tion in the transport through a DQD with one extra

electron, and through a TQD with two extra electrons

in the system. The quantum dot arrays are subjected

to two different external magnetic fields: an inhomoge-

neous dc field, which produces different Zeeman splittings

in the dot, and a time dependent ac field, that rotates

the electron spin in one dot, when the resonance condi-

tion ω = ∆Z is fulfilled. For the DQD, we have ana-

lyzed both off-resonance and resonance conditions of the

ac field with either one of the Zeeman splittings. Our

results show that ac magnetic fields produce strongly

spin-polarized current through a DQD depending on the

detuning of the energy levels in the dots and on the res-

onance conditions.

Finally, we have proposed a TQD in series as both

spin-polarizer and spin-inverter.

TQD different Zeeman splittings in the sample combined

with a resonant ac frequency give way to spin-polarized

currents. In addition, spin-polarized incoming current

can be achieved, and thus the spin-polarizing mechanism

can be extended to a spin-inversion mechanism. Our re-

sults show that dc and ac magnetic fields combined with

gate voltages allow one to manipulate the current spin-

polarization through DQDs and TQDs which are then

able to work as a spin-filter and spin-inverter.

In spintronic devices at the nanometer scale an en-

vironment of nuclei introduces additional spin-flip pro-

cesses that can lower the efficiency of the desired mecha-

nism. In our set-up, we do not expect spin-flip processes

due to hyperfine interaction to influence drastically on

the results, because hyperfine spin-flip times are usually

much longer than typical tunneling times in quantum dot

arrays, especially in finite magnetic fields, where the hy-

perfine interaction is an inelastic process.

Therefore, the systems presented in this work are

promising candidates for spintronic devices.

As in a DQD, in a

Acknowledgements

We are grateful to R. Sánchez, C. Creffield, J. Sabio

and S. Kohler for helpful discussions and critical read-

ing of the manuscript. We acknowledge financial sup-

port through grant MAT2008-02626 (MEC), from JAE

(CSIC)(M.B.) and from ITN no. 234970 (EU).

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