ac-Driven double quantum dots as spin pumps and spin filters.
ABSTRACT We propose and analyze a new scheme of realizing both spin filtering and spin pumping by using ac-driven double quantum dots in the Coulomb blockade regime. By calculating the current through the system in the sequential tunneling regime, we demonstrate that the spin polarization of the current can be controlled by tuning the parameters (amplitude and frequency) of the ac field. We also discuss spin relaxation and decoherence effects in the pumped current.
arXiv:cond-mat/0502440v1 [cond-mat.mes-hall] 18 Feb 2005
AC-driven double quantum dots as spin pumps and spin filters.
Ernesto Cota1, Ram´ on Aguado2and Gloria Platero2
1-Centro de Ciencias de la Materia Condensada - UNAM, Ensenada, Mexico.
2-Instituto de Ciencia de Materiales, CSIC, Cantoblanco, Madrid,28049, Spain.
(Dated: February 2, 2008)
We propose and analyze a new scheme of realizing both spin filtering and spin pumping by using ac-
driven double quantum dots in the Coulomb blockade regime. By calculating the current through
the system in the sequential tunneling regime, we demonstrate that the spin polarization of the
current can be controlled by tuning the parameters (amplitude and frequency) of the ac field. We
also discuss spin relaxation and decoherence effects in the pumped current.
PACS numbers: 85.75.-d, 73.23.Hk, 73.63.Kv
The emerging field of spintronics aims at creating de-
vices based on the spin of electrons . One of the most
important requirements for any spin-based electronics is
the ability to generate a spin current. Proposals for gen-
erating spin-polarized currents include spin injection by
using ferromagnetic metals  or magnetic semiconduc-
tors . Alternatively, one may use quantum dots (QDs)
as spin filters or spin pumps [4, 5]. For QD spin filters,
dc transport through few electron states is used to obtain
spin-polarized currents as demonstrated experimentally
by Hanson et al  following the proposal of Recher et al
. Spin current rectification has also been realized .
The basic principle of spin pumps is, on the other hand,
closely related to that of charge pumps. In a charge pump
a dc current is generated by combining ac driving with ei-
ther absence of inversion symmetry in the device, or lack
of time-reversal symmetry in the ac signal. The range of
possible pumps includes turnstiles, adiabatic pumps or
nonadiabatic pumps based on photon-assisted tunneling
(PAT) [9, 10].
In this Letter we propose and analyze a new scheme
of realizing both spin filtering and spin pumping by using
a double quantum dot (DQD), in the Coulomb block-
ade regime, with time-dependent gate voltages and in
the presence of a uniform magnetic field. The periodic
variation of the gate potentials allows for a net dc cur-
rent through the device even with no dc voltage ap-
plied [11, 12]: if the system is driven at a frequency
(or subharmonic) corresponding to the energy difference
between two time-independent eigenstates, the electrons
become completely delocalized [13, 14]. If the left reser-
voir (chemical potential µL) can donate electrons to the
left dot (at a rate ΓL) and the right reservoir (chemical
potential µR) can accept electrons from the right dot (at
a rate ΓR) the system will then pump electrons from left
to right, even when there is no dc bias applied, namely
µL= µR(Fig. 1a). Starting from this pumping principle
our device has two basic characteristics: i) If the process
involves two-particle states, the pumped current can be
completely spin-polarized even if the contact leads are
not spin polarized and ii) the pumping can occur either
through singlet (Fig. 1b) or triplet (Fig. 1c) states de-
pending on the applied frequency, such that the degree of
spin polarization can be tuned by means of the ac field.
For example, if one drives the system (initially prepared
in a state with n = nL+nR= 3 electrons: |L =↓↑,R =↑?)
at a frequency corresponding to the energy difference be-
tween the singlets in both dots, the electron with spin
↓ becomes delocalized in the DQD system. If now the
chemical potential for taking ↓ (↑) electrons out of the
right dot is above (below) µR, a spin-polarized current
is generated. The above conditions for the chemical po-
tentials can be achieved by breaking the spin-degeneracy
through a Zeeman term ∆z = gµBB, where B is the
external magnetic field (which is applied parallel to the
sample in order to minimize orbital effects), g is the ef-
fective g-factor and µBthe Bohr magneton.
