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
PACS numbers: 72.25.Dc, 73.21.La
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
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
B(t) + HT+ HLeads,
where the individual terms are
Uiˆ ni↑ˆ ni↓+1
V ˆ niˆ nj
iσˆ cjσ+ ˆ c†
lkσˆ clσ+ ˆ c†
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-
[∆iSzi+ Bac(cos(ωt)Sxi+ sin(ωt)Syi)],
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:
σσ′ ˆ c†
iσσσσ′ˆ ciσ′ the spin operator of the ith
iSzi]) into the ro-
[(∆i− ?ω)Szi+ BacSxi].
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-
˙ ρmn(t) = − i?m|[H0
− 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
Γkn). The transition rates are calculated using Fermi’s
Γl[f(Em− En− µl)δNm,Nn+1+
(1 − f(Em− En− µl))δNm,Nn−1],
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 =
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,↓?)
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,↓?
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
I↑= Γ?ρ|↑,0,↑?+ ρ|↓,0,↑?
I↓= Γ?ρ|↑,0,↓?+ ρ|↓,0,↓?
The spin polarization of the current is defined as
P =I↑− I↓
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.
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
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
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
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?±=
√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:
| ↑2? + | ↓2?)
ac+ δ2± δ?B2
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
12) + (Γ2+ δ2)(Γ2+ 10t2
ac+ Γ2+ δ2)
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
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.
12+ 4ǫ2+ δ2
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.
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-
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 ∆∗
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
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).
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-
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
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).
1D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120
2E. Cota, R. Aguado, and G. Platero, Phys. Rev. Lett. 94,
107202 (2005); R. Sánchez, E. Cota, R. Aguado, and G.
Platero, Phys. Rev. B 74, 035326 (2006).
3F.H.L. Koppens, C. Buizert, K.J. Tielrooij, I.T. Vink, K.C.
Nowack, T. Meunier, L.P. Kouwenhoven, and L.M.K. Van-
dersypen, Nature 442, 766 (2006).
4H.-A. Engel and D. Loss, Phys. Rev. B 65, 195321 (2002).
5R. Sánchez, C. López-Monís, and G. Platero, Phys. Rev.
B, 77, 165312 (2008).
6M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99,
7J. Danon and Y. V. Nazarov, Phys. Rev. Lett. 100, 056603
8M. Busl, R. Sánchez, and G. Platero, Phys. Rev. B(R)
81, 121306 (2010); M. Busl, R. Sánchez, and G. Platero,
Journal of Physics, Conference Series, 245, 012016 (2010).
9K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, L. M.
K. Vandersypen, Science 318, 1430 (2007).
10E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus,
M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 99,
246601 (2007); M. Pioro-Ladrière, T. Obata, Y. Tokura,
Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S.
Tarucha, Nature Phys. 4, 776 (2008).
11M. Busl, R. Sánchez, and G. Platero, Physica E, 42, 830
12W. A. Coish and D. Loss, Phys. Rev. B 75, 161302(R)
13S.M. Huang, Y. Tokura, H. Akimoto, K. Kono, J. J. Lin, S.
Tarucha, and K. Ono, Phys Rev. Lett. 104, 136801 (2010).
14D. Schröer, A. D. Greentree, L. Gaudreau, K. Eberl, L. C.
L. Hollenberg, J. P. Kotthaus, and S. Ludwig, Phys. Rev.
B 76, 075306 (2007).
15M. C. Rogge and R. J. Haug, Phys. Rev. B 77, 193306
16L. Gaudreau, S. A. Studenikin, A. S. Sachrajda, P. Za-
wadzki, A. Kam, J. Lapointe, M. Korkusinski, and P.
Hawrylak, Phys. Rev. Lett. 97, 036807 (2006); G. Granger,
L. Gaudreau, A. Kam, M. Pioro-Ladrière, S. A. Stu-
denikin, Z. R. Wasilewski, P. Zawadzki, and A. S. Sachra-
jda, Phys. Rev. B 82, 075304 (2010).
17M. Korkusinski, I. Puerto Gimenez, P. Hawrylak, L. Gau-
dreau, S. A. Studenikin, and A. S. Sachrajda, Phys. Rev. B
75, 115301 (2007); F. Delgado, Y.-P. Shim, M. Korkusin-
ski, and P. Hawrylak, Phys. Rev. B 76, 115332 (2007).
18B. Michaelis, C. Emary and C. W. J. Beenakker, Europhys.
Lett. 73, 677 (2006); C. Emary, Phys. Rev. B 76, 245319
19C. Pöltl, C. Emary and T. Brandes, Phys. Rev. B 80,
20F. Delgado, Y.-P. Shim, M. Korkusinski, L. Gaudreau, S.
A. Studenikin, A. S. Sachrajda, and P. Hawrylak Phys.
Rev. Lett. 101, 226810 (2008).
21R.Žitko, J. Bonča, A. Ramšak, and T. Rejec, Phys. Rev.
B 73, 153307 (2006); T. Kuzmenko, K. Kikoin, and Y.
Avishai, Phys. Rev. B 73, 235310 (2006); E. Vernek, C. A.
Büsser, G. B. Martins, E. V. Anda, N. Sandler, and S. E.
Ulloa, Phys. Rev. B 80, 035119 (2009).
22A. Vidan, R.M. Westervelt, M. Stopa, M. Hanson, and
A.C. Gossard, App. Phys. Lett. 85, 3602 (2004).
23T. Kostyrko and B. R. Bulka, Phys. Rev. B 79, 075310
24Daniel S. Saraga and Daniel Loss, Phys. Rev. Lett. 90,
25Y. Tokura, T. Kubo, Y.-S. Shin, K. Ono, S. Tarucha, Phys-
ica E, 42, 994 (2010).