Spin-polarized transport through weakly coupled double quantum dots in the
Coulomb-blockade regime
Ireneusz Weymann1, ∗
Department of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
(Dated: October 29, 2018)
We analyze cotunneling transport through two quantum dots in series weakly coupled to external
ferromagnetic leads. In the Coulomb blockade regime the electric current flows due to third-order
tunneling, while the second-order single-barrier processes have indirect impact on the current by
changing the occupation probabilities of the double dot system. We predict a zero-bias maximum in
the differential conductance, whose magnitude is conditioned by the value of the inter-dot Coulomb
interaction. This maximum is present in both magnetic configurations of the system and results
from asymmetry in cotunneling through different virtual states. Furthermore, we show that tunnel
magnetoresistance exhibits a distinctively different behavior depending on temperature, being rather
independent of the value of inter-dot correlation. Moreover, we find negative TMR in some range
of the bias voltage.
PACS numbers: 72.25.Mk, 73.63.Kv, 85.75.-d, 73.23.Hk
I. INTRODUCTION
Transport properties of double quantum
dots (DQDs) have recently attracted much
interest.1,2,3,4,5,6,7,8,9,10,11,12,13 This is because these
structures are ideal systems to study the fundamental
interactions between electrons and spins.14,15,16,17,18
Double quantum dots exhibit a variety of interesting
effects, such as for example current rectification due to
the Pauli spin blockade,1,10,11,12,13 negative differential
conductance,3 formation of molecular states,7 spin
pumping,8,9 etc. Furthermore, double quantum dots are
also being considered for future applications in quantum
computing.19 Experimentally, such systems may be
realized for example in lateral and vertical semiconduc-
tor quantum dots.1,3,5,20 Another implementation of
DQDs are single wall metallic carbon nanotubes with
top gate electrodes, which enable changing of charge
on each dot separately, as well as the intrinsic DQD
parameters.7,21,22,23
The goal of this paper is to analyze transport prop-
erties of DQDs weakly coupled to ferromagnetic leads
in the Coulomb blockade regime. When the leads are
ferromagnetic, transport strongly depends on the mag-
netic configuration of the system, giving rise to tunnel
magnetoresistance (TMR), spin accumulation, exchange
field, etc.24,25,26,27,28,29,30,31,32 In the Coulomb blockade
regime the electric current flows due to higher-order tun-
neling processes (cotunneling), while the first-order tun-
neling processes (sequential tunneling) are exponentially
suppressed.31,33,34,35 The problem of spin-polarized co-
tunneling has been so far addressed mainly in the case
of single quantum dots.31,32,36,37,38,39,40,41 For example,
it was shown that tunnel magnetoresistance exhibits dis-
tinctively different behavior depending on the number of
electrons on the dot.31 Moreover, the zero-bias anomaly
was found in the differential conductance when magnetic
moments of the leads form antiparallel configuration.36
Another interesting behavior was predicted for quantum
dots coupled to ferromagnetic leads with non-collinear
alignment of magnetizations – the exchange field was
found to increase the differential conductance for certain
non-collinear configurations, as compared to the parallel
one.32,42 On the other hand, it was shown experimentally
for nonmagnetic systems that the conductance of quan-
tum dots in the cotunneling regime may serve as a handle
to determine the spectroscopic g-factor.43
In the case of double quantum dots considered in this
paper, in the Coulomb blockade regime the electric cur-
rent flows due to third-order tunneling processes, while
the single-barrier second-order processes together with
third-order processes determine the double dot occupa-
tion probabilities. Assuming that double quantum dot
is occupied by two electrons in equilibrium, one on each
dot, we calculate the differential conductance G and tun-
nel magnetoresistance TMR. We show that differential
conductance exhibits a maximum at the zero bias. We
further distinguish two different mechanisms leading to
this new zero-bias anomaly. The first one is an asym-
metry in cotunneling through different virtual states of
the DQD system, which leads to an enhancement of G at
zero bias. Such asymmetry is induced by a finite value
of the inter-dot Coulomb interaction. This mechanism
is rather independent of magnetic configuration of the
system. The second mechanism leading to the zero-bias
maximum in differential conductance is the interplay be-
tween spin accumulation and third-order tunneling pro-
cesses carrying the current. This mechanism does depend
on the magnetic configuration of the system and, as we
show in the sequel, is found to be more efficient in the
antiparallel configuration. We also analyze the behavior
of TMR and show that the TMR exhibits a maximum
at zero bias, which strongly depends on the temperature.
Furthermore, the TMR may become negative in some
range of the bias voltage.
Finally, we note that there are several experimental re-
alizations of single quantum dots attached to ferromag-
http://arxiv.org/abs/0704.1213v1
netic leads,44,45,46,47,48,49,50,51,52,53,54 while experimental
data on spin-polarized transport through double quan-
tum dots is lacking. We believe that the results presented
in this paper will be of assistance in discussing future ex-
periments.
II. MODEL AND METHOD
The schematic of a double quantum dot coupled to fer-
romagnetic leads is shown in Fig. 1. It is assumed that
the magnetizations of the leads are oriented collinearly,
so that the system can be either in the parallel or antipar-
allel magnetic configuration. The Hamiltonian Ĥ of the
DQD system is given by, Ĥ = ĤL+ĤR+ĤDQD+ĤT. The
first two terms describe noninteracting itinerant electrons
in the leads, Ĥj =
εjkσc
cjkσ for the left (j = L)
and right (j = R) lead, where εjkσ is the energy of an
electron with the wave vector k and spin σ in the lead
j, and c
(cjkσ) denotes the respective creation (anni-
hilation) operator. The double dot is described by the
Hamiltonian
ĤDQD =
j=L,R
εjnjσ +
j=L,R
Ujnj↑nj↓
+ U ′
nLσnRσ′ , (1)
with njσ = d
jσdjσ , where d
jσ (djσ) is the creation (an-
nihilation) operator of an electron with spin σ in the left
(j = L) or right (j = R) quantum dot, and εj is the cor-
responding single-particle energy. The Coulomb interac-
tion on the left (right) dot is described by UL (UR). The
last part of ĤDQD corresponds to the inter-dot Coulomb
correlation, whose strength is given by U ′. As we are in-
terested in the low bias voltage regime where the system
is in the Coulomb blockade, it is justifiable to assume
that the energy level of each dot is independent of the
bias voltage. For the sake of clarity of further discussion
we also assume εL = εR ≡ ε and UL = UR ≡ U .
We note that in a general case, the exchange interac-
tion between spins in the two dots may lead to the forma-
tion of singlet and triplet states.7 However, this exchange
interaction was found to be rather small as compared to
the other energy scales,1 and thus, following previous the-
oretical works,6,55 we will neglect it.
Tunneling processes between the two dots and elec-
trodes are described by the Hamiltonian,
ĤT =
j=L,R
djσ + h.c.
LσdRσ + h.c.
where tj denotes the tunnel matrix elements between the
jth lead and the jth dot, and t describes the hopping be-
tween the two quantum dots. Coupling of the jth dot
to the jth lead can be expressed as Γσj = 2π|tj |
2ρσj , with
ρσj being the spin-dependent density of states of the cor-
responding lead. With the definition of the spin polar-
LL U+ε RR U+ε
FIG. 1: (color online) Schematic of a double quantum dot
coupled to ferromagnetic leads. The magnetic moments of
the leads can form either parallel or antiparallel configuration.
The system is symmetrically biased.
ization of lead j, pj = (ρ
j − ρ
j )/(ρ
j + ρ
j ), the cou-
pling can be expressed as, Γ
j = Γj(1 ± pj), with
Γj = (Γ
j + Γ
j )/2. Here, Γ
j and Γ
j describe the
coupling of the jth dot to the spin-majority and spin-
minority electron bands of lead j, respectively. As re-
ported in Ref. 43, typical values of the coupling strength
are of the order of tens of µeV. In the following, we as-
sume symmetric couplings, ΓL = ΓR ≡ Γ/2, and equal
spin polarizations of the leads, pL = pR ≡ p.
