# Charge and Spin Thermoelectric Transport in Benzene-Based Molecular Nano-
Junctions: A Quantum Many-Body Study

Parbati Senapati  Department of Physics, Indian Institute of Technology Patna,
Bihta, Bihar, 801106, India  Prakash Parida<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Patna, Bihta, Bihar, 801106, India

###### Abstract

Within the Coulomb blockade regime, our study delves into the charge, spin,
and thermoelectric transport characteristics in a benzene-based molecular
nano-junction using Pauli master equation and linear response theory. The
charge- and spin-transport studies show strong negative differential
conductance features in the current-voltage ($I-V$) characteristics for the
ortho and meta connections of electrodes on either side. Contrarily, the para-
connection displays Coulomb staircase behavior. Exploring spin current
behavior in the presence of spin-polarized electrodes or external Zeeman
field, we establish a methodology that facilitates precise control over the
specific spin flow. Various charge and spin thermoelectric transport
coefficients have been studied with varying chemical potentials. We focus on
spin-polarized- conductance, the Seebeck coefficient, and the figure of merit.
By adjusting electrode polarization or employing an external magnetic field,
we achieve an impressive peak value for the spin thermoelectric figure of
merit, approximately 4.10. This outcome underscores the strategic value of
harnessing both spin-polarized electrodes and external magnetic fields within
the domain of spin caloritronics.

##  I Introduction

Molecular electronics is a branch of science that studies the electronic and
thermal transport properties of circuits which use individual molecules as
basic building blocks. Due to its potential applications in nanoscale
electronic devices like transistors, rectifiers, sensors, and switches [1, 2,
3, 4], this field of study has garnered a lot of interest from both the
research community and industry level. For example, the possibility of
investigating electronic and thermal conduction at the smallest scales, where
the usual size of molecules is between 1 and 10 nm, could result in a higher
packing density of devices, which would have benefits for cost, efficiency,
and power consumption. The intermolecular interactions that could be used in
nanoscale self-assembly technology, possibly leading to low-cost
manufacturing, are another appealing feature. Additionally, chemists have
access to a wide variety of molecular structures, which allows them to
logically and deliberately alter the properties of molecules. Molecular
junctions are also ideal systems for investigating the fundamental concepts
underlying the mechanisms of charge transfer followed by heat (energy)
transfer.

In recent years, researchers have been working on developing high-performance
thermoelectric materials and equipment that can recycle waste heat into
electricity [5, 6, 7, 8, 9, 10]. The benefits of thermoelectric (TE) devices,
such as their compact size, lack of mechanical moving parts, durability, and
ability to produce electricity at low heat gradients, have sparked a lot of
interest. Many past attempts have been made to increase thermoelectric
performances through experimentation and theoretical work [11, 12, 13, 14, 15,
16, 17, 18, 19] using quantum dots and molecules in the Coulomb blockade [20,
21, 22] and Kondo regimes [23, 24, 25, 26]. The thermoelectric performance is
crucial for describing a thermoelectric device that is important to sustain
and limits TE device applicability [27, 28, 29]. The thermoelectric figure of
merit (ZT) determines the performance of a thermoelectric device,
$ZT=S^{2}G_{V}T/K_{t}$, where T is the temperature, S is the Seebeck
coefficient (thermopower), $G_{V}$ is the electrical conductance, and $K_{t}$
is the total thermal conductance given by $K_{t}=(k_{e}+k_{ph})$,
$k_{e}(k_{ph})$ is the electron (phonon) contribution to $K_{t}$. Since $ZT$
can be increased by increasing the power factor ($S^{2}G_{V}$) or lowering the
thermal conductivity, a high-performance thermoelectric material should have
both high thermopower and electrical conductivity and low thermal
conductivity. However, having a high $ZT$ in natural materials appears
challenging. Several factors impede the rise of $ZT$. First, the Wiedemann-
Franz law [30] is obeyed in typical solids, which means that an increase in
electric conductivity leads to an increase in thermal conductivity. Second,
the Mott relation [31] states that an increase in electrical conductivity will
likely decrease the Seebeck coefficient. Bulk materials in general show low
$ZT$. The advancement of nanotechnology has made it possible to enhance the
$ZT$ by reducing the dimensional complexity of bulk structures, and it
increases the power factor [32, 33, 34, 35, 36]. Engineering the bulk
structure at the nanoscale can induce different aspects regarding the
electronic density of states and the transmission coefficient, which help the
electrons to pass through a device. Moreover, The Wiedemann-Franz law and
other classical conclusions, such as the Mott relation, may not hold because
of the quantum phenomena that have evolved in nanostructure materials. The
development of the nano-thermoelectric device has sparked interest in this
area [37, 38, 39].

Furthermore, the thermoelectric properties of nanostructure materials can be
controlled by varying the gate voltage. As a result, it paves the way for
discovering valuable thermoelectric devices on a new and vast scale. In 1993,
Hicks and Dresselhaus were the first to exploit low-dimensional structure
materials to attain high $ZT$ [40]. Many theoretical models for thermoelectric
transfer via quantum point contacts, quantum dots, and other strongly
correlated nanostructures have been proposed [41, 42]. Tagani et al.
investigate the thermopower of double quantum dot systems that are weakly
coupled to metal electrodes using a density matrix approach and observe that
an increase in Coulomb repulsion enhances the $ZT$ by lowering the bipolar
effect. Additionally, they analyzed how interdot tunnelling affects the figure
of merit [43]. Wierzbicki et al. theoretically investigated the thermal
transport via a double quantum dot in the linear response domain. They found
that interference effects dramatically increased thermopower and
thermoelectric efficiency [44]. Natalya demonstrates how coulomb interactions
cause considerable augmentation of the $S$ and $ZT$ in numerous quantum dots
in a serial configuration [45].

Additionally, the exploration and control of spin-polarized current through
nanostructures has also attracted a lot of theoretical [46, 47] and
experimental [48, 49] attention in recent years due to its significance for
our understanding of basic quantum physics and its numerous applications,
including spintronics, nanoelectronics, and quantum computation [50, 51].
Researchers are investigating the prospect of using electron spin rather than
its charge as the basis of new electronic devices known as “spintronics” in an
effort to increase computational power and speed. The study of the interaction
between spin effects and heat transfer has garnered a lot of attention as
spintronics has advanced. It is demonstrated that the relative magnetic
configurations of ferromagnetic electrodes affect the thermoelectric effects.
A spin-Seebeck effect analogue to the charge-Seebeck effect was recently
proposed [52, 53]. The experimental technique used for studying spin transport
includes scanning probe spin thermoelectric microscopy (SPSTM), which operates
by utilizing an STM with a magnetic tip and detects electron spin orientations
by sensing the local density of states as the tip nears a sample surface.
Applying a thermal gradient induces a spin-dependent Seebeck effect,
generating a spin-related thermoelectric voltage. This technique maps these
signals with high spatial resolution, investigating spin transport properties
at the nanoscale[54]. An experimental measurement of the spin voltage produced
by a temperature gradient in a ferromagnetic $Ni_{81}Fe_{19}$ film has also
been performed [55]. The observed spin thermopower in this bulk sample,
however, is so negligible that it might be dominated by the associated charge
thermopower, which is several orders of magnitude more. After that, Dubi and
DiVentra [56] investigated a quantum dot in contact with two ferromagnetic
electrodes that were held at various temperatures and discovered that the spin
thermopower can be as great as the charge thermopower and even exceed it in
magnitude. The quantum dot needs to be trapped between two ferromagnetic leads
in their setup for it to be subjected to a significant amount of Zeeman
splitting, which could be limited in terms of practical applicability.

Not limited to quantum-dot nano-junctions, researchers also have
experimentally investigated thermoelectric transport properties of various
molecular bridges. Many experimental techniques have been employed to study
thermoelectric transport in such molecular junctions. In the scanning
tunneling microscope break junction (STMBJ) technique, a molecular monolayer
is anchored with a bias voltage applied. The tip approaches the substrate,
captures molecules, moves away, measures conductance changes, forming
electrode-molecule-electrode junctions and providing statistical conductance
data[9]. To measure thermopower, a temperature difference is induced between
the substrate and tip[57]. The mechanically controllable break junction (MCBJ)
method, similar to STMBJ, adjusts the gap using a piezoelectric actuator,
capturing molecules and integrating a heater for thermopower measurement[58,
59]. In thermal conductance assessment, the scanning thermal probe microscopy
(SThM) technique, part of atomic force microscopy (AFM), is widely employed,
especially for studying thermal conductance in self-assembled monolayers
(SAMs)[60, 61]. These studies are aimed at revealing the potential of
different molecular systems for applications in thermoelectric devices.
Diverse examples include organic molecules, inorganic nanowires, single-
molecule junctions, as well as other organic and inorganic structures [62, 63,
64, 65]. This systematic approach enables researchers to assess and predict
the thermoelectric performance of a range of molecular architectures,
providing valuable insights into their potential applications in energy
conversion and electronic devices.

