# Polarized currents in Coulomb blockade and Kondo regimes without magnetic
fields

Anh T. Ngo  Department of Physics and Astronomy, and Nanoscale and Quantum
Phenomena Institute, Ohio University, Athens, Ohio 45701-2979  Edson Vernek
Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena
Institute, Ohio University, Athens, Ohio 45701-2979  Instituto de Física -
Universidade Federal de Uberlândia, Uberlândia, MG 38400-902, Brazil  Sergio
E. Ulloa  Department of Physics and Astronomy, and Nanoscale and Quantum
Phenomena Institute, Ohio University, Athens, Ohio 45701-2979

###### Abstract

We present studies of the Coulomb blockade and Kondo regimes of transport
through a quantum dot connected to current leads through spin-polarizing
quantum point contacts (QPCs). This structure, arising from the effect of
lateral spin-orbit fields defining the QPCs, results in spin-polarized
currents even in the absence of external magnetic fields and greatly affects
the correlations in the dot. Using equation-of-motion and numerical
renormalization group calculations we obtain the conductance and spin
polarization for this system under different parameter regimes. We find that
the system exhibits spin-polarized conductance in both the Coulomb blockade
and Kondo regimes, all in the absence of applied magnetic fields. We analyze
the role that the spin-dependent tunneling amplitudes of the QPC play in
determining the charge and net magnetic moment in the dot. These effects,
controllable by lateral gate voltages, may provide an alternative approach for
exploring Kondo correlations, as well as possible spin devices.

###### pacs:

72.15.Qm, 72.25.-b, 72.10.-d, 73.23.Hk

##  I Introduction

Electronic transport in semiconducting nanostructures is studied both as it
has great potential applications in spintronics, 5  and because of its
exquisite control of parameters, which allow insightful probes into
fundamental physical phenomena. Electrons in such systems experience
externally controlled confining environments that result in strong Coulomb
interactions with other electrons. As such, quantum dot (QD) structures
provide well-characterized and defined systems for studying quantum many-body
physics. QDs may also allow the realization of solid state quantum computation
devices as well as spintronic semiconductor devices with unprecedented
functionalities. QDref

Manipulation of spin-polarized current sources is crucial in spintronics. This
typically requires efficient spin injection into conventional semiconductors.
The difficulties with spin injection from ferromagnetic metal leads has
stimulated extensive efforts to produce spin polarized currents out of
unpolarized sources. Schmidt  In this context, the Rashba spin-orbit (RSO)
coupling mechanism provides a basis for possible device applications. Rashba
This coupling, arising from interfacial structure asymmetries, depends on the
materials used as well as on the confinement geometry of the structures. 6 ; 9
Most interestingly, the Rashba effect allows external tunability, which has
been studied experimentally in QDs 7  and quantum point contacts. Bird ; 8

In this paper we study the electronic transport through a quantum dot
connected to polarizing quantum point contacts (QPCs) in both the Coulomb
blockade (CB) and Kondo regimes. Due to strong spin-orbit interactions, 8 ; 9
QPCs can exhibit spin-dependent hybridization of the QD states with the leads,
without applied magnetic fields, opening the possibility for generating spin-
polarized transport in an all-electrical setup. These effects are controllable
by lateral gate voltages applied on QPCs, resulting in spatially asymmetric
structures, as in recent experiments.8  Using the equation-of-motion technique
and numerical renormalization group (NRG) calculations we obtain the
electronic Green’s function, conductance and spin polarization in different
system regimes. Our results show that both the CB and Kondo regimes exhibit
non-zero spin-polarized conductance in this system. We analyze how the spin-
dependent hybridization of the QPC modifies the charge accumulation in the
dot, as well as the density of states (spectral functions) of the system.
Interestingly, we find that the polarizing QPCs produce spin polarization and
split DOS in the Kondo regime akin to that reported for current injection from
ferromagnetic leads, Martinek  although here it occurs for unpolarized
reservoirs. Our theoretical studies suggest that these effects could be
accessible in experiments and result in future spintronic devices.

