# Four terminal quantum dot as an efficient rectifier of heat and charge
currents

Karol I. Wysokiński  Institute of Physics, M. Curie-Skłodowska University, pl.
M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland

###### Abstract

We propose an efficient method of heat rectification in a simple system
consisting of a quantum dot asymmetrically coupled to four mutually
perpendicular electrodes. In such a device the Hall-like charge and heat
currents appear in response to the voltage bias or temperature difference
between one pair of electrodes. Even though both longitudinal (along the bias)
and Hall-like (perpendicular to the bias) currents are rectified under
appropriate conditions, the rectification factor is typically much bigger for
the latter currents. This is true for heat and charge flow. The Hall-like
currents are predicted to exist in linear as well as non-linear transport
regimes and require broken mirror symmetry but not time reversal symmetry. The
linear effect exists only in geometry which breaks two inversion symmetries
along two pairs of electrically coupled terminals. The proposed system is
attainable within current technology and provides novel platform of
simultaneous heat and charge management at the nanoscale.

##  I Introduction

A rectifier is a device in which the magnitude of the charge or heat current
depends on the sign of the electric or thermal bias. Efficient heat current
rectifiers are among the most desirable devices at the nanoscale as they allow
for heat management in miniaturised electronic devices. Various geometries and
devices have been proposed to reach the goal. An early proposal of nanoscale
heat rectification has appeared in the context of molecular electronics [1]
and is still debated in the literature [2, 3].

In recent years an increased interest is observed in theoretical analysis and
experimental studies of thermal diodes [4]. Besides molecules including those
with negative U centers [5], many systems with tuned quantum properties have
been proposed as possible rectifiers. These include inter alia recently
analysed two terminal junctions with bath particles obeying different particle
exchange statistics [6] and the use of quantum entanglement as a possible
tools to enhance rectification properties [7]. Also various novel solid state
systems and materials have been put forward. These include heterostructures,
functionally graded or phase changing normal materials, superconductors, etc.
as reviewed recently [8, 9].

In the context of this paper, the rectifying devices based on quantum dots are
of special interest [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
They consist of a single or more quantum dots coupled to two external
reservoirs. Probably the first experimental demonstration of rectifying
properties of a two terminal quantum dot is that in Ref. [24] inferred from
the asymmetric line shape of the thermopower. Rectification of both charge and
heat currents in the Coulomb blockade regime in the system with two quantum
dots has been studied in [20]. In [23] the authors concentrate on heat
rectification through quantum dots in the Coulomb blockade regime using master
equation approach. They considered two-terminal and four-terminal devices. In
the latter case, two coupled quantum dots form a main nanoscopic element in
which each of the individual dots is contacted by two separate terminals.

Quantum dots play an important role in novel electronic devices like single
electron transistors [25], heat nano-engines [26] and many more, including
building blocks of quantum computers [27, 28]. Quantum dots with large
charging energy U, coupled to external metallic leads behave like magnetic
impurities in noble metals [29, 30] and at low temperature show Kondo effect
[31, 32].

The device we are proposing consists of a single quantum dot tunnel coupled to
four normal electrodes in a cross geometry as shown in Fig. (1). Application
of the voltage or thermal bias along $LR$ electrodes (i.e. x direction)
results in a simultaneous flow of (longitudinal) charge and heat currents
between $L$ and $R$ electrodes and also between $U$ and $D$ electrodes (Hall-
like currents) if the system breaks mirror symmetry(-ies). A device with non-
symmetrical couplings works as an efficient rectifier. The rectification is
observed for both currents and directions. If the interaction $U$ of electrons
is non-negligible, as it is usually the case in small structures, and if
working temperature is low enough, the device allows to study the effect of
Kondo correlations on longitudinal and Hall \- like currents.

Four terminal nano-junctions with similar geometry have been studied
previously. General symmetry properties of phase coherent transport in the
presence of magnetic flux were derived in [33]. The same geometry was
theoretically analysed in the context of Hall - like currents and resistances
[34, 35] taking into account polarisation of electrons and strong spin-orbit
interaction in the central region. Wei et al. studied planar four terminal
system [36] and found non-linear Hall effect induced by the dipole of Berry
curvature. The hybrid structure consisting of a quantum dot contacted by
tunnel barriers to four electrodes including two superconducting and two
normal leads, allows [37] the control of super-current flowing between a pair
of superconducting electrodes by the bias voltage applied to normal
electrodes. The cross geometry of the four terminal nano-junction containing
an asymmetric prism -like scatterer has been studied experimentally [38]. Even
though our system is different from the experimental one it features similar
antisymmetric dependence of the four terminal resistance on the current as
discussed below.

We shall analyse two possible boundary conditions: open with floating
electrodes $U$ and $D$ and open with flow of Hall-like current. There is no
Hall voltage [39] under open boundary conditions. However, the Hall-like
currents perpendicular to the direction of the bias exist under closed
boundary conditions. These conditions are necessary but not sufficient for the
observation of Hall-like currents. Breaking of single or two mirror symmetries
is needed. The system with broken symmetries shows rectification properties.
The rectification efficiency may attain its maximal possible value equal
unity. Tuning the system to such hot spots allows perfect rectification of
charge or heat. We hope this property can be verified experimentally as the
fabrication of the proposed devices is possible by the present-day technology.