Many theoretical techniques and formalisms have been used for understanding
the transport characteristics of single molecules connected to electrodes. The
study of transport at the nanoscale has standardised numerical approaches
based on the combination of non-equllibrium Green’s-function (NEGF) methods
with tight-binding models or density functional theory (DFT) [66, 67]. These
techniques are suitable for analyzing quantum transport in molecular bridges
that are strongly coupled to electrodes. In a recent study, Gallego et al.
investigated the enhancement of thermoelectric properties in single organic
radical molecules through a combined approach of theoretical analysis and
experimental validation. The study showcased simultaneous improvements in both
the Seebeck coefficient and electrical conductance within Blatter radicals
[68]. Several groups also have recently considered the thermoelectric
potential of conjugated monocyclic compounds like benzene for discovering
interference effects in strong coupling regimes. Soto-Gómez et al.
investigated the electrical and thermal transport properties of a single-
molecule junction of Catechol using real-space renormalized Green’s functions
in a tight-binding approximation. Their findings indicate the potential
utility of catechol as both a conductive and thermoelectric molecule, opening
avenues for innovative applications in molecular electronic devices[65]. Zhang
et al. explored the thermoelectric properties of single molecular junctions
using a two-level model based on electron–phonon interactions and NEGF
formalism. They investigated the potential for achieving high-efficiency
thermoelectric devices by engineering the energy level splitting in the
molecular junction[69].

Using Green’s function technique, Haidong et al. studied the thermoelectric
properties of a benzene molecule strongly connected to two metal leads. They
found that the magnetic flux significantly impacts the thermoelectric
properties. Additionally, they demonstrate how this theoretical benzene model
may be applied to high-efficiency thermoelectric devices and has good thermal
engine features [70]. In the linear response regime, Sartipi et al. suggested
the thermoelectric transport through a benzene molecule strongly connected
with three metallic terminals and explored possible conductance and
thermopower coefficients. Furthermore, a significant improvement in the figure
of merit is seen in the three-terminal configuration with tunable temperature
differences. The results indicate that the third terminal model can
effectively increase efficiency at maximum output power compared to the two-
terminal model [71].

However, alternative approaches are needed to describe transport through a
molecule that is weakly coupled to leads. Because of the importance of the
Coulomb interaction in these systems, it is usual to use a Pauli master
equation (PME) for the reduced density matrix in the Coulomb blockade regime.
For instance, Hettler et al. used an electronic structure calculation to
create an effective interacting Hamiltonian for the benzene orbitals. They
used a rate equation technique to determine the I-V properties of the
associated molecular junction [72]. Begemann et al. investigated the transport
properties of a benzene-based single electron transistor. They analyzed how
the interplay between orbital symmetry and Coulomb interaction manifests
itself as destructive interference involving orbitally degenerate states,
resulting in selective conductance suppression and negative differential
conductance when the contacts are switched from para to meta configuration
[73, 74].

As discussed above, all previous theoretical studies on transport properties
of different molecular bridges either explored only charge transport using
both generalized quantum master equation (QME) and NEGF methods [72, 73, 74,
75], or charge thermoelectric proporties in strong coupling regime using NEGF
method [70, 71]. To the best of our knowledge, till date, the thermoelectric
transport properties of any kind of molecular bridges in the Coulomb blockade
(weak coupling) regime where manybody effect is dominant, has not been
explored. Choosing benzene molecule as a simplistic yet pivotal toy model, for
the first time, we have explored thermoelectric transport in molecular nano-
junctions. The versatility of the benzene molecular junction enables us to
explore diverse connections (ortho, meta, and para) with electrodes. This
selection has yielded valuable insights into distinctive properties associated
with each connection.

Further, the choice of the weak coupling regime for studying charge and
thermoelectric transport properties in molecular junctions is motivated by
computational and analytical tractability. In the weak coupling limit, the
interaction between the molecular system and the electrodes is assumed to be
relatively weak compared to other energy scales in the system. This
simplification facilitates the use of perturbative methods and allows for more
straightforward mathematical treatment. Also understanding and controlling the
staircase behavior in weak coupling is crucial for practical applications of
single-electron transistors, enabling advancements in ultra-sensitive sensing,
quantum computing, and energy-efficient electronics by precise manipulation of
individual electrons. Moreover, in the strong coupling regime, the dominant
effects in transport behavior result from the synergy of electrodes,
molecules, and their interface. These components collectively shape the
overall behavior. However, in a weak coupling regime, the entirety of
transport behaviors predominantly originates from the molecule itself. Changes
in electrodes or the interface have comparatively minimal impact in such
scenarios.

In this article, using Pauli master equation (PME) formalism, for the first
time, we explore the spin- and thermoelectric transport properties in benzene
molecular bridges weakly coupled to electrodes on both sides. This method is
also extendable and applicable to other molecular systems to study the
transport properties in a weak coupling regime. The paper is structured as
follows. In section II, we describe the model Hamiltonian and theoretical
formulaton for transport studies in our system. In the next section, we
present our numerical results and offer rigorous discussions on it. Our
results show many interesting transport characteristics of benzene molecular
junction in response to spin-polarized electrodes and Zeeman field. The last
section includes the final conclusions.

##  II Theoretical Formalisms

###  II.1 Model

We model the benzene molecule using the extended Hubbard model. The benzene
molecule is weakly coupled to electrodes at ortho, meta and para connections
as shown in Fig. 1. We start by considering only the localized
$\mathrm{P_{z}}$ orbitals (one per carbon atom) in an interacting Hamiltonian
of isolated benzene. Considering the nearby electron-electron interactions and
hopping between nearest neighbour sites, the most general form of Hamiltonian
for benzene is as follows:

$\displaystyle H_{0}=\sum_{i=1,\sigma}^{N}\
\epsilon_{i\sigma}a^{{\dagger}}_{i\sigma}a_{i\sigma}+\sum_{i\sigma}-(t_{i,i+1}a^{{\dagger}}_{i\sigma}a_{i+1,\sigma}+h.c.)$
$\displaystyle+U\sum_{i=1}^{N}(n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2})$
$\displaystyle+V\sum_{i=1}^{N}(n_{i\uparrow}+n_{i\downarrow}-1)(n_{i+1\uparrow}+n_{i+1\downarrow}-1)$
(1)

Where $\epsilon_{i\sigma}$=$\epsilon_{i}-\hat{\sigma}2\mu_{B}B$ is the on-site
energy for various atomic sites whose degeneracies are lifted by an external
magnetic field B (Zeeman splitting), $\mu_{B}$ is the Bohr magneton,
$\hat{\sigma}$=+(-) for $\sigma$=$\uparrow$($\downarrow$), $\epsilon_{i}$ is
set to zero for all sites, $t_{i,i+1}$ is the hopping intensity between
nearest neighbour sites ($i$ to $i+1$), $U$ is the Hubbard interaction term,
and $V$ denotes nearest neighbour coulomb repulsion. $a_{i\sigma}^{\dagger}$
($a_{i\sigma}$) is the creation (annihilation) operator of spin $\sigma$ in
the $i^{th}$ ($i=1$ to $6$) site and
$n_{i\sigma}$=$a_{i\sigma}^{\dagger}$$a_{i\sigma}$. Here, mechanical
oscillations are disregarded, and all atoms are assumed in their equilibrium
positions. The parameter for carbon-carbon hopping terms is assumed to be 2.5
eV for benzene. According to information from the literature, the Hubbard
electron-electron interaction for the electron at the carbon sites and the
nearest neighbour coulomb repulsions is assumed to be 10 eV and 6 eV
respectively[73].

Figure 1:  Representation of the benzene molecule attached to the electrode at
(a) ortho (1-6), (b) meta (1-5), (c) para (1-4) connection. Blue blocks
represent the electrodes that are weakly coupled to the molecule.

###  II.2 Method

####  II.2.1 Pauli master equation

For the description of the benzene molecule weakly coupled to electrodes, we
utilize the total Hamiltonian $H=H_{0}+H_{el}+H_{T}$, where $H_{0}$ comprising
the interacting Hamiltonian for isolated benzene (described in Eq. 1), the
Hamiltonian for the electrodes as $H_{el}$ = $\sum_{\alpha
k\sigma}\epsilon_{k\sigma}c^{{\dagger}}_{\alpha k\sigma}c_{\alpha k\sigma}$,
where $\epsilon_{k\sigma}$ denotes the energy of the electron with wave vector
$k$, spin $\sigma$ in the electrode $\alpha$=L/R, $c^{{\dagger}}_{\alpha
k\sigma}(c_{\alpha k\sigma}$) are creation (annihilation) operator, and the
Hamiltonian $H_{T}$ = $\sum_{\alpha k\sigma}(b_{\alpha
k\sigma}a^{{\dagger}}_{\alpha\sigma}c_{\alpha k\sigma}+b^{\star}_{\alpha
k\sigma}c^{{\dagger}}_{\alpha k\sigma}a_{\alpha\sigma}$) representing the
coupling between electrodes and molecule. Because of the weak coupling to the
electrodes, we can assume that the potential drop is all concentrated at the
electrode-molecule interface and has no effect on the molecule itself. The
preferred method for addressing dynamics in weak coupling is the Liouville
equation approach. The quantum Liouville equation for the reduced density
matrix is given by $\frac{\partial\rho}{\partial
t}=\frac{i}{\hbar}[\rho,H]+\mathcal{C}\\{\rho\\}$, where $\rho$ is the reduced
density operator and $\mathcal{C}\\{\rho\\}$ describes the dissipative part of
the dynamics after eliminating electrode memory effect by Markovian
approximation (temporal coarse-graining)[76].