Our main findings can be summarized in Fig. 2 where
we present a plot of the pumped current as a function of
the applied frequency for a particular choice of parame-
ters. Importantly, the current presents a series of peaks
which are uniquely associated with a definite spin po-
larization: the pumped current is 100% spin-down (up)
polarized for ω = ω↓(ω = ω↑). The other peaks corre-
spond to subharmonics of the energy difference between
the relevant states, see below. This particular example
illustrates how by tuning the external ac field one can
operate the driven DQD as a bipolar spin filter with no
dc voltage applied.
Formalism.– Our system consists of an asymmetric
DQD connected to two reservoirs kept at the chemical
potentials µα, α = L,R. Using a standard tunneling
Hamiltonian approach, we write for the full Hamiltonian
Hl+ HDQD+ HT, where Hl =?
describes the leads and HDQD= HL
describes the DQD. It is assumed that only one orbital
in the left dot participates in the spin-polarized pump-
ing process whereas two orbitals in the right dot (energy
separation ∆ǫ) have to be considered. The isolated left
dot is thus modelled as a one–level Anderson impurity:
right dot is modelled as: HR
dex i = 0,1 denotes the two levels. In practice, we take
LσdLσ+ULnL↑nL↓, whereas the isolated
σ,σ′nR0σnR1σ′) + JS0S1. The in-
FIG. 1: (Color online) Schematic diagram of the double dot
electron pump. a) Pumping through one-particle states: the
current is spin-unpolarized. b) Pumping through two particle
states with ?ω↓= ES0
ergies of the singlets). If the chemical potentials fulfil the con-
ditions µ1,0(2,1) < µL, µ2,0(1,2) > µR and µ2,1(1,2) < µR
(see text) the resulting pumped current is spin-down polar-
ized. c) Pumping involving a triplet (ET+
this case the pumped current is spin-up polarized. Note that
during the pumping process only electrons with spin down
(case b) or up (case c) become delocalized by the microwaves.
Dashed arrows denote delocalized spins whereas solid arrows
denote spins that remain localized on each dot.
Rare the en-
R)in the right dot. In
is included in the charging energies UR > UL. Experi-
mentally, this asymmetry can be realized by making the
right dot smaller. Si= (1/2)?
spins of the two levels. As a consequence of Hund’s rule,
the intra–dot exchange, J, is ferromagnetic (J < 0) such
that the singlet |S1? = (1/√2)(d†
is higher in energy than the triplets |T+? = d†
|T0? = (1/√2)(d†
ET+= ∆ǫ + UR− J/4. Finally, we consider the case
where ∆ǫ > ∆z+J/4 such that the triplet |T+? is higher
in energy than the singlet |S0? = (1/√2)(d†
scribes tunneling between dots. The tunneling between
R0= 0 (E↓
R0= ∆z), so all the asymmetry
Riσσσσ′dRiσ′ are the
R1↑)|0? and |T−? =
R1↓|0?. Due to the Zeeman splitting ET−> ET0>
LσdRiσ+ h.c.) de-
leads and each QD is described by the perturbation HT =
ΓL,R= 2πDL,R|VL,R|2are the tunneling rates. It is as-
sumed that the density of states in both leads DL,Rand
the tunneling couplings are energy-independent.
We study the system by a reduced density matrix
(RDM), ρ = TrLχ, where χ is the full density matrix,
and TrL is the trace over the leads. The dynamics of
the RDM is formulated in terms of the eigenstates and
eigenenergies of each isolated QD. We concentrate on the
Coulomb blockade regime (with up to two electrons per
dot, which defines a basis of 20 states) and study the se-
quential tunneling regime (Born-Markovapproximation).
For example, the diagonal elements of the RDM read
˙ ρss= −i
where Wij are the transition rates (calculated using a
standard Fermi Golden Rule).
sider an ac field acting on the dots, such that the sin-
gle particle energy levels become ǫL(R) → ǫL(R)(t) =
tude and frequency, respectively, of the applied field. We
include spin relaxation and decoherence phenomenolog-
ically in the corresponding elements of the equation for
the RDM. Relaxation processes are described by the spin
relaxation time T1= (W↑↓+W↓↑)−1, where W↑↓and W↓↑
are spin-flip relaxation rates fulfilling a detailed balance.
