Transport through a double-quantum-dot system with noncollinearly polarized leads

Abstract

We investigate linear and nonlinear transport in a double quantum dot system weakly coupled to spin-polarized leads. In the linear regime, the conductance as well as the nonequilibrium spin accumulation are evaluated in analytic form. The conductance as a function of the gate voltage exhibits four peaks of different heights with mirror symmetry with respect to the charge neutrality point. As the polarization angle is varied, due to exchange effects, the position and shape of the peaks change in a characteristic way, which preserves the electron-hole symmetry of the problem. In the nonlinear regime, various spin-blockade effects are observed. Moreover, negative differential conductance features occur for noncollinear magnetizations of the leads. In the considered sequential tunneling limit, the tunneling magnetoresistance (TMR) is always positive with a characteristic gate voltage dependence for noncollinear magnetization. If a magnetic field is added to the system, the TMR can become negative.

Full text

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Transport through a double-quantum-dot system with noncollinearly polarized leads
R. Hornberger, S. Koller, G. Begemann, A. Donarini, and M. Grifoni
Institut für Theoretische Physik, Universität Regensburg, 93035 Regensburg, Germany
?Received 29 November 2007; revised manuscript received 21 February 2008; published 12 June 2008?
We investigate linear and nonlinear transport in a double quantum dot system weakly coupled to spin-
polarized leads. In the linear regime, the conductance as well as the nonequilibrium spin accumulation are
evaluated in analytic form. The conductance as a function of the gate voltage exhibits four peaks of different
heights with mirror symmetry with respect to the charge neutrality point. As the polarization angle is varied,
due to exchange effects, the position and shape of the peaks change in a characteristic way, which preserves the
electron-hole symmetry of the problem. In the nonlinear regime, various spin-blockade effects are observed.
Moreover, negative differential conductance features occur for noncollinear magnetizations of the leads. In the
considered sequential tunneling limit, the tunneling magnetoresistance ?TMR? is always positive with a char-
acteristic gate voltage dependence for noncollinear magnetization. If a magnetic field is added to the system,
the TMR can become negative.
DOI: 10.1103/PhysRevB.77.245313PACS number?s?: 73.63.Kv, 72.25.?b, 73.23.Hk, 85.75.?d
I. INTRODUCTION
Spin-polarized transport through nanostructures is attract-
ing increasing interest due to its potential application in
spintronics1,2as well as in quantum computing.3Down-
scaling magnetoelectronic devices to the nanoscale implies
that Coulomb interaction effects become increasingly
important.4,5In particular, the interplay between spin-
polarization and Coulomb blockade can give rise to a com-
plex transport behavior in which both the spin and the charge
of the “information carrying” electron play a role. This has
been widely demonstrated by many experimental studies on
single-electron transistors ?SETs? with ferromagnetic leads,
with central element being either a ferromagnetic particle,6–8
normal metal particles,9,10a two-level artificial molecule,11a
C60molecule,12or a carbon nanotube,13showing the increas-
ing complexity and variety of the investigated systems. Ini-
tially, the theoretical work was mainly focused on the differ-
ence in the transport properties for parallel or antiparallel
magnetizations in generic spin-valve SETs.14–24More re-
cently, the interplay between spin and interaction effects for
noncollinear magnetization configurations attracted quite
some interest both in systems with a continuous energy
spectrum,25–28as well as in single-level quantum dots,29–36
many-level nanomagnets,37and in carbon nanotube quantum
dots.38In the noncollinear case, a much richer physics is
expected than in the collinear one. For example, two separate
exchange effects have to be taken into account. On the one
hand, there is the nonlocal interface exchange, in scattering
theory for noninteracting systems described by the imaginary
part of the spin-mixing conductance,39which in the context
of current-induced magnetization dynamics acts as an effec-
tive field.40Such an effective field has been experimentally
found to strongly affect the transport dynamics in spin valves
with MgO tunnel junctions.41This effect has also been re-
cently involved to explain negative tunneling magnetoresis-
tance ?TMR? effects in carbon nanotube spin valves13and
called spin-dependent interface phase shifts.22,42The second
exchange term is an interaction-dependent exchange effect
due to virtual tunneling processes that is absent in noninter-
acting systems.25,28,30,38This latter exchange effect is poten-
tially attractive for quantum information processing since it
allows to switch on and off magnetic fields in arbitrary di-
rections just by a gate electric potential.
Recently, there has been increasing interest in double-
quantum-dot systems ?that can be realized, e.g., in semicon-
ducting structures43or carbon nanotubes44? as tunable sys-
tems attractive for studying fundamental spin correlations. In
fact, the exchange Coulomb interaction induces a singlet-
triplet splitting, which can be used to perform logic gates.45
Moreover, Coulomb interaction together with the Pauli prin-
ciple can be used to induce spin-blockade when the two elec-
trons have triplet correlations.46–49The Pauli spin-blockade
effect can be used to obtain a spin-polarized current even in
the absence of spin-polarized leads; it requires a strong
asymmetry between the two on-site energies of the left and
right dots.
So far, transport through a double-dot ?DD? system
with spin-polarized leads has been addressed in few
theoretical23,24,50and experimental11works, for the case of
collinearly polarized leads only. While Ref. 23 addresses ad-
ditional Pauli spin-blockade regimes when one lead is half-
metallic and one is nonmagnetic, Ref. 24 focuses on the
effects of higher order processes in symmetric DD systems,
which can, e.g., yield a zero bias anomaly or a negative
tunneling magnetoresistance. In Ref. 11, Coulomb blockade
spectroscopy is used to measure the energy difference be-
tween symmetric and antisymmetric molecular states and to
determine the spin of the transferred electron.
In this work, we investigate spin-dependent transport in
the so-far unexplored case of a DD system connected to
leads with arbitrary polarization direction. Specifically, we
focus on the low transparency regime where a weak coupling
between the DD and the leads is assumed. Our model takes
into account interface reflections as well as exchange effects
due to the interactions and relevant for noncollinear polariza-
tion. We focus on the case of a symmetric DD, so that rec-
tification effects induced by Pauli spin-blockade are ex-
cluded. In the linear transport regime, the conductance is
calculated in closed analytic form. This yields four distinct
resonant tunneling regimes, but due to the electron-hole sym-
PHYSICAL REVIEW B 77, 245313 ?2008?
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metry of the DD Hamiltonian, each possess a symmetric mir-
ror with respect to the charge neutrality point. However, by
applying an external magnetic field, this symmetry is broken,
which can lead to negative tunneling magnetoresistance fea-
tures. Finally, in the nonlinear regime, some excitation lines
can be suppressed for specific polarization angles, and nega-
tive differential features also occur.
The method developed in this work to investigate charge
and spin transport is based on the Liouville equation for the
reduced density matrix ?RDM? in lowest order in the reflec-
tion and tunneling Hamiltonians. The obtained equations of
motion are fully equivalent to those that could be obtained
by using the Green’s function method30,51in the same weak-
tunneling limit. The advantage of our approach is that it is, in
our opinion, easier to understand and to apply for newcom-
ers, as it is based on standard perturbation theory and does
not require knowledge of the nonequilibrium Green’s func-
tion formalism.
The paper is organized as follows. In Sec. II, we introduce
the model system for the ferromagnetic DD single-electron
transistor. In Sec. III, the coupled equations of motion for the
elements of the DD reduced density matrix are derived.
Readers not interested in the derivation of the dynamical
equations can directly go to Secs. IV and V, where results for
charge and spin transfer in the linear and nonlinear regimes,
respectively, are discussed. Finally, we present results for the
transport characteristics in the presence of an external mag-
netic field in Sec. VI. Conclusions are drawn in Sec. VII.
II. MODEL
We consider a two-level DD, or a single molecule with
two localized atomic orbitals, attached to ferromagnetic
source and drain contacts and with a capacitive coupling to a
lateral gate electrode. The system is described by the total
Hamiltonian,
Hˆ= Hˆ?+ Hˆs+ Hˆd+ HˆT+ HˆR,
?1?
accounting for the DD Hamiltonian, the source ?s? and drain
?d? leads, and the tunneling ?T? and reflection ?R? Hamilto-
nians. The two contacts are considered to be magnetized
along an arbitrary but fixed direction determined by the mag-
netization vectors m ??. The two magnetization axes enclose
an angle ???0°,180°? ?see Fig. 1?. The spin quantization
axis z ??in lead ? is parallel to the magnetization m ??of the
lead. The majority of electrons in each contact will then be in
the spin-up state. The Hamiltonians Hˆs,Hˆdthat model the
source and drain contacts ??=s,d? read as follows:
Hˆ?=?
k??