The reduced dynamics is often approximated by a generalized QME, encompassing
both diagonal and off-diagonal elements of the density matrix to account for
coherences between different charge states. However, in a weak coupling
regime, the validity of the PME[77, 78] is constrained to considering only the
diagonal elements of the density matrix in the eigenbasis of the Hamiltonian
while neglecting the off-diagonal terms. The PME describes the time evolution
of the probabilities of occupation of the eigenstates of the Hamiltonian
(diagonal elements of the density matrix)[56, 72]. The standard approach is to
take into account the steady-state solution , which reduces the problem to an
algebraic system of linear equations, the solution to which is the population
of the molecule, or the kernel of the rate matrix. In non-stationary
scenarios, the application of the PME would lead to a violation of current
continuity[76, 79, 80]. Moreover, the justification for using the PME is
specifically limited to very small devices [81, 76]. Given the molecular scale
of our system, employing the PME is not only appropriate but also valid in the
weak-coupling limit i.e., $\Gamma_{0}<<k_{B}T$, (where $\Gamma_{0}$ is the
coupling strength of electrodes taken to be $1~{}meV$). The weak coupling
implies that the symmetry of the molecule, as well as the structure of the
Hamiltonian (Eq. (1)), will remain unchanged.

PME can be comprehended as an outcome of first-order time-dependent
perturbation theory (Fermi’s golden rule) or as a solution derived from a
many-body Schrödinger equation [82]. The significant advantage of utilizing
PME lies in their ability to model extremely complex systems with relative
simplicity. These systems would otherwise demand significantly more
computational effort or alternative approaches. Moreover, employing PME
enables the observation of numerous remarkable phenomena exhibited by such
complex systems. We use the exact diagonalization (ED) approach to diagonalize
$H_{0}$ (6-sites Hamiltonian with $4^{6}$, i.e., 4096 basis) within the PME
formalism, while the electrodes are considered as electronic reservoirs and
are described by their Fermi distributions. Diagonalization of $H_{0}$ yields
many-body eigenstates $|s>$ and related eigen energies $E_{s}$. Electron
transfer in molecules and electrodes is exclusively considered within the rate
matrix $W$, the elements of which are the rates of transition between many
body eigenstates of the reduced system. The PME (taking diagonal element of
density matrix ($\rho_{ss}=P_{s}$)) are as follows[83]:

$\displaystyle\frac{dP_{s}}{dt}=\sum_{s^{\prime}}(W_{{s^{\prime}}\rightarrow
s}P_{s^{\prime}}-W_{{s}\rightarrow{s^{\prime}}}P_{s}).$ (2)

Where $W_{s^{\prime}\rightarrow s}$ ($W_{{s}\rightarrow{s^{\prime}}}$) denotes
the rate of transition from the many-body Fock state $s^{\prime}$ to $s$ ($s$
to $s^{\prime}$), which differs by one electron and $P_{s}(P_{s^{\prime}})$ is
the probability that the system will be in the many-body state
$s(s^{\prime})$. The transition rates are as follows:

$\displaystyle
W_{{s^{\prime}}\rightarrow{s}}^{L+}=\Gamma_{L\sigma}f_{L}(E_{s}-E_{s^{\prime}})\sum_{\sigma}|<s|a^{\dagger}_{1\sigma}|s^{\prime}>|^{2}$
(3) $\displaystyle
W_{{s^{\prime}}\rightarrow{s}}^{R+}=\Gamma_{R\sigma}f_{R}(E_{s}-E_{s^{\prime}})\sum_{\sigma}|<s|a^{\dagger}_{N\sigma}|s^{\prime}>|^{2}$
(4)

The corresponding equation for $W_{{s}\rightarrow{s^{\prime}}}^{L-}$ and
$W_{{s}\rightarrow{s^{\prime}}}^{R-}$ obtained by replacing
$f_{L,R}(E_{s}-E_{s^{\prime}})$ by $(1-f_{L,R}(E_{s}-E_{s^{\prime}}))$,where
$f_{L/R}$ is the Fermi function for left/right electrode. In this case, $+/-$
denotes the creation/destruction of an electron within the molecule due to
electron migration from/to the left ($L$)/right ($R$) electrodes.
$\Gamma_{L\sigma}$ and $\Gamma_{R\sigma}$ are spin dependent electrode-
molecule coupling strength for left and right electrodes. Additionally, we
assumed that the creation and annihilation occur only at the sites directly
connected to the electrodes. The total transition rate will be calculated by
summing four terms
($W_{{s}\rightarrow{s^{\prime}}}=W_{{s}\rightarrow{s^{\prime}}}^{L+}+W_{{s}\rightarrow{s^{\prime}}}^{R+}+W_{{s}\rightarrow{s^{\prime}}}^{L-}+W_{{s}\rightarrow{s^{\prime}}}^{R-}$).
One can figure out the population of many-body states by solving the steady
state PME, $\frac{dP_{s}}{dt}=\sum_{s^{\prime}}(W_{{s^{\prime}}\rightarrow
s}P_{s^{\prime}}-W_{{s}\rightarrow{s^{\prime}}}P_{s})=0$. This equation is in
the form of a homogeneous linear system (AB=0) that cannot be solved. So, we
have used $\mathrm{\sum_{s}{P_{s}}=1}$ to eliminate one row/column of the
matrix, which helps to reformulate the eigenvector problem into an
inhomogeneous linear system (AB=X). This inhomogeneous linear sets of
equations can be solved by using linear algebraic methods.

####  II.2.2 Charge, spin and heat Currents

At a steady state, the current coming from the molecule to one electrode is
exactly cancelled by the current flowing from the other into the molecule.
However, if only the current between the molecule and one of the electrodes is
considered, the expression does not vanish, and we may depict the actual
electric(charge) current flowing through the system by,

$\displaystyle
I_{\alpha}=\frac{e}{\hbar}\sum_{s,s^{\prime}}(W_{{s^{\prime}}\rightarrow
s}^{\alpha+}P_{s^{\prime}}-W_{{s}\rightarrow{s^{\prime}}}^{\alpha-}P_{s})$ (5)

Similarly, the energy/heat current is also computed as;

$\displaystyle
Q_{\alpha}=\frac{1}{\hbar}(E_{s}-E_{s^{{}^{\prime}}})\sum_{s,s^{\prime}}(W_{{s^{\prime}}\rightarrow
s}^{\alpha+}P_{s^{\prime}}-W_{{s}\rightarrow{s^{\prime}}}^{\alpha-}P_{s})$ (6)

where $\alpha=L/R$. Additionally, the current flowing from the electrode is
the sum of current flowing in the spin up and spin down channel. The charge
current can be written as, $I_{C}=I_{\alpha\uparrow}+I_{\alpha\downarrow}$.
While the spin current flowing from electronic reservoirs (electrodes) are
determined by the corresponding spin-polarized charge current
i.e.,$I_{S}=I_{\alpha\uparrow}-I_{\alpha\downarrow}$.

####  II.2.3 Thermoelectric formulations

For thermoelectric investigations, a linear response regime is employed. In
this case, we presume that the left lead is slightly hotter than the right
one, i.e. $T_{L}=T_{R}+\Delta T$ and $\mu_{L}=\mu_{R}-e\Delta V$, then the
charge and heat current is given by,

$\displaystyle I_{\alpha}=G_{V}\Delta V+G_{T}\Delta T$ (7) $\displaystyle
Q_{\alpha}=M\Delta V+K\Delta T$ (8)

Where $G_{V}$ and $G_{T}$ are electrical conductance and thermal coefficient,
respectively. Then we obtain $G_{V}$ and $G_{T}$ by putting
$I_{L}=\frac{1}{2}(I_{L}-I_{R})$ and expanding the Fermi-Dirac distribution
function as $f_{L}(x)=f_{R}(x)-\frac{(x-\mu)}{T}f^{{}^{\prime}}(x)\Delta
T+e\Delta Vf^{{}^{\prime}}(x)$. where $f^{\prime}(x)=\frac{\partial
f(x)}{\partial x}$ .

$\displaystyle
G_{V}=\frac{e^{2}}{2\hbar}\sum_{s,s^{\prime}}\sum_{\sigma}\Gamma_{L\sigma}f^{\prime}(E_{s}-E_{s^{\prime}})|<s|a^{\dagger}_{1\sigma}|s^{\prime}>|^{2}$
$\displaystyle(P_{s}+P_{s^{\prime}})$ (9) $\displaystyle
G_{T}=\frac{-e}{2\hbar}\sum_{s,s^{\prime}}\sum_{\sigma}\Gamma_{L\sigma}f^{\prime}(E_{s}-E_{s^{\prime}})\frac{((E_{s}-E_{s^{\prime}})-\mu)}{T}$
$\displaystyle|<s|a^{\dagger}_{1\sigma}|s^{\prime}>|^{2}(P_{s}+P_{s^{\prime}})$
(10)

Under the condition of charge current $I_{\alpha}=0$, thermopower can be
calculated as $S=\frac{G_{T}}{G_{V}}$. The magnitude and sign of the
thermopower are influenced by asymmetry in the distribution of electrons near
the chemical potential. Using the Onsager relation, electron thermal
conductance can be written as [84],

$\displaystyle k_{e}=\frac{-Q_{\alpha}}{\Delta T}|_{I_{\alpha}=0}$
$\displaystyle=K-S^{2}G_{V}T$ (11)

Where,

$\displaystyle
K=\frac{1}{2\hbar}\sum_{s,s^{\prime}}\sum_{\sigma}\Gamma_{L\sigma}f^{\prime}(E_{s}-E_{s^{\prime}})\frac{((E_{s}-E_{s^{\prime}})-\mu)^{2}}{T}$
$\displaystyle|<s|a^{\dagger}_{1\sigma}|s^{\prime}>|^{2}(P_{s}+P_{s^{\prime}})$
(12)

Then the figure of merit can be written as,
$ZT=\frac{S^{2}G_{V}T}{k_{e}+k_{ph}}$ .The phonon thermal conductance can be
expressed as $k_{ph}=3k_{0}$ ($k_{0}=(\pi^{2}k_{B}^{2}T)/3h$ is the quantum
thermal conductance).