A lower bound for the spin relaxation time T1 of 50µs
with a field B = 7.5T has been obtained recently 
for a single electron in a QD using energy spectroscopy
and relaxation measurements. In the following, we fo-
cus on zero temperature results such that W↓↑= 0 and
thus T1= W−1
ence time) describes intrinsic spin decoherence. We take
T2= 0.1T1in all the calculations .
In practice, we integrate numerically the dynamics of
the RDM in the chosen basis. In particular, all the results
shown in the next paragraphs are obtained by letting the
system evolve from the initial state | ↓↑,↑? until a sta-
tionary state is reached. The dynamical behavior of the
system is governed by rates which depend on the elec-
trochemical potentials of the corresponding transitions.
The electrochemical potential µ1(2),i(N1,N2) of dot L(R)
is defined as the energy needed to add the N1(2)th elec-
tron to energy level i of dot L(R), while having N2(1)
electrons on dot R(L) . The current from left to right
is: IL→R(t) = ΓR
IR→L. Here, states |s? are such that the right dot is oc-
cupied. For ease in the notation, we take from now on
? = e =1, such that VAC, ω, etc, have units of energy.
Results.– A calculation of the stationary current, for
each direction of spin, namely Itot=?
tion of ω (and fixed intensity VAC = V0
the results shown in Fig. 2. The main peak of I↓(con-
In addition we con-
cosωt, where eVAC and ω are the ampli-
↑↓. The rate T2−1(T2is the spin decoher-
sρss(t), with a similar expression for
σ=↑,↓Iσas a func-
AC= 0.7), gives
FIG. 2: (Color online) Pumped current as a function of the ac
frequency. The spin-down component (solid line) shows three
peaks corresponding to one (ω = ω↓ = 0.3), two (ω = 0.15)
and three (ω = 0.1) photon processes, respectively.
spin-up component (dashed line) shows a main one-photon
resonance at ω = ω↑ = 0.7 and up to six more satel-
lites corresponding to multiphoton processes.
ΓL = ΓR = 0.001, t = 0.005, UL = 1.0, UR = 1.3, J = 0.2,
µL = µR = 1.31, ∆z = 0.026, ∆ǫ = 0.45, VAC = V0
(in meV) correspond to typical values in GaAs QDs. In par-
ticular, the Zeeman splitting corresponds to a magnetic field
B ≈ 1T. Inset: same as in main figure but with a lower
intensity VAC = V0
AC/5 = 0.14.
tinuous line) occurs at ω = ω↓≡ E(↑,↓↑)−E(↓↑,↑)= UR−
UL= 0.3 (see caption) At this frequency I↑≈ 0 (dashed
line) demonstrating the efficiency of the spin-polarized
pump. For this particular case, pumping of spin-down
electrons occurs as one electron with spin ↓ becomes de-
localized (via a one photon process) between both dots.
If the chemical potential for taking ↓ electrons out of the
right dot fulfils µ2,0(1,2) = UR− E↑> µRwhile, on the
other hand, the chemical potential for taking ↑ electrons
out of the right dot fulfils µ2,1(1,2) = UR−E↓< µRthe
resulting pumped current is spin-down polarized.
emphasize again that this pumping of spin polarized (↓)
electrons is realized with unpolarized leads. Such spin-
polarized current is obtained either through the sequence
frequency to ω = ω↓/2 = 0.15, there is a second peak
(corresponding to the absorption of two photons) in the
spin down current.A three-photon process occurs at
ω = ω↓/3 = 0.1.
⇒ (↓↑,↑) or, alternatively,
⇒ (↓↑,↑). Reducing the
By increasing the frequency to ω = ω↑ ≡ E(↓,↑↑)−
E(↓↑,↑)= ∆ǫ + UR− UL− J/4 = 0.7 (see Fig. 2), a
current peak with spin up polarization appears. In this
case, the pumped current occurs as one electron with
spin ↑ becomes delocalized between the states (↓↑,↑) and
(↓,↑↑). This spin up electron subsequently decays to the
right reservoir which leads to a pumped current through
the sequence (↓↑,↑)
that the spin polarization, defined as
⇒ (↓↑,↑) or (↓↑
⇒ (↓↑,↑). At ω↑, I↓≈ 0 such
P(ω,VAC) ≡I↑− I↓
has been completely reversed by tuning the frequency of
the ac field, namely P(ω↑,V0
Note that the energy difference between both processes,
ω↑−ω↓= ∆ǫ−J/4 corresponds to the energy difference
between the triplet excited state and the singlet ground
state in the right dot at zero magnetic field.