In this paper we analyze spin-dependent transport
through double quantum dot in the case of the Coulomb
blockade regime. We assume that in equilibrium each
dot is singly occupied, so that there are two electrons
in the DQD system. This transport regime can be real-
ized for example in lateral quantum dots3,5,20 or in single
wall carbon nanotubes with top gate electrodes.7,21,22,23
In such devices by changing the respective gate voltages
one can tune the charge on each dot separately and also
change the strength of the coupling t between the two
dots. Furthermore, we also note that in DQDs the on-
level interaction U is usually larger than the inter-dot
interaction U ′. In the case where the DQD is doubly
occupied and U > U ′, the system can be in four differ-
ent states |χ〉, namely | ↑↑〉 = | ↑〉| ↑〉, | ↑↓〉 = | ↑〉| ↓〉,
| ↓↑〉 = | ↓〉| ↑〉, | ↓↓〉 = | ↓〉| ↓〉, where the first (sec-
ond) ket corresponds to the left (right) dot. The occu-
pation of the other two-particle states |d0〉 = | ↑↓〉|0〉
and |0d〉 = |0〉| ↑↓〉 is suppressed due to large on-level
interaction on the dots.
In the Coulomb blockade the charge fluctuations are
suppressed and the system is in a well-defined charge
state. As a consequence, all tunneling processes leading
to a change of the DQD charge state are exponentially
suppressed. The current can thus flow due to higher-
order tunneling processes (cotunneling) through virtual
states in the double quantum dot.31,33,34,35 The lowest-
order processes which give a dominant contribution to
electric current flowing through the DQD structure in the
case of Coulomb blockade are the third-order tunneling
processes. Generally, the rate for an nth-order (n ≥ 2)
tunneling from lead j to lead j′ associated with a change
of the double dot state from χ into χ′ is given by35
jj′ =
v1,v2,...,vn−1
|HT|Φv1〉〈Φv1 |HT|Φv2〉 × · · · × 〈Φvn−1 |HT|Φ
(εi − εv1)(εi − εv2)× · · · × (εi − εvn−1)
δ(εi − εf ) , (3)
where εi and εf denote the energies of initial and final
states and |Φ
j 〉 is the state of the system with an electron
in the lead j and the double dot in state |χ〉, while |Φvn〉
denotes a virtual state of the DQD system and εvn its en-
ergy. From the above expression one can determine the
third-order (n = 3) tunneling rates that give the main
contribution to electric current. We note that there are
also tunneling events that do not affect the DQD charge
state but can have an influence on transport. These are
the second-order processes which take place through a
single tunnel barrier, either left or right. Such single-
barrier processes contribute to the electric current in an
indirect way, namely by changing the occupation prob-
abilities and this way the current. The rate of single-
barrier second-order cotunneling is given by Eq. (3) for
n = 2 and j = j′. It is also worth noting that among dif-
ferent higher-order tunneling events one can distinguish
the elastic (non-spin-flip) and inelastic (spin-flip) pro-
cesses. The former ones change the state of the double
dot (χ 6= χ′), while the latter ones do not (χ = χ′).
L R L R
L RL Rσ
FIG. 2: (color online) Examples of possible tunneling pro-
cesses through double quantum dot in the Coulomb blockade
regime. The third-order process (a) from left to right lead,
where | ↑↓〉 is the initial state (1) and | ↑↑〉 is the final state
(7), takes place via five virtual states (2)-(6). This process
contributes directly to the current flowing through the sys-
tem. The second-order process through the left barrier (b),
with | ↑ σ〉 (| ↓ σ〉) being the initial (1) [finite (4)] state, where
σ =↑, ↓, affects the occupations of the double quantum dot.
This process takes place via two virtual states (2)-(3).
Examples of possible processes in the case of the
Coulomb blockade regime are shown in Fig. 2. The upper
part of the figure presents a third-order process from the
left to right lead which contributes to electric current.
This is an inelastic process which leads to a change of
the double dot state from | ↓↑〉 to | ↑↑〉. It takes place
through five virtual states, as sketched in Fig. 2a. On the
other hand, the bottom part of Fig. 2 displays a single-
barrier second-order process, occurring via two virtual
states. This process does not contribute to electric cur-
rent but affects the DQD occupation probabilities. The
process shown in Fig. 2b takes place through the left
barrier and changes the double dot state from | ↑ σ〉 into
| ↓ σ〉, with σ =↑, ↓. To make the discussion more trans-
parent, in Appendix we present the explicit formulas for
the rates corresponding to the two processes shown in
Fig. 2.
By calculating all the second-order single-barrier and
third-order rates one can determine the occupation prob-
abilities from the following stationary master equation
jj + γ
(3)χ→χ
jj + γ
, (4)
where Pχ denotes the probability for the double dot to be
in state |χ〉. The occupations are fully determined with
the aid of the normalization condition,
Pχ = 1. The
third-order current flowing through the system from the
left to right lead is then given by
I = e
LR − γ
(3)χ→χ
. (5)
We note that generally the use of the master equation
approach may lead to wrong results in the regime where
the level renormalization effects or the effects due to ex-
change field become important, i.e. close to resonance or
for noncollinear magnetic configurations.29,30,31,32 How-
ever, in the following we consider only the case of deep
Coulomb blockade and collinear configurations, which
justifies the employed approach.36,39,40
III. RESULTS AND DISCUSSION
We first present the results for cotunneling through
double quantum dots coupled to nonmagnetic leads, p =
0. Next, we analyze the case when the leads are fer-
romagnetic (p > 0) and the system can be either in
parallel or antiparallel magnetic configuration. The cur-
rent flowing through the system depends then on the
magnetic configuration giving rise to tunnel magnetore-
sistance. The TMR is qualitatively defined as24,25,28
TMR = IP/IAP−1, where IP (IAP) is the current flowing
through the system in the parallel (antiparallel) magnetic
configuration.
A. DQD coupled to nonmagnetic leads
The differential conductance G of the DQD coupled to
nonmagnetic leads as a function of the bias voltage for
several values of the inter-dot interaction parameter U ′ is
shown in Fig. 3. In the case of negligible U ′, the differen-
tial conductance exhibits a smooth parabolic dependence
on the bias voltage. However, for a finite inter-dot cor-
relation, there is a maximum in G at zero bias. As one
can see in the figure, the magnitude of this maximum in-
creases with increasing U ′. Furthermore, when increasing
U ′, the minimum of G at V = 0 splits into two minima,
separated by the zero-bias peak.
In order to understand the mechanism leading to such
behavior, we note that in the spinless case, p = 0, the
occupation probabilities do not depend on the applied
voltage and are equal to 1/4, i.e. each of the four DQD
states, | ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉, is equally occupied. Fur-
thermore, the single-barrier second-order processes which
provide a channel for spin relaxation in the dots do not
have any influence on transport either. As a consequence,
the zero-bias maximum results from intrinsic dependence
of third-order tunneling rates on the value of the inter-
dot correlation parameter U ′.
The electric current flows through the DQD system
due to third-order processes which involve correlated
tunneling through virtual states of the system. More
specifically, these virtual states include the single-particle
states, |σ0〉 and |0σ〉, the two-particle states, |0d〉 and
|d0〉, and the three-particle states, |σd〉 and |dσ〉. In
equilibrium, the energy of these virtual states is respec-
tively given by ε1 = ε, ε2 = 2ε+U , and ε3 = 3ε+U+2U
On the other hand, the energy of the initial state is εi =
2ε+U ′. Consequently, the resolvents that determine the
rates, see Eq. (3), are given by (εi − ε1)
−1 = (ε+U ′)−1,
(εi−ε2)
−1 = (U ′−U)−1, (εi−ε3)
−1 = (−ε−U −U ′)−1.