##  III Numerical Results and Discussion

Our work introduces a novel dimension by pioneering the investigation of
charge and spin thermoelectric transport within benzene molecular junctions
using the PME method. In our quest to comprehend these transport mechanisms,
we delve into the analysis of charge and spin current transport within the
junction. While many previous research has delved into this realm, our
approach distinguishes itself through the incorporation of uniquely formulated
equations and parameters. This strengthens the robustness of our research
endeavor.

###  III.1 Charge and spin Transport

In our investigation, we have meticulously examined both charge and spin
transport properties within the benzene molecule, encompassing ortho, meta,
and para connections. This molecule has been weakly coupled to electrodes
positioned on either side. To ensure the molecule remained in its six-electron
ground state with no bias, we skilfully tuned the chemical potential of the
electrodes. This adjustment of the chemical potential prevented any net charge
transfer between the molecule and the electrodes, maintaining the desired
neutral charge state and facilitating a zero-bias environment. Employing the
PME approach, we have conducted a comparative analysis of the current-voltage
($I-V$) characteristics across three distinct connections of the benzene
molecule. These connections represent the ortho, meta, and para
configurations. Notably, our analysis is solely focused on the neutral (six-
electron) or anionic (seven electron) charge states of the benzene molecule.
The nature of current-voltage characteristics in nanoscale systems is
intricately influenced by a multitude of factors. Among these, the electronic
structure of the constituent molecule stands out as a pivotal determinant.
This electronic arrangement governs the availability and movement of charge
carriers, thereby shaping the overall transport behavior. Equally vital is the
underlying physics that composes charge and energy flow within the system.
Quantum mechanical effects, electronic interactions, and coupling between the
molecule and the electrodes collectively contribute to the observed current-
voltage traits. Furthermore, the profile of the potential drop experienced
across the molecule as it spans the gap between the electrodes plays a
critical role. This electrostatic potential landscape directly affects the
energy levels and pathways available to charge carriers, thereby influencing
the flow of current and the resulting voltage response.

In summary, the intricate interplay of these factors—the electronic structure
of the molecule, the underlying physical mechanisms, and the electrostatic
potential distribution—exerts a profound influence on the observed current-
voltage characteristics in nanoscale systems. Understanding and controlling
these elements are paramount in designing and tailoring the functionality of
such systems for a diverse range of applications.

To elucidate the origins of all observed current-voltage ($I-V$)
characteristics, our analysis delves into several crucial factors. These
encompass a detailed examination of the charge density distribution across all
molecular sites from which we gain insights into how charge carriers are
distributed throughout the molecule, the eigenstates corresponding to the
6-electron ground state ($\phi_{6}^{gs}$), and the 7-electron ground state
with both spin-up ($\phi_{7\uparrow}^{gs}$) and spin-down
($\phi_{7\downarrow}^{gs}$) configurations which provides a comprehensive view
of the energy landscape and the accessible electronic transitions and lastly
the occupation probabilities of all relevant electronic states, a
comprehensive grasp of how charge carriers populate the available energy
levels within the system. A schematic depiction of the energy landscape
associated with the transition from $6e^{-}\rightarrow 7e^{-}$ states in
benzene is presented in Fig. S1, available in the supporting information (SI).
This transition represents the lowest energy excitations in low-bias
scenarios.

####  III.1.1 Case of Normal electrode
($\Gamma_{L\sigma}$=$\Gamma_{R\sigma}$=$\Gamma_{0}$, B=0)

Figure 2:  (a), (b) and (c) represents the charge ($I_{C}$) and spin ($I_{S}$)
current of ortho, meta and para connection of benzene respectively, (d), (e)
and (f) represents the respective probabilities of occupied many-body
electronic states for the normal electrode connections. The probabilities of
occupied electronic state are represented by a green line for $\phi_{6}^{gs}$
(singlet), a magenta line for $\phi_{7\downarrow}^{gs}$ (doublet) and a blue
line for $\phi_{7\uparrow}^{gs}$ (doublet).

Fig. 2 depicts the charge ($I_{C}$) and spin ($I_{S}$) current of benzene
molecular junction as a function of bias voltage with three different
configurations (i.e., ortho, meta, and para) of normal electrode (NE)
connection. With the unique geometric arrangements and electronic properties
associated with each configuration, we anticipate the emergence of intriguing
and distinctive patterns in the way current responds to the applied bias
voltage. It is noticed that the ortho $(1-6)$ and meta $(1-5)$ connections of
benzene exhibit remarkable negative differential conductance (NDC)-like
features in $I_{C}$. The distinct nature of this behavior is visually depicted
in Fig. 2 (a) and 2 (b), where an increase in bias leads to the abrupt decline
of current. The emergence of NDC can be directly attributed to the distinct
coupling geometry present in the ortho and meta connections of the benzene
molecule, as thoroughly investigated in our study. In these specific molecular
arrangements, the intricate interplay between the molecular structure and the
applied bias voltage gives rise to a remarkable phenomenon. As the bias
voltage increases, this unique geometry enforces a nonconducting state that
becomes increasingly prominent beyond a specific threshold voltage.
Essentially, this particular coupling geometry controls the flow of charge
through the molecular junction. This discovery underscores the connection
between molecular architecture and electronic behavior, emphasizing how
specific configurations can generate intriguing and impactful transport
characteristics.

Understanding the pivotal role played by molecular geometry in shaping
electronic transport not only advances our theoretical insights but also opens
doors to designing tailored molecular devices with precise and desirable
electronic functionalities-a promising frontier in the realm of molecular
electronics. Similar NDC characteristics for benzene have already been
reported in the literature, including radiative relaxation to a blocking state
[72] and an interference-induced blocking state [74, 73]. NDC has also been
demonstrated experimentally for molecular junctions with a wide range of
molecules, including polyporphyrine oligomers, nitroamine-functionalized
benzene, and azobenzene[85, 86, 87]. Unlike the NDC-like features observed in
the ortho and meta connections, the current-voltage ($I-V$) characteristics of
the para ($1-4$) connection of the benzene molecule displays a distinctive
staircase-like behavior, as illustrated in Fig. 2 (c). This staircase-like
response reinforces the relation between the molecular structure, electronic
states, and bias voltage.

The analysis of occupation probabilities of various many-body states within
the ortho connection of benzene, as depicted in Fig. 2 (d), provides valuable
insights into transport behavior. At bias voltages ranging from $-0.2$ to
$0.2$ V, the occupation probabilities predominantly favour the 6-electron
ground state ($\phi_{6}^{gs}$), which is a singlet, while other states remain
nearly unpopulated. This bias range is insufficient to induce an extra
electron into the molecule, thus maintaining a stable neutral charge state. As
a consequence, no current flows through the molecular bridge, and the system
operates within the Coulomb blockade regime. As the bias voltage increases,
both in the negative and positive regimes, the probability of occupying the
7-electron ground states with spin up ($\phi_{7\uparrow}^{gs}$) and spin down
($\phi_{7\downarrow}^{gs}$) rises. We note here that 7-electron ground state
is a 4-fold degenerate state: $\phi_{7\uparrow}^{gs}$ (doublet) and
$\phi_{7\downarrow}^{gs}$ (doublet). This increased probability renders these
states energetically accessible for electron transitions from the 6-electron
state to the 7-electron state, facilitating a current flow. Remarkably, in the
bias range of $1.18$ to $2.12$ V, a distinct change occurs in the occupation
probabilities. While the probability of $\phi_{7\downarrow}^{gs}$ state
diminishes, that of $\phi_{7\uparrow}^{gs}$ state continues to rise. This
variation in probabilities correlates with the emergence of NDC in the
current-voltage characteristics. Importantly, in this region, the probability
of $\phi_{7\uparrow}^{gs}$ state dominates over $\phi_{6}^{gs}$ and
$\phi_{7\downarrow}^{gs}$ states, signifying that $\phi_{7\uparrow}^{gs}$
state predominantly contributes to the NDC behavior observed in the positive
bias regime of the ortho connection of benzene. Conversely, the origin of NDC
in the negative bias regime ($-1.29$ to $-2.10$ V) is distinct. Here, the
probability of $\phi_{7\downarrow}^{gs}$ state predominates, leading to the
occurrence of NDC. The spin current, as shown in Fig. 2 (a), demonstrates a
negative value in the positive bias regime due to the dominance of the current
contributed by the down electron ($I_{\downarrow}$) over the current from the
up electron ($I_{\uparrow}$). Conversely, in the negative bias region,
$I_{\uparrow}$ significantly outweighs $I_{\downarrow}$, resulting in a
negative spin current for that regime.