Reducing the frequency to ω = ω↑/2 and ω = ω↑/3
and so on, peaks corresponding to absorption of up to
seven photons appear. Note that each of these peaks
has a different width. This remarkable fact can be at-
tributed to a renormalization of the inter-dot hopping
induced by the ac potential [9, 11]. At the frequency
(or subharmonic) corresponding to the energy difference
between two levels, the Rabi frequency becomes renor-
malized by the ac potential as ΩRabi=2tLRJN(VAC/ω),
JN is the Bessel function of order N (=number of pho-
tons absorbed). The width of the peaks is given by the
coupling to the leads provided that ΓL,R > ΩRabi. By
contrast, if ΓL,R< ΩRabithe width of the current peak
is determined by ΩRabi . Since ΩRabi depends on the
number of photons in a nonlinear fashion (through the
dependence of JN on the ratio VAC/ω), it follows that
the widths of the peaks change in a non trivial way as
a function of ω. A similar nonlinear dependence of the
height of the peaks as a function of the ratio VAC/ω is
expected. We illustrate this nontrivial behavior in the in-
set of Fig. 2 where we explore the low intensities regime
AC/5=0.14). In general, the trend we obtain is
consistent with previous analytical estimations .
Another interesting feature of the spin pump is that
there are frequencies where the one-photon process cor-
responding to pumping of ↓ electrons can overlap with
multiphoton processes corresponding to pumping of ↑
electrons. Thus, at these frequencies the current is no
longer fully spin-polarized. One can use this to modify
the polarization of the current by changing VAC(at fixed
ω). We illustrate this with Fig. 3, where the parameters
are chosen such that the N=1 peak of I↓is centered at
the same frequency (ω = ω↓= 0.3) as the N=2 peak of I↑.
At this frequency, the spin polarization can be tuned by
modifying the intensity of the ac potential (Fig.3, inset).
This result, together with those shown in Fig. 2, demon-
strate that the spin polarization of the current P(ω,VAC)
can be fully manipulated by tuning either the frequency
or the intensity of the external ac field.
Finally, it is important to note also that, contrary to
the case for spin-down pumping, the pumping of spin
up electrons leaves the double dot in the excited state
| ↓,↑?. This makes the spin-up current extremely sen-
sitive to spin relaxation processes. If the spin ↓ decays
AC) = 1 = −P(ω↓,V0
0.2 0.40.60.8 1.01.2
FIG. 3: (Color online) Pumped current as a function of the ac
frequency. Same parameters as Fig. 2 except ∆ǫ = 0.35. The
interesting feature here is the overlap between the the one-
photon absorption peak of I↓ (solid line) and the two-photon
absorption peak of I↑(dashed line) at ω = ω↓= 0.3. The inset
shows the spin polarization P(ω,VAC) versus the ac intensity
VAC for fixed frequency ω = ω↓ = 0.3 demonstrating the
possibility of controlling the spin polarization of the current
by tuning the intensity of the ac field.
before the next electron enters into the left dot, namely
if W↑↓ ? ΓL, a spin-down current appears through the
and the pumping cycle is no longer 100% spin-up po-
larized. We study this effect in Fig. 4, where we plot
the main PAT peak at ω↑ = 0.6 for increasing W↑↓.
As one expects, the peak broadens as W↑↓ increases.
The full widths (FWHM) of the resonances are plotted
as a function of W↑↓ in the inset. For large intensities
(VAC = V0
AC, circles) the FWHMs grow in a nonlinear
fashion. This is reminiscent of the so-called saturation
regime, a well known phenomenon in the context of op-
tical Bloch equations . Note, however, that there are
three energy scales involved now in the dynamics of the
density matrix, ΩRabi, Γ and W↑↓, such that other sources
of nonlinearity (like the ones described when discussing
Fig. 2) cannot be completely ruled out . In order
to minimize nonlinear effects we investigate the low in-
tensity regime (VAC= V0
AC/5, squares) where we expect
a FWHM dominated by decoherence. The behavior is
now linear with a slope which, interestingly, approaches
FWHM∼ 20W↑↓= 2/T2. Thus, experiments along these
lines would complement the information about decoher-
ence extracted from other setups .