The third-order tunneling processes take place via two
consecutive virtual states. Thus, the rate is propor-
tional to the product of two resolvents, depending on
virtual states being involved in a process. Generally,
one can distinguish four different contributions – the
first one involves two single-particle states, (ε + U ′)−2,
the second one involves one single-particle and one two-
-6 -4 -2 0 2 4 6
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 3: (color online) Differential conductance as a function
of the bias voltage for different inter-dot interaction parame-
ter U ′, as indicated in the figure. The other parameters are
kBT = 0.1Γ, ε = −10Γ, U = 20Γ, t = Γ, and p = 0.
particle state, [(ε+U ′)(U ′−U)]−1, the third contribution
comes from one two-particle and one three-particle state,
[(U − U ′)(ε + U + U ′)]−1, and the last one involves two
three-particle states, (ε + U + U ′)−2. After a crude es-
timation, one can see from the above formulas that by
increasing U ′, the contribution coming from the first two
resolvents is increased, the third one is roughly constant,
while that of the last resolvent is decreased. This gener-
ally leads to an asymmetry of cotunneling through dif-
ferent virtual states. Such asymmetry gives rise to an
enhancement of the conductance through the system by
increasing the rate of processes occurring via one-particle
and two-particle DQD states. As a result, with increas-
ing U ′, a maximum develops in the differential conduc-
tance at the zero bias, see Fig. 3. On the other hand,
for a given value of U ′, the differential conductance de-
creases with increasing the bias voltage and reaches a
minimum at |eV | ≈ 2U ′. At this bias voltage the effect
of finite inter-dot Coulomb interaction is compensated by
the transport voltage, and the differential conductance
reaches minimum, which is present on both sides of the
zero-bias anomaly. Moreover, although the position of
the two minima depends on the inter-dot correlation, its
value is rather independent of U ′, see Fig. 3.
When considering the case of linear response, zero tem-
perature and negligible inter-dot correlation, the mini-
mum value of the differential conductance can be ap-
proximated by the following formula
e2t2Γ2
ε(ε+ U)
(ε+ U)2
. (6)
For the parameters assumed to calculate Fig. 3, from the
above formula one finds, G = 0.45× 10−3 e2/h, which is
in good agreement with numerical results.
-6 -4 -2 0 2 4 6
1.0 b
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 4: (color online) Bias dependence of the differential con-
ductance in the parallel (a) and antiparallel (b) magnetic con-
figurations for different inter-dot interaction parameter U ′, as
indicated. The other parameters are kBT = 0.1Γ, ε = −10Γ,
U = 20Γ, t = Γ, and p = 0.5.
B. DQD coupled to ferromagnetic leads
If the leads are ferromagnetic (p 6= 0), the single-
barrier second-order processes start to influence trans-
port by affecting the DQD occupation probabilities.
Transport characteristics are then a result of the inter-
play between processes driving the current and processes
leading to spin relaxation in the dots. First, we note that
the rate of single-barrier processes is proportional to tem-
perature, while that of third-order processes depends on
the applied bias voltage, see Eqs. (A1) and (A2). This
will give rise to interesting phenomena, depending on the
relative ratio of the second-order and third-order pro-
cesses, as will be discussed in the following.
In Fig. 4 we show the bias dependence of the differ-
ential conductance for the parallel and antiparallel mag-
netic configurations of the system for several values of the
inter-dot interaction parameter U ′. First of all, it can be
seen that the value of G at the zero bias increases with
increasing the inter-dot correlation. This is a general fea-
ture which is present in both magnetic configurations of
the system and gives rise to the zero-bias maximum, see
Fig. 4a and b. The mechanism leading to such behavior
was already discussed in the nonmagnetic case, i.e. a fi-
nite value of U ′ results in increased cotunneling through
one-particle and two-particle virtual states, which in turn
leads to an enhancement of the differential conductance
at the zero bias.
Another feature visible in the case of ferromagnetic
leads is that even for negligible U ′ there is a small max-
imum in G at the zero bias, irrespective of magnetic
configuration of the system. This maximum bears a re-
semblance to the zero-bias anomaly found in the case of
single quantum dots.36 However, in single quantum dots
the maximum is present only in the antiparallel configu-
ration, while in the case of double quantum dots, inter-
estingly, the zero-bias peak is present in both magnetic
configurations, see Fig. 4a and b. In order to understand
this behavior we note that when there is a finite bias
voltage applied to the system, a nonequilibrium spin ac-
cumulation can build up in the DQD. More precisely, for
positive bias voltage in the parallel configuration one ob-
serves unequal occupation of singlet states, P|↓↑〉 > P|↑↓〉,
while triplets are roughly equally occupied (no spin ac-
cumulation), P|↑↑〉 ≈ P|↓↓〉. On the other hand, in the
antiparallel configuration there is unequal occupation of
triplet states (spin accumulation), P|↓↓〉 > P|↑↑〉, whereas
singlets are equally occupied, P|↓↑〉 ≈ P|↑↓〉. It is fur-
ther interesting to realize that for positive bias voltage
main contribution to the current comes from third-order
tunneling processes having the initial state | ↑↓〉 for the
parallel and | ↑↑〉 for the antiparallel magnetic configura-
tion. Thus, with increasing the bias voltage (V > 0), the
contribution coming from those processes is decreased,
leading to a decreased conductance. As a consequence,
one observes a maximum at the zero bias even in the case
of U ′ = 0, see Fig. 4.
The zero-bias maximum in differential conductance is
therefore a result of superposition of two different ef-
fects. The first one concerns the asymmetry of cotun-
neling through virtual states, which is induced by a fi-
nite value of the inter-dot Coulomb interaction. Whereas
the second one is associated with unequal occupation of
the corresponding DQD states, which results from spin-
dependent tunneling rates.
When the DQD is coupled to ferromagnetic leads, an
important role is played by the single-barrier second-
order processes – they do not contribute to the current,
but lead to the spin relaxation in the double dot system.
In order to gain more intuitive understanding of the dis-
cussed phenomena, in the following we present a crude
quantitative analysis of the processes determining trans-
port behavior. When considering the low temperature
limit and assuming U = −2ε, U ′ = 0, the rate of single-
barrier second-order processes can be approximated by39
|σχ〉→|σ̄χ〉
4kBTΓ
. (7)
On the other hand, we note that generally the fastest
third-order processes are the ones leading to the change
of the dot state from |σσ̄〉 into |σ̄σ〉. With the same
assumptions as made above, one can approximate the
rate of such processes by the following formula
|σσ̄〉→|σ̄σ〉
jj′ ≈
16|eV |t2Γ2
. (8)
-6 -4 -2 0 2 4 6
0.6 b
T = 0.2Γ
T = 0.1Γ
T = 0.05Γ
T = 0.02Γ
T = 0.01Γ
FIG. 5: (color online) Bias dependence of the differential con-
ductance in the parallel (a) and antiparallel (b) magnetic con-
figurations for different temperatures and for U ′ = 2Γ. The
other parameters are the same as in Fig. 4.
The above expressions show explicitly that the relative
ratio of both processes depends on the internal system
parameters as well as the temperature and applied bias
voltage. Furthermore, one can now roughly estimate the
bias voltage at which the corresponding second-order and
third-order processes become comparable, it is given by
|eV | ≈
. (9)
This formula will be helpful in discussing the temperature
dependence of transport characteristics.
The influence of temperature on the bias dependence of
differential conductance in both magnetic configurations
is shown in Fig. 5. One can see that with increasing
thermal energy, the width of the zero-bias peak is in-
creased, while the maximum value of G for V = 0 stays
rather unchanged. This is due to the fact that by raising
the temperature, one increases the role of single-barrier
second-order processes, see Eq. (7), giving rise to faster
spin relaxation. Spin relaxation in turn leads to a de-
crease in the spin accumulation induced in the system.39
Therefore, the temperature effects on the differential con-
ductance are more visible in the antiparallel configuration
than in the parallel one. By decreasing T , the relative
role of second-order processes is decreased, which leads
to larger spin accumulation, P|↓↓〉 > P|↑↑〉. This in turn
gives rise to an increased and more robust drop of the
differential conductance with the bias voltage, see for ex-
ample the curves for kBT = 0.2Γ and kBT = 0.01Γ in
-6 -4 -2 0 2 4 6
-0.04
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 6: (color online) Bias dependence of the TMR for differ-
ent inter-dot interaction parameter U ′ and for kBT = 0.1Γ.
The other parameters are the same as in Fig. 4.