Analyzing the occupation probability distribution within the context of the
meta-connection of benzene (illustrated in Fig. 2 (e)) yields insightful
observations about its transport behavior. At very low bias ranges, the
$\phi_{6}^{gs}$ state becomes predominantly occupied. As the bias voltage is
applied in both the positive and negative regimes, the probabilities
associated with $\phi_{7\uparrow}^{gs}$ and $\phi_{7\downarrow}^{gs}$ states
rise, rendering them energetically accessible for electron transitions and
subsequently enabling the flow of current. However, a notable transition
becomes evident in the positive bias regime within the range of $1.13$ to
$2.23$ V. As bias increases within this window, the probability associated
with the $\phi_{7\downarrow}^{gs}$ state diminishes, while that of the
$\phi_{7\uparrow}^{gs}$ state experiences a rapid increase. This pronounced
shift in probabilities is intimately tied to the emergence of NDC in $I_{C}$.
The behavior of the spin current exhibits distinct characteristics in the
positive and negative bias regimes, primarily due to the fact that the current
associated with the down electron ($I_{\downarrow}$) outweighs the current
attributed to the up electron ($I_{\uparrow}$) in both these regimes.

Figure 3:  The charge density distribution of 6-electron ground state
($\phi_{6}^{gs}$), spin up and spin down of 7-electron ground state
($\phi_{7\uparrow}^{gs}$ and $\phi_{7\downarrow}^{gs}$) over the sites of
benzene.

Fig. 2 (f) provides a visual representation of the probability analysis for
the para connection of benzene. At bias voltages ranging from $-0.4$ to $0.4$
V, the 6-electron ground states ($\phi_{6}^{gs}$) are fully occupied, placing
the system within the Coulomb blockade region. Consequently, during this bias
range, the system remains nonconductive, yielding no current flow. However, as
the bias voltage increases beyond this range, an intriguing transition occurs.
Both the 7-electron ground states with spin up ($\phi_{7\uparrow}^{gs}$) and
spin down ($\phi_{7\downarrow}^{gs}$) become increasingly active in the
transport process. As the probabilities associated with these states rise and
the occupancy of $\phi_{6}^{gs}$ decreases, the conditions become favorable
for the transition of an electron from a 6-electron state to a 7-electron
state. This transition facilitates the onset of current flow through the
system. Furthermore, with a continued increase in bias voltage, the transport
process is influenced by the involvement of higher electronic states, leading
to a steep rise in current. Notably, in this case, the observed spin current
is minimal. This is attributed to the nearly equal contributions of the
currents associated with spin up ($I_{\uparrow}$) and spin down
($I_{\downarrow}$), resulting in a balanced overall spin current behavior. By
delving into the occupation probability analysis and spin current behaviors
within the ortho, meta, and para-connection of benzene, we gain a
comprehensive understanding of the interplay between electronic states, bias
voltage variations, and transport phenomena. This knowledge contributes to a
deeper comprehension of the phenomenon of NDC as well as the staircase
behaviors, enriching our understanding of molecular electronic transport
dynamics.

The distribution of charge density across all sites of the benzene molecule
has been thoroughly investigated for both the $\phi_{6}^{gs}$ state and, the
$\phi_{7\uparrow}^{gs}$ and $\phi_{7\downarrow}^{gs}$ states, as depicted in
Fig. 3. This analysis provides crucial insights into the population of
molecular sites and their propensity to facilitate the creation and
annihilation of electrons, ultimately driving charge transport. The
significant influence of the onsite Hubbard electron-electron interaction
leads to a single electron occupying each site within the $\phi_{6}^{gs}$
state. To comprehensively grasp the implications of this charge density
distribution and gain a deeper understanding of the nature of the $I-V$
characteristics of the molecules, let us delve into an interpretation. For
simplicity, we will assume that the electrochemical potential at the left
electrode diminishes with a bias of $\mu-V/2$ and increases with a bias of
$\mu+V/2$. As the bias increases, it can be seen that transition
$6e^{-}\leftrightarrow 7e^{-}$ occurs which implies that an electron is
created at the right electrode resulting in a current flow. Again, when the
bias increases, the transition towards a more excited state may occur.
However, in our study, we are only concerned with low-bias characteristics.

In the $\phi_{7\downarrow}^{gs}$ state, the electrons remain almost equally
distributed over all the atomic sites. Consequently, an electron jumps from
the $\phi_{6}^{gs}$ to the $\phi_{7\downarrow}^{gs}$ and has been created at
the right electrode with an increase in bias. As a result, the current
increases significantly. But the $\phi_{7\uparrow}^{gs}$ exhibit unequal
charge density distribution of electrons in ortho and meta connection which is
not favourable for electron transition. Therefore, the current diminishes as
bias rises in the positive regime, leading to an NDC in $I-V$ characteristics.
Again, in the ortho connection, there is an NDC in a negative bias regime
because a little bit of difference has been noticed in the
$\phi_{7\downarrow}^{gs}$ state. In the para connections,
$\phi_{7\uparrow}^{gs}$ as well as $\phi_{7\downarrow}^{gs}$ state have almost
equally distributed electrons in all atomic sites, giving a step like $I-V$.
As a result, the population and depopulation of the anionic states, where the
charge distribution patterns are distinctive and important for current flow,
produce an exhilarating non-linear behaviour of current-voltage
characteristics.

Figure 4:  The charge ($I_{C}$) and spin ($I_{S}$) current as a function of
the bias voltage in the ortho, meta and para connection of benzene molecule at
different spin polarization $p$.

####  III.1.2 Case of ferromagnetic electrode
($\Gamma_{\alpha\uparrow}$$\neq$$\Gamma_{\alpha\downarrow}$)

In this section, we delve into the impact of ferromagnetic electrodes (FE)
weakly coupled to distinct configurations of the benzene molecule. The
presence of ferromagnetism in the electrodes introduces a spin-dependent
coupling strength, denoted as $\Gamma_{\alpha\sigma}$, where $\alpha$
represents the electrode and $\sigma$ denotes the spin orientation. This
coupling strength is elucidated through a spin-polarization parameter $p$,
formulated as
$p=\frac{\Gamma_{\alpha\uparrow}-\Gamma_{\alpha\downarrow}}{\Gamma_{\alpha\uparrow}+\Gamma_{\alpha\downarrow}}$
[88]. The spin-dependent nature of these coupling strengths plays a crucial
role in the electron transport properties of the molecular junction. This
coupling introduces an intriguing avenue to explore spin-related effects and
their interplay with molecular configurations and bias voltage. Further
analysis will shed light on the interdependence between these factors,
offering insights into the intricate dynamics of charge and spin transport in
these systems.

In our study, we have exclusively considered the parallel configuration of
electrodes, characterized by $\Gamma_{L\uparrow(\downarrow)}=(1\pm
p)\Gamma_{0}$ and $\Gamma_{R\uparrow(\downarrow)}=(1\pm p)\Gamma_{0}$. Fig. 4
illustrates the charge and spin currents of the ortho, meta, and para
connections of the benzene molecule as a function of bias voltage, considering
different degrees of polarization. Fig. 4 (a), (c), and (e) present the charge
current behavior, revealing that as the polarization $p$ increases, there are
subtle or even negligible changes in the peak height/position of the current
curves across the three distinct benzene connections. However, as expected and
shown in Fig. 4 (b), (d), and (f), the behavior of the spin current undergoes
a drastic change with increasing polarization. Notably, the spin current
displays a rapid ascent as polarization increases. This trend signifies that
the absolute value of the spin-up current ($I_{\uparrow}$) consistently
surpasses the absolute value of the spin-down current ($I_{\downarrow}$). This
phenomenon can be attributed to the accumulation of electrons with a specific
spin direction $\sigma$ on the atomic sites of the molecule within specific
bias windows. These distinctive outcomes in the spin current ($I_{S}$) can
also be understood in the context of the values of $\Gamma_{L\sigma}$ and
$\Gamma_{R\sigma}$, which play a significant role in shaping the overall
transport behavior. Taking all these factors into consideration, it becomes
evident that, as polarization increases, the localization of spin-up electrons
onto the molecular atomic sites becomes progressively more pronounced compared
to spin-down electrons [89]. The interplay of polarization, electrode-molecule
coupling, and spin characteristics leads to the fascinating and complex spin-
related behaviors observed in the current-voltage characteristics.

####  III.1.3 Case of normal electrode with external magnetic field (B)

Figure 5:  The charge ($I_{C}$) and spin ($I_{C}$) current as a function of
the bias voltage in the ortho, meta and para connection of benzene molecule at
different magnetic field $B$.

In this section, we shift our focus to examine the influence of an external
magnetic field on the weakly coupled normal electrodes connected to the
molecule. The presence of an external magnetic field leads to the splitting of
the molecular energy levels, thereby facilitating the selective transport of
pure spin-up ($\uparrow$) or spin-down ($\downarrow$) electrons through the
system. This effect can introduce significant spin-related phenomena and
behaviors that contribute to the overall transport characteristics in the
molecular setup.

Fig. 5 displays the charge current ($I_{C}$) and spin current ($I_{S}$) as
functions of bias voltage (V) in the ortho, meta, and para connections of the
benzene molecule under varying external magnetic fields. In Fig. 5 (a), (c),
and (e), the charge current responses exhibit subtle shifts as the external
magnetic field strength is increased. This phenomenon can be attributed to the
energy level adjustments prompted by changes in the magnetic field. When the
magnetic field strength is changed, the energy levels experience shifts,
causing them to move away from the bias window through which the current
flows. Consequently, different bias voltage windows might be necessitated to
facilitate current flow under different magnetic field conditions. In
addition, the behavior of specific energy levels within the molecular system
is crucial to understand the effects of the external magnetic field. At the
$\phi_{6}^{gs}$ energy level, the total spin is zero. Consequently, this
energy level remains unaffected by alterations in the magnetic field (charge
density plot for the various magnetic fields shown in Fig. S2 in SI). However,
the introduction of a magnetic field leads to the disruption of 4-fold
degeneracy within the 7-electron ground state, $\phi_{7}^{gs}$ (Table S1
displays the energy level splitting in the $7e^{-}$ state with variations in
the magnetic field has been provided in SI). This disruption is evident in the
form of an energy level splitting, resulting in the emergence of two distinct
doublet states: $\phi_{7\downarrow}^{gs}$($7e^{-}$ ground state) and
$\phi_{7\uparrow}^{1es}$ ($7e^{-}$ 1st excited state). The extent and
direction of this energy level splitting depend on the orientation of the
external magnetic field.

