Spinpolarized currents in double and triple quantum dots driven by ac magnetic fields
ABSTRACT We analyze transport through both a double quantum dot and a triple quantum dot with inhomogeneous Zeeman splittings in the presence of crossed dc and ac magnetic fields. We find that strongly spinpolarized current can be achieved by tuning the relative energies of the Zeemansplit levels of the dots, by means of electric gate voltages: depending on the energylevel detuning, the double quantum dot works either as spinup or spindown 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.
 [Show abstract] [Hide abstract]
ABSTRACT: In this paper, spinfiltering properties of transmitted electrons through a quantum ring in the presence of Rashba spinorbit interaction and magnetic flux are studied. To investigate the effects of coupling between the leads and ring on the spinfiltering, the Smatrix method is used. It is shown that by tuning the Rashba spinorbit strength and the magnetic flux, the quantum ring can act as a perfect spinfilter with high efficiency. The spinfiltering can be changed from spin up to spin down and vice versa by changing the Rashba strength when the magnetic flux is held constant or by changing the magnetic flux when the Rashba strength is held constant. In addition, the effect of the angle between the leads on spinfiltering properties is taken into account and the angles at which the spinfiltering can occur are determined. The spinfiltering can take place in narrow ranges of electron energy for weak coupling, while for strong coupling it can take place in a wide range of electron energy.Journal of Applied Physics 01/2013; 113(4):044316. · 2.21 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we study the spindependent electron transport properties of graphene npn junction in the presence of Rashba spinorbit interaction using transfer matrix method. It is found that for a graphene npn junction, the spinresolved electron conductance depends on the strength of Rashba spinorbit interaction and the builtin potential of graphene junctions. For an appropriate value of Rashba strength, perfect electron spininversion with high conductance can take place when the graphene junction is biased with sufficient builtin potential. In this case, the graphene junction can be used as a spininverter which is an important device in spintronic. Finally, the spindependent conductance for a graphene nnn junction is investigated and compared with that of npn junction.Journal of Applied Physics 11/2012; 112(10). · 2.21 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study spinresolved electron transport in a double quantum ring in the presence of Rashba spinorbit interaction and a magnetic flux using quantum waveguide theory. We show that, at the proper values of the system parameters such as the Rashba coupling constant, the radius of the rings, and the angle between the leads, the double quantum ring can act as a perfect electron spininverter with very high efficiency. Also, the double quantum ring can work as a spin switch. The spin polarization of transmitted electrons can be controlled and changed from −1 to +1 by using a magnetic flux.Journal of Applied Physics 01/2013; 113:014303. · 2.21 Impact Factor
Page 1
arXiv:1010.2729v2 [condmat.meshall] 27 Nov 2010
Spinpolarized 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 spinpolarized current can be achieved by tuning the relative energies of the Zeemansplit
levels of the dots, by means of electric gate voltages: depending on the energy level detuning, the
double quantum dot works either as spinup or spindown filter. We show that a triple quantum
dot in series under crossed dc and ac magnetic fields can act not only as spinfilter but also as
spininverter.
PACS numbers: 72.25.Dc, 73.21.La
I.INTRODUCTION
A key aim in spintronics is the realization of spinbased
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 spinorbit 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 microferromagnet 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 threespin
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, socalled
“dark states”8,18,19and AharonovBohm oscillations8,20
have been studied. TQDs have been used as a testing
ground for Kondo physics21and have been proposed as
current rectifiers22,23and spinentanglers.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 spinflip 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 spinpolarized currents
and thus for the design of spinfilters and spininverters.
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
Page 2
2
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
spinpolarized 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
interdot 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
zdirection, and an ac field Bacapplied in xydirection:
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 timedependent
Hamiltonian can be transformed by means of a unitary
transformation5,8U(t) = exp(−i[ωt?
tating reference frame. The resulting timeindependent
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 BornMarkov
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+
Page 3
3
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,RγL,R2are 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(ρ0,↑?+ ρ0,↓?)
(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 twoelectron basis for the
TQD contains fifteen twoelectron states, and one zero
and six oneelectron 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
• 1electron states: ↑,0,0?, ↓,0,0?
• 2electron 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 spinresolved 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 spinlevels 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 socalled 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 zdirection 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 spinlevel 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 spinlevels 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
Page 4
4
For certain detunings, this will lead to spinpolarized 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 twolevel systems coupled by t12. In
a twolevel 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 spinlevels 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 spinmixing 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 spinpolarized: 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: Spinresolved 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
spinlevels 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 spinfilter, 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 spinfilter, 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
Page 5
5
FIG. 4: (Color online) Polarization P versus t12 , Bac and δ
in acdriven DQD at detuning ǫ = ±δ/2 for ω = ∆1: Left
and middle panel: For both small t12 and Bac, spinpolarized
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 sidepeaks 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 sidepeaks 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 (lightcolored regions), which are spinpolarized in
opposite direction (cf. previous section). Right side: ω = ∆2:
Current flows only around δ = ǫ = 0 (lightcolored 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 sidepeaks appear.
In summary, at ω ≈ ∆1, dot 2 can always act as a
spinfilter. 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 spinpolarized 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 spinpolarized 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 spinpolarized sidepeaks arise.
D.Nonresonant 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
Page 6
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tion, namely ∆∗
this case, the mixing of the spinstates 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 currentpeak. At all other
detunings ǫ, spin bottleneck avoids the formation of po
larized sidepeaks. In Fig. 6 we show the total current
I (upper left) and spinresolved 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 offresonant 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
sidepeaks 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 acdriven 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 setup — for
example, considering dc magnetic fields in xdirection25
— would only produce polarized currents for ∆1< ∆2.
1= (∆1− ∆2)/2, and ∆∗
2= −∆∗
1. In
E.A triple quantum dot as spininverter
Now we want to implement the spintronic functional
ity of the spinfilter device towards a spininverter, and
to this end we consider a TQD. Our goal is to produce
spinpolarized 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 spinresolved 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 spinpolarizer, 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 spinpolarizer and spininverter, 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 spinpolarization,
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 spinresolved 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 spindirection, 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 offresonant 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 spinfilter 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 spinresolved 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 spinup (case I) or spindown (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 offresonance and resonance conditions of the
ac field with either one of the Zeeman splittings. Our
results show that ac magnetic fields produce strongly
spinpolarized 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
spinpolarizer and spininverter.
TQD different Zeeman splittings in the sample combined
with a resonant ac frequency give way to spinpolarized
currents. In addition, spinpolarized incoming current
can be achieved, and thus the spinpolarizing mechanism
can be extended to a spininversion 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 spinfilter and spininverter.
In spintronic devices at the nanometer scale an en
vironment of nuclei introduces additional spinflip pro
cesses that can lower the efficiency of the desired mecha
nism. In our setup, we do not expect spinflip processes
due to hyperfine interaction to influence drastically on
the results, because hyperfine spinflip 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 MAT200802626 (MEC), from JAE
(CSIC)(M.B.) and from ITN no. 234970 (EU).
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Supplementary resources (1)

PhysRevB.82.205304