The paper is organized as follows. First, we present the main features of the
current polarization in quantum point contact systems with lateral spin-orbit,
obtained by scattering matrix methods in section II. The next section
describes the model of a quantum dot connected to current leads via polarizing
QPCs. In section III.1 we use the equation-of-motion (EOM) technique to obtain
the dot Green’s function and discuss numerical results in both the CB and
Kondo regimes. In section III.2 we revisit the problem using the numerical
renormalization group approach, which provides the most reliable information
in the Kondo regime for realistic system parameters. The paper concludes with
final remarks and summary in section IV.

##  II Polarizing QPC

The effective electric field in the $z$ direction that creates a two-
dimensional electron gas (2DEG) confined to the $x$-$y$ plane results in the
well-known Rashba spin orbit interaction, Rashba ; 6

$\displaystyle
H_{SO}^{R}=\frac{\alpha}{\hbar}\left(\sigma_{x}P_{y}-\sigma_{y}P_{x}\right).$
(1)

Here, $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices, and $\alpha$ is Rashba
spin orbit coupling constant, which is proportional to the field and is
therefore material and structure dependent. Electrons on this 2DEG entering a
quantum dot, pass through QPCs defined via a confining potential $V(x,y)$
which can be thought as made of two parts: $U(y)+V_{b}(x,y)$, where $U(y)$
defines a hard-wall potential of width $W$ (related in the experiment to the
side-wall etching defining the structure) outlining the overall channel
structure, while $V_{b}(x,y)$ is the potential generating the QPC barrier, and
effectively modulated by the side gate potentials of the structure. 8  We
model such barrier by 12

$\displaystyle V_{b}(x,y)=\frac{1}{2}V_{g}\left(1+\cos\frac{\pi
x}{L_{x}}\right)+\frac{1}{2}m\omega^{2}\bar{y}^{2}\Theta(\bar{y})$ (2)

with $\bar{y}=y-y_{s}$, and

$\displaystyle y_{s}=W_{1}\left(1-\cos\frac{\pi x}{L_{x}}\right)\,,$ (3)

where $\Theta(x)$ is the step function, $m$ is the effective mass of the
electrons, $L_{x}$ is the unit length of the structure in the $x$ direction
(along the current direction) and $\omega$ is the confinement potential
frequency. Notice that this potential form is asymmetric in the $y$ direction,
to reflect an essential ingredient in the experiments: the QPC potential must
lack $y$-reflection symmetry in order to generate the polarizing effect along
the $z$-direction. 12 ; 12b ; 12c  In fact, the fields forming $V_{b}$
generate a spin-orbit coupling given by 6

$\displaystyle V_{SO}^{\beta}=-\frac{\beta}{\hbar}\nabla
V_{b}\cdot(\mathbf{\hat{\sigma}}\times\mathbf{\hat{P}}),$ (4)

where $\beta$ is material-specific. Notice that $\nabla V_{b}$ lies in the
$x$-$y$ plane, so that the barrier fields induce a lateral spin-orbit
coupling. The total Hamiltonian of the QPC will then be given by,

$\displaystyle
H=\frac{P_{x}^{2}+P_{y}^{2}}{2m}+H_{SO}^{R}+V(x,y)+V_{SO}^{\beta}.$ (5)

Figure 1: (color online) Total and spin-dependent conductances for an
asymmetric QPC as function of Rashba spin-orbit coupling $\alpha$, obtained
from a scattering matrix approach.9  The interplay of vertical and lateral SO
effects may result in large asymmetry for the up and down spin components for
realistic structure parameters, even in the tunneling regime shown here
($G_{\sigma}\ll 1$).