Fig. 5 (b), (d), and (f) display the spin current responses as a function of
bias voltage for three distinct benzene configurations. Notably, these results
underscore the remarkable ability to manipulate the transport of a specific
type of electron, either spin $\uparrow$ or spin $\downarrow$, by adjusting
the strength of the external magnetic field. When applying magnetic fields of
$B=2T$ and $5T$, a clear trend emerges: the spin-down current
($I_{\downarrow}$) surpasses the spin-up current ($I_{\uparrow}$). This
observation signifies that a greater number of spin-down electrons become
localized at specific atomic sites within the molecule, thus contributing to
the dominant current flow of spin $\downarrow$ electrons. Intriguingly, when a
stronger magnetic field, $B=10T$, is introduced, a fascinating outcome occurs.
All three benzene configurations exhibit the flow of pure spin-down current
($I_{\downarrow}$). This distinctive behavior can be comprehended through the
concept of energy levels that align with the specific bias window enabling
current conduction. Under this strong magnetic field, the eigenstate
associated with $\phi_{7\downarrow}^{gs}$ remains within this bias window,
whereas the $\phi_{7\uparrow}^{1es}$ state experiences a greater shift,
causing it to fall outside this bias window. Consequently, the system
exclusively facilitates the flow of spin-down electrons through the molecular
junction. This capability to selectively manipulate and control the flow of a
specific spin current holds significant promise, especially in the context of
spintronic applications. The introduction of an external magnetic field offers
a versatile tool to tailor the transport characteristics of molecular systems,
which can potentially revolutionize the development of advanced spintronic
devices. Indeed, the findings strongly suggest that utilizing an external
magnetic field to modulate the behavior of a weakly coupled molecule to a
normal electrode can offer distinct advantages compared to the alternative of
switching to a spin-polarized electrode (more flexibility in adjusting the
strength and orientation of the magnetic field).

###  III.2 Charge thermoelectric transport

We then investigate the thermoelectric transport properties of the meta
connected benzene with varying temperature and the chemical potential. Because
the position of $\mu$ is a crucial ingredient in understanding different
transport across molecules. It is especially interesting since changing the
$\mu$ by doping, adding a side group, a gate field, or using another contact
material can optimise molecular characteristics. In a small system, we have
also verified and previously published research [20] findings that changing
the $\mu$ of the electrodes and connecting a gate voltage in the channel
(molecule) produce the same results. Here we investigate all results by
varying chemical potential. In the following section, we delve into the
thermoelectric properties of the meta connection of benzene. This emphasis is
placed due to the fact that the results from the other two connections, namely
ortho and para, yield nearly indistinguishable outcomes. Consequently, our
focus remains directed towards explaining the fundamental physics underlying
these phenomena and exploring the potential utility of benzene molecular
junctions in upcoming fields such as electronics, optoelectronics, and
thermoelectrics.

The electrical conductance of benzene molecule as a function of $\mu$ at
different temperature is shown in the left panel of Fig. 6. As all
calculations were performed with a zero bias, the transition between the
lowest energy levels (i.e., only the ground state with $N$ and $(N\pm 1)$
particles with significant probabilities of occupied different many-body
states) is important for conductance. The electrical conductance ($G_{V}$)
peaks occur when the energy required for an electronic transition,
specifically from the $Ne^{-}$ state to the $(N+1)$ $e^{-}$ state, aligns with
the chemical potential of the electrode. The $G_{V}$ shows four sharp peaks
(two peaks in negative and two peaks in positive $\mu$ (found also in Fig. S3,
in SI)) corresponding to resonance in meta-connected benzene. The first peak
appears at $\mu$ $\approx$ -$7.84$ $eV$. As the $\mu$ increases, the second
peak appears arround $-5.05$ $eV$. A positive change in $\mu$ resulting in the
emergence of the third peak in conductance at $\approx$ $5.05$ $eV$ (which is
the transition energy between $6e^{-}$ to $7e^{-}$ states,
$\delta\varepsilon$= $E_{7}-E_{6}$ = $5.05$ $eV$). At a very low-temperature
limit, as the Fermi-Dirac distribution function approaches a step function,
its derivative ($f^{\prime}(\delta\varepsilon)$) exhibits a sharp peak (Dirac
delta type) when the energy level resonant with chemical potential
($E\approx\mu$). In our case, $f^{\prime}(\delta\varepsilon)$ is expected to
show a sharp peak at $\mu$=$\delta\varepsilon=5.05~{}eV$. When this sharp peak
aligns with the energy levels associated with electronic transitions in a
system, it can lead to peak in electrical conductance. In Fig. S5 ( found in
SI), $G_{V}$ shows a peak at $\mu$= $5.05$ $eV$, where
$f^{\prime}(\delta\varepsilon)$ also exhibits a sharp peak. Moreover, when
$f^{\prime}(\delta\varepsilon)$ is at its maximum, we noticed that the
probabilies of $6e^{-}$ ($P_{6}$) and probabilies of $7e^{-}$ ($P_{7}$) states
are equal i.e. $P_{6}$=$P_{7}$=0.5. At $\mu$= $5.05$ $eV$, the only transition
possible from 6$\rightarrow$7 indicating that only $6e^{-}$ and $7e^{-}$
states are populated. Solving $\frac{dP_{s}}{dt}=0$, further confirms that
only $P_{6}$ and $P_{7}$ are dominated. According to the detailed balance
condition, $P_{6}$ and $P_{7}$ are equal probable and emerges peaks in
conductance. The final peak appears when the transition from $7e^{-}$ to
$8e^{-}$ configuration occurs, where the probability of 7- and 8-electron
configurations is dominated.

Figure 6:  Electrical conductance ($G_{V}$) (left panel) and thermal
coefficient ($G_{T}$) (right panel) as a function of chemical potential
($\mu$) at different temperatures.

Indeed, in the proximity of the resonance points, the populated states mainly
involve the $N$ and $N\pm 1$ particle configurations. This phenomenon is
closely tied to the conductance behavior of benzene, where specific energy
transitions become dominant, leading to pronounced conductance peaks.
Additionally, the presence of Coulomb blockade regions between these peaks is
a noteworthy feature. An intriguing and notable observation is the symmetry
exhibited by the conductance peaks concerning the chemical potential. This
symmetry indicates that changes in the chemical potential result in a
proportional adjustment of the conductance peaks, maintaining a consistent
pattern. This intriguing symmetry emphasizes the relationship between
electronic energy levels, chemical potential, and the resulting conductance
characteristics. Such symmetry could hold significant implications for
designing and optimizing molecular electronic devices based on benzene and
similar molecular systems.

Additionally, we observed that the conductance resonance peaks diminish with
wider intensity as the temperature rises (due to the influences of Fermi-Dirac
distribution) which suggests that the resistance of the system rises. At low
temperatures, the Fermi-Dirac distribution results in a sharp occupancy of
energy levels up to the chemical potential. When the temperature rises, the
broader distribution of energy levels means that a wider range of energy
states is populated by electrons. This leads to a situation where the
conductance resonance peaks, which were initially sharp at specific energy
levels, become less distinct and spread out over a broader energy range.
Indeed, a significant change in the conductance characteristics of a material
or system can have a substantial impact on its thermoelectric properties,
specifically the thermopower (also known as the Seebeck coefficient) and the
figure of merit (ZT). When the thermopower increases and combines with
appropriate electrical conductivity and low thermal conductivity, the ZT value
can be enhanced. This results in improved thermoelectric efficiency, making
the material more suitable for applications like waste heat recovery or power
generation.

Figure 7:  Electronic thermal conductance ($k_{e}$) (left panel) and Seebeck
coefficient ($S$) (right panel) as a function of chemical potential ($\mu$) at
different temperatures.

The right panel of Fig. 6 depicts the behaviour of the thermal coefficient as
a function of $\mu$ at different temperatures. It has been noted that $G_{T}$=
0 for some $\mu$ (-7.84,-5.05,0,5.05 and 7.84 eV). First, when $G_{V}$ attains
the peak value, $G_{T}$ and $S$ are zero because an electron transition occurs
at that specific $\mu$ (as discussed above), resulting in an electrical
current but no net energy transport. Here the probability of being in the $N$
and $N+1$ electron states will be equal to each other and the transition
energy for both electron configurations exactly matches with $\mu$ of the
electrode. The temperature gradient does not significantly produce the charge
current at resonance energies. Additionally, in benzene, there is no $G_{T}$
in the plateau region (where there is no electrical conductance) of $G_{V}$,
indicating that there won’t be any current due to temperature gradient if
there is no carrier to flow. The $G_{T}$ changes sign at all resonance points
indicating that both type of carriers i.e., electrons and holes are involved
in the transport mechanism. The conductance is caused by the transfer of holes
when $G_{T}$ is positive, and when the conductance is caused by the transfer
of electrons, $G_{T}$ yields negative values. Finally, we also observe that,
as the temperature rises, the width of the $G_{T}$ expands, but the intensity
of the $G_{T}$ decreases due to the temperature dependence on Fermi-Dirac
distribution.