The organisation of the paper is as follows. In Section (II) we describe the
studied device and its modelling. Two possible boundary conditions are
discussed in (II.1) and (II.2). The results of numerical calculations
presented in Section (III) are followed by the concluding Section (IV). Some
detailed calculations of currents flowing in the system and the on-dot Green
function are relegated to the Appendices (A) and (B).

Figure 1: (color online) The four terminal quantum dot geometry used in this
work. The couplings $\Gamma^{\lambda}$ between the central dot and the
terminals ($\lambda=L,R,U,D$) take on arbitrary values. The external leads can
differ by the chemical potential $\mu_{\lambda}$ and/or temperature
$T_{\lambda}$.

##  II The geometry, currents and boundary conditions

The time reversal symmetric nano-structure consisting of a quantum dot tunnel-
coupled to four normal leads is illustrated in Fig. (1). The quantum dot is
understood here as the small grain with quantized spectrum, which can be
modelled by a single level. Electrodes or terminals are macroscopic objects
characterised by temperature $T\lambda=T+\Delta T_{\lambda}$. They may be
electrically biased ($eV_{\lambda}$) with chemical potential
$\mu_{\lambda}=\mu+eV_{\lambda}$, where $\mu$ is an equilibrium value of the
chemical potential and $T$ the equilibrium temperature common to all
electrodes. We assume $\mu=0$.

The Hamiltonian describing the system under consideration is a standard
single-impurity Anderson model

$\displaystyle{H}$ $\displaystyle=$
$\displaystyle\sum_{\lambda{k}\sigma}\varepsilon_{\lambda{k}}n_{\lambda{k}\sigma}+\sum_{\sigma}\varepsilon_{d\sigma}n_{\sigma}+U_{d}n_{\uparrow}n_{\downarrow}$
(1) $\displaystyle+$
$\displaystyle\sum_{\lambda{k}\sigma}\left({V}_{\lambda{k}\sigma}c^{\dagger}_{\lambda
k\sigma}d_{\sigma}+{V}_{\lambda{k}\sigma}^{*}d^{\dagger}_{\sigma}c_{\lambda{k}\sigma}\right),$

where
$n_{\lambda{k}\sigma}=c^{\dagger}_{\lambda{k}\sigma}c_{\lambda{k}\sigma}$ and
$n_{\sigma}=d^{\dagger}_{\sigma}d_{\sigma}$ denote particle number operators
for the leads and the dot, respectively. The operators $c^{\dagger}_{\lambda
k\sigma}(d^{\dagger}_{\sigma})$ create electrons in respective states
$\lambda{k}\sigma$ $(\sigma)$ in the lead $\lambda$ (on the dot). The energies
of the leads are measured from their chemical potentials $\mu_{\lambda}$,
$\varepsilon_{\lambda k}=\varepsilon_{0\lambda k}-\mu_{\lambda}$, with the
dependence of $\varepsilon_{0\lambda k}$ on $\lambda$ allowing for a different
spectrum in each of the leads. $U_{d}$ denotes the energy cost of placing two
electrons on a quantum dot. In this work we neglect spin dependence of the on
dot energy $\varepsilon_{d\sigma}=\varepsilon_{d}$ and electron hoping
amplitudes $V_{\alpha k\sigma}=V_{\alpha k}$. These approximations are relaxed
in hybrid system with normal and ferromagnetic electrodes or in the presence
of magnetic field. To study the transport properties we employ the non-
equilibrium Greens function technique (for details see Appendix (A) and (B)).

###  II.1 Absence of Hall voltage in systems with open boundary conditions

We start by assuming the bias $B$ (this denotes either voltage $V$ or
temperature $\Delta T$) between $L$ and $R$ electrodes. For open boundary
conditions chemical potentials $\mu_{U}$ and $\mu_{D}$ are obtained by
requiring that the currents in those electrodes vanish: $I_{U}=0$ and
$I_{D}=0$. Using equations (7) and (11) to calculate the currents and
neglecting spin dependence it is easy to show that under arbitrary voltage
bias $V_{LR}$ the vanishing of charge currents in the $U$ and $D$ electrodes
requires

$F_{D}-F_{U}=0=\int\frac{dE}{2\pi}(f_{D}(E)-f_{U}(E))N(E).$ (2)

For equal temperatures ($T_{D}=T_{U}$) the only solution is $\mu_{D}=\mu_{U}$
and no voltage appears along ($UD$)-direction, regardless the parameters of
the model. This is valid for arbitrary voltages and temperatures, $i.e.$ in
linear and non-linear regimes. Thus under open boundary conditions there is no
’Hall voltage’ in the system and thus no true Hall effect. This, however, does
not preclude flow of the Hall currents.

###  II.2 The Hall-like current in the four terminal QD with closed boundary
conditions

The situation is completely different if one applies closed boundary
conditions and allows for a current flow along y-direction. We bias a system
along x-direction with $V=V_{R}-V_{L}$ or $\Delta T=T_{R}-T_{L}$ and calculate
the current flow along the $UD$-direction. It has to be noted that voltage
bias induces both, heat $J^{Q}$ and charge $I$ currents along $LR$ and $UD$
and the same is true for temperature bias. The currents are calculated as
$I^{LR}=(I^{L}-I^{R})/2$ and $I^{UD}=(I^{U}-I^{D})/2$ for charge and
$J^{LR}_{Q}=(J^{L}_{Q}-J^{R}_{Q})/2$, $J^{UD}_{Q}=(J^{U}_{Q}-J^{D}_{Q})/2$ for
a heat flow.