Using a scattering-matrix formalism to study the spin-dependent electron
transport in this QPC, 11 ; 9  we find that a net spin-polarized conductance
is produced only for $y$-asymmetric potentials. 9  We can calculate the
conductance of the structure, assuming that the SO coupling $\alpha$ and
$V(x,y)$ are zero at the source and drain 2DEG reservoirs, while both of these
terms in the Hamiltonian are turned on in the QPC region. Typical structure
parameters in experiments can be cast in terms of characteristic length and
energy scales, $L_{0}=32.5$ nm, and $E_{0}=3.12$ meV, with
$\alpha_{0}=E_{0}L_{0}=1.0\times 10^{-12}$ eVm, a typical value of spin-orbit
coupling. Using $W=L_{x}=2L_{0}$ and $W_{1}=0.6L_{0}$ as width/length of the
confining potential, with $\omega=6\times 10^{13}$s-1, and $\beta=0.97\times
10^{-16}$m2, gives results as shown in Fig. 1. This figure shows spin-
dependent conductances as functions of Rashba coupling $\alpha$, for a
potential barrier which is near its conduction onset (or “pinch-off”, as
controlled by the value of $V_{g}$); arrows indicate the results of spin up
and down conductances. We see that conductances $G_{\uparrow}$ and
$G_{\downarrow}$ can be very different from each other, even in the tunneling
regime (where each $G_{\sigma}\ll 1$) in which the QPC would operate to create
a quantum dot in the 2DEG. We stress that for these realistic values of
structure parameters, one obtains non-zero spin polarization even when there
is no external magnetic field and the injection is unpolarized. This
interesting result can be understood from anticrossing features in the subband
energy structure in the channel region defining the QPC. 8 ; 9  The spin
mixings and avoided crossings generate spin rotation as electrons pass through
the narrow constriction of the QPC, and can generate large values of the ratio
$G_{\uparrow}/G_{\downarrow}$, even in the tunneling regime. Two of these QPCs
can then be used to define the QD and result in interesting charging and
conductance regimes, as we will see below.

##  III Quantum dot with polarizing QPCs

In order to address the transport through a quantum dot formed with polarizing
QPCs, we consider the single impurity Anderson model given by the following
Hamiltonian:

$\displaystyle H=\sum_{\ell k\sigma}\varepsilon_{\ell k}c^{\dagger}_{\ell
k\sigma}c_{\ell
k\sigma}+\sum_{\sigma}\varepsilon_{d}c^{\dagger}_{d\sigma}c_{d\sigma}+Un_{d\uparrow}n_{d\downarrow}$
$\displaystyle+\sum_{\ell k\sigma}t_{\sigma}(c^{\dagger}_{d\sigma}c_{\ell
k\sigma}+c^{\dagger}_{\ell k\sigma}c_{d\sigma}),$ (6)

where $c^{\dagger}_{d\sigma}(c_{d\sigma})$ is the creation (annihilation)
operator of an electron of spin $\sigma$ in the dot. The quantities
$\varepsilon_{\ell k}$, $\varepsilon_{d}$ are the energies of the electrons in
the $\ell^{\it th}$ conduction band channels ($\ell=L,R$) and the single local
energy level in the dot, respectively. $U$ is the Coulomb repulsion between
electrons occupying the QD with
$n_{d\sigma}=c^{\dagger}_{d\sigma}c_{d\sigma}$, while $t_{\sigma}$ represents
the lead-QD hybridization occurring via tunneling through the QPC, and which
is assumed to be $k$-independent. The density of states for conduction
electrons in each lead is taken to be constant,
$\rho_{L}(\varepsilon)=\rho_{R}(\varepsilon)\equiv\rho=(1/2D)\Theta(D-|\varepsilon|)$,
where $D$ is the conduction band halfwidth (hereafter taken as our energy
unity).

The theoretical description of such quantum dot system, especially in the
strong correlations regime, has been greatly developed over the years. Mahan
Techniques of note include quantum Monte-Carlo, 2  equations of motion for the
Green’s functions,3  and the numerical renormalization group approach.costi
In what follows, we explore the role that SO interactions play on the Coulomb
blockade and Kondo regimes of transport of the QD, utilizing equations of
motion and numerical renormalization group formalisms.