Materials with low thermal conductance are used to create a compelling and
reliable thermoelectric device. Fig. 7(left panel) displays the electronic
thermal conductance as a function of $\mu$ at different T. At low temperatures
($T\approx 2K$) [90], the peaks of $G_{V}$ and $k_{e}$ appear when the
transition energy overlaps with the resonant energy; as a result, there is
some heat and charge current transfer due to electrons. When resonance
approaches, the charge current starts to flow and develop a peak in electric
conductance. The electron contribution to heat transfer is then primarily due
to electrons resonantly tunneling through the system, and heat conductance
approximately follows the charge conductance. However, at high temperatures,
the difference between $G_{V}$ and $k_{e}$ becomes apparent because as the
temperature rises, the thermal distribution of electrons becomes broader.
Because of this distribution, tunneling electrons contribute differently to
charge and heat conductance due to distinct energy-dependent weighting (energy
of a tunneling electron is insignificant for charge current but crucial for
heat transfer). Since we calculate all of the results here at high
temperatures, $k_{e}$ behaves differently than $G_{V}$, and the peaks of
thermal conductance are merged to generate some smaller peaks in $k_{e}$ of
benzene, as shown in Fig. 7 (left panel). In the coulomb blockade region, the
charge and heat conductances are suppressed because of no carrier flow. As the
thermal conductance of benzene is less, it should be more efficient and
reliable for future thermoelectric device applications. Moreover, we also
observed that, as the temperature rises, the electronic thermal conductance
also rises steeply.

Figure 8:  Thermoelectric figure of merit (ZT) as a function of chemical
potential ($\mu$) at different temperatures.

The $\mu$ and $T$ dependence of the thermopower is shown in the right panel of
Fig. 7. The saw-tooth pattern (2D pattern is shown in Fig. S4 in SI) of the
thermopower indicates a change in the electron population of the molecule. The
sawtooth pattern observed in the Seebeck coefficient as a function of chemical
potential can also be attributed to discrete electronic energy levels within
the molecular junction. As the chemical potential varies, electronic states
become successively occupied or unoccupied, leading to changes in the charge
carrier concentration and, consequently, the Seebeck coefficient. The discrete
nature of the energy levels results in abrupt shifts in the Seebeck
coefficient, creating a sawtooth pattern. The thermopower sign at different
$\mu$ determines the type of dominant charge carriers (hole or electrons). The
positive (negative) thermopower suggests that holes (electrons) are
responsible for the current. When the $\mu$ crosses the symmetry point
($\mu$=$0$) or resonant quantized levels, the thermopower (or Seebeck
coefficient) changes magnitude and sign alternately, exhibiting oscillating
behavior with $\mu$. In Fig. 7, S changes the sign near the symmetry point and
reaches a sharp maximum on one side of the symmetry point and a minimum on the
other. When $\mu$ deviates further from the symmetry point, S changes sign
once more. The Coulomb blockade effect is responsible for the underlying
physics and this dependence is consistent with other research insights [20,
21, 91, 90]. Therefore, $S$ is zero in these energies (symmetry point and
resonant quantized levels). As $\mu$ increases and approaches the resonance,
electrons tunnel from the higher temperature electrode to the lower
temperature electrode, causing a voltage drop under the condition of
diminishing current. As a result of this voltage drop, the thermopower S is
positive (S is negative in units of $k_{B}/e$ due to the negative charge of
the electrons). At resonance, the induced voltage drop is zero and the
thermopower vanishes because electrons can tunnel to the molecule from the
colder and hotter electrodes without requiring a temperature gradient. When
$\mu$ is less than the resonance, the net electron flow is from right to left
and the sign of S changes. The thermopower vanishes and changes sign again in
the middle of the Coulomb blockade regime, as previously mentioned. The other
resonance has been also explaining in the similar way. In the symmetry points,
electrons and holes have the same weight but opposite signs in the transport
mechanism, so the net current is zero. In other words, electrons and holes
carry the charge in opposite ways. There is non-zero thermopower at some
specific $\mu$, indicating that only one type of dominant charge carrier has
participated in transport.

The probabilities of occupied many-body electronic states near the $\mu$ are
crucial factors in determining the thermoelectric properties of a system. The
Seebeck coefficient is influenced by the asymmetry in the probabilities of
occupying different many-body electronic states with respect to $\mu$. An
asymmetric distribution of occupied states can lead to a non-zero or even high
Seebeck coefficient. In Fig. S3 (found in SI), we observed that, at $\mu$ =
$5.05~{}eV$, $G_{V}$ attains a peak where $G_{T}$ and $S$ are zero. This
occurs when $\delta\varepsilon=\mu$, and P6 and P7 are equally probable. The
condition $\delta\varepsilon=\mu$ indicates resonance, where the energy
required for an electronic transition matches the chemical potential. When
this alignment occurs, electrons can effectively transition between the
$6e^{-}$ and $7e^{-}$ states. However, since $P_{6}$ and $P_{7}$ are equal,
there is no net energy transport. The absence of energy transport can be
attributed to the fact that, on average, the energy gained during transitions
from $6e^{-}$ to $7e^{-}$ is balanced by the energy lost during the reverse
transitions from $7e^{-}$ to $6e^{-}$. Consequently, while there is electron
transport, there is no net energy transfer, leading to $G_{T}=0$ and $S=0$ at
$\mu$ = $5.05~{}eV$. The widening peak in the Seebeck coefficient with
increasing temperature is attributed to the thermal broadening of energy
levels, influenced by the Fermi-Dirac distribution. This thermal broadening
effect highlights the intricate interplay between temperature, energy
distribution, and the resulting thermoelectric properties in the studied
system.

Finally, we investigate the thermoelectric performance of the molecular
junction. As illustrated in Fig. 8, the figure of merit is zero at resonances
(because $ZT$ is directly dependent on $S$) and coulomb blockade region. Due
to the inverse relationship between conductance and thermopower, the figure of
merit exhibits fluctuating behaviour. Furthermore, because $S$ and $G_{V}$
decrease with increasing temperature, $ZT$ decreases with broader peaks shown
in Fig. 8. The distinctive advantage inherent to molecular systems is their
ability to make a negligible phononic contribution to thermal conductivity
($k_{ph}$). This unique characteristic endows these systems with the ability
to predominantly influence thermal performance through electronic thermal
conductivity ($k_{e}$). $ZT$ has the maximum peak value at low temperatures
when $k_{e}$ is very low. As benzene have lower thermal conductance, it
exhibits more enhanced figure of merit which is applicable for thermoelectric
performance. It was demonstrated that the activated contribution to $k_{ph}$
can significantly boost thermoelectric performance at relatively low
temperatures. Consequently, this study underlines the promising prospects of
utilizing benzene molecular junctions to pave the way for groundbreaking
advancements in thermoelectric devices. By capitalizing on the enhanced figure
of merit and manipulating the interplay between electronic and phononic
contributions to thermal conductivity across distinct temperature ranges,
these molecular systems hold immense potential for revolutionizing
thermoelectric technologies.

###  III.3 Spin thermoelectric transport

In this section, we have investigated the spin thermoelectric transport in our
molecular junction (meta-connection) in a Zeeman-type external magnetic field
as well as switching the electrode from normal to ferromagnetic. This
intricate configuration possesses the distinctive capability of facilitating
pure spin thermoelectric transport phenomena within the molecular junction. In
order to calculate the spin-Seebeck coefficient, we consider a system with
both an infinitesimal temperature bias and a spin-voltage bias. The charge and
spin currents are described as $I=I_{\uparrow}+I_{\downarrow}$, and
$I_{s}=I_{\uparrow}-I_{\downarrow}$ (note: they have the same dimensions). In
linear response, the spin current is given by $I_{s}=G_{V}^{s}\Delta
V+G_{T}^{s}\Delta T$, where the thermal response coefficient $G_{T}^{s}$ is
connected to the fact that a temperature gradient can cause both a spin flow
and an energy flow. When we set $I_{s}$ to zero, we get the spin-Seebeck
coefficient $S_{s}=\frac{G_{T}^{s}}{G_{V}^{s}}$. The spin figure of merit can
be defined as $Z_{s}T=\frac{G_{V}^{s}S_{s}T}{k_{e}^{s}+k_{ph}}$. The spin
conductance $G_{V}^{s}$ may be negative; hence the absolute value is used. In
the same way that charge transfer is expected to be good, a system with
$Z_{s}T>1$ is expected to be a good heat-to-spin-voltage converter.

Figure 9:  (a) Spin conductance $G_{V}^{s}$, (b) spin Seebeck coefficient
$S_{s}$ (c) spin thermal coefficient $G_{T}^{s}$, and (d) spin thermoelectric
figure of merit $Z_{s}T$ as a function of chemical potential ($\mu$) at
different polarization (p) in meta-connected benzene.