The necessary condition for the existence of currents along unbiased $UD$
direction is broken mirror symmetry between U and D electrodes. This is
realised by assuming different couplings to the dot. For symmetric system with
$\Gamma^{U}=\Gamma^{D}$ the Hall - like currents vanish, $I^{UD}=0$
($J^{UD}_{Q}=0$). This is easily seen assuming isothermal conditions
($T_{\lambda}=T$ for all $\lambda$) and linear response regime (small voltage
$V=V_{L}-V_{R}$). For symmetric distribution of voltages $V_{L/R}=\pm V/2$ at
the corresponding terminals, one expands the Fermi functions

$f_{\lambda}(E)=f_{0}(E)+f_{0}^{\prime}(E)(-eV_{\lambda}),$ (3)

and using Eq. (7) gets the Hall-like charge current

$I^{UD}=\frac{4e^{2}}{h}\frac{(\Gamma^{U}-\Gamma^{D})[\Gamma^{L}-\Gamma^{R}]}{\sum_{\lambda}\Gamma^{\lambda}}F_{0}^{\prime}V.$
(4)

where $F_{0}^{\prime}=\int\frac{dE}{2\pi}(-\frac{\partial f_{0}(E)}{\partial
E})N(E)$. Both currents vanish for $\Gamma^{L}=\Gamma^{R}$ and/or
$\Gamma^{U}=\Gamma^{D}$. This condition is relaxed in the non-linear regime
and breaking the mirror symmetry along $UD$ is enough to get the Hall-like
currents.

##  III The results

The asymmetry of couplings plays an important role and in most cases we shall
characterise it by a single parameter $\alpha$, which defines anisotropy of
our system. It may take arbitrary positive value but we shall study a few
representative values only. In most studied cases we assume simple asymmetry
with $\Gamma^{R}=\Gamma^{D}=\alpha\Gamma_{0}$ and
$\Gamma^{L}=\Gamma^{U}=\Gamma_{0}$, where $\Gamma_{0}$ is our energy unit. We
also assume Boltzmann constant, Planck constant and the electron charge as
$k_{B}=\hbar=e=1$. Thus energy E and other parameters like $T$, $V$
and$\varepsilon_{d}$ are all measured in units of $\Gamma_{0}$.

Figure 2: (color online) Charge current $I^{UD}$ induced by voltage between
$LR$ terminals is a symmetric function of $V$ as visible in panel (a). The
same is true for heat current which is also symmetric function of voltage
(panel b). Insets show the corresponding currents along $LR$ direction. These
currents are linear functions at small voltages with non-linear behaviour
observed at high bias. Other parameters $T=1$, $\varepsilon_{d}=-4$ and
$U=12$.

###  III.1 The currents

Contrary to the linear regime in which existence of Hall-like currents require
breaking of two mirror symmetries, in the non-linear regime both charge
$I^{ij}$ and heat $J^{ij}_{Q}$ currents flow between $ij=LR$ as well as
$ij=UD$ electrodes if only a single mirror symmetry is broken along $UD$
direction. This is illustrated in figure (2) for voltage bias and three values
$\Gamma^{D}=0.5,1.5,2$ with all other couplings equal to $\Gamma_{0}=1$. In
accord with Eq. (4) for $\Gamma^{L}=\Gamma^{R}$ the linear contributions
vanish and the Hall currents are (at least) quadratic functions of voltage for
small $V$ . Simultaneously the longitudinal currents (both $I^{LR}$ and
$J^{LR}_{Q}$ ) are linear functions of $V$ for $V\rightarrow 0$ with well
visible departures from linearity at larger voltages (see insets in Fig. (2) a
and b). This agrees with general analytical results of Section (II.2).
Thermally induced currents (not shown) exhibit the same behaviour, namely the
currents perpendicular to the bias are quadratic functions of $\Delta T$ for
small $\Delta T$. These currents do not appear in the linear order if the
system breaks the single mirror symmetry only. In full analogy to the voltage
bias the longitudinal thermally induced currents are linear functions of
$\Delta T$ for $\Delta T\rightarrow 0$, with departures from linearity at
elevated $\Delta T$ values.

For the particular set of parameters the $UD$ heat current is positive for
$\Gamma^{D}>1$ and negative for $\Gamma^{D}<1$. This is true independently of
the bias as visible from panel (b) in Fig. (2). Similar symmetry is valid for
heat currents along $LR$, which for $\Gamma^{D}>1$ are of opposite sign to the
currents corresponding to $\Gamma^{D}<1$. The flow towards $U$ is either
blocked or facilitated.

The response of the system to applied bias quite generally depends on its
symmetry and the set of parameters. To illustrate this in Fig. (3) we show
thermally induced charge currents in panel (a) and heat currents induced by
the voltage in panel (b). Panel (a) shows the currents in system at
temperature $T=2$ with the quantum dot tuned to $\varepsilon_{d}=-9$, and for
two values of anisotropy $\alpha=0.5,1.5$ and $U=12$. The currents flowing
along the $UD$ are about an order of magnitude smaller then those along $LR$
direction. However, one notices that the charge Hall - like current for
$\alpha=0.5$ vanishes for $\Delta T/2\approx 0.4$ but is finite for $\Delta
T/2\approx-0.4$. This is a bias for which the rectification factor is maximal
(=1) as we shall see in the following section. In panel (b) we show Hall -
like heat currents vs. the voltage. Important fact to note is the non-
monotonous dependence of currents on the bias and the points were they vanish.
These special points are marked by asterisks.