###  III.1 Equation of motion approach and numerical results

To calculate the charge and conductance of the system we calculate Green’s
functions (GFs), which allow us to take into account the correlations induced
by the Coulomb interaction in the QD. The retarded double-time Green’s
functions are defined as ($\hbar=1$)13

$i\langle\langle
A;B\rangle\rangle=\int_{-\infty}^{\infty}\langle[A(\tau),B(0)]_{+}\rangle\Theta(\tau)e^{-i\omega\tau}d\tau,$
(7)

where $A$ and $B$ are generic fermionic operators, $[A,B]_{+}$ indicates their
anticommutator and $\langle\cdots\rangle$ indicates the thermodynamic average
for $T>0$, or the ground state expectation value for $T=0$. The GF can be
obtained using equation of motion (EOM) techniques, so that

$\displaystyle\omega\langle\langle
A;B\rangle\rangle=\langle[A,B]_{+}\rangle+\langle\langle[A,H];B\rangle\rangle,$
(8)

where $[A,B]$ represents a commutator. Iteration of this formula generates a
hierarchy of expressions, starting with the local one-particle GF as

$\left(\omega-\varepsilon_{d}-\sum_{k}\frac{\tilde{t}_{\sigma}^{2}}{\omega-\varepsilon_{k}}\right)\langle\langle
c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle=1+U\langle\langle
c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle,$ (9)

where $\tilde{t}_{\sigma}=\sqrt{2}t_{\sigma}$ and $\bar{\sigma}=-\sigma$. The
new (higher order) GF on the right hand side of Eq. (9) can also be determined
from (8), giving

$\displaystyle(\omega-\varepsilon_{d}-U)\langle\langle
c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle=\langle
n_{d\bar{\sigma}}\rangle+\tilde{t}_{\sigma}\sum_{k}\left(\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle\right).$
(10)

####  III.1.1 Coulomb blockade regime

Although the EOM in Eq. (10) is exact, a solution of the impurity GF requires
a procedure to truncate and/or decouple the higher order terms appearing on
the right hand side of (10). A solution that captures the Coulomb blockade
physics is given by the Hubbard-I approximation:14

$\displaystyle\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
n_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle,$ (11)

which allows one to write

$\displaystyle G_{d\sigma}(\omega)\equiv\langle\langle
c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle_{\omega}=\frac{G^{0}_{d\sigma}(\omega)}{1-G^{0}_{d\sigma}(\omega)\tilde{t}^{2}_{\sigma}\tilde{g}(\omega)},$
(12)

where $G^{0}_{d\sigma}(\omega)=\frac{1-\langle
n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}}+\frac{\langle
n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}-U}$ is the local GF in the
“atomic” approximation (the exact result for $t_{\sigma}=0$), and
$\tilde{g}(\omega)=\sum_{k}(\omega-\epsilon_{k})^{-1}$ is the non-interacting
GF of the leads. The DOS of the system (proportional to the imaginary part of
$G_{d\sigma}$) contains two Hubbard peaks of width proportional to
$\Gamma_{\sigma}=\pi t^{2}_{\sigma}/D$, resulting in the broadening of the
poles of $G^{0}_{d\sigma}$. The spectral weights of these peaks are controlled
by the dot level occupancy with opposite spin, and caused by the Coulomb
interaction in the dot. Notice that the SO-induced polarization of the QPC
results in different peak widths for the different spins. The Hubbard-I
approximation (11) is known to be valid for a large $U/\Gamma$ ratio, when the
Hubbard subbands are well separated in energy scale. Mahan  It is the simplest
scheme which describes correlated electrons, although, since it ignores the
Kondo effect, it is a reasonable description only at temperatures higher than
the Kondo scale ($T\gg T_{K}$–see next section).

Figure 2: (color online) Occupancies for spin-up $\langle
n_{d\uparrow}\rangle$, spin-down $\langle n_{d\downarrow}\rangle$ and total
spin $\langle n_{d\uparrow}\rangle+\langle n_{d\downarrow}\rangle$ vs.
$\varepsilon_{d}$ at zero temperature. Parameters used are
$\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $U=0.3$, with $D=1$.
Notice asymmetry in $\langle n_{d\sigma}\rangle$ on each side of the plateau.
Figure 3: (color online) Spin-dependent conductance and polarization as
function of $\varepsilon_{d}$ at zero temperature. The other parameters are as
in Fig. 2.