??k??− ???c?k??
†
c?k??,
?2?
where c?k??
create, respectively annihilate, electrons with momentum k
and spin ??in lead ?. The electrochemical potentials ??
=?0?+eV?contain the bias voltages Vsand Vdat the left and
right lead with Vs−Vd=Vbias. There is no voltage drop within
the DD. We denote in the following ?k??−??ª??k??.
Tunneling processes into and out of the DD are described
by HˆT. We denote with d???
tion operators in the DD. We assume that tunneling only can
happen between a contact and the closest dot, so that we can
use the convention that the lead indices ?=s,d correspond to
?=1,2 for the DD. With t?the tunneling amplitude, we find
HˆT=?
?k??
The so-called reflection Hamiltonian HˆRincludes reflec-
tion events at the lead-molecule interface.28,38For strongly
shielded leads, the overall effect is the occurrence of a small
energy shift ?Rinduced by the magnetic field in the contacts
and built up during several cycles of reflections at the bound-
aries. It reads
HˆR= − ?R ?
?=s,d
†
and c?k??are electronic lead operators. They
†
, d???the creation and destruc-
?t?d???
†
c?k??+ t?
?c?k??
†
d????.
?3?
?d?↑?
†
d?↑?− d?↓?
†
d?↓??.
?4?
Finally, the DD Hamiltonian needs to be specified. As the
spin quantization axis of the DD, z ??, we choose the direction
perpendicular to the plane spanned by z ?sand z ?d?Ref. 30? ?see
Fig. 2?. The two remaining basis vectors x ??and y ??are along
z ?s+z ?d, respectively along z ?s−z ?d. The matrices that math-
ematically describe the above transformations read
?2?+ e+i?/4
− e+i?/4
Ms↔?=
1
+ e−i?/4
+ e−i?/4? = Md↔?
?
.
?5?
We express the DD Hamiltonian in the localized basis
such that, e.g., ?+,−? describes a state with a spin-up electron
on site 1 and a spin-down electron on site 2 ?with spin di-
rections expressed in the spin-coordinate system of the DD?.
Such state can be obtained by applying creation operators on
the vacuum state, i.e., ?+,−?=d1↑
†d2↓
†?0?. In general, the order-
gate
gate
Θ
FIG. 1. ?Color online? Schematic picture of the model: a double-
quantum-dot system attached to polarized leads. The significance of
the on-site and intersite interactions U and V, respectively, is de-
picted. The source and drain contacts are polarized and the direction
of the magnetizations m ??is indicated by the arrows.
FIG. 2. ?Color online? The spin quantization axis of the double-
dot, z?, is chosen to be perpendicular to the plane spanned by the
magnetization directions m ?s, m ?din the leads. The latter enclose an
angle ?.
HORNBERGER et al.
PHYSICAL REVIEW B 77, 245313 ?2008?
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Page 3

ing of the creation operators is defined as d1↑
The DD Hamiltonian then reads
†d2↑
†d1↓
†d2↓
†?0?.
Hˆ?=?
???
+?? −U
??d???
2− V??
†
d???+ b?
??
?d1??
†
d2??+ d2??
†
d1???
?=1
2
?
??
d???
†
d???+ U?
?=1
2
n?↑?n?↓?
+ V?n1↑?+ n1↓???n2↑?+ n2↓??,
?6?
where the spin index ? indicates that the operators are ex-
pressed in the spin-coordinate system of the DD. The tunnel-
ing coupling between the two sites is b, while U and V are
on-site and intersite Coulomb interactions. In the following,
we consider a symmetric DD with equal on-site energies ?1
=?2. Thus, we can incorporate the on-site energies in the
parameter ? proportional to the applied gate voltage Vgate.
To understand transport properties of the two-site system
in the weak-tunneling regime, we have to analyze the eigen-
states of the isolated interacting system. These states, which
are expressed in terms of the localized states, and the corre-
sponding eigenvalues are listed in Table I.52Table I also
indicates the eigenvalues of the total spin operator. The
ground states of the DD with odd particle number are spin
degenerate. In contrast, the ground states with even particle
number have total spin S=0 and are not degenerate. In the
case of the two-particle ground state, the parameters ?0and
?0determine whether the electrons prefer to pair in the same
dot or are delocalized over the DD structure. Since the eigen-
states are normalized to one, the condition ?0
The energy difference between the S=0 ground state and the
triplet is given by the exchange energy,
2+?0
2=1 holds.
J =1
2?? − U + V? = 2?b??R +?1 + R2?,
where ?=4?b??1+R2and R=?V−U?/?4?b??.
Besides the triplet, one observes the presence of
higher two-particle excited states with total spin S=0.
Finally, we remark that HˆTand HˆRcontain operators of the
DD, d???
leads, while Hˆ?is already expressed in terms of DD opera-
tors d???
system of the DD.
†
and d???, in the spin-coordinate systems of the
†
and d???with spin expressed in the coordinate
III. DYNAMICAL EQUATIONS FOR THE REDUCED
DENSITY MATRIX
In this section, we shortly outline how to derive the equa-
tion of motion for the RDM to lowest nonvanishing order in
the tunneling and reflection Hamiltonians. The method is
based on the well known Liouville equation for the total
density matrix in lowest order in the tunneling and reflection
Hamiltonians. Equations of motion for the reduced density
matrix are obtained upon performing the trace over the lead
degrees of freedom,53yielding, after standard approxima-
tions, Eqs. ?13? and ?14? below. In the case of spin-polarized
leads, however, it is convenient to express the equations of
motion for the RDM in the basis that diagonalizes the iso-
lated system’s Hamiltonian and in the system’s spin quanti-
zation axis. After rotation from the leads’quantization axis to
the DD one we obtain ?Eq. ?21??, which forms the basis of all
the subsequent analysis.
Let us start from the Liouville equation for the total den-
sity matrix ? ˆI?t? in the interaction picture,
i?d? ˆI?t?
dt
= ?HˆT
I?t? + HˆR
I?t?,? ˆI?t??,
?7?
with HˆTand HˆRtransformed into the interaction picture by
HˆT/R
indicates the time at which the perturbation is switched on.
Integrating Eq. ?7? over time and inserting the obtained ex-
pression in the right-hand side of Eq. ?7? one equivalently
finds
I
?t?=ei/??Hˆ?+Hˆs+Hˆd??t−t0?HˆT/Re−i/??Hˆ?+Hˆs+Hˆd??t−t0?, where t0
? ˙ˆI?t? = −i
??HˆR
I?t?,? ˆI?t0?? −i
??HˆT
I?t?,? ˆI?t0?? −
1
?2?
t0
t
dt??HˆT
I?t?
+ HˆR
I?t?,?HˆT
I?t?? + HˆR
I?t??,? ˆI?t????.
?8?
TABLE I. Eigenstates of the double-dot system and correspond-
ing eigenvalues and parity. In the limit ?b?→? where the interdot
hopping is unhindered, R→0 and ?0→?0. For ?b?→0, i.e., no in-
terdot hopping takes place, we find that R→+? and ?0→1, ?0
→0 if U?V; state ?2? then becomes degenerate to ?2??0??, forming
a Heitler–London state. In turn, if U?V, then R→−? and ?0→0,
?0→1.
AbbreviationState EigenvalueSpin
?0??0,0?
00
?1e??
1
?2???,0?+?0,???
??+b
1/2
?1o??
?2?
1
?2???,0?−?0,???
?0
?2??+,−?+?−,+??
+
?+,+?
1
?2??+,−?−?−,+??
?−,−?
1
?2??2,0?−?0,2??
?0
?2??+,−?+?−,+??
−
1
?2??2,??+??,2??
1
?2??2,??−??,2??
?2,2?
in terms of R=?U−V?/?4?b??:
?=4?b??1+R2,
??−b
2?U+V−??
1/2
0
?0
?2??2,0?+?0,2??
2??+1
?2??1??
?2??0??
?2??−1??
?2??
?2??
2??+V
1
2??+U
2?U+V+??
0
0
?0
?2??2,0?+?0,2??
2??+1
?3o??
?3e??
?4?