Figure 3: (color online) In panel (a) we show thermally induced longitudinal
charge currents $I^{LR}$ (solid lines) and Hall-like currents $I^{UD}$ (lines
with symbols) for two values of asymmetry $\alpha=0.5$ and $\alpha=1.5$ as
functions of the temperature difference. In the panel (b) the voltage
dependent Hall - like heat currents $J^{UD}_{Q}$ are shown together with
points marked with * where the currents vanish for one sign of voltages and
take on non-zero values for opposite sign. In both panels $U=12$.

###  III.2 Rectification

If mirror symmetries along $LR$ and $UD$ directions are broken both charge and
heat currents in two pairs of electrodes depend on the sign of bias $B=V$ or
$B=\Delta T$, i.e. we find $I^{LR}(B)\neq I^{LR}(-B)$ and $I^{UD}(B)\neq
I^{UD}(-B)$. The same is true for heat currents $J^{LR}_{Q}(B)$ and
$J^{UD}_{Q}(B)$. To quantify the rectification efficiency one introduces
special parameter called rectification coefficient. One possible definition is

$R^{ij}=\frac{|(|I^{ij}(B)|-|I^{ij}(-B)|)|}{[|I^{ij}(B)|+|I^{ij}(-B)|]}$ (5)

With this definition the rectification factor ranges from 0 to 1. The former
means no rectification, while the latter perfect rectification. Similar
rectification coefficients are defined for heat currents $J^{Q}(B)$ and we
denote them by $R^{ij}_{Q}$ in the following.

Perfect rectification is expected in cases when the current vanishes for one
sign of the bias and attains finite value for the other. The goal is to tune
the parameters of the system to such ’hot spots’. In the panel (b) of Fig. (3)
such hot spots have been marked by asterisks. The rectification coefficient
takes on the maximal possible value at those points. In general it is
difficult to tune the system and find vanishing of charge currents. However,
panel (a) of this figure shows points in which Hall-like charge currents
vanish for specific values of $\Delta T$. For $\alpha=0.5$ the $I^{UD}$
current vanishes for $\Delta T/2$ close to 0.4, while for $\alpha=1.5$ the
Hall - like charge current nearly vanishes for $\Delta T/2$ close to 2. In
both cases the currents are relatively large for opposite sign of temperature
bias.

The rectification coefficients for thermally induced charge and heat currents
flowing along $LR$ and $UD$ directions are plotted in Fig. (4). Panel (a)
shows $R^{ij}$ for charge currents, while panel (b) $R^{ij}_{Q}$ for heat
currents. The rectification of both currents flowing along the bias direction
($I^{LR}$ and $J^{LR}_{Q}$) is relatively small of order of a few percent. For
the parameters of the model presented in the figure, the coefficient $R^{UD}$
is of the same order or slightly higher then $R^{LR}$. Its maximal value is
about 10%. However, the heat rectification coefficient
$R^{UD}_{Q}>>R^{LR}_{Q}$ and reaches values higher then 50%. It monotonously
increases with $\Delta T$.

Temperature plays an important role in our interacting system, as the Kondo
effect sets in at low temperature. To see this we show in Fig. (5) results
obtained for the same system as in Fig. (4) for two vastly different
temperatures. We plot rectification factors $R^{ij}$ (main panel) and the
charge currents $I^{ij}$ (inset) vs. voltage $V$ for two values of
temperature. For lower temperature $T=0.01$ the Kondo resonances are expected
to appear in the density of states. The signature of the Kondo effect is
visible in the inset as a strongly increased conductance (the curves with
symbols correspond to low temperature) at zero voltage. Strong non-linearities
at elevated voltages combined with well visible asymmetry of $I^{UD}$ current
(at low $T=0.01$) for voltage around $|V^{*}|\approx 1.2$ result in
$|I^{UD}(V^{*})|=|I^{UD}(-V^{*})|$ and vanishing of $R^{UD}(V^{*})$ as well as
its non-monotonous dependence on voltage. At high temperature $T=1$ there is
no Kondo effect and one obtains continuous increase of rectification with $V$.
The rectification of longitudinal current $R^{LR}$ is affected by the Kondo
effect only quantitatively. For the particular set of parameters its low
temperature value is roughly doubled with respect to higher temperature but
remains low, at the level of a few percent.

The general observation is that typically the rectification factors for
longitudinal currents are small of the order of a few percent. This agrees
with previous systematic calculations of the rectification factor in [23] in a
two terminal single level quantum dot. The rectification factors were small of
order of a few percent like those for longitudinal currents in the present
work. In our four terminal geometry the rectification factor for the Hall-like
currents is usually much higher and may be as large 100%.

In view of the observation [23] that the rectification coefficient for a two
level quantum dot attains large value in the two terminal system it would be
of interest to extend our four terminal model to two level quantum dot and to
study how the longitudinal and Hall - like currents and their rectification
factors are affected. This will be studied in a future work.

Figure 4: (color online) Rectification ratios $R^{ij}$ of charge (a) and heat
$R^{ij}_{Q}$ (b) currents as a function of $\Delta T/2$ for a few values of
$\alpha$, voltage $V=0$, temperature $T=1$, $\varepsilon_{d}=-4$ and $U=12$.
Figure 5: (color online) Rectification ratios $R^{ij}$ (main panel) and
currents $I^{ij}$ (inset) as a function of voltage for two temperatures $T=1$
(solid lines) and $T=0.01$ (lines with symbols). Other parameters
$\varepsilon_{d}=-4$ and $U=12$.