Fig. 5. As a consequence, with decreasing temperature,
the value of the differential conductance at the minimum
is decreased and the width of the zero-bias peak becomes
smaller – the two minima in G appear at smaller bias
voltage. This is due to the fact that the relative ra-
tio of the second-order and third-order processes changes
with changing T and, consequently, the bias voltage at
which the rates of these two processes are comparable is
changed, see Eq. (9). The dependence of the differential
conductance on temperature in the parallel configuration
is less pronounced than in the antiparallel configuration
because for the parallel configuration the single-barrier
spin-flip processes only slightly affect the DQD occupa-
tions. This results from the fact that in the parallel
configuration there is a left-right symmetry between the
couplings to the spin-majority and spin-minority electron
subbands.36
We also note that in the spinless case discussed in pre-
vious subsection the single-barrier second-order processes
do not affect transport in any way, and the occupations
of all DQD states are equal. Therefore, the differential
conductance only slightly depends on temperature.
In Fig. 6 we present the TMR as a function of the
bias voltage for several values of the inter-dot correla-
tion parameter. First of all, it can be seen that for low
bias voltages tunnel magnetoresistance is only slightly af-
fected by the inter-dot interaction. This is due to the fact
that the asymmetry in tunneling through virtual states
induced by finite value of U ′ changes transport charac-
teristics in both magnetic configurations in a similar way,
see Fig. 4. As a consequence, the TMR, which reflects
the difference between the parallel and antiparallel mag-
netic configuration, is roughly independent of the value
of inter-dot correlation.
Another interesting feature visible in Fig. 6 is the sign
change of the TMR – with increasing the bias voltage,
tunnel magnetoresistance decreases from a maximum at
the zero bias to a minimum, at which TMR changes sign
and becomes negative. At this bias voltage conductance
in the parallel configuration is smaller than in the an-
-6 -4 -2 0 2 4 6
T = 0.2Γ
T = 0.1Γ
T = 0.05Γ
T = 0.02Γ
T = 0.01Γ
FIG. 7: (color online) Tunnel magnetoresistance as a function
of the bias voltage for different temperatures and for U ′ = 2Γ.
The other parameters are the same as in Fig. 4.
tiparallel configuration. This seemingly counterintuitive
fact can be understood when one takes into account the
effect of second-order processes giving rise to spin re-
laxation. As already mentioned, in the parallel configu-
ration one finds, P|↑↓〉 6= P|↓↑〉, while in the antiparallel
configuration one has, P|↑↑〉 6= P|↓↓〉. Spin relaxation pro-
cesses decrease the spin accumulation in the antiparallel
configuration, which leads to an enhancement of the dif-
ferential conductance, see Fig. 5b. On the other hand,
in the parallel configuration the DQD occupations only
slightly depend on second-order processes. As a conse-
quence, if the spin relaxation processes are sufficiently
fast, γ(2)
|σχ〉→|σ̄χ〉
jj & γ
(3)|σσ̄〉→|σ̄σ〉
jj′ , one observes negative
TMR effect.
In Fig. 7 we display the TMR effect as a function
of the bias voltage for different temperatures. First of
all, one can see that TMR exhibits a nontrivial depen-
dence on temperature. This is because by changing T ,
one effectively changes the amount of processes leading
to spin relaxation which affect spin accumulation and,
thus, conductance in the antiparallel configuration. For
low temperatures, second-order processes are suppressed
and TMR becomes positive in the whole range of the bias
voltage with a minimum at the zero bias, see the curve
for kBT = 0.01Γ in Fig. 7. On the other hand, for higher
temperatures the rate of single-barrier second-order pro-
cesses is increased, which gives rise to two minima in the
TMR separated by the zero-bias maximum, see Figs. 6
and 7. Moreover, at these minima TMR changes sign
and becomes negative. We note that the negative TMR
was also observed in single quantum dots in the limit of
fast spin relaxation in the dot.39
Finally, we present analytical formulas approximat-
ing tunnel magnetoresistance in the most characteristic
transport regimes. For |eV | ≫ kBT , the TMR can be
expressed as
TMR =
1− p2
53− 3p2(1 + 7p2 − p4)
(5 + 3p2)(3 − p2)2
, (10)
where we have assumed the symmetric Anderson model
for each dot, U = −2ε, and U ′ = 0. This formula ap-
proximates the TMR in the zero temperature limit, i.e.
in the absence of second-order processes. On the other
hand, the linear response TMR calculated with the same
assumptions can be approximated by
TMR =
1− p2
13 + 3p4
(9 + p2)(5 + 3p2)
. (11)
IV. CONCLUDING REMARKS
We have considered cotunneling transport through
double quantum dots in series weakly coupled to fer-
romagnetic leads. In the Coulomb blockade regime the
current flows through the system due to third-order tun-
neling processes. We have also taken into account the
single-barrier second-order processes which do not con-
tribute to the current but affect the DQD occupation
probabilities.
We have shown that the differential conductance ex-
hibits a maximum at the zero bias, irrespective of mag-
netic configuration of the system. This anomalous be-
havior results from the superposition of two different ef-
fects. The first effect is associated with asymmetry of
cotunneling through different virtual states which can
be induced by the inter-dot Coulomb interaction. The
second mechanism results from the interplay of single-
barrier second-order processes leading to spin relaxation
and the third-order tunneling processes contributing to
the current. The first mechanism does not depend on
the value of spin polarization of the leads, the second
one, on the contrary, results from the spin dependency of
tunneling rates.
We have also analyzed the temperature dependence of
transport characteristics. By changing thermal energy,
one effectively changes the rate of the second-order pro-
cesses, i.e. the amount of spin relaxation processes. We
have shown that the width of the zero-bias maximum
in the differential conductance increases with increasing
temperature. This effect is most visible in the antiparal-
lel configuration, which is due to the fact that in the an-
tiparallel configuration spin relaxation decreases the spin
accumulation induced in the DQD system, while occupa-
tions in the parallel configuration only slightly depend on
the spin relaxation.
Furthermore, we have also shown that TMR exhibits
a nontrivial dependence on temperature. For low tem-
peratures, the TMR exhibits a minimum at the zero
bias. However, for higher temperatures this minimum
splits into two minima separated by a maximum at the
zero bias. At the these minima tunnel magnetoresistance
changes sign and becomes negative.
Acknowledgments
We acknowledge discussions with J. Barnaś. This
work, as part of the European Science Foundation EU-
ROCORES Programme SPINTRA, was supported by
funds from the Ministry of Science and Higher Educa-
tion as a research project in years 2006-2009 and the EC
Sixth Framework Programme, under Contract N. ERAS-
CT-2003-980409, and the Foundation for Polish Science.
APPENDIX A: EXAMPLES OF COTUNNELING
RATES
In the following we present the explicit formulas for the
third-order and second-order tunneling rates correspond-
ing to processes shown in Fig. 2a and b. To determine the
rate γ(3)
|↓↑〉→|↑↑〉
LR one needs to find the initial and final
energies of the whole process, as well as the energies of
the virtual states, as sketched in Fig. 2a. Then, by cal-
culating the respective energy differences and plugging
them into Eq. (3), one finds
|↓↑〉→|↑↑〉
dωf+(ω)f−(ω + µL − µR)
(ω + µL − ε− U − U ′)2
U − U ′
ω + µL − ε− U − U ′
ω + µL − ε− U ′
,(A1)
where f+ is the Fermi function and f− = 1 − f+. On
the other hand, the single-barrier second-order rate for
the process shown in Fig. 2b can be found in a similar
way. This rate is given by
|↑σ〉→|↓σ〉
dωf+(ω)f−(ω)
ω + µL − ε− U − U ′
ω + µL − ε− U ′
. (A2)
We note that in the simplest approximation38 for the
Coulomb blockade regime one can pull out the resolvents
in front of the integrals. Then, one arrives at the follow-
ing integral,
dωf+(ω)f−(ω+ ξ), where ξ = µL − µR or
ξ = 0, correspondingly, which can be easily calculated.35
As a consequence, one can see that the rate of single-
barrier second-order processes is proportional to temper-
ature T , whereas that of the third-order processes de-
pends on the bias voltage V .
∗ Electronic address: weymann@amu.edu.pl
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  We analyze cotunneling transport through two quantum dots in series weakly
coupled to external ferromagnetic leads. In the Coulomb blockade regime the
electric current flows due to third-order tunneling, while the second-order
single-barrier processes have indirect impact on the current by changing the
occupation probabilities of the double dot system. We predict a zero-bias
maximum in the differential conductance, whose magnitude is conditioned by the
value of the inter-dot Coulomb interaction. This maximum is present in both
magnetic configurations of the system and results from asymmetry in cotunneling
through different virtual states. Furthermore, we show that tunnel
magnetoresistance exhibits a distinctively different behavior depending on
temperature, being rather independent of the value of inter-dot correlation.