3??+U+2V+b
3??+U+2V−b
4??+2U+4V
1/2
1/2
0
?0=
1
?2
1
?1+R2−R?1+R2
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The time evolution of the RDM,
? ˆ?
I?t? ª Trleads?? ˆI?t??,
?9?
is now formally obtained from Eq. ?8? by tracing out the lead
degrees of freedom. To proceed, we make the following stan-
dard approximations.
?i? The leads are considered as reservoirs of noninteract-
ing electrons that stay in thermal equilibrium at all times. In
fact, we only consider weak tunneling and, therefore, the
influence of the DD on the leads is marginal. Hence, we can
approximatively factorize the density matrix of the total sys-
tem as
? ˆI?t? ? ? ˆ?
I?t?? ˆs? ˆd,
?10?
where ? ˆsand ? ˆdare time independent and given by the usual
thermal equilibrium expression for the contacts ? ˆs/d=e−?Hˆs/d
where ? is the inverse temperature and Zs/dare the partition
sums over all states of lead s/d.
?ii? We consider the lowest nonvanishing order in HˆT/R.
?iii? We apply the Markov approximation, i.e., in the in-
tegral in Eq. ?8?, we replace ? ˆ?
words, it is assumed that the system loses all memory of its
past due to the interaction with the lead electrons.
Furthermore, being interested in the long term behavior of
the system only, we send t0→−?. We finally obtain the gen-
eralized master equation ?GME? for the reduced density ma-
trix,
Zs,d,
I?t?? with ? ˆ?
I?t?. In other
? ˙ˆ?
I?t? = −i
?Trleads?HˆR
?2?
0
I?t?,? ˆ?
I?t?? ˆs? ˆd?
−
1
?
dt? Trleads??HˆT
I?t?,?HˆT
I?t − t??,? ˆ?
I?t?? ˆs? ˆd???.
?11?
A. Contribution from the tunneling Hamiltonian
In the following, we derive the explicit expression for the
GME in the basis of the isolated DD. For simplicity, we omit
the contribution of the reflection Hamiltonian in a first in-
stance. When we shall have obtained the final form of the
GME due to the tunneling term, we will see that it is easy to
insert the contribution from the reflection Hamiltonian. Let
us then start from the tunneling Hamiltonian in the interac-
tion picture,
I?t? =?
i,j
HˆT
?k???
+ H.c.,
t?c?k??
†
?d????ij?i??j?exp?i??i− ?j+ ??k???t/??
?12?
where ?d????ij=?i?d????j? and ?d???
electron annihilation and creation operators in the spin-
quantization axis of lead ? expressed in the basis of
the energy eigenstates of the quantum dot system. To
simplify Eq. ?11?, standard approximations are invoked.
?i? The first one is the secular approximation: fast oscil-
lations in time average out in the stationary limit we
are interested in and thus can be neglected. Together with
the relation Trleads?? ˆs? ˆdc?k??
where f????k?? is the Fermi function, and the cyclic proper-
ties of the trace we get
†
?ij=?i?d???
†
?j? are the
†
c??k????=?kk?????????f????k??,
? ˆ˙?
I?t? = −
1
?2?
0
?
dt??
?k??
?t??2??
ilm
f????k????d????il?d???
†
?lm?i??m?? ˆ?
I?t?exp?i??m− ?l+ ??k???t?/??
+?
ilm
+?
ilm
+?
ilm
−?
iljm
−?
iljm
−?
iljm
−?
iljm
?1 − f????k?????d???
†
?il?d????lm?i??m?? ˆ?
I?t?exp?− i??l− ?m+ ??k???t?/??
f????k???? ˆ?
I?t??d????il?d???
†
?lm?i??m?exp?− i??i− ?l+ ??k???t?/??
?1 − f????k????? ˆ?
I?t??d???
†
?il?d????lm?i??m?exp?+ i??l− ?i+ ??k???t?/??
?1 − f????k?????d????ij? ˆ?
I?t?jl?d???
†
?lm?i??m?exp?+ i??m− ?l+ ??k???t?/??
f????k????d???
†
?ij? ˆ?
I?t?jl?d????lm?i??m?exp?− i??l− ?m+ ??k???t?/??
?1 − f????k?????d????ij? ˆ?
I?t?jl?d???
†
?lm?i??m?exp?− i??i− ?j+ ??k???t?/??
I?t?jl?d????lm?i??m?exp?+ i??j− ?i+ ??k???t?/???.
f????k????d???
†
?ij? ˆ?
?13?
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?ii? For the second approximation, we notice that we wish
to evaluate single components ?n?? ˆ?
system’s energy eigenbasis. Therefore, we assume that the
DD is in a pure charge state with a certain number of elec-
trons N and energy EN. In fact, in the weak-tunneling limit,
the time between two tunneling events is longer than the
I?m? of the RDM in the
time where relaxation processes happen. That is, we can
neglect matrix elements between states with different num-
ber of electrons and only regard elements of ? ˆ?
nect states with same electron number N and same energy
EN. So, we can divide ? ˆ?
and ENand find
I, which con-
Iinto submatrices labeled with N
? ˙nm
ENN?t? = −?
??
???
? ?
l,l???N−1?
, ?
j??ENN?
, ?
h,h???N+1?
??t??2?
?a? +?f???h− ?j?D?????h− ?j? +i
?b? +??1 − f???j− ?l??D?????j− ?l? −i
?c? +?f???h− ?j?D?????h− ?j? −i
?d? +??1 − f???j− ?l??D?????j− ?l? +i
??d?k
f???k?D?????k?
?k− ?h+ ?j??d????nh?d???
?k− ?j+ ?l??d???
?k− ?h+ ?j??nj
?k− ?j+ ?l??nj
†
?hj?jm
ENN?t?
??d?k
?1 − f???k??D?????k?
†
?nl?d????lj?jm
ENN?t?
??d?k
f???k?D?????k?
ENN?t??d????jh?d???
†
?hm
??d?k
?1 − f???k??D?????k?
ENN?t??d???
†
?jl?d????lm
?e? − 2?1 − f???h− ?j??D?????h− ?j??d????nh??d???
†
?hm?h?h
ElN−1?t???.
EhN+1?t?
?f? − 2f???j− ?l?D?????j− ?l??d???
†
?nl??d????lm?l?l
?14?
In Eq. ?14?, we used the notation ?nm
By convention, ??l,l?,?j,?h,h?? means that in each line
??a?–?f??, we sum over the indices occurring in this line only.
Notice that the sum over j is restricted to states of energy
Ej=EN=En=Em. For the states with N?1 electrons, we have
to sum over all energies; therefore, we indexed the density
matrix with Eh=Eh?in line ?e? and El=El?in line ?f?. Further,
wereplaced the sumover
→?d??k??D??????k???, where D??????k??? denotes the den-
sity of states in lead ? for the spin direction ??, and applied
the following useful formula:
?d??k??G???k????
0
= ??G?E? ? i???d??k??
ENNª?n?? ˆ?
I,ENN?m?.
k
byanintegral,
?k
t
dt?e??i/?????k??−E?t?
G???k???
???k??− E?,
?15?
where the prime at the integral denotes Cauchy’s prin-
cipal-partintegration. In
=D??????k???f?
simplify the notations, we replaced ??k??by ?kin Eq. ?14?.
ourcase,
−=1−f?. In order to
G???k???
????k??? with f?
+=f?and f?
B. Transformation into the spin-coordinate system of the
double dot
In Sec. II, we introduced the transformation rules for
changing from the lead spin coordinates ??into the DD spin
coordinates ??. These rules give
?
d?↓?
with ?sª−?
Thus, Eq. ?14? can be easily expressed in the DD spin
quantization axis. For example, it holds
d?↑?
†
† ?=
1
?2?
+ e−i??/2
− e−i??/2
+ e+i??/2
+ e+i??/2??
d?↑?
d?↓?
†
† ?,
?16?
2, ?dª+?
2.
?
??
D???d??
†d??=1
2?D?↑?+ D?↓???
??
??????d???
†
d???
+1
2?D?↑?− D?↓???
??
????−??
?
d???
†
d?−??,
?17?
where we introduced
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???????ª?
2?
1
??= ???
??? = ↓
ei??
e−i?? ??= ↓
??= ↑
??? = ↑.?
??E? ??? ??? ,?
For later convenience, we also define
F??????
?
ª1
D?+?f?
D?+?f?
??E? + D?−?f?
??E? − D?−?f?
??E?
??= ???
and its related principal-part integral,
?E? ª??d?F??????