###  III.3 Four terminal resistance in the non-linear regime

For many terminal nano-junctions with currents $I^{ij}$ and voltages $V^{kl}$
measured between various pairs of electrodes one defines [33] the four
terminal resistances $R_{ij,kl}=\frac{V^{kl}}{I^{ij}}$. The voltages are
applied between terminals $k$ and $l$ and currents measured between terminals
$i$ and $j$. Here we shall calculate resistances for a special couplings which
break a mirror symmetry along $y$ direction only. If we assume couplings to
$L,R$ and $U$ terminals equal to $\Gamma_{0}$ and only the coupling to
$\Gamma^{D}=(0.5,1.5,2)\Gamma_{0}$ different from others, the resulting
symmetry of the model resembles that of the nano-junction earlier studied
experimentally [38]. These authors considered a four terminal structure with
asymmetric triangular prism-like scatterer placed in its centre. The scatterer
effectively blocked the flow of charge from/to one of the terminals. In our
case it is the coupling $\Gamma^{D}$ which effectively blocks (if $<1$) the
current from/to this terminal.

It should be recalled that in the linear transport regime the resistances
$R_{ij,kl}$ do not depend on the current. In the non-linear regime the Hall-
like current $I^{UD}$ depends on the longitudinal current $I^{LR}$ and so does
the resistance $R_{UD,LR}=\frac{V^{LR}}{I^{UD}}=R_{UD,LR}(I^{LR})$. Due to the
vanishing of the Hall-like currents at some voltages in our set-up, the four
terminal resistance $R_{UD,LR}$ is not well defined at those points. That is
why we show in figure (6) the inverse resistance
$R^{-1}_{UD,LR}=\frac{I^{UD}}{V^{LR}}$ as a function of the current
$I=I^{LR}$. It is seen that the resistance obeys a symmetry
$R_{UD,LR}(I)=-R_{UD,LR}(-I)$. Interestingly, similar symmetry of the four
terminal resistance has been earlier observed [38] in the non-linear ballistic
transport in the (already mentioned) four terminal micro-junction with
triangular asymmetric (prism-like) scatterer. The absence of voltage $V^{UD}$
between $U$ and $D$ electrodes in our model and its presence in experimental
setup is a main reason of the perfect symmetry $R_{UD,LR}(I)=-R_{UD,LR}(-I)$
in our case and only an approximate one in the nano-junction [38].

Inverse resistances $R^{-1}_{UD,LR}$ calculated in the Kondo regime
characterised by the parameters $\varepsilon_{d}=-4$, $U=12$ and $T=0.01$
display the same perfect antisymmetric dependence (not shown) on the
longitudinal current $I^{LR}$. Needless to say that this is true for the
mirror symmetry broken along $UD$ direction in a similar way as in experiment
[38].

Figure 6: (color online) Broken mirror symmetry along vertical direction only
$\Gamma^{R}=\Gamma^{L}=\Gamma^{U}=1$ and $\Gamma^{D}=0.5,1.5,2$ results in the
four terminal resistivity antisymmetric with respect to the current flowing
along horizontal direction. This resembles experimental result [38] found in
the non-linear transport regime of nano-junction with artificial asymmetric
scatterer.

###  III.4 Gate voltage dependence and the effect of interactions

In the non-linear regime the density of states (DOS) defined in (11) and
entering formula (9) is known to depend crucially on the interactions $U$
between carriers, voltages $V_{\lambda}$ and temperatures $T_{\lambda}$ of the
leads. We limit the studies of this section to isothermal condition when all
leads have the same temperature $T_{\lambda}=T$. If temperature is low enough
and the on-dot level $\varepsilon_{d}$ is slightly below the chemical
potential(s) the Kondo peak(s) appears in the density of states of the
interacting quantum dot. With chemical potentials $\mu=0$ at electrodes $U$
and $D$, $\mu_{L/R}=\mu\pm eV/2$ at the left/right electrode one expects three
Kondo peaks pinned to the chemical potentials of the electrodes. They are
visible in Fig. (9) shown in Appendix (B). Various curves in the figure
correspond to different values of $\Gamma^{R}=\Gamma^{D}=\alpha\Gamma_{0}$
couplings with $\Gamma^{L}=\Gamma^{U}=\Gamma_{0}$. For $\varepsilon_{d}$
outside the Kondo regime DOS around $E=0$ changes smoothly with $\alpha$ and
voltage $V$ (not shown).

In figure (7) we show the currents $I^{LR}$ (main panel) and $I^{UD}$ (inset)
as function of dot energy $\varepsilon_{d}$ at temperatures below ($T=0.01$)
and above ($T=0.5$) the Kondo temperature. The dot energy can be easily
changed by gate voltage. The system breaks mirror symmetry with respect to
both $LR$ and $UD$ directions as we have assumed
$\Gamma^{L}=\Gamma^{U}=\Gamma_{0}$ and $\Gamma^{R}=\Gamma^{D}=1.5\Gamma_{0}$.
At lower temperature one observes signatures of the Kondo effect. These are
peaks for those gate voltages for which one expects the Kondo resonances in
the density of states (c.f. Fig. (9)). The effect is rather small due to the
fact that we integrate the density of states over a range ($-V/2,V/2$) around
$E=0$. In this energy window there are three Kondo peaks including one well
pronounced and voltage independent at $E=0$. It is bounded with the chemical
potential $\mu=0$ of both $U$ and $D$ electrodes. Interestingly, the effect of
Kondo correlations is more pronounced in the Hall current $I^{UD}$ as compared
to longitudinal one $I^{LR}$.