In our investigation, an intriguing observation emerged: a spin thermoelectric
effect manifested even in the absence of spin-polarized electrodes. Initially,
we noticed that the ground state would lead to an equal electron distribution,
as evidenced by the charge density plot in Fig. 3. Remarkably, we
strategically selected a parameterized system that yielded a perceptible spin-
thermoelectric effect, all without resorting to alterations in the
ferromagnetic electrodes or the application of an external magnetic field.
This intriguing outcome strongly hints at the intricate interplay between
charge distribution, electronic configuration, and spin-related phenomena. The
localized or delocalized nature of electronic states within the connections of
benzene might be pivotal in generating such spin-dependent transport behavior,
even without explicit spin polarization of electrodes. Our carefully designed
parameterized system have the different delocalization of electronic charge
state in $\phi_{7\uparrow}^{gs}$ and $\phi_{7\downarrow}^{gs}$ in meta-
connected benzene ( Fig. 3), showcasing this unexpected spin-thermoelectric
effect, underscores the inherent propensity of the system. The absence of
external influences, such as magnetic fields, further emphasizes the unique
attributes of the system. Additional simulations or experiments with varying
system parameters could unveil valuable insights into the underlying
mechanisms governing this spin thermoelectric effect. This discovery opens
intriguing possibilities, potentially paving the way for innovative energy
conversion strategies and novel functionalities by harnessing spin-related
phenomena in thermoelectric applications.

Subsequently, we employ ferromagnetic electrodes to delve into the realm of
spin thermoelectric transport within benzene molecular junctions. These
ferromagnetic electrodes, coupled with the molecule, serve as efficient
electron reservoirs exhibiting pronounced spin polarization. Fig. 9 offers a
comprehensive depiction of all thermoelectric coefficients, revealing their
dependencies on $\mu$ across various polarizations (p). Given our prior
analysis, we shift our focus in this section towards examining the involved
influence of spin-polarized electrodes on the spin thermoelectric effect. We
can see in Fig. 9 (a), (c), and (d) that $G_{V}^{s}$, $G_{T}^{s}$, and
$Z_{s}T$ strongly vary with the spin polarization of electrodes and increase
with p, whereas the spin Seebeck coefficient (Fig. 9(b)) is unaffected by p.
The spin-dependent coupling strength is a multiplicative constant.
Consequently, as polarization increases, both $G_{V}^{s}$ and $G_{T}^{s}$
experience proportional increase, given their role in governing interactions
between particle spins and external influences. The parameter $S_{s}$ is
intricately linked to the $\frac{G_{T}^{s}}{G_{V}^{s}}$ratio. So, when
polarization increases, the relationship between $G_{V}^{s}$ and $G_{T}^{s}$
remains consistent. This leads to the observation that alterations in
polarization do not impact the value of $S_{s}$. Moreover, the spin Seebeck
coefficient is a material-specific property that is determined by the
material’s intrinsic properties and the temperature gradient applied. It
doesn’t depend on the spin polarization of the electrodes in the same way that
other thermoelectric coefficients might, because it is more related to the
intrinsic properties of the material and its response to temperature gradients
rather than the specifics of the electrode configuration. Furthermore, the
sign of $G_{T}^{s}$ and $S_{s}$ is opposite to the sign of $G_{T}$ and $S$ as
the voltage induced in the channel corresponding to down spin is higher than
up spin. Hence, a significant discovery emerges: the strategic integration of
a higher spin-polarized electrode offers distinct advantages in the realm of
spin caloritronics, leading to a notable rise in the value of $Z_{s}T$
(maximum value of $Z_{s}T$ with $p=1$ is 4.09) as the degree of spin
polarization (p) increases (shown in Fig. S6 in SI). The maximum value of
$Z_{s}T$ is notably high compared to to earlier studies on quantum dots i.e.,
around 0.17 [56], 0.6 [90]. This underscores the inherent benefits associated
with enhancing spin polarization, ultimately amplifying the effectiveness and
potential of spin-based caloritronic processes.

Figure 10:  (a) Spin conductance $G_{V}^{s}$, (b) spin Seebeck coefficient
$S_{s}$, (c) spin thermal coefficient $G_{T}^{s}$, and (d) spin thermoelectric
figure of merit $Z_{s}T$ as a function of chemical potential ($\mu$) at
different $B$ in meta-connected benzene.

We also explore the influence of an external magnetic field on the spin
thermoelectric properties. Fig. 10 depicts the thermoelectric coefficients as
a function of $\mu$ at various magnetic field (B). Initially, upon introducing
a magnetic field of strength 2T, a noteworthy observation emerged. The
resulting maximal value of $Z_{s}T$ approximated 4.10, which aligning
remarkably well with the $Z_{s}T$ value witnessed in exclusively pure spin-
polarized electrodes ($p=0$). In contrast to spin polarized electrodes, we
observed an interesting trends here, i.e. in Fig. 10 (a), (c), and (d), we see
that the maximum value of $G_{V}^{s}$, $G_{T}^{s}$, and $Z_{s}T$ has been
saturated with increasing magnetic field (also shown in Fig. S7 in SI). The
saturation of these parameters suggests that while magnetic fields can
certainly influence spin thermoelectric properties, there exists an upper
threshold beyond which their impact on transport behavior becomes less
pronounced. As the magnetic field increases, the energy levels can become more
broadened due to the Zeeman effect. Strong magnetic fields can enhance the
filtering of spins with certain orientations, leading to a polarization of the
spin current. In our case, when B=10T, the occupancy of the spin-down level
increases and tunneling processes involving the spin-down channel become more
efficient. This means that electrons with a spin-down orientation find it
easier to tunnel through the molecular junction. This is a significant
observation, as it indicates a preference for electron transport with a
specific spin orientation under the influence of the external magnetic field.
However, this effect can saturate as the magnetic field becomes stronger and
all available spins of a particular orientation are already aligned and
accounted for. This saturation phenomenon serves as a critical factor in
comprehending the relation between magnetic fields and the thermoelectric
response of molecular systems.
One thing we also noticed is that the peak positions shift when the magnetic
field changes in comparison to the other cases mentioned above. In the
presence of an external magnetic field, the energy levels associated with
different spin orientations experience splitting. Consequently, the electronic
structure of the molecular system is modified, leading to shifts in the
positions of peaks and valleys in transport properties. In conclusion, our
investigation has illuminated the advantageous role of spin-polarized
electrodes as well as the application of an external magnetic field within the
realm of spin caloritronics. This discovery holds promise for future
innovations in spin-based energy conversion and manipulation technologies. By
using spin-polarized electrodes, we achieve efficient spin filtering, allowing
only specific spin orientations to pass through the molecular junction. This
can lead to a higher degree of spin polarization in the charge carriers,
enhancing the potential for exploiting spin-dependent thermoelectric effects.
External magnetic fields offer the advantage of direct control over the spin
dynamics within the molecular junction. This control can be valuable for
investigating the interplay between spin states and thermoelectric properties.

##  IV Conclusion and outlook

Our investigation into charge transport properties within benzene molecular
junctions across different connections (ortho, meta, and para) has yielded
insightful findings. Notably, ortho and meta connections exhibit prominent
negative differential conductance within specific voltage ranges, holding
potential for novel single-molecule electronic and thermionic devices. Prior
to the implementation of spin-polarized electrode, our study uncovers a
noteworthy phenomenon: the influence of spin current even in the absence of an
external magnetic field. This intriguing behavior emerges due to our precisely
designed parameterized regime, resulting in discernible charge localization
and delocalization in the anionic charge states. Our study also explores the
effect of ferromagnetic electrodes and external Zeeman field on spin current.
Applying a strong external magnetic field to the system enables precise
control of specific spin transport (spin up or spin down). This proves more
advantageous for achieving pure spin current flow compared to adjusting spin-
polarized electrodes, as it offers greater flexibility in tuning the strength
and orientation of the magnetic field.

Further, our exploration extends to investigate the thermoelectric transport
properties of the molecular junction in the linear response regime. The
observed trends in response to varying chemical potentials showcasing low
thermal and high electrical conductance signify a promising avenue for
refining thermoelectric capabilities. Important spin thermoelectric
coefficients (i.e., spin conductance, spin Seebeck coefficient and the spin
figure of merit) with an external magnetic field as well as spin-polarized
electrodes have been studied. By tuning electrode polarization or applying an
external magnetic field, we achieve a remarkable maximum spin thermoelectric
figure of merit value, approximately 4.10. This value is notably high compared
to previous studies involving quantum dots. These results underscore the
strategic advantage of utilizing both spin-polarized electrodes and external
magnetic fields in the realm of spin caloritronics.

While single-molecule devices within a weak coupling regime have limitations,
it’s crucial to explore their behavior, especially in scenarios where
electron-electron correlation dominates. In such instances, the weak coupling
regime remains applicable despite its limitations. In a recent study, Li et
al. highlighted the limitations of the mean-field-based DFT+NEGF approach,
attributing its failure to address strong correlation effects in the spin-
crossover molecule. They instead utilized a Coulomb blockade model,
characterized by weak coupling, which effectively aligned with the
experimental results[92]. Fu et al. investigated the transport behavior of
single-molecule Field-Effect Transistors (FETs), providing detailed insights
into phenomena such as Coulomb blockade, the Kondo effect, and electron–phonon
coupling. The study also explored diverse applications of single-molecule
FETs, including the regulation of quantum interference, spin effects,
thermoelectric effects, and superconductivity[93]. Ongoing research in the
molecular-scale led to promising applications in various areas such as single-
molecule transistors, spintronics devices, thermoelectric nanogenerators,
molecular-scale temperature sensors, miniature cooling devices, quantum
computing thermal management, and numerous others[1, 94, 95]. In conclusion,
our study not only contributes crucial insights into the intricate interplay
of charge, spin, and thermal transport at the molecular scale but also paves
the way for exploring other molecular junctions for potential thermoelectric
applications.

## acknowledgement

PS acknowledges DST-INSPIRE (IF190005) for financial support. PP thanks DST-
SERB for ECRA project (ECR/2017/003305).

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