P??????
??
????? − E?−1.
C. Contribution from the reflection Hamiltonian
In order to give the full expression for the GME in the
system’s eigenbasis, we need to compute the contribution
from the reflection Hamiltonian in Eq. ?11?. In analogy to
what we did to evaluate the contribution from the tunneling
Hamiltonian, we must first transform HˆRinto the interaction
picture and then perform the secular approximation to get rid
of the time dependence. To start, we express HˆRin the DD
spin quantization basis,
HˆR
I= − ?R?
?
?
j??N?
l??N−1?
?
??????
???????
?
d??jl
†
d??lj
??j??j?.
?18?
The commutator is easily evaluated to be
−i
?Trleads?HˆR
I,? ˆ?
I?t??s?d?
= −i
??
?
?R ?
j??N??
l??N−1??
??????
??j??j??.
???????
?
?d??jl
†
d??lj
??j?
??j?? ˆ?
I?t? − ? ˆ?
I?t?d??jl
†
d??lj
?19?
In order to include this commutator in the master equation
?Eq. ?14??, let us introduce the following abbreviation:
R??????=
1
?t??2?R????↑????↓+ ???↓????↑?.
?20?
Now, we can add R??????in Eq. ?14? in lines ?b? and ?d? to
find the final form of the complete master equation in the DD
spin-coordinate system. It reads
? ˙nm
ENN?t? = −?
??
?=s,d
?t??2?
??,???
? ?
l,l???N−1?
, ?
j??ENN?
, ?
h,h???N+1?
??h− ?j???d????nh?d????
??j− ?l? + R?????????d???
??h− ?j???nj
??j− ?l? + R?????????nj
??
?a? + ????????F??????
??F??????
?c? + ????????F??????
??F??????
+
??h− ?j? +i
?P??????
+†
?hj?jm
ENN?t?
?b? + ???????
−
??j− ?l? −i
??P??????
−
†
?nl?d?????lj?jm
ENN?t?
+
??h− ?j? −i
?P??????
+
ENN?t??d????jh?d????
†
?hm
?d? + ???????
−
??j− ?l? +i
??P??????
−
ENN?t??d???
†
?jl?d?????lm
?e? − 2???????F??????
−
??h− ?j??d????nh??h?h
EhN+1?t??d????
†
?hm
?f? − 2???????
?
F??????
+
??j− ?l??d???
†
?nl??l?l
ElN−1?t??d?????lm.
?21?
D. Current formula
We now observe that Eq. ?21? can be recast in the follow-
ing Bloch–Redfield form:
? ˙nm
ENN?t? = −?
jj?
Rnmjj?
NN
?jj?
ENN?t? +?
hh?
Rnmhh?
NN+1?hh?
EhN+1?t?
+?
ll?
Rnmll?
NN−1?ll?
ElN−1?t?,
?22?
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where the sums in Eq. ?22? run over states with fixed particle
number: j,j????ENN??, h,h????N+1??, l,l????N−1??. The
Redfield tensors are given by ??=s,d? ?Ref. 38?
=?
l?or h?
Rnmjj?
NN
??
??mj????,nhhj
?+?NN+1+ ??,nllj
?+?NN−1? + ?nj???,j?hhm
?−?NN+1
+ ??,j?llm
?−?NN−1??,
?23?
Rnmkk?
NN?1=?
?
???,k?mnk
?+?NN?1+ ??,k?mnk
?−?NN?1?,
?24?
where the quantities ??,njjk
Eq. ?21?. They are
???NN?1can be easily read out from
???????
?i
?P??????
???????
?i
??P??????
?nl?d?????l?k?.
With the stationary density matrix ? ˆ?st
current ?through lead ?=s/d=?? follows from
I = 2?e Re?
n,n?,j
We numerically solve Eq. ?22? and use the result to evaluate
the current flowing through the DD, as will be Secs. V–VII.
At low-bias voltages, however, we can make some further
approximations to arrive at an analytical formula for the
static dc.
??,nhh?k
???NN+1= ?
?????????t??2?F??????
??h− ?k???d????nh?d????
?t??2?F??????
??k− ?l? + R????????
+
??h− ?k?
+†
?h?k?,
??,nll?k
???NN−1= ?
????????
?
−
??k− ?l?
−
??d???
†
I
being known, the
N?
???,njjn?
?+?NN+1− ??,njjn?
?+?NN−1??n?n,st
EnN.
?25?
IV. LOW-BIAS REGIME
A. General considerations
A low-bias voltage ensures that merely one channel is
involved with respect to transport properties. Here, we focus
on gate voltages that align charge states N and N+1. More-
over, we can focus on density matrix elements that involve
the energy ground states EN
we shall use the following compact notations:
?0?and EN+1
?0?only. In the following,
? ˆ?
I,EN
?0?Nª ? ˆ?
?N?,
?n?? ˆ?
?N??m? = ?nm
?N?.
?26?
Evaluation of the current requires the knowledge of ? ˆ?
and ? ˆ?
are obtained from Eq. ?21? or, equivalently, from Eq. ?22?. In
the low-bias regime, this task is simplified since ?i? terms
which try to couple states with particle numbers unlike N and
N+1 can be neglected; ?ii? we can reduce the sums over h,h?
?N?
?N+1?, i.e., a solution of the set of coupled equations that
in the equation for ? ˆ˙?
? ˆ˙?
other transitions are exponentially suppressed by the Fermi
function. Notice, however, that these two approximations are
not appropriate for the principal-part terms since they are not
energy conserving. The resulting equations for ? ˆ?
?Eqs. ?B1? and ?B2?, respectively? can be found in Appendix
B. In the following, we shall apply those equations to derive
an analytical expression for the conductance in the four dif-
ferent resonant charge state regimes possible in a DD system,
i.e.,
?N?and over l,l? in the equation for
?0?and EN+1
?N+1?to energy-ground states EN
?0?because all the
?N?and ? ˆ?
?N+1?
N = 0 ↔ N = 1,
N = 1 ↔ N = 2,
N = 2 ↔ N = 3,
N = 3 ↔ N = 4.
?27?
In all of the four cases, we get a system of five coupled
equations involving diagonal and off-diagonal elements of
the RDM. The matrix elements of the dot operators between
the involved states entering these equations are given in Ap-
pendix A. Before going into the details of these equations, it
is instructive to analyze the structure and the physical sig-
nificance of the involved RDM elements.
B. Elements of the reduced density matrix
N=0. In the case of an empty system, we have only one
density matrix element in the corresponding block with fixed
particle number N=0, i.e.,
?00
?0??t? = :W0,
?28?
describing the probability to find an empty double-dot sys-
tem.
N=1. In this case, we have four eigenstates for the system,
where the two even ones build the degenerate ground state
and the two odd ones are excited states ?see Table I?. In the
low-bias regime, we only need to take into account transi-
tions between ground states. Therefore, we have to deal with
the 2?2 matrix,
?
?1e↓1e↑
The total occupation probability for one electron is
W1ª W1↑+ W1↓.
The meaning of the off-diagonal elements, the so-called co-
herences, becomes clear if we regard the average spin in the
system,
?1e↑1e↑
?1?
?1?
?1e↑1e↓
?1e↓1e↓
?1?
?1? ?= :?
W1↑
w1e−i?1
w1ei?1
W1↓?.
?29?
?30?
Si
?1?=1
2Tr??i
Pauli? ˆ?
?1??t??,
?31?
where i=x,y,z and ?i
yields
Pauliare the Pauli spin matrices. This
Sx
?1?= w1cos ?1,
Sy
?1?= − w1sin ?1,
?32?
Sz
?1?=1
2?W1↑− W1↓?.
?33?
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N=2. For the case N=2, we actually have six different
eigenstates, but only one of them, ?2?, is a ground state ?with
spin S=0?, see Table I. Only this ground state must be con-
sidered in the low-bias regime, yielding
?22
?2??t? = :W2.
?34?
This element describes the probability to find a dot with two
electrons.
N=3. In this case, we have again four eigenstates for the
system, whereas the two odd ones build the degenerate
ground state and the two even ones are excited. In the low-
bias regime, we only need to deal with the 2?2 matrix in-
volving the three-particle ground states,
?
?3o↓3o↑
The total occupation probability for three electrons is
W3ª W3↑+ W3↓.
As for the case N=1, the off-diagonal elements yield infor-
mation on the average spin Si
tem through the following relations:
?3o↑3o↑
?3?