The occupancies of spin $\uparrow$ and $\downarrow$ are calculated self-
consistently from the equation

$\displaystyle\langle n_{d\sigma}\rangle=\int
f(\omega)\left(-\frac{1}{\pi}\text{Im}\left[G_{d\sigma}(\omega)\right]\right)d\omega\,,$
(13)

where $f(\omega)$ is the Fermi function. It is clear that when the QPCs are
not polarizing, $t_{\uparrow}=t_{\downarrow}$, the occupancy curves for spin-
up and down coincide and a plateau of width $\sim U$ appears when the QD level
moves below the Fermi level ($\varepsilon_{d}+U\gtrsim
E_{F}\gtrsim\varepsilon_{d}$). This situation changes when the QPCs are
polarized (see Fig. 2), as the different $t_{\sigma}$ result in
$\Gamma_{\uparrow}\neq\Gamma_{\downarrow}$, which in turn produce different
$\langle n_{d\uparrow}\rangle$ and $\langle n_{d\downarrow}\rangle$,
especially on both sides of the plateau.

The zero-bias conductance is calculated using a Landauer formula generalized
for interacting systems 15

$\displaystyle G_{\sigma}=\frac{e^{2}}{h}\Gamma_{\sigma}\int
d\epsilon\frac{\partial
f(\epsilon)}{\partial\epsilon}\,\text{Im}[G_{d\sigma}(\epsilon)],$ (14)

for symmetric coupling of the leads for each spin. One can also calculate the
polarization factor

$\displaystyle\eta=\frac{G_{\uparrow}-G_{\downarrow}}{G_{\uparrow}+G_{\downarrow}},$
(15)

which gives a measure of current polarization in the system. Figure 3 shows
the spin-dependent conductances and polarization for the system in the
Hubbard-I approximation. As $\langle n_{d\uparrow}\rangle\neq\langle
n_{d\downarrow}\rangle$ in general (except at the particle-hole symmetry
point, $\varepsilon_{d}=-U/2$), the conductance per spin $G_{\sigma}$ are also
different. Notice that the spin-dependent conductance peaks are very
asymmetric and non-Lorentzian, due to the peculiar behavior of the occupancies
and their different up and down-spin couplings. As we consider here the case
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$, one clearly sees that generally
$G_{\uparrow}>G_{\downarrow}$ over the entire range of $\varepsilon_{d}$
values. As a consequence, there is a net up-spin polarization ($\simeq 60$%)
and conductance around the resonant peaks, with the latter reaching $\simeq
0.3(e^{2}/h)$.

####  III.1.2 Kondo regime

To study the low-temperature behavior of the system within the EOM we need to
consider higher order GFs in Eq. (10). Using Lacroix’s approach, 3  one can
obtain relations for the three GFs on the right side of (10) as:

$\displaystyle(\omega-\varepsilon_{k\sigma})\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=$
$\displaystyle\langle[c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c_{k\sigma}n_{d\bar{\sigma}};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(16) $\displaystyle=$ $\displaystyle\tilde{t}_{\sigma}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\tilde{t}_{\bar{\sigma}}\sum_{k^{\prime}}[\langle\langle
c_{k\sigma}c^{\dagger}_{d\bar{\sigma}}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$
$\displaystyle(\omega-\varepsilon_{d\sigma}+\varepsilon_{d\bar{\sigma}}-\varepsilon_{k\bar{\sigma}})\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(17) $\displaystyle=\langle
c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}\rangle+\tilde{t}_{\bar{\sigma}}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[-\tilde{t}_{\bar{\sigma}}\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle+\tilde{t}_{\sigma}\langle\langle
c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$

and

$\displaystyle(\omega-\varepsilon_{d\sigma}-\varepsilon_{d\bar{\sigma}}+\varepsilon_{k\bar{\sigma}}-U)\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(18) $\displaystyle=\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle-\tilde{t}_{\bar{\sigma}}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[\tilde{t}_{\sigma}\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle-\tilde{t}_{\bar{\sigma}}\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\sigma}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle].$