In the upper panel of colour coded Fig. (8) we show the dependence of
longitudinal $I^{LR}$ charge current on $\delta=\varepsilon_{d}+U/2$ and
voltage $V$. Lower panel shows similar dependence for the Hall - like charge
current $I^{UD}$. Both currents are calculated for temperature in the Kondo
regime ($T=0.01$) and for $U=12$. Note, the same scale for both panels,
however, with Hall-like current multiplied by the factor of 10.

Figure 7: (color online) The dependence of the longitudinal $I^{LR}$ (main
panel) and transverse $I^{UD}$ (inset) currents on the detuning
$\delta=\varepsilon_{d}+U/2$ (can be changed by the gate voltage) for system
with $\Gamma^{L}=\Gamma^{U}=\Gamma_{0}$ and
$\Gamma^{R}=\Gamma^{D}=1.5\Gamma_{0}$ (i.e. $\alpha=1.5$), $U=16$, voltage
bias $V=4$ and for two temperatures $T=0.01,0.5$. Signatures of the Kondo
effect visible as non smooth dependence of the currents on $\delta$ are
observed for $T=0.01$.

Figure 8: (color online)The maps show the charge currents on the plane
($\delta,V$ ). Upper panel illustrates longitudinal current
$I^{LR}(\varepsilon_{d},V)$, while lower panel the Hall - like current
$I^{UD}(\delta,V)$. We assumed $U=12$ and temperature $T=0,0.01$.

##  IV Summary and conclusion

We have studied the non-linear transport characteristics of a system
consisting of strongly interacting quantum dot coupled to two pairs of normal
leads arranged in a cross geometry. The voltage or thermal bias applied to one
pair of the leads induces the (heat and charge) current flow between both
pairs of terminals, provided the quantum dot is non-symmetrically coupled.
Different couplings between the leads and the dot break mirror symmetries of
the device. In the linear regime observation of two mutually perpendicular
(charge and heat) currents in response to the appropriate bias requires
breaking of both symmetries. Beyond linear regime, breaking the mirror
symmetry along $UD$ is enough. All currents are rectified in a system breaking
both mirror symmetries, with the rectification factors of Hall - like currents
typically much bigger then that of longitudinal currents. Interestingly the
rectification factor for heat typically exceed that for charge.

If only the symmetry between Hall leads (perpendicular to those biased) is
broken the resulting non-linear four terminal resistance
$R_{ij,kl}=\frac{V^{kl}}{I^{ij}}$ is an antisymmetric function of the
longitudinal current $I=I^{LR}$, i.e. $R_{UD,LR}(I)=-R_{UD,LR}(-I)$. This is
in qualitative agreement with experimental data [38] on non-linear transport
in four terminal micro-junction with asymmetric prism - like scatterer. The
notable difference is that in our system the anti-symmetry is exact while in
experiment it is approximate. It is important to recall that in the linear
regime $R_{ij,kl}$ does not depend on the current $I=I^{ij}$.

We want also to underline that the analysed device with a cross geometry
enables the study of the Kondo effect on both longitudinal and Hall - like
currents. It turns out that the Kondo resonance appearing in the density of
states at low temperature affects the longitudinal currents in the four
terminal system to lesser extend then in the corresponding two terminal
quantum dot geometry. Additionally in our geometry the signatures of the Kondo
effect in the Hall -like current are more pronounced then in the longitudinal
one.

Our results demonstrate a new route to achieve the efficient rectification of
longitudinal and Hall - like (heat and charge) currents. The proposed system
is attainable within current technology and provides a novel platform of
simultaneous heat and charge management at the nanoscale.

###### Acknowledgements.

The work reported here has been supported by the M. Curie-Skłodowska
University and the National Science Center, Poland (“Weave” programme) through
grant no. 2022/04/Y/ST3/00061.

##  Appendix A The current in the system

Calculating the currents we follow the general definitions relying on equation
of motion of the number and energy operators [40, 41]. The required Green
functions are obtained in a standard way [42, 43, 44]. Additionally we are
assuming the wide band limit when the couplings
$\Gamma^{\lambda}_{\sigma}(E)=2\pi\sum_{k}|V_{\lambda
k\sigma}|^{2}\delta(E-\varepsilon_{\lambda k})=\Gamma^{\lambda}_{\sigma}$ do
not depend on energy. However, we keep here the spin dependence of adequate
parameters. In this limit one finds the exact relation (for details see [44,
45])

$\displaystyle\langle d^{\dagger}_{\sigma}d_{\sigma}\rangle$ $\displaystyle=$
$\displaystyle-i\int\frac{dE}{2\pi}G_{\sigma}^{<}(E)$ $\displaystyle=$
$\displaystyle
i\int\frac{dE}{2\pi}\frac{\sum_{\lambda}\Gamma_{\sigma}^{\lambda}f_{\lambda}(E)}{\sum_{\lambda}\Gamma_{\sigma}^{\lambda}}[G^{r}_{\sigma}(E)-G^{a}_{\sigma}(E)],$

which allows to write the charge and heat currents flowing out of the
$\lambda$ electrode as