?3?
?3o↑3o↓
?3o↓3o↓
?3?
?3? ?= :?
W3↑
w3e−i?3
w3ei?3
W3↓?.
?35?
?3?=1
2Tr??i
Pauli? ˆ?
?3??t?? in the sys-
Sx
?3?= w3cos ?3,
Sy
?3?= − w3sin ?3,
?36?
Sz
?3?=1
2?W3↑− W3↓?.
?37?
N=4. Finally, if the double quantum dot is completely filled
with four electrons, we only have one nondegenerate state.
Correspondingly, there is only one relevant RDM matrix el-
ement,
?44
?4??t? = :W4,
?38?
describing the probability to find four electrons in the sys-
tem. The total spin is S=0.
Hence, we see that in all of the four cases ?Eq. ?27??, we
get a system of five equations with the five independent
physical quantities WN,WN+1and Sx
?i?,Sy
?i?,Sz
?i?with i=1 or 3.
C. Conductance formula
We shall exemplarily present results for the resonant
transition N=1↔N=2. For the other transitions, similar
considerationsapply. The
W1,W2,Sx
and ?36? to the density matrix elements of ? ˆ?
?B1? and ?B2? and Table III, and with W1=1−W2we finally
obtain the following:
quantities ofinterestare
?1?,Sy
?1?,Sz
?1?, which are related through Eqs. ?32?
?1?. From Eqs.
W˙1= −?
??
?=s,d
?t??2k+
2?2F?↓↓
+??2?W1− 4F?↓↓
−??2?W2
− 4F?↑↓
+??2?S??1?· m ???,
?39?
S?˙?1?= −?
??
?=s,d
?t??2k+
2?2F?↑↑
+??2?S??1?− ?F?↑↓
+??2?W1
− 2F?↑↓
? S??1??.
−??2?W2?m ??+
2
?k+
2P???1,?E2? − E1
?0??m ??
?40?
We have introduced the notation 4k?
nonvanishing principle-value factors P?↑↓
parameter R?↑↓have been merged to the following compact
form:
2ª??0??0?2. All of the
?
and the reflection
P???1,?E2? − E1
?0?? ª −1
2?P?↓↑
−??1? + R?↓↑? − k+
2P?↑↓
+??2?
+1
4P?↑↓
+??2?− ?1e? −1
4P?↑↓
+??2?− ?1e?
− k−
2P?↑↓
+??2?− ?1e?,
?41?
where we introduced the chemical potential ?N+1=EN+1
−EN
gies. We notice that the set of coupled equation ?Eq. ?39? and
?40?? for the evolution of the populations and of the spin
accumulation has a similar structure to that reported in Refs.
28, 30, and 38 for a single-level quantum dot, a metallic
island, and a single-walled carbon nanotube, respectively.
Some prefactors and the argument of the principal-part
terms, however, are DD specific. In particular, as in Refs. 28
and 30, we clearly identify a spin precession term originating
from the combined action of the reflection at the interface
and the interaction. The associated effective exchange split-
ting is ?B1, where ?=−g?Bis the gyromagnetic ratio and
B?1ª2
?
?0?
?0?and ?E2? denotes the four different two-particle ener-
??
?t??2P?m ??
?42?
is the corresponding effective exchange field. We now focus
on the stationary limit. In the absence of the precession term,
the spin accumulation has only a Sy
to our particular choice of the spin quantization axis, Sx
=0 holds. The exchange field tilts the accumulated spin out
of the magnetizations’ plane and gives rise to a nonzero Sz
component proportional to B1and Sy
To get further insight in the spin dynamics, we observe
that since we are looking at the low-bias regime, we can
linearize the Fermi function f?in the bias voltage, i.e.,
?1?component since, due
?1?
?1?
?1?.
TABLE II. Matrix elements for the N=0↔N=1 transition in-
duced by operators d?↑and d?↓, ?=1,2.
0↔1
?1e↓?
0
1
?2
0
1
?2
?1e↑?
1
?2
0
1
?2
0
?1o↑?
1
?2
0
−1
?2
0
?1o↓?
0
1
?2
0
−1
?2
d1↑:?0?
d1↓:?0?
d2↑:?0?
d2↓:?0?
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f???? = ?1 + e???+eV???−1? f???„1 − f?− ??e?V?….
By introducing the polarization of the contacts,
?43?
p???? ª
D?↑???? − D?↓????
D?↑???? + D?↓????,
factors as
we can express the F??????
?
F?↑↑
???? ?1
2D????f?????1 ? f????e?V??,
F?↑↓
???? = p????F?↑↑
????,
where D?=D?↑?+D?↓?. It is also sufficient for our calcula-
tions to regard the density of states as a constant quantity,
D????=D?. Consequently, the polarization is also constant,
p????=p?. Finally, we focus in the following on the symmet-
ric case where both leads have the same properties, which, in
particular, means that tunneling elements, polarizations, den-
sity of states, and reflection amplitude are equal,
t1= t2= :t,
p1= p2= :p,
D1= D2= :D,
R1??−??= R2??−??= :R.
?44?
Upon introducing the linewidth ?=2?
G12=I12/Vbiasfor the resonant regime N=1↔N=2 reads
G12??? =?
2e2?k+
f?− ?2? + 1
??1 −
Similarly, we find for an arbitrary resonance ?i=0,1,2,3?,
?D?t?2, the conductance
2f??2?f?− ?2?
p2sin2??
2?
2?2cos2??
1 + ?B1/f??2?2?k+
2??. ?45?
Gii+1??? =?
2e2???i + 1?d†?i??2f??i+1?f?− ?i+1?
1 + f„?− 1?i?i+1…?1 −
p2sin2??
2?
1 + ?Bi+1/f??− 1?i+1?i+1?2???i + 1?d†?i??2?2cos2??
2??,
?46?
where ??i+1?d†?i?? is a shortcut notation for the nonvanishing
matrix elements ??Ei+1
II–V. It holds ??1?d??
=??3?d??
Moreover,
principal-part contributions and the ones coming from the
reflection Hamiltonian in the effective magnetic fields,
?0?i+1?d??
†?0??=??4?d??
†?Ei
?0?i?? calculated in Tables
†?3??=1/?2 and ??2?d??
wegathered
†?1??
†?2??=k+.togetherthe
B?2= B?1,
B?3= B?4ª2
??
?
?t??2P????4,E3
?0?− ?E2??m ??.
The latter are defined in terms of the following function:
P????4,E3
?0?− ?E2?? ª −1
2?P?↓↑
+??4? + R?↓↑? − k+
2P?↑↓
−??3?
+1
4P?↑↓
−??3o− ?2?? −1
4P?↑↓
−??3o− ?2??
− k−
2P?↑↓
−??3o− ?2??.
Moreover, a closer look to Eq. ?46? shows that its angular
dependence is strongly coupled to the square of the ratio
??Bi?/???f??−1?i?i??, which is the effective exchange split-
ting rescaled by the coupling and the Fermi function. The
ratio occurs in the denominators, and its value depends on
the gate voltage. As the change of Biunder the variation in
the gate voltage is comparatively small, the factor dominat-
ing the gate voltage evolution is the Fermi function. This
accounts for the population of the dot: only if a nonzero spin
is present ?i.e., odd filling: one or three electrons? the effec-
tive magnetic field can have an influence. That is why cor-
respondingly the renormalized effective exchange splitting
vanishes for even fillings, namely, below 0↔1 and 2↔3,
respectively, above the 1↔2 and 3↔4 resonances. This can
nicely be seen from Fig. 3 ?remember that Vgate?−?, so “be-
low” means larger, “above” means smaller ??, where the four
different factors ??Bi?2/?f??−1?i?i???2are plotted. The
curves belonging to the resonances involving the half-filling
do not immediately go to zero but show a more complex
behavior with some small intermediate peaks due to the in-
fluence of the various excited states present for a two-
electron population of the dot. As we expect, G01??? and
G34??? ?G12??? and G23???, respectively? are mirror symmet-
ric with respect to each other when the gate voltage is varied.