Moreover, we find negative TMR in some range of the bias voltage.

Spin-polarized transport through weakly coupled double quantum dots in the
Coulomb-blockade regime
Ireneusz Weymann1, ∗
Department of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
(Dated: October 29, 2018)
We analyze cotunneling transport through two quantum dots in series weakly coupled to external
ferromagnetic leads. In the Coulomb blockade regime the electric current flows due to third-order
tunneling, while the second-order single-barrier processes have indirect impact on the current by
changing the occupation probabilities of the double dot system. We predict a zero-bias maximum in
the differential conductance, whose magnitude is conditioned by the value of the inter-dot Coulomb
interaction. This maximum is present in both magnetic configurations of the system and results
from asymmetry in cotunneling through different virtual states. Furthermore, we show that tunnel
magnetoresistance exhibits a distinctively different behavior depending on temperature, being rather
independent of the value of inter-dot correlation. Moreover, we find negative TMR in some range
of the bias voltage.
PACS numbers: 72.25.Mk, 73.63.Kv, 85.75.-d, 73.23.Hk
I. INTRODUCTION
Transport properties of double quantum
dots (DQDs) have recently attracted much
interest.1,2,3,4,5,6,7,8,9,10,11,12,13 This is because these
structures are ideal systems to study the fundamental
interactions between electrons and spins.14,15,16,17,18
Double quantum dots exhibit a variety of interesting
effects, such as for example current rectification due to
the Pauli spin blockade,1,10,11,12,13 negative differential
conductance,3 formation of molecular states,7 spin
pumping,8,9 etc. Furthermore, double quantum dots are
also being considered for future applications in quantum
computing.19 Experimentally, such systems may be
realized for example in lateral and vertical semiconduc-
tor quantum dots.1,3,5,20 Another implementation of
DQDs are single wall metallic carbon nanotubes with
top gate electrodes, which enable changing of charge
on each dot separately, as well as the intrinsic DQD
parameters.7,21,22,23
The goal of this paper is to analyze transport prop-
erties of DQDs weakly coupled to ferromagnetic leads
in the Coulomb blockade regime. When the leads are
ferromagnetic, transport strongly depends on the mag-
netic configuration of the system, giving rise to tunnel
magnetoresistance (TMR), spin accumulation, exchange
field, etc.24,25,26,27,28,29,30,31,32 In the Coulomb blockade
regime the electric current flows due to higher-order tun-
neling processes (cotunneling), while the first-order tun-
neling processes (sequential tunneling) are exponentially
suppressed.31,33,34,35 The problem of spin-polarized co-
tunneling has been so far addressed mainly in the case
of single quantum dots.31,32,36,37,38,39,40,41 For example,
it was shown that tunnel magnetoresistance exhibits dis-
tinctively different behavior depending on the number of
electrons on the dot.31 Moreover, the zero-bias anomaly
was found in the differential conductance when magnetic
moments of the leads form antiparallel configuration.36
Another interesting behavior was predicted for quantum
dots coupled to ferromagnetic leads with non-collinear
alignment of magnetizations – the exchange field was
found to increase the differential conductance for certain
non-collinear configurations, as compared to the parallel
one.32,42 On the other hand, it was shown experimentally
for nonmagnetic systems that the conductance of quan-
tum dots in the cotunneling regime may serve as a handle
to determine the spectroscopic g-factor.43
In the case of double quantum dots considered in this
paper, in the Coulomb blockade regime the electric cur-
rent flows due to third-order tunneling processes, while
the single-barrier second-order processes together with
third-order processes determine the double dot occupa-
tion probabilities. Assuming that double quantum dot
is occupied by two electrons in equilibrium, one on each
dot, we calculate the differential conductance G and tun-
nel magnetoresistance TMR. We show that differential
conductance exhibits a maximum at the zero bias. We
further distinguish two different mechanisms leading to
this new zero-bias anomaly. The first one is an asym-
metry in cotunneling through different virtual states of
the DQD system, which leads to an enhancement of G at
zero bias. Such asymmetry is induced by a finite value
of the inter-dot Coulomb interaction. This mechanism
is rather independent of magnetic configuration of the
system. The second mechanism leading to the zero-bias
maximum in differential conductance is the interplay be-
tween spin accumulation and third-order tunneling pro-
cesses carrying the current. This mechanism does depend
on the magnetic configuration of the system and, as we
show in the sequel, is found to be more efficient in the
antiparallel configuration. We also analyze the behavior
of TMR and show that the TMR exhibits a maximum
at zero bias, which strongly depends on the temperature.
Furthermore, the TMR may become negative in some
range of the bias voltage.
Finally, we note that there are several experimental re-
alizations of single quantum dots attached to ferromag-
http://arxiv.org/abs/0704.1213v1
netic leads,44,45,46,47,48,49,50,51,52,53,54 while experimental
data on spin-polarized transport through double quan-
tum dots is lacking. We believe that the results presented
in this paper will be of assistance in discussing future ex-
periments.
II. MODEL AND METHOD
The schematic of a double quantum dot coupled to fer-
romagnetic leads is shown in Fig. 1. It is assumed that
the magnetizations of the leads are oriented collinearly,
so that the system can be either in the parallel or antipar-
allel magnetic configuration. The Hamiltonian Ĥ of the
DQD system is given by, Ĥ = ĤL+ĤR+ĤDQD+ĤT. The
first two terms describe noninteracting itinerant electrons
in the leads, Ĥj =
εjkσc
cjkσ for the left (j = L)
and right (j = R) lead, where εjkσ is the energy of an
electron with the wave vector k and spin σ in the lead
j, and c
(cjkσ) denotes the respective creation (anni-
hilation) operator. The double dot is described by the
Hamiltonian
ĤDQD =
j=L,R
εjnjσ +
j=L,R
Ujnj↑nj↓
+ U ′
nLσnRσ′ , (1)
with njσ = d
jσdjσ , where d
jσ (djσ) is the creation (an-
nihilation) operator of an electron with spin σ in the left
(j = L) or right (j = R) quantum dot, and εj is the cor-
responding single-particle energy. The Coulomb interac-
tion on the left (right) dot is described by UL (UR). The
last part of ĤDQD corresponds to the inter-dot Coulomb
correlation, whose strength is given by U ′. As we are in-
terested in the low bias voltage regime where the system
is in the Coulomb blockade, it is justifiable to assume
that the energy level of each dot is independent of the
bias voltage. For the sake of clarity of further discussion
we also assume εL = εR ≡ ε and UL = UR ≡ U .
We note that in a general case, the exchange interac-
tion between spins in the two dots may lead to the forma-
tion of singlet and triplet states.7 However, this exchange
interaction was found to be rather small as compared to
the other energy scales,1 and thus, following previous the-
oretical works,6,55 we will neglect it.
Tunneling processes between the two dots and elec-
trodes are described by the Hamiltonian,
ĤT =
j=L,R
djσ + h.c.
LσdRσ + h.c.
where tj denotes the tunnel matrix elements between the
jth lead and the jth dot, and t describes the hopping be-
tween the two quantum dots. Coupling of the jth dot
to the jth lead can be expressed as Γσj = 2π|tj |
2ρσj , with
ρσj being the spin-dependent density of states of the cor-
responding lead. With the definition of the spin polar-
LL U+ε RR U+ε
FIG. 1: (color online) Schematic of a double quantum dot
coupled to ferromagnetic leads. The magnetic moments of
the leads can form either parallel or antiparallel configuration.
The system is symmetrically biased.
ization of lead j, pj = (ρ
j − ρ
j )/(ρ
j + ρ
j ), the cou-
pling can be expressed as, Γ
j = Γj(1 ± pj), with
Γj = (Γ
j + Γ
j )/2. Here, Γ
j and Γ
j describe the
coupling of the jth dot to the spin-majority and spin-
minority electron bands of lead j, respectively. As re-
ported in Ref. 43, typical values of the coupling strength
are of the order of tens of µeV. In the following, we as-
sume symmetric couplings, ΓL = ΓR ≡ Γ/2, and equal
spin polarizations of the leads, pL = pR ≡ p.