$\displaystyle I_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{2e}{\hbar}\int\frac{dE}{2\pi}\sum_{\sigma}\Gamma_{\sigma}^{\lambda}$
(7) $\displaystyle\times$
$\displaystyle\frac{\sum_{\lambda^{\prime}}\Gamma^{\lambda^{\prime}}_{\sigma}(f_{\lambda^{\prime}}(E)-f_{\lambda}(E))}{\sum_{\lambda^{\prime}}\Gamma^{\lambda^{\prime}}_{\sigma}}\mathrm{Im}G_{\sigma}^{r}(E),$
$\displaystyle J_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{2e}{\hbar}\int\frac{dE}{2\pi}\sum_{\sigma}\Gamma_{\sigma}^{\lambda}(E)(E-\mu_{\lambda})$
(8) $\displaystyle\times$
$\displaystyle\frac{\sum_{\lambda^{\prime}}\Gamma^{\lambda^{\prime}}_{\sigma}(f_{\lambda^{\prime}}(E)-f_{\lambda}(E))}{\sum_{\lambda^{\prime}}\Gamma^{\lambda^{\prime}}_{\sigma}}\mathrm{Im}G_{\sigma}^{r}(E).$

These expressions can be used for calculating the currents in an arbitrary
system consisting of the central dot and several terminals under appropriate
boundary conditions. Note, the Fermi-Dirac distribution function
$f_{\lambda}(E))$ depends on the electrode $\lambda$ $via$ its chemical
potential $\mu_{\lambda}=\mu+eV_{\lambda}$ or applied voltage $V_{\lambda}$
and temperature $T_{\lambda}$. The common value of the chemical potential and
temperature of the system in equilibrium is denoted $\mu,T$. It is convenient
to express the currents in terms of auxiliary function

$F_{\lambda\sigma}=\int\frac{dE}{2\pi}f_{\lambda}(E)N_{\sigma}(E),$ (9)
$F_{\lambda\sigma}^{Q}=\int\frac{dE}{2\pi}(E-\mu_{\lambda})f_{\lambda}(E)N_{\sigma}(E),$
(10)

respectively. The auxiliary function

$N_{\sigma}(E)=-\frac{1}{\pi}ImG^{r}(E,\\{V_{\lambda}\\})$ (11)

denotes density of states (DOS) on the dot for spin $\sigma$ electrons. In
general this quantity depends on the voltages $\\{V_{\lambda}\\}$ and
temperatures $T_{\lambda}$ of all terminals.

Figure 9: (color online)The evolution of the on-dot density of states plotted
as a function of energy with changing the parameter $\alpha$ describing the
couplings between the dot and right ($R$) and down ($D$) terminals with
$\Gamma^{R}=\Gamma^{D}=\alpha\Gamma_{0}$, $\Gamma^{L}=\Gamma^{U}=\Gamma_{0}$.
The parameters take on the following values: the on-dot energy
$\varepsilon_{d}=-5.0$, the source-drain voltage $V=2$, temperature $T=0.01$
and the interaction $U=12$. Note that the region of width $V$ around the
chemical potential $\mu=0$ is important as it contributes to the currents in
particular terminals.

##  Appendix B The Green function

For completeness we recall the formulae for the on-dot Green function which
qualitatively correctly describe the Kondo effect. One uses standard equation
of motion technique[45, 43, 44] and finds the Green function

$\langle\langle
d_{\sigma}|d^{\dagger}_{\sigma}\rangle\rangle_{E}=\frac{1+I_{d}(E)[\langle
n_{\bar{\sigma}}\rangle+b_{1\bar{\sigma}}-b_{2\bar{\sigma}}]}{E-\varepsilon_{\sigma}-\Sigma_{0\sigma}+\Sigma_{t}(E)},$
(12)

where

$\Sigma_{t}(E)=I_{d}(E)[\Sigma_{1}^{T}+\Sigma_{2}^{T}-(b_{1\bar{\sigma}}-b_{2\bar{\sigma}})\Sigma_{0\sigma}],$
(13)

$I_{d}(E)=\frac{U}{E-\varepsilon_{\sigma}-U-\Sigma_{0\sigma}-\Sigma_{\sigma}^{(1)}-\Sigma_{\sigma}^{(2)}}.$
(14)

The various pieces of the self-energy are supplemented by the inverse life-
times $i\gamma_{\bar{\sigma}_{1}/2}$ of the single particle $\bar{\sigma}$,
respectively two-particle $2$ state and read