This in turn reflects the electron-hole symmetry of the DD
Hamiltonian. The parameters of the figures are chosen to be
as follows ?b?0?:
kBT = 4 ? 10−2?b?,
?? = 4 ? 10−3?b?,
U = 6?b?,
V = 1.6?b?,
?47?
and p=0.8, R=0.05D. As expected, the peaks are mirror
symmetric with respect to the half-filling gate voltage. No-
tice also the occurrence of different peak heights, both in the
parallel and in the antiparallel case. For both polarizations,
the principal-part terms entering Eq. ?46? vanish, the spin
accumulation is entirely in the magnetization plane, and the
peak ratio is solely determined by the ratio of the ground
state overlaps 2/k+
2. For polarization angles ??0,?, the ra-
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tio is also determined by the nontrivial angular and voltage
dependence of the effective exchange fields. Finally, as ex-
pected from the conductance formula Eq. ?46?, the conduc-
tance is suppressed in the antiparallel compared to the paral-
lel case. The four conductance peaks are plotted as a function
of the gate voltage in Fig. 4 for the polarization angles ?
=0 and ?=?, top and bottom figures, respectively. These
features of the conductance are nicely captured by the color
plot of Fig. 5, where numerical results for the conductance
plotted as a function of gate voltage and polarization angle
are shown. The conductance suppression nearby ?=? is
clearly seen.
In the following, we analyze in detail the single resonance
transitions. Due to the mirror symmetry, it is convenient to
investigate together the N=0↔N=1, N=3↔N=4 and N
=1↔N=2, N=2↔N=3 resonances. We use the convention
that for a fixed resonance, the parameter ?=0 when ?N+1
=0.
1. Resonant regimes N=0^N=1 and N=3^N=4
The expected mirror symmetry of G01and G34is shown in
Fig. 6, where the conductance peaks are plotted for different
polarization angles ? of the contacts. Notice that the analyti-
cal expressions ?Eq. ?46?? ?continuous lines? perfectly match
the results obtained from a numerical integration of the mas-
ter equation ?Eq. ?21?? with the current formula ?Eq. ?25??.
We also can see that the maxima of the conductance decrease
with ? growing up to ?. It can be shown that the peaks for
?=0 and ?=? lie at the same value of ? because the effec-
tive fields Biexactly vanish due to trigonometrical prefac-
tors. In other words, virtual processes captured in the effec-
tive fields Bido not play a role in the collinear cases. For
noncollinear configurations, however, the peak maxima are
shifted toward the gate voltages where an odd population of
the dot dominates because there the effective exchange field
can act on the accumulating spin and make it precess, which
eases tunneling out. These findings are in agreement with
results obtained for a single-level quantum dot,30a metallic
island,28and carbon nanotubes.38To quantify the relative
magnitude of the current for a given polarization angle ?
with respect to the case ?=0, we introduce the angle-
dependent TMR as
TMRNN+1??,?? = 1 −GN,N+1??,??
GN,N+1?0,??.
For the 0↔1 transition, it reads
TABLE III. Matrix elements for the N=1↔N=2 transition induced by d?↑
?2??sz??, with sz=0,?1, specifies which one of the triplet elements is addressed.
†and d?↓
†, ?=1,2. The notation
1↔2
?1e↓?
?0+?0
2
0
1
2
0
1
2
−?0+?0
2
0
?1e↑?
0
?1o↑?
0
−1
?2
0
0
0
0
?1o↓?
−?0+?0
2
0
−1
0
1
2
−?0−?0
2
0
d1↑
†:
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
1
?2
0
0
0
0
2
d1↓
†:
−?0−?0
2
0
1
2
0
−1
?0−?0
2
0
−1
?2
0
0
0
0
?0−?0
2
0
−1
0
−1
?0+?0
2
0
−1
?2
0
0
0
0
0
0
1
?2
0
0
0
0
2
−1
0
0
?2
22
d2↑
†:
?0+?0
2
0
−1
0
−1
−?0+?0
2
0
?0−?0
2
0
−1
0
1
2
?0+?0
2
0
22
2
d2↓
†:
−?0−?0
2
0
−1
0
1
2
?0−?0
2
−?0+?0
2
0
−1
0
−1
−?0−?0
2
0
0
0
0
22
−1
0
0
?2
−1
0
0
?2
2
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TMR01=
p2sin2??
2/f2?− ?1??2?cos2??
2?
1 + ?B1
2?.
?48?
Hence, the TMR vanishes for ?=0 and takes the constant
value,
TMR01??,?? = p2,
at ?=?. For the remaining polarization angles, ??0 and
???, the TMR is gate voltage dependent and positive. The
behavior of the TMR as a function of the gate voltage is
shown in Fig. 7. To understand the gate voltage dependence
of the TMR at noncollinear angles, we have to remember
that the dot is depleted with increasing ?. For the 0↔1 tran-
sition, this means that at positive ?, the dot is predominantly
empty, so that an electron that enters the dot also leaves it
fast. In this situation, the TMR is finite and its value depends
in a complicated way on the amplitude of the exchange field.
At negative ?, the DD is predominantly occupied with an
TABLE IV. Matrix elements for the N=2↔N=3 transition governed by d?↑and d?↓, ?=1,2.
2↔3
?3o↓?
0
?3o↑?
?0+?0
2
0
−1
0
−1
−?0+?0
2
0
?3e↑?
?0−?0
2
0
−1
0
1
2
?0+?0
2
0
?3e↓?
0
?2g?
d1↑:
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
?2g?
?2??+1??
?2??0??
?2??−1??
?2??
?2??
0
0
1
?2
0
0
0
0
1
?2
0
0
22
2
d1↓:
−?0−?0
2
0
−1
0
1
2
?0−?0
2
0
−?0+?0
2
0
−1
0
−1
−?0−?0
2
0
1
?2
0
0
0
0
1
?2
0
0
0
0
22
2
d2↑:
−?0−?0
2
0
−1
0
−1
?0−?0
2
0
?0−?0
2
0
1
2
0
−1
?0+?0
2
0
−1
?2
0
0
0
0
0
0
1
?2
0
0
0
0
2
−1
0
0
?2
22
d2↓:
?0+?0
2
0
−1
0
1
2
−?0+?0
2
−?0+?0
2
0
1
2
0
1
2
−?0−?0
2
1
?2
0
0
0
0
2
TABLE V. Matrix elements for the N=3↔N=4 transition in-
duced by the operators d?↑
†
and d?↓
†, ?=1,2.
3↔4
?3o↓?
1
?2
0
−1
?2
0
?3o↑?
0
1
?2
0
−1
?2
?3e↑?
0
−1
?2
0
−1
?2
?3e↓?
−1
?2
0
−1
?2
0
d1↑
d1↓
d2↑
d2↓
†:?2,2?
†:?2,2?
†:?2,2?
†:?2,2?
0
2
4
6
8
10
-20 -15-10 -50510 1520
(γBi)/(¯ hΓf((−1)iµi))
ξ [|b|]
Resonance
0↔ 1
1↔ 2
2↔ 3
3↔ 4
FIG. 3. ?Color online? Gate voltage dependence of the renormal-
ized effective exchange splitting entering the conductance formula
?Eq. ?46??. Notice the mirror symmetry of the 0↔1 with the 3↔4
curve and of the 1↔2 with the 2↔3 one.
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Page 12

electron that can now interact with the exchange field, which
makes the spin precess and thus eases tunneling out of the
dot. Consequently, GNN+1??,???GNN+1?0,?? and the TMR
vanishes. Finally, Fig. 8 illustrates the angular dependence of
the normalized conductance for three different values of the
gate voltage. We detect a common absolute minimum for the
conductance at ?=?, i.e., transport is weakened in the anti-
parallel case. The width of the curves is dependent on the
renormalized effective exchange ??Bi?/??f?. The larger its
value, the narrower the curves because the spin precession
can equilibrate the accumulated spin for all angles but ?
=?. Notice again the equivalence of the curves belonging to
?=?2?b? for the 1↔2 resonance to the curves with ?
=?2?b? for the 3↔4 resonance.
2. Resonant regimes N=1^N=2 and N=2^N=3
For the resonant 1↔2 and 2↔3 transitions, qualitatively
analogous results as for the 0↔1 and 3↔4 transitions are
found. Thus, exemplarily, we only show the angular depen-
dence of the normalized conductance in Fig. 9, showing the
expected absolute conductance minimum at ?=0.
V. NONLINEAR TRANSPORT
In this section, we present the numerical results deduced
from the general master equation ?Eq. ?21?? combined with
the current formula ?Eq. ?25??. We show the differential con-
ductance
dI
dV??,Vbias? for the three distinct angles ?=0, ?