In this paper we analyze spin-dependent transport
through double quantum dot in the case of the Coulomb
blockade regime. We assume that in equilibrium each
dot is singly occupied, so that there are two electrons
in the DQD system. This transport regime can be real-
ized for example in lateral quantum dots3,5,20 or in single
wall carbon nanotubes with top gate electrodes.7,21,22,23
In such devices by changing the respective gate voltages
one can tune the charge on each dot separately and also
change the strength of the coupling t between the two
dots. Furthermore, we also note that in DQDs the on-
level interaction U is usually larger than the inter-dot
interaction U ′. In the case where the DQD is doubly
occupied and U > U ′, the system can be in four differ-
ent states |χ〉, namely | ↑↑〉 = | ↑〉| ↑〉, | ↑↓〉 = | ↑〉| ↓〉,
| ↓↑〉 = | ↓〉| ↑〉, | ↓↓〉 = | ↓〉| ↓〉, where the first (sec-
ond) ket corresponds to the left (right) dot. The occu-
pation of the other two-particle states |d0〉 = | ↑↓〉|0〉
and |0d〉 = |0〉| ↑↓〉 is suppressed due to large on-level
interaction on the dots.
In the Coulomb blockade the charge fluctuations are
suppressed and the system is in a well-defined charge
state. As a consequence, all tunneling processes leading
to a change of the DQD charge state are exponentially
suppressed. The current can thus flow due to higher-
order tunneling processes (cotunneling) through virtual
states in the double quantum dot.31,33,34,35 The lowest-
order processes which give a dominant contribution to
electric current flowing through the DQD structure in the
case of Coulomb blockade are the third-order tunneling
processes. Generally, the rate for an nth-order (n ≥ 2)
tunneling from lead j to lead j′ associated with a change
of the double dot state from χ into χ′ is given by35
jj′ =
v1,v2,...,vn−1
|HT|Φv1〉〈Φv1 |HT|Φv2〉 × · · · × 〈Φvn−1 |HT|Φ
(εi − εv1)(εi − εv2)× · · · × (εi − εvn−1)
δ(εi − εf ) , (3)
where εi and εf denote the energies of initial and final
states and |Φ
j 〉 is the state of the system with an electron
in the lead j and the double dot in state |χ〉, while |Φvn〉
denotes a virtual state of the DQD system and εvn its en-
ergy. From the above expression one can determine the
third-order (n = 3) tunneling rates that give the main
contribution to electric current. We note that there are
also tunneling events that do not affect the DQD charge
state but can have an influence on transport. These are
the second-order processes which take place through a
single tunnel barrier, either left or right. Such single-
barrier processes contribute to the electric current in an
indirect way, namely by changing the occupation prob-
abilities and this way the current. The rate of single-
barrier second-order cotunneling is given by Eq. (3) for
n = 2 and j = j′. It is also worth noting that among dif-
ferent higher-order tunneling events one can distinguish
the elastic (non-spin-flip) and inelastic (spin-flip) pro-
cesses. The former ones change the state of the double
dot (χ 6= χ′), while the latter ones do not (χ = χ′).
L R L R
L RL Rσ
FIG. 2: (color online) Examples of possible tunneling pro-
cesses through double quantum dot in the Coulomb blockade
regime. The third-order process (a) from left to right lead,
where | ↑↓〉 is the initial state (1) and | ↑↑〉 is the final state
(7), takes place via five virtual states (2)-(6). This process
contributes directly to the current flowing through the sys-
tem. The second-order process through the left barrier (b),
with | ↑ σ〉 (| ↓ σ〉) being the initial (1) [finite (4)] state, where
σ =↑, ↓, affects the occupations of the double quantum dot.
This process takes place via two virtual states (2)-(3).
Examples of possible processes in the case of the
Coulomb blockade regime are shown in Fig. 2. The upper
part of the figure presents a third-order process from the
left to right lead which contributes to electric current.
This is an inelastic process which leads to a change of
the double dot state from | ↓↑〉 to | ↑↑〉. It takes place
through five virtual states, as sketched in Fig. 2a. On the
other hand, the bottom part of Fig. 2 displays a single-
barrier second-order process, occurring via two virtual
states. This process does not contribute to electric cur-
rent but affects the DQD occupation probabilities. The
process shown in Fig. 2b takes place through the left
barrier and changes the double dot state from | ↑ σ〉 into
| ↓ σ〉, with σ =↑, ↓. To make the discussion more trans-
parent, in Appendix we present the explicit formulas for
the rates corresponding to the two processes shown in
Fig. 2.
By calculating all the second-order single-barrier and
third-order rates one can determine the occupation prob-
abilities from the following stationary master equation
jj + γ
(3)χ→χ
jj + γ
, (4)
where Pχ denotes the probability for the double dot to be
in state |χ〉. The occupations are fully determined with
the aid of the normalization condition,
Pχ = 1. The
third-order current flowing through the system from the
left to right lead is then given by
I = e
LR − γ
(3)χ→χ
. (5)
We note that generally the use of the master equation
approach may lead to wrong results in the regime where
the level renormalization effects or the effects due to ex-
change field become important, i.e. close to resonance or
for noncollinear magnetic configurations.29,30,31,32 How-
ever, in the following we consider only the case of deep
Coulomb blockade and collinear configurations, which
justifies the employed approach.36,39,40
III. RESULTS AND DISCUSSION
We first present the results for cotunneling through
double quantum dots coupled to nonmagnetic leads, p =
0. Next, we analyze the case when the leads are fer-
romagnetic (p > 0) and the system can be either in
parallel or antiparallel magnetic configuration. The cur-
rent flowing through the system depends then on the
magnetic configuration giving rise to tunnel magnetore-
sistance. The TMR is qualitatively defined as24,25,28
TMR = IP/IAP−1, where IP (IAP) is the current flowing
through the system in the parallel (antiparallel) magnetic
configuration.
A. DQD coupled to nonmagnetic leads
The differential conductance G of the DQD coupled to
nonmagnetic leads as a function of the bias voltage for
several values of the inter-dot interaction parameter U ′ is
shown in Fig. 3. In the case of negligible U ′, the differen-
tial conductance exhibits a smooth parabolic dependence
on the bias voltage. However, for a finite inter-dot cor-
relation, there is a maximum in G at zero bias. As one
can see in the figure, the magnitude of this maximum in-
creases with increasing U ′. Furthermore, when increasing
U ′, the minimum of G at V = 0 splits into two minima,
separated by the zero-bias peak.
In order to understand the mechanism leading to such
behavior, we note that in the spinless case, p = 0, the
occupation probabilities do not depend on the applied
voltage and are equal to 1/4, i.e. each of the four DQD
states, | ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉, is equally occupied. Fur-
thermore, the single-barrier second-order processes which
provide a channel for spin relaxation in the dots do not
have any influence on transport either. As a consequence,
the zero-bias maximum results from intrinsic dependence
of third-order tunneling rates on the value of the inter-
dot correlation parameter U ′.
The electric current flows through the DQD system
due to third-order processes which involve correlated
tunneling through virtual states of the system. More
specifically, these virtual states include the single-particle
states, |σ0〉 and |0σ〉, the two-particle states, |0d〉 and
|d0〉, and the three-particle states, |σd〉 and |dσ〉. In
equilibrium, the energy of these virtual states is respec-
tively given by ε1 = ε, ε2 = 2ε+U , and ε3 = 3ε+U+2U
On the other hand, the energy of the initial state is εi =
2ε+U ′. Consequently, the resolvents that determine the
rates, see Eq. (3), are given by (εi − ε1)
−1 = (ε+U ′)−1,
(εi−ε2)
−1 = (U ′−U)−1, (εi−ε3)
−1 = (−ε−U −U ′)−1.