$b_{1\bar{\sigma}}(E)=\int\frac{d\varepsilon}{2\pi}\frac{\sum_{\lambda}\Gamma^{\lambda}_{\bar{\sigma}}f_{\lambda}(\varepsilon)\langle\langle
d_{\bar{\sigma}}|d^{\dagger}_{\bar{\sigma}}\rangle\rangle^{a}_{\varepsilon}}{E-\varepsilon-\varepsilon_{1}+i\tilde{\gamma}^{\bar{\sigma}}_{1}},$
(15)
$b_{2\bar{\sigma}}(E)=\int\frac{d\varepsilon}{2\pi}\frac{\sum_{\lambda}\Gamma^{\lambda}_{\bar{\sigma}}f_{\lambda}(\varepsilon)\langle\langle
d_{\bar{\sigma}}|d^{\dagger}_{\bar{\sigma}}\rangle\rangle^{a}_{\varepsilon}}{E+\varepsilon-\varepsilon_{2}+i\tilde{\gamma}_{2}},$
(16)
$\Sigma^{T}_{1\sigma}(E)=\int\frac{d\varepsilon}{2\pi}\frac{\sum_{\lambda}\Gamma^{\lambda}_{\bar{\sigma}}f_{\lambda}(\varepsilon)[1+\frac{i}{2}\Gamma_{\bar{\sigma}}\langle\langle
d_{\bar{\sigma}}|d^{\dagger}_{\bar{\sigma}}\rangle\rangle^{a}_{\varepsilon}]}{E-\varepsilon-\varepsilon_{1}+i\tilde{\gamma}^{\bar{\sigma}}_{1}},$
(17)
$\Sigma^{T}_{2\bar{\sigma}}(E)=\int\frac{d\varepsilon}{2\pi}\frac{\sum_{\lambda}\Gamma^{\lambda}_{\bar{\sigma}}f_{\lambda}(\varepsilon)[1-\frac{i}{2}\Gamma_{\bar{\sigma}}\langle\langle
d_{\bar{\sigma}}|d^{\dagger}_{\bar{\sigma}}\rangle\rangle^{r}_{\varepsilon}]}{E+\varepsilon-\varepsilon_{2}+i\tilde{\gamma}_{2}}.$
(18)

In the above we have introduced
$\varepsilon_{1}=\tilde{\varepsilon}_{\sigma}-\tilde{\varepsilon}_{\bar{\sigma}}$,
and
$\varepsilon_{2}=\tilde{\varepsilon}_{\sigma}+\tilde{\varepsilon}_{\bar{\sigma}}+U$.
The subscripts 1 and 2 refer to the excited 1- and 2-electron states of the
dot, respectively. The symbols $a/r$ denote advanced/retarded Green function.
The self-consistency requires that input dot occupation $\langle
n_{\bar{\sigma}}\rangle$ equals that obtained from $G_{\bar{\sigma}}^{r}(E)$
in the consecutive iteration step with a given accuracy. As noted earlier,
there exists exact relation

$\langle n_{\sigma}\rangle=\int
dE\frac{\sum_{\lambda}\Gamma_{\sigma}^{\lambda}f_{\lambda}(E)}{\sum_{\lambda}\Gamma_{\sigma}^{\lambda}}(-\frac{1}{\pi})ImG_{\sigma}^{r}(E),$
(19)

valid for energy independent couplings;
$\Gamma_{\sigma}^{\lambda}(E)\equiv\Gamma^{\lambda}_{\sigma}$. If this
condition is violated, as it might be the case in graphene [46, 47, 48],
hybrid systems with one (or both) of the electrodes being a superconductor,
$e.g.$, d-wave [49] one, other approaches are needed.

The inverse lifetimes $\tilde{\gamma}_{\alpha}$ of the excited states
$\alpha=|\sigma\rangle,|\uparrow,\downarrow\rangle$ stem from higher order
processes [41, 45]. They can be calculated up to the desired order $via$ the
generalized Fermi rule as

$\tilde{\gamma}_{\alpha}=2\pi\sum_{|f\rangle}|\langle
T(E_{\alpha})\rangle|^{2}\delta(E_{\alpha}-E_{f}),$ (20)

with $T(E)=\hat{V}+\hat{V}g(E)\hat{V}+\cdots$ the scattering matrix, where
$\hat{V}$ denotes the part of the Hamiltonian describing the coupling between
quantum dot and reservoirs. In the discussed approach, one also replaces
$\varepsilon_{d}$ by $\tilde{\varepsilon}_{d}$, to be calculated self-
consistently from

$\tilde{\varepsilon}_{d}=\varepsilon_{d}+\Sigma_{1}^{T}(\tilde{\varepsilon}_{d})+\Sigma_{2}^{T}(\tilde{\varepsilon}_{d}).$
(21)

Finally, the self-energies $\Sigma_{\sigma}^{(1,2)}$ are equal to
$\Sigma_{0\sigma}$ for $i\tilde{\gamma}^{\alpha}_{1,2}=i0^{+}$; however, for
arbitrary values of $i\tilde{\gamma}^{\alpha}_{1,2}$ they have to be
calculated directly from

$\Sigma^{(1,2)}_{\sigma}(E)=\sum_{\lambda}\Gamma^{\lambda}_{\bar{\sigma}}\int\frac{d\varepsilon}{2\pi}\frac{1}{E\mp\varepsilon-\varepsilon_{1,2}+i\tilde{\gamma}^{\bar{\sigma}}_{1,2}}.$
(22)

The on-dot density of states is obtained from the retarded Green function. It
is defined by Eq. (11) and shown in figure (9) as a function of energy for a
number of values of the asymmetry factor $\alpha$ (c.f its definition in
Section (III) affecting couplings between the dot and leads. One observes
three Abrikosov-Suhl resonances pinned to Fermi levels of the left $\mu_{L}$,
right $\mu_{R}$ and up and down $\mu_{U}=\mu_{D}=\mu=0$ electrodes. Terminal
$R$ with $\mu_{R}=-1.0$ is decoupled from the dot for $\alpha=0$ so the
corresponding density of states shows two Kondo resonances: one at $\mu_{U}=0$
and other at $\mu_{L}=+1.0$. With increasing $\alpha$ the third Kondo
resonance appears around energy $E=-1=\mu_{R}$. It has to be noted that the
Kondo peak at zero energy and two such structures at $V_{L/R}$ make the
dependence of currents on gate voltage more complicated in comparison to the
two terminal case as the central resonance is always present and only its
weight changes with parameters.

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