0.08
0.06
0.04
0.02
-8 -6 -4-2 0246
8
Conductance G[e2/h]
ξ[|b|]
resonance
regime
1 ↔ 02 ↔⇐ 13 ↔ 24 ↔ 3
0.01
0.02
0.03
-8-6-4 -20246
8
Conductance G[e2/h]
ξ[|b|]
FIG. 4. ?Color online? Conductance at low bias for the parallel
case ?=0 ?top? and antiparallel case ?=? ?bottom?. Notice the
different peak heights and the mirror symmetry with respect to the
half-filling value ?=0. The conductance in the antiparallel configu-
ration is always smaller than that in the parallel one.
0
0.02
0.04
0.06
0.08
Θ[rad]
ξ[|b|]
0
π
2
π
3
2π
2π
-6
-4
-2
0
2
4
6
G[e2/h]
FIG. 5. ?Color online? Conductance as a function of the polar-
ization angle and of the gate voltage. The minimal conductance
peaks occur as expected at ?=?.
0.08
0.06
0.04
0.02
20-2
Conductance G01[e2/h]
ξ[|b|]
Θ = 0.0π
Θ = 0.5π
Θ = 0.7π
Θ = 0.9π
Θ = 1.0π
0.08
0.06
0.04
0.02
20-2
Conductance G34[e2/h]
ξ[|b|]
Θ = 0.0π
Θ = 0.5π
Θ = 0.7π
Θ = 0.9π
Θ = 1.0π
FIG. 6. ?Color online? The conductance G01??? ?upper figure?
?G34??? ?lower figure?? vs gate voltage for different polarization
angles. The mirror symmetry of the conductance peaks for the
0↔1 and 3↔4 transitions is clearly observed. Notice the excellent
agreement between the prediction of the analytical formula ?Eq.
?46?? ?continuous lines? and the results of a numerical integration of
Eq. ?21? with Eq. ?25? ?symbols?.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2
0-2
TMR01
ξ[|b|]
Θ = 0.0π
Θ = 0.5π
Θ = 0.7π
Θ = 0.9π
Θ = 1.0π
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
20-2
TMR34
ξ[|b|]
Θ = 0.0π
Θ = 0.5π
Θ = 0.7π
Θ = 0.9π
Θ = 1.0π
FIG. 7. ?Color online? TMR for the 0↔1 ?upper figure? and
3↔4 ?lower figure? transitions vs gate voltage. The TMR is always
positive and for collinear lead magnetizations, ?=0 and ?=?, in-
dependent of gate voltage.
HORNBERGER et al.
PHYSICAL REVIEW B 77, 245313 ?2008?
245313-12

Page 13

=?
tively. The results confirm the electron-hole symmetry and
the symmetry upon bias voltage inversion I??,Vbias?=−I??,
−Vbias?. In all of the three cases, we can nicely see the ex-
pected three closed and the two half-open diamonds, where
the current is blocked and the electronic number of the
double-dot system stays constant. At higher bias voltages,
the contribution of excited states is manifested in the appear-
ance of several excitation lines. One clearly sees that transi-
2, and ?=?, see Fig. 10, top, middle, and bottom, respec-
tion lines present in the parallel case are absent in the anti-
parallel case. Moreover, in the case of noncollinear
polarization, ?=?/2, negative differential conductance
?NDC? is observed.
In the following, we want not only to explain the origin of
these two features, but alongside also give another example
for spin-blockade effects, which play a decisive role in the
DD physics. As a starting point, we plot in Fig. 11 ?top? the
current through the system for the three different angles ?
=?0,?
voltages. We recognize that for eVBias?2.4?b?, the current is
Coulomb blocked in all of the three cases. In this configura-
tion, exactly one electron stays in the double-dot. From about
eVbias?2.4?b?, the channel where the ground state energies
?1and ?2are degenerate opens ??1e??↔?2? transition? and
the current begins to flow. With increasing bias, more and
more transport channels become energetically favorable. In
particular, for all the polarization angles ?, we observe two
consecutivestepscorresponding
?1e??↔?2?? transitions. The latter, occurring at about eVbias
=4?b?, involves the excited two-particle triplet states ?2??Sz??.
The next excitation step, indicated with a circle in Fig. 11
?top?, belongs to the ?1o??↔?2?? transition. The associated
2,?? at a fixed gate voltage ?=4?b? and positive bias
to
?0?↔?1e??
and
0.4
0.6
0.8
1
2π
3π
2
π
π
2
0
G01(Θ)/G01(Θ = 0)
Angle Θ[rad]
ξ = −2
ξ = 0
ξ = +2
0.4
0.6
0.8
1
2π
3π
2
π
π
2
0
G34(Θ)/G34(Θ = 0)
Angle Θ[rad]
ξ = −2
ξ = 0
ξ = +2
FIG.8.
?Coloronline?
G01???/G01?0??upperfigure?
?G34???/G34?0? ?lower figure?? vs polarization angle. For all the
three chosen values of the gate voltage, the curve displays an abso-
lute minimum at ?=?. Notice the overall agreement of the analyti-
cal predictions ?Eq. ?46?? given by the continuous curves with out-
comes of a numerical solution of the master equation ?Eq. ?21??
together with Eq. ?25? ?symbols?.
0.4
0.6
0.8
1
2π
3π
2
π
π
2
0
G12(Θ)/G12(Θ = 0)
Angle Θ[rad]
ξ = −2
ξ = 0
ξ = +2
0.4
0.6
0.8
1
2π
3π
2
π
π
2
0
G23(Θ)/G23(Θ = 0)
Angle Θ[rad]
ξ = −2
ξ = 0
ξ = +2
FIG.9.
?Coloronline?
G12???/G12?0??upperfigure?
?G23???/G23?0? ?lower figure?? vs polarization angle. Notice the
overall agreement of the analytical predictions ?Eqs. ?45? and ?46??
?continuous lines? with the data ?symbols? coming from numerical
solutions of the equations for the reduced density matrix.
eVbias[|b|]
-12
-8
-4
0
4
8
12
0.04
0.06
0.08
0.02
0
eVbias[|b|]
-12
-8
-4
0
4
8
12
dI
dV[e2/h]
ξ[|b|]
eVbias[|b|]
-8-6 -4-202468
-12
-8
-4
0
4
8
12
FIG. 10. ?Color online? Differential conductance
allel ?=0 ?top?, perpendicular ?=?/2 ?middle?, and antiparallel
?=? ?bottom? configurations. The two half diamond and three dia-
mond regions correspond to bias and gate voltage values where
transport is Coulomb blocked. The excitation lines, where excited
states start to contribute to resonant transport, are clearly visible in
all of the three cases. However, a negative differential conductance
is observed in the perpendicular case, while some excitation lines
are absent in the antiparallel configuration.
dI
dVfor the par-
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Page 14

line is missing for the antiparallel configuration, as well as
the lines corresponding to ?1o??↔?2??, ?1e??↔?2??,
?2?↔?3o??, and ?1e??↔?2??. Crucially, in all of these tran-
sitions, a two-particle state with total spin zero is involved.
In order to explain the absence of these lines, let us, e.g.,
focus on the first missing step corresponding to the
?1o??↔?2?? resonance. In the parallel case ?say, both con-
tacts polarized spin-up?, there is always an open channel cor-
responding to the situation in which the spin in the DD is
antiparallel to that in the leads ?i.e., ?1o−??. In the antiparallel
case ?say, source polarized spin-up and drain polarized spin-
down?, originally a spin-down might be present in the dot.
An electron that enters the DD from the source must then be
spin-up ?in order to form the ?2?? state?, but as the drain is
down-polarized, it will be the spin-down electron that leaves
the DD, which corresponds to a spin flip. Now, the presence
of a spin-up electron in the DD prevents a majority ?another
spin-up? electron from the source to enter the DD, such that
we end up in a blocking state. The transition is hence forbid-
den.
A similar yet different spin-blockade effect determines the
occupation probabilities for the triplet state ?Fig. 11, bottom?.
Naturally, for all angles, the probability to be in the triplet
state increases above the resonance at eVbias=4?b?, but inter-
estingly, such probability is largest in the antiparallel case.