Summary and experimental accessibility.–In summary,
we have proposed and analyzed a new scheme of realizing
both spin filtering and spin pumping by using ac-driven
double quantum dots coupled to unpolarized leads. Our
results demonstrate that the spin polarization of the cur-
rent can be manipulated (including fully reversing) by
just tuning the parameters of the ac field. Our results
0.560.58 0.600.62 0.64
FIG. 4: (Color online) Pumped current near resonance ω =
ω↑= 0.6 for different relaxation rates. Inset: FWHM of the
total current as a function of relaxation rate, for strong (black
dots) and weak (red squares) field intensity
also show that the width in frequency of the spin-up
pumped current gives information about spin decoher-
ence in the quantum dot. We finish by mentioning that
our proposal is within reach with today’s technology for
high-frequency experiments in quantum dots [10, 13, 14].
Indeed, PAT with two-electron spin states has recently
been reported .
We thank Wilfred van der Wiel for his help. Work
supported by Programa de Cooperaci´ on Bilateral CSIC-
CONACYT, by grant DGAPA-UNAM 114403-3 (E.C),
by the EU grant HPRN-CT-2000-00144 and by the Min-
isterio de Ciencia y Tecnolog´ ıa of Spain through grant
MAT2002-02465 (R. A. and G. P.) and the ”Ram´ on y
Cajal” program (R. A.).
 S. A. Wolf et al., Science 294, 1488 (2001).
 M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790
(1985); F. J. Jedema et al., Nature(London) 410, 345
 R. Fiederling et al., Nature (London) 402, 787 (1999);
Y. Ohno et al., Nature (London) 402, 790 (1999).
 E. R. Mucciolo et al., Phys. Rev. Lett. 89, 146802 (2002);
Susan K. Watson et al., Phys. Rev. Lett. 91, 258301
(2003); M. G. Vavilov et al., cond-mat/0410042.
 T. Aono, Phys. Rev. B 67, 155303 (2003); Qing-feng Sun
et al., Phys. Rev. Lett. 90, 258301 (2003); E. Cota et al.,
Nanotechnology 14, 152-156 (2003).
 R. Hanson et al. Phys. Rev. B, 70, 241304 (2004).
 P. Recher et al. Phys. Rev. Lett., 85, 1962 (2000).
 K. Ono et al. Science 297, 1313 (2002); A. C. Johnson
et al. cond-mat/0410679.
 G. Platero and R. Aguado, Physics Reports, 395, 1
 W. G. van der Wiel et al., ’Photon assisted tunneling in
quantum dots’, in: I.V. Lerner, et al. (Eds.), Strongly
Correlated Fermions and Bosons in Low-dimensional
Disordered Systems, Kluwer Academic Publishers, pp.
 C. A. Stafford and N. S. Wingreen Phys. Rev. Lett. 76,
 B. L. Hazelzet, M. R. Wegewijs, T. H. Stoof, and Yu. V.
Nazarov, Phys. Rev. B 63, 165313 (2001).
 T. H. Oosterkamp et al., Nature (London) 395, 873
 J. R. Petta et al., cond-mat/0408139.
 R. Hanson et al., Phys. Rev. Lett. 91, 196802 (2003).
 A detailed study of spin relaxation and decoherence in a
GaAs quantum dot due to spin-orbit interaction can be
found in, Vitaly N. Golovach et al., Phys. Rev. Lett. 93,
 Robert W. Boyd, Non-linear Optics (Academic Press,
 A detailed analytical study of the dynamics of the ef-
fective few level problem (to be published elsewhere) is
needed in order to substantiate these arguments.
 See also, Hans-Andreas Engel and Daniel Loss, Phys.
Rev. Lett., 86, 4648 (2001).
 T. Kodera, W. G. van der Wiel, K. Ono, S. Sasaki, T.
Fujisawa, and S. Tarucha, Physica E, 22, 518 (2004).