Following the decoupling procedure in Ref. 3, , each GF of the type
$\langle\langle A^{*}BC,D^{*}\rangle\rangle$ is replaced by

$\displaystyle\approx\langle A^{*}B\rangle\langle\langle
C,D^{*}\rangle\rangle-\langle A^{*}C\rangle\langle\langle
B,D^{*}\rangle\rangle,$ (19)

resulting in an equation for the dot GF given by

$\displaystyle G_{d\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\left[U(\omega)-\langle n_{d\bar{\sigma}}\rangle-
B_{\bar{\sigma}}(\omega)-B_{\bar{\sigma}}(\omega_{1})\right]$ (20)
$\displaystyle\times\left\\{U(\omega)[\omega-\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)]+[B_{\bar{\sigma}}(\omega)+B_{\bar{\sigma}}(\omega_{1})\right]\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)+A_{\bar{\sigma}}(\omega_{1})\\}^{-1}\,,$

where
$\Sigma_{\sigma}(x)=\sum_{k}|\tilde{t}_{\sigma}|^{2}/(x-\varepsilon_{k\sigma})$,
$U(\omega)=[U-\omega+\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)+\Sigma_{\bar{\sigma}}(\omega)-\Sigma_{\bar{\sigma}}(\omega_{1})]/U$
and $\omega_{1}=-\omega+\varepsilon_{d\sigma}+U$. The functions
$A_{\sigma}(\omega)$ and $B_{\sigma}(\omega)$ are given by

$\displaystyle B_{\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d\omega^{\prime}f(\omega^{\prime})\left[G_{d\sigma}(\omega^{\prime})\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-G^{*}_{d\sigma}(\omega^{\prime})\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right]$
$\displaystyle A_{\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d\omega^{\prime}f(\omega^{\prime})\left[\left(1+G_{d\sigma}(\omega^{\prime})\Sigma_{\sigma}(\omega^{\prime})\right)\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-\left(1+G^{*}_{d\sigma}(\omega^{\prime})\Sigma^{*}_{\sigma}(\omega^{\prime})\right)\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right],$
(21)

and have to be calculated self-consistently. In the limit
$U\rightarrow\infty$, the GF (20) acquires a simpler form,

$\displaystyle G_{d\sigma}(\omega)=\frac{1-\langle n_{d\bar{\sigma}}\rangle-
B_{\bar{\sigma}}(\omega)}{\omega-\varepsilon_{d\sigma}-(1-B_{\bar{\sigma}}(\omega))\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)}.$
(22)

Figure 4: (color online) Density of states in QD as a function of $\omega$ in
the Kondo regime. Parameters used are $T=10^{-7}$, $\varepsilon_{d}=-0.136$,
$\Gamma_{\uparrow}=0.06$ and $\Gamma_{\downarrow}=0.03$ ($D=1$). Sharp feature
near the Fermi level ($\omega=0$) is the signature of the Kondo screening,
although asymmetric here for the different spin species.