The third-order tunneling processes take place via two
consecutive virtual states. Thus, the rate is propor-
tional to the product of two resolvents, depending on
virtual states being involved in a process. Generally,
one can distinguish four different contributions – the
first one involves two single-particle states, (ε + U ′)−2,
the second one involves one single-particle and one two-
-6 -4 -2 0 2 4 6
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 3: (color online) Differential conductance as a function
of the bias voltage for different inter-dot interaction parame-
ter U ′, as indicated in the figure. The other parameters are
kBT = 0.1Γ, ε = −10Γ, U = 20Γ, t = Γ, and p = 0.
particle state, [(ε+U ′)(U ′−U)]−1, the third contribution
comes from one two-particle and one three-particle state,
[(U − U ′)(ε + U + U ′)]−1, and the last one involves two
three-particle states, (ε + U + U ′)−2. After a crude es-
timation, one can see from the above formulas that by
increasing U ′, the contribution coming from the first two
resolvents is increased, the third one is roughly constant,
while that of the last resolvent is decreased. This gener-
ally leads to an asymmetry of cotunneling through dif-
ferent virtual states. Such asymmetry gives rise to an
enhancement of the conductance through the system by
increasing the rate of processes occurring via one-particle
and two-particle DQD states. As a result, with increas-
ing U ′, a maximum develops in the differential conduc-
tance at the zero bias, see Fig. 3. On the other hand,
for a given value of U ′, the differential conductance de-
creases with increasing the bias voltage and reaches a
minimum at |eV | ≈ 2U ′. At this bias voltage the effect
of finite inter-dot Coulomb interaction is compensated by
the transport voltage, and the differential conductance
reaches minimum, which is present on both sides of the
zero-bias anomaly. Moreover, although the position of
the two minima depends on the inter-dot correlation, its
value is rather independent of U ′, see Fig. 3.
When considering the case of linear response, zero tem-
perature and negligible inter-dot correlation, the mini-
mum value of the differential conductance can be ap-
proximated by the following formula
e2t2Γ2
ε(ε+ U)
(ε+ U)2
. (6)
For the parameters assumed to calculate Fig. 3, from the
above formula one finds, G = 0.45× 10−3 e2/h, which is
in good agreement with numerical results.
-6 -4 -2 0 2 4 6
1.0 b
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 4: (color online) Bias dependence of the differential con-
ductance in the parallel (a) and antiparallel (b) magnetic con-
figurations for different inter-dot interaction parameter U ′, as
indicated. The other parameters are kBT = 0.1Γ, ε = −10Γ,
U = 20Γ, t = Γ, and p = 0.5.
B. DQD coupled to ferromagnetic leads
If the leads are ferromagnetic (p 6= 0), the single-
barrier second-order processes start to influence trans-
port by affecting the DQD occupation probabilities.
Transport characteristics are then a result of the inter-
play between processes driving the current and processes
leading to spin relaxation in the dots. First, we note that
the rate of single-barrier processes is proportional to tem-
perature, while that of third-order processes depends on
the applied bias voltage, see Eqs. (A1) and (A2). This
will give rise to interesting phenomena, depending on the
relative ratio of the second-order and third-order pro-
cesses, as will be discussed in the following.
In Fig. 4 we show the bias dependence of the differ-
ential conductance for the parallel and antiparallel mag-
netic configurations of the system for several values of the
inter-dot interaction parameter U ′. First of all, it can be
seen that the value of G at the zero bias increases with
increasing the inter-dot correlation. This is a general fea-
ture which is present in both magnetic configurations of
the system and gives rise to the zero-bias maximum, see
Fig. 4a and b. The mechanism leading to such behavior
was already discussed in the nonmagnetic case, i.e. a fi-
nite value of U ′ results in increased cotunneling through
one-particle and two-particle virtual states, which in turn
leads to an enhancement of the differential conductance
at the zero bias.
Another feature visible in the case of ferromagnetic
leads is that even for negligible U ′ there is a small max-
imum in G at the zero bias, irrespective of magnetic
configuration of the system. This maximum bears a re-
semblance to the zero-bias anomaly found in the case of
single quantum dots.36 However, in single quantum dots
the maximum is present only in the antiparallel configu-
ration, while in the case of double quantum dots, inter-
estingly, the zero-bias peak is present in both magnetic
configurations, see Fig. 4a and b. In order to understand
this behavior we note that when there is a finite bias
voltage applied to the system, a nonequilibrium spin ac-
cumulation can build up in the DQD. More precisely, for
positive bias voltage in the parallel configuration one ob-
serves unequal occupation of singlet states, P|↓↑〉 > P|↑↓〉,
while triplets are roughly equally occupied (no spin ac-
cumulation), P|↑↑〉 ≈ P|↓↓〉. On the other hand, in the
antiparallel configuration there is unequal occupation of
triplet states (spin accumulation), P|↓↓〉 > P|↑↑〉, whereas
singlets are equally occupied, P|↓↑〉 ≈ P|↑↓〉. It is fur-
ther interesting to realize that for positive bias voltage
main contribution to the current comes from third-order
tunneling processes having the initial state | ↑↓〉 for the
parallel and | ↑↑〉 for the antiparallel magnetic configura-
tion. Thus, with increasing the bias voltage (V > 0), the
contribution coming from those processes is decreased,
leading to a decreased conductance. As a consequence,
one observes a maximum at the zero bias even in the case
of U ′ = 0, see Fig. 4.
The zero-bias maximum in differential conductance is
therefore a result of superposition of two different ef-
fects. The first one concerns the asymmetry of cotun-
neling through virtual states, which is induced by a fi-
nite value of the inter-dot Coulomb interaction. Whereas
the second one is associated with unequal occupation of
the corresponding DQD states, which results from spin-
dependent tunneling rates.
When the DQD is coupled to ferromagnetic leads, an
important role is played by the single-barrier second-
order processes – they do not contribute to the current,
but lead to the spin relaxation in the double dot system.
In order to gain more intuitive understanding of the dis-
cussed phenomena, in the following we present a crude
quantitative analysis of the processes determining trans-
port behavior. When considering the low temperature
limit and assuming U = −2ε, U ′ = 0, the rate of single-
barrier second-order processes can be approximated by39
|σχ〉→|σ̄χ〉
4kBTΓ
. (7)
On the other hand, we note that generally the fastest
third-order processes are the ones leading to the change
of the dot state from |σσ̄〉 into |σ̄σ〉. With the same
assumptions as made above, one can approximate the
rate of such processes by the following formula
|σσ̄〉→|σ̄σ〉
jj′ ≈
16|eV |t2Γ2
. (8)
-6 -4 -2 0 2 4 6
0.6 b
T = 0.2Γ
T = 0.1Γ
T = 0.05Γ
T = 0.02Γ
T = 0.01Γ
FIG. 5: (color online) Bias dependence of the differential con-
ductance in the parallel (a) and antiparallel (b) magnetic con-
figurations for different temperatures and for U ′ = 2Γ. The
other parameters are the same as in Fig. 4.
The above expressions show explicitly that the relative
ratio of both processes depends on the internal system
parameters as well as the temperature and applied bias
voltage. Furthermore, one can now roughly estimate the
bias voltage at which the corresponding second-order and
third-order processes become comparable, it is given by
|eV | ≈
. (9)
This formula will be helpful in discussing the temperature
dependence of transport characteristics.
The influence of temperature on the bias dependence of
differential conductance in both magnetic configurations
is shown in Fig. 5. One can see that with increasing
thermal energy, the width of the zero-bias peak is in-
creased, while the maximum value of G for V = 0 stays
rather unchanged. This is due to the fact that by raising
the temperature, one increases the role of single-barrier
second-order processes, see Eq. (7), giving rise to faster
spin relaxation. Spin relaxation in turn leads to a de-
crease in the spin accumulation induced in the system.39
Therefore, the temperature effects on the differential con-
ductance are more visible in the antiparallel configuration
than in the parallel one. By decreasing T , the relative
role of second-order processes is decreased, which leads
to larger spin accumulation, P|↓↓〉 > P|↑↑〉. This in turn
gives rise to an increased and more robust drop of the
differential conductance with the bias voltage, see for ex-
ample the curves for kBT = 0.2Γ and kBT = 0.01Γ in
-6 -4 -2 0 2 4 6
-0.04
 U' = 0
 U' = Γ
 U' = 2Γ
 U' = 3Γ
FIG. 6: (color online) Bias dependence of the TMR for differ-
ent inter-dot interaction parameter U ′ and for kBT = 0.1Γ.
The other parameters are the same as in Fig. 4.