This is due to the fact that a majority spin in the parallel
configuration ?spin-up? can be easily transmitted through the
DD via the triplet states ?2??1?? or ?2??0??. In the antiparallel
case, however, a blocking state establishes ?say, again source
polarized spin-up and drain polarized spin-down?. Let ini-
tially a spin-down electron be present on the DD. From the
source electrode, most likely a majority electron ?polarized
spin-up? will enter the dot. Now, just as in the previous case,
the consecutive tunneling event will cause a spin flip in the
DD because the spin-down electron ?majority electron of the
drain? will leave the dot. So, the DD is finally in a spin-up
state, and once the next majority spin-up electron from the
source enters, the DD ends up in the triplet state ?2??+1?? and
will remain there for a long time due to the fact that the
majority spins in the drain are down polarized. Hence, the
triplet state ?2??+1?? acts as a trapping state.
Notice that the two distinct spin-blockade effects are dif-
ferent from the Pauli spin-blockade discussed in the DD
literature.46–49Moreover, the second effect, relying on the
existence of degenerate triplet states, is also different from
the spin-blockade found in Ref. 30 for a single-level quan-
tum dot.
Finally, let us turn to the negative differential conduc-
tance, which occurs for noncollinearly polarized leads ?see
the dashed blue lines in Fig. 11? and which we find to be-
come more evident for higher polarizations ?not shown?. By
neglecting the exchange field, we would just expect the mag-
nitude of the current for the noncollinear polarizations to lie
somewhere in between the values for the parallel and the
antiparallel current because the noncollinear polarization
could, in principle, be rewritten as a linear combination of
the parallel and the antiparallel configuration. Now, the effect
of the exchange is to cause precession and therewith equili-
bration of the accumulating spin, which corresponds to shift-
ing the balance in favor of the parallel configuration, i.e.,
enhancing the current. The decisive point is that the ex-
change field is not only gate dependent but also bias voltage
dependent and reaches a minimum around eVbias?8?b?. This
explains the decrease of the current up to this point. After-
ward, the influence of the spin precession regains weight.
The same consideration applies for the other NDC regions
observed in Fig. 10, e.g., in the gate voltage region ??2?b?
involving the N=0↔N=1 transition, as described in Ref.
30.
VI. EFFECTS OF AN EXTERNAL MAGNETIC FIELD
In this section, we wish to discuss the qualitative changes
brought by an external magnetic field applied to the DD.
Specifically, the magnetic field is assumed to be parallel to
the magnetization direction of the drain. For simplicity, we
focus on the experimental standard case of parallel and anti-
parallel lead polarizations and of low-bias voltages. Then,
the magnetic field causes an energy shift ?EZeemandepend-
ing on whether the electron spin is parallel or antiparallel,
respectively, to it. For collinear polarization angles, the
principal-part contributions vanish, and the equations for the
RDM are easily obtained. We exemplarily report results for
0↔1 and 1↔2 transitions. Let us then consider the param-
eter regime nearby the 0↔1 resonance and setup a system of
three equations with three unknown variables, W0, W1↑, and
W1↓. The first equation corresponds to the normalization con-
dition W1↑+W1↓+W0=1. The remaining equations are the
equations of motion for W1↑/↓, which can be written as
W˙1↑= −?
??
?=s,d
?t??2?F?↑
−??1↑?W1↑− F?↑
+??1↑?W0?, ?49?
0
0.2
0.4
0.6
0.8
1
1.2
??
??
??
8
??
024610121416
Current I[eΓ]
eVbias[|b|]
Θ = 0
Θ =π
Θ = π
2
0
0.2
0.4
0.6
0.8
1
0246810121416
Occ. probability triplet states
eVbias[|b|]
Θ = 0
Θ =π
Θ = π
2
FIG. 11. ?Color online? Current ?top? and triplet occupation
?bottom? for the two collinear ??=0,?=?? cases and the perpen-
dicular case ??=?
currence of a pronounced negative differential conductance feature
for perpendicular polarization ?=?/2.
2? at a fixed gate voltage ?=4?b?. Notice the oc-
HORNBERGER et al.
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245313-14

Page 15

W˙1↓= −?
??
?=s,d
?t??2?F?↓
−??1↓?W1↓− F?↓
+??1↓?W0?, ?50?
where
?1↑/↓= ?1? EZeeman,
?E?=D???f??E?. D?↑/↓=D???. On the other hand,
Ds↑/↓=Ds?sand Dd↑/↓=Dd?din the antiparallel case.
Upon considering symmetric contacts ?t1=t2=t, D1=D2
=D?, we find the following in the parallel case:
G01?? = 0?
=?e2
8?f?− ?1↑?f?− ?1↓?
?51?
and F???
?
?
p?f??1↑? − f??1↓?? + f??1↑? + f??1↓?
f?− ?1↑?f??1↓? + f?− ?1↑?f?− ?1↓? + f??1↑?f?− ?1↓?.
?52?
For the antiparallel case, we obtain
G01?? = ??
= G01?? = 0?
1 − p2?f??1↑? + f??1↓??
p?f??1↑? − f??1↓?? + f??1↑? + f??1↓?.
?53?
Analogously, we find the following for the 1↔2 transition:
G12?? = 0?
=?e2k+
2?
2
f??2↑?f??2↓?
?
p?f?− ?2↓? − f?− ?2↑?? + f?− ?2↑? + f?− ?2↓?
f?− ?2↑?f??2↓? + f?− ?2↑?f?− ?2↓? + f??2↑?f?− ?2↓?,
?54?
G12?? = ??
= G12?? = 0?
?
1 − p2?f?− ?2↑? + f?− ?2↓??
p?f?− ?2↓? − f?− ?2↑?? + f?− ?2↑? + f?− ?2↓?.
?55?
The remaining resonances are analogously calculated. Figure
12 shows the four conductance resonances for the parallel
and antiparallel configurations. Strikingly, the applied mag-
netic field breaks the symmetry between tunneling regimes
0↔1 and 3↔4, as well as between 1↔2 and 2↔3 reso-
nances in case of parallel contact polarizations. The reason
for this behavior is the following: in the low-bias regime,
transitions between ground states dominate transport. In par-
ticular, the magnetic field removes the spin degeneracy of
?1e?? and ?3o?? states, such that states with spin aligned to
the external magnetic field are energetically favored. There-
fore, the transport electron in tunneling regime 0↔1 is a
majority spin carrier. For the case 3↔4, however, two of the
three electrons of the ground state ?3o↑? have spin-up, such
that the fourth electron that can be added to the DD has to be
a minority spin carrier. Therefore, the conductance gets di-
minished with respect to the 0↔1 transition, and the mirror
symmetry present in the zero field case is broken. Analo-
gously, the broken symmetry in the case of the 1↔2 and
2↔3 transitions can be understood. Correspondingly, the
TMR can become negative for values of the gate voltages
around the 2↔3 and 3↔4 resonances. We observe that a
negative TMR has recently been predicted in Ref. 42 for the
case of a single impurity Anderson model with orbital and
spin degeneracies ?see Fig. 13?. In that work, a negative
TMR arises due to the assumption that multiple reflections at
the interface cause spin-dependent energy shifts. In our ap-
proach, however, where the contribution from the reflection
Hamiltonian is treated to the lowest order, see Eq. ?19?, such
spin-dependent energies originate from the magnetic-field-
induced Zeeman splitting.
VII. CONCLUSIONS
In summary, we have evaluated linear and nonlinear trans-
port through a double-quantum-dot ?DD? coupled to polar-
ized leads with arbitrary polarization directions. Due to
strong Coulomb interactions, the DD operates as a single-
electron transistor, a F-SET, at low enough temperatures. A
detailed analysis of the current-voltage characteristics of the
DD and comparison with results of previous studies on other
0.08
0.06
0.04
0.02
-6 -4 -20246
Conductance G [e2/h]
ξ[|b|]
0 ↔ 1
1 ↔ 2
2 ↔ 3
3 ↔ 4
FIG. 12. ?Color online? Conductance vs gate voltage for parallel
?continuous line? and antiparallel ?dashed lines? contact configura-
tions and Zeeman splitting EZeeman=0.05?b?. The magnetic field
breaks the mirror symmetry with respect to the gate voltage in the
parallel configuration.
0.4
0.2
0.0
-
-0.2
-0.4
-8 -40
6
48
TMR
ξ[|b|]
FIG. 13. ?Color online? TMR vs gate voltage in the presence of
an external magnetic field. In contrast to the zero field case, the
TMR can become negative in the vicinity of the 2↔3 and 3↔4
resonances.
TRANSPORT THROUGH A DOUBLE-QUANTUM-DOT SYSTEM…
PHYSICAL REVIEW B 77, 245313 ?2008?
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