In the following, we solve numerically for the spectral function (DOS) in the
Kondo regime by the self-consistent iteration of Eqs. (13) and (22). Figure 4
shows the DOS vs. $\omega$ at low temperature ($T=10^{-7}$) for both spin
orientations (and total), when the QD electron level $\varepsilon_{d}$ is
taken to be at $-0.136$. The three curves exhibit Kondo resonance peaks near
the Fermi level ($\omega\simeq 0$), in addition to a much broader peak at
$\omega\simeq\varepsilon_{d}$. 3  Since the hybridization between leads and QD
are spin dependent in this case, the DOS clearly splits into two different
components for spin up and down. The DOS for spin down is shifted upwards in
energy with respect to the spin up component, resulting in a lower occupancy
for the down spin. Figure 5 shows indeed the occupancy curves for $\langle
n_{d\uparrow}\rangle$, $\langle n_{d\downarrow}\rangle$ and total $\langle
n\rangle$ vs. $\varepsilon_{d}$ at a given temperature ($T=10^{-7}$).
Qualitatively similar to the results in the Coulomb blockade regime, the
occupancy curves show that $\langle n_{d\uparrow}\rangle>\langle
n_{d\downarrow}\rangle$ for $\omega\gtrsim-0.05$, while the relation is
reversed for smaller $\omega$ values, reflecting the asymmetry introduced by
the polarizing QPCs, through $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$.
Notice also that the plateaus at 0.5 are not as well defined here, due to the
enhanced spin and charge fluctuations in the Kondo regime. The spin-orbit
effect can be understood qualitatively as arising from a shift in the dot
level (as well as depending on the occupancy factors): since the effective
level position of the electron with spin $\sigma$ is given by
$\varepsilon_{d\sigma}\approx\varepsilon_{d}+\text{Re}\Sigma^{{}^{\prime}}_{\sigma}(\omega)$,
where $\Sigma^{{}^{\prime}}_{\sigma}(\omega)\propto t_{\sigma}^{2}$ is the
self-energy, and thus naturally causes the spin-dependent occupancy seen in
the figure.

Figure 5: (color online) Occupation in the QD as function of $\epsilon_{d}$.
Parameters used as in Fig. 4.  Figure 6: (color online) (a) Conductance, and
(b) polarization and difference between spin up and down conductance, as
functions of $\epsilon_{d}$. Parameters as in Fig. 4.

Figure 6(a) shows the spin-dependent conductance curves $G_{\sigma}$ vs.
$\varepsilon_{d}$. Several features are noteworthy. As $\varepsilon_{d}$
changes, the spin-dependent conductances exhibit the anticipated peaks at low
temperature, with spin up conductance dominating (naturally, as
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$). Figure 6(b) shows the difference
between spin up and down conductances as well as the spin polarization vs.
$\varepsilon_{d}$. In this regime, the difference between the conductance for
the two spin orientations reaches $\simeq 0.4(e^{2}/h)$. Correspondingly, the
net spin polarization reaches $\eta\simeq 60$%.

Let us now analyze in more detail the effect of temperature on the conductance
of the system and especially its drop for $\varepsilon_{d}\ll 0$. The
conductance curves in Fig. 6(a) are highly asymmetric about the Fermi energy
and vanish rapidly away from it. This vanishing for very negative values of
$\varepsilon_{d}$ is due to the the Kondo temperature, $T_{K}$, becoming
smaller than the temperature of the system. Figure 7 presents the total zero
bias conductance as function of $\varepsilon_{d}$ for several values of
temperature. As $T$ is lowered, the total conductance increases for a given
$\varepsilon_{d}$, and the width of the conductance peak increases in
$\varepsilon_{d}$. This explicitly reflects the existence of the Kondo
resonant peak in the spectral function, and how the system will reach the
unitary limit of conductance for $T=0$ for $\varepsilon_{d}$ well below the
Fermi level.

Figure 7:  (color online) Total conductance as function of $\varepsilon_{d}$
for several temperatures $T$. Notice that $T_{K}$ is strongly suppressed for
more negative $\varepsilon_{d}$ values, which lowers the conductance at a
given $T$. Parameters as in Fig. 4.

Our discussions above are applied to the case of infinite $U$, where the EOM
method gives qualitatively accurate results in the Kondo regime. Extension of
this approach for finite $U$ is known to be problematic, including its failure
to exhibit a Kondo resonance at the particle-hole symmetry point
($\varepsilon_{d}=-U/2$). 18  To carry out our study in the finite $U$ case,
we use instead the essentially exact numerical renormalization group approach,
as we discuss in the following section.