Fig. 5. As a consequence, with decreasing temperature,
the value of the differential conductance at the minimum
is decreased and the width of the zero-bias peak becomes
smaller – the two minima in G appear at smaller bias
voltage. This is due to the fact that the relative ra-
tio of the second-order and third-order processes changes
with changing T and, consequently, the bias voltage at
which the rates of these two processes are comparable is
changed, see Eq. (9). The dependence of the differential
conductance on temperature in the parallel configuration
is less pronounced than in the antiparallel configuration
because for the parallel configuration the single-barrier
spin-flip processes only slightly affect the DQD occupa-
tions. This results from the fact that in the parallel
configuration there is a left-right symmetry between the
couplings to the spin-majority and spin-minority electron
subbands.36
We also note that in the spinless case discussed in pre-
vious subsection the single-barrier second-order processes
do not affect transport in any way, and the occupations
of all DQD states are equal. Therefore, the differential
conductance only slightly depends on temperature.
In Fig. 6 we present the TMR as a function of the
bias voltage for several values of the inter-dot correla-
tion parameter. First of all, it can be seen that for low
bias voltages tunnel magnetoresistance is only slightly af-
fected by the inter-dot interaction. This is due to the fact
that the asymmetry in tunneling through virtual states
induced by finite value of U ′ changes transport charac-
teristics in both magnetic configurations in a similar way,
see Fig. 4. As a consequence, the TMR, which reflects
the difference between the parallel and antiparallel mag-
netic configuration, is roughly independent of the value
of inter-dot correlation.
Another interesting feature visible in Fig. 6 is the sign
change of the TMR – with increasing the bias voltage,
tunnel magnetoresistance decreases from a maximum at
the zero bias to a minimum, at which TMR changes sign
and becomes negative. At this bias voltage conductance
in the parallel configuration is smaller than in the an-
-6 -4 -2 0 2 4 6
T = 0.2Γ
T = 0.1Γ
T = 0.05Γ
T = 0.02Γ
T = 0.01Γ
FIG. 7: (color online) Tunnel magnetoresistance as a function
of the bias voltage for different temperatures and for U ′ = 2Γ.
The other parameters are the same as in Fig. 4.
tiparallel configuration. This seemingly counterintuitive
fact can be understood when one takes into account the
effect of second-order processes giving rise to spin re-
laxation. As already mentioned, in the parallel configu-
ration one finds, P|↑↓〉 6= P|↓↑〉, while in the antiparallel
configuration one has, P|↑↑〉 6= P|↓↓〉. Spin relaxation pro-
cesses decrease the spin accumulation in the antiparallel
configuration, which leads to an enhancement of the dif-
ferential conductance, see Fig. 5b. On the other hand,
in the parallel configuration the DQD occupations only
slightly depend on second-order processes. As a conse-
quence, if the spin relaxation processes are sufficiently
fast, γ(2)
|σχ〉→|σ̄χ〉
jj & γ
(3)|σσ̄〉→|σ̄σ〉
jj′ , one observes negative
TMR effect.
In Fig. 7 we display the TMR effect as a function
of the bias voltage for different temperatures. First of
all, one can see that TMR exhibits a nontrivial depen-
dence on temperature. This is because by changing T ,
one effectively changes the amount of processes leading
to spin relaxation which affect spin accumulation and,
thus, conductance in the antiparallel configuration. For
low temperatures, second-order processes are suppressed
and TMR becomes positive in the whole range of the bias
voltage with a minimum at the zero bias, see the curve
for kBT = 0.01Γ in Fig. 7. On the other hand, for higher
temperatures the rate of single-barrier second-order pro-
cesses is increased, which gives rise to two minima in the
TMR separated by the zero-bias maximum, see Figs. 6
and 7. Moreover, at these minima TMR changes sign
and becomes negative. We note that the negative TMR
was also observed in single quantum dots in the limit of
fast spin relaxation in the dot.39
Finally, we present analytical formulas approximat-
ing tunnel magnetoresistance in the most characteristic
transport regimes. For |eV | ≫ kBT , the TMR can be
expressed as
TMR =
1− p2
53− 3p2(1 + 7p2 − p4)
(5 + 3p2)(3 − p2)2
, (10)
where we have assumed the symmetric Anderson model
for each dot, U = −2ε, and U ′ = 0. This formula ap-
proximates the TMR in the zero temperature limit, i.e.
in the absence of second-order processes. On the other
hand, the linear response TMR calculated with the same
assumptions can be approximated by
TMR =
1− p2
13 + 3p4
(9 + p2)(5 + 3p2)
. (11)
IV. CONCLUDING REMARKS
We have considered cotunneling transport through
double quantum dots in series weakly coupled to fer-
romagnetic leads. In the Coulomb blockade regime the
current flows through the system due to third-order tun-
neling processes. We have also taken into account the
single-barrier second-order processes which do not con-
tribute to the current but affect the DQD occupation
probabilities.
We have shown that the differential conductance ex-
hibits a maximum at the zero bias, irrespective of mag-
netic configuration of the system. This anomalous be-
havior results from the superposition of two different ef-
fects. The first effect is associated with asymmetry of
cotunneling through different virtual states which can
be induced by the inter-dot Coulomb interaction. The
second mechanism results from the interplay of single-
barrier second-order processes leading to spin relaxation
and the third-order tunneling processes contributing to
the current. The first mechanism does not depend on
the value of spin polarization of the leads, the second
one, on the contrary, results from the spin dependency of
tunneling rates.
We have also analyzed the temperature dependence of
transport characteristics. By changing thermal energy,
one effectively changes the rate of the second-order pro-
cesses, i.e. the amount of spin relaxation processes. We
have shown that the width of the zero-bias maximum
in the differential conductance increases with increasing
temperature. This effect is most visible in the antiparal-
lel configuration, which is due to the fact that in the an-
tiparallel configuration spin relaxation decreases the spin
accumulation induced in the DQD system, while occupa-
tions in the parallel configuration only slightly depend on
the spin relaxation.
Furthermore, we have also shown that TMR exhibits
a nontrivial dependence on temperature. For low tem-
peratures, the TMR exhibits a minimum at the zero
bias. However, for higher temperatures this minimum
splits into two minima separated by a maximum at the
zero bias. At the these minima tunnel magnetoresistance
changes sign and becomes negative.
Acknowledgments
We acknowledge discussions with J. Barnaś. This
work, as part of the European Science Foundation EU-
ROCORES Programme SPINTRA, was supported by
funds from the Ministry of Science and Higher Educa-
tion as a research project in years 2006-2009 and the EC
Sixth Framework Programme, under Contract N. ERAS-
CT-2003-980409, and the Foundation for Polish Science.
APPENDIX A: EXAMPLES OF COTUNNELING
RATES
In the following we present the explicit formulas for the
third-order and second-order tunneling rates correspond-
ing to processes shown in Fig. 2a and b. To determine the
rate γ(3)
|↓↑〉→|↑↑〉
LR one needs to find the initial and final
energies of the whole process, as well as the energies of
the virtual states, as sketched in Fig. 2a. Then, by cal-
culating the respective energy differences and plugging
them into Eq. (3), one finds
|↓↑〉→|↑↑〉
dωf+(ω)f−(ω + µL − µR)
(ω + µL − ε− U − U ′)2
U − U ′
ω + µL − ε− U − U ′
ω + µL − ε− U ′
,(A1)
where f+ is the Fermi function and f− = 1 − f+. On
the other hand, the single-barrier second-order rate for
the process shown in Fig. 2b can be found in a similar
way. This rate is given by
|↑σ〉→|↓σ〉
dωf+(ω)f−(ω)
ω + µL − ε− U − U ′
ω + µL − ε− U ′
. (A2)
We note that in the simplest approximation38 for the
Coulomb blockade regime one can pull out the resolvents
in front of the integrals. Then, one arrives at the follow-
ing integral,
dωf+(ω)f−(ω+ ξ), where ξ = µL − µR or
ξ = 0, correspondingly, which can be easily calculated.35
As a consequence, one can see that the rate of single-
barrier second-order processes is proportional to temper-
ature T , whereas that of the third-order processes de-
pends on the bias voltage V .
∗ Electronic address: weymann@amu.edu.pl
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http://arxiv.org/abs/cond-mat/0611336