###  III.2 NRG results for finite U

For the finite-$U$ case we study the spin polarized conductance using the
standard numerical renormalization group approach.Wilson75 ; costi  In this
case, unlike the previous infinite-$U$ case, the processes involving double
occupied states are naturally present in the dynamics of the system, and allow
for a reliable description of the low-energy behavior. We set $U=0.5$ and
$T=0$. Figure 8 depicts the local density of states calculated with NRG for
the same system parameters as before, $\Gamma_{\uparrow}=0.06$,
$\Gamma_{\downarrow}=0.03$, and with $\epsilon_{d}=-0.2$. This value of
$\epsilon_{d}$ corresponds to a situation where the system is away from the
particle-hole symmetric point ($=-U/2$), more suitable to compare to the
previous infinite-$U$ calculation (where the system never reaches the p-h
symmetry point). Notice that there is a strong spin asymmetry in the DOS; on
the negative side of the $\omega$-axis, the DOS for spin up (solid curve)
presents a peak near $-0.2$, corresponding to the energy of the local bare
orbital $\varepsilon_{d}$, slightly shifted by the real part of the proper
self-energy. On the positive side of the $\omega$-axis, the peak near
$\epsilon_{d}+U$ would result in the succeeding CB peak, which appears only
for the spin-down DOS (dashed curve). In this regime as well, the fact that
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$ favors the spin-up occupancy to the
detriment of the spin-down occupancy. One also notices the important peaks
close to the Fermi level, signature of the Kondo effect, which are slightly
split away and suppressed by a seemingly effective magnetic field induced by
the spin-asymmetric coupling to the leads. This phenomenon is akin to the
suppression discussed by Martinek et al.,Martinek  in the context of a QD
coupled to ferromagnetic leads. Notice, however, that no external
magnetization is present in our system and that the polarization is only
arising from the QPCs and due to the lateral SO interaction.

Figure 8: (color online) Density of states in the QD as function of energy
obtained from NRG calculations. Polarizing QPCs generate effective splitting
of the DOS for different spins. System parameters used are $U=0.5$,
$\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $D=1$, and $T=0$.

Analogously to the infinite-$U$ case, the spin-asymmetry discussed above
induces spin polarized transport in the system. In Fig. 9 we show the
conductance as a function of $\varepsilon_{d}$ for the same parameters as Fig.
8. Notice that away from the p-h symmetric point ($\varepsilon_{d}=-0.25$) the
conductance for spin up is much larger than for spin down, resulting in a
sizeable polarization ($\eta\simeq 70$%), as shown by the (green) dotted
curve. At the p-h symmetric point the conductance for both spins reaches the
unitary limit and $\eta\rightarrow 0$; this is consistent with the restoration
of the Kondo state of the system at the p-h point for ferromagnetic leads.
Sindel

Figure 9: (color online) Total conductance as function of $\varepsilon_{d}$
obtained from NRG calculations. Same system parameters as in Fig. 8.

##  IV Summary

In summary, we have investigated the spin-dependent transport properties of
quantum dot structures with polarizing quantum point contacts. We have shown
that as QPCs can generate finite spin-polarized currents, due to the
combination of lateral and perpendicular spin-orbit interactions, they also
induce current polarization in quantum dots made with these QPCs. Using
equation-of-motion techniques and numerical renormalization group
calculations, we obtained the electronic Green’s function, conductance and
spin polarization in different parameter regimes. Our results demonstrate that
both in the Coulomb blockade and Kondo regimes, the quantum dot exhibits non-
zero spin-polarized conductance, even when the injection is unpolarized and
there are no applied magnetic fields. The spin-dependent coupling is shown to
give rise to nontrivial effects in the density of states of the single QD,
resulting in strong modification of the charge distribution in the system.
Most importantly, these effects are controllable by lateral gate voltages
applied to the QPCs, and together with the ability to create quantum dots,
they provide a new approach for exploring spintronic devices, spin polarized
sources and spin filters.

##  V Acknowledgements

We thank helpful discussions with P. Debray and N. Sandler, as well as
financial support from CNPq, CAPES, and FAPEMIG in Brazil, and NSF-PIRE, and
NSF-MWN/CIAM in the US.

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