UNB]State University of Campinas, Applied Physics Department , Campinas
$13083-970$, SP, Brazil UNB]University of Brasília, Institute of Physics and
International Center of Physics, Brasília $70919$-$970$, DF, Brazil UNB]State
University of Campinas and Center for Computational Engineering and Sciences,
Campinas $13083-970$, SP, Brazil UNB]University of Brasília, Institute of
Physics and International Center of Physics, Brasília $70919$-$970$, DF,
Brazil UNIR]Physics Department, Federal University of Rondônia, $76801-974$,
Porto Velho, Brazil

# On the mechanical, thermoelectric, and excitonic properties of Tetragraphene
monolayer

Raphael M. Tromer  [  L. A. Ribeiro Júnior  [  Douglas S. Galvão  [  Alexandre
C. Dias  [  Elie A. Moujaes  [<EMAIL_ADDRESS>
###### Abstract

Two-dimensional carbon allotropes have attracted much attention due to their
extraordinary optoelectronic and mechanical properties, which can be exploited
for energy conversion and storage applications. In this work, we use density
functional theory simulations and semi-empirical methods to investigate the
mechanical, thermoelectric, and excitonic properties of Tetrahexcarbon (also
known as Tetragraphene). This quasi-2D carbon allotrope exhibits a combination
of squared and hexagonal rings in a buckled shape. Our findings reveal that
tetragraphene is a semiconductor material with a direct electronic bandgap of
$2.66\text{\,}\mathrm{eV}$. Despite the direct nature of the electronic band
structure, this material has an indirect exciton ground state of
$2.30\text{\,}\mathrm{eV}$, which results in an exciton binding energy of
$0.36\text{\,}\mathrm{eV}$. At ambient temperature, we obtain that the lattice
thermal conductivity ($\kappa_{L}$) for tetragraphene is approximately 118
$\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$. Young’s modulus and the shear
modulus of tetragraphene are almost isotropic, with maximum values of
$286.0\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ and
$133.7\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$, respectively, while exhibiting
a very low anisotropic Poisson ratio value of $0.09\text{\,}$.

##  1 Introduction

Two-dimensional (2D) carbon-based materials have attracted considerable
attention due to their extraordinary physical and chemical properties, which
have been exploited in flat electronics applications.1, 2 This fact is partly
due to the revolution in materials science created by the advent of
graphene.1, 2 Graphene has unique features combining high electrical
conductivity, robust mechanical strength, and excellent thermal properties.3,
4 These characteristics allow various applications, from organic electronics 3
to energy storage 5 and sensor technologies.3, 6

In addition to graphene, many other 2D carbon structures, including
graphyne,7, 8, 9 biphenylene monolayers,10, 11 amorphous carbon,12, 13 and
fullerene networks,14, 15, 16 were inspected, most of which have been
successfully synthesized. Despite these accomplishments, the search for novel
carbon allotropes continues with an emphasis on structures that could overcome
some specific limitations associated with graphene, mainly its zero electronic
bandgap, which limits its use in digital electronic devices.

Computational methodologies for material design have become an important and
reliable tool in research for various compelling reasons. Theoretical
predictions of novel nanostructured materials can precede their synthesis by
many years or even decades. Some examples include the biphenylene and the
fullerene networks, which were theoretically predicted in 1987 17 and 1990,18
respectively. In this sense, the ability to simulate and predict material
properties through computational models enables virtual experiments, thereby
speeding up the exploration and advancement of novel materials.

Recently, a novel quasi-2D carbon nanostructure named tetrahexcarbon was
introduced by Ram and Mizuseki.19 This structure exhibits a combination of
squared and hexagonal rings in a buckled shape (see Figure 1). Due to this
combination, an alternative name for the structure (tetragraphene 20) has been
proposed, and we adopted it in the present study. Tetragraphene contains
carbon atoms in $sp^{2}$ and $sp^{3}$ hybridizations. It is a semiconductor
material with a direct electronic bandgap of $3.70\text{\,}\mathrm{eV}$. It
also presents high electronic mobility ($1.46\times 10^{-4}$cmV-1s-1) and
relatively low cohesion energy.19

Investigations into the electronic properties of tetragraphene-like structures
have revealed metallic and semiconductor characteristics, with bandgap values
ranging from $0.82\text{\,}$ up to $4.4\text{\,}\mathrm{eV}$, depending on
structural parameters.20 Studies on the mechanical properties of
tetragraphene, using density functional theory (DFT) approaches, predicted
values of Young’s moduli along the zigzag and armchair directions of
$280\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}288\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$.19,
20, 21, 22 They also exhibit an intrinsic negative Poisson’s ratio and
predicted rupture stresses and strains along zigzag and armchair directions of
$40\text{\,}\mathrm{\char 37\relax}50\text{\,}\mathrm{\char 37\relax}$ and
$33\text{\,}\mathrm{\char 37\relax}40\text{\,}\mathrm{\char 37\relax}$,
respectively.21 Based on these unique characteristics, we comprehensively
explored tetragraphene’s thermoelectric and excitonic properties to address
its potential to serve as an active layer in thermal transport and energy
conversion applications. To our knowledge, no studies have been reported on
these relevant aspects of tetragraphene.

In this work, we use DFT simulations to investigate the mechanical,
thermoelectric, and excitonic properties of tetragraphene. Through our
computational approach, we obtained valuable information about this material’s
electronic and structural characteristics. In addition, we have also inspected
optical and vibrational properties, along with a discussion of their Raman
spectra.

##  2 Theoretical Methods

###  2.1 Computational details

The electronic band structure and density of states (DOS) were acquired using
the plane wave-based Quantum Espresso (QE) code.23 The structural geometry was
optimized using a variable cell relaxation procedure until the force acting on
each atom is less than
${10}^{-4}\text{\,}\mathrm{R}\mathrm{y}\mathrm{/}\mathrm{a}\mathrm{u}$ and the
pressure is not larger than $0.16\text{\,}\mathrm{kbar}$. Scalar relativistic
projector augmented wave (PAW) potentials 24, 25 within the Perdew-Burke-
Ernzerhof (PBE) parametrization 26 in the Generalized Gradient approximation
(GGA) 27 were used for the exchange-correlation functional. A cutoff energy of
$60\text{\,}\mathrm{R}\mathrm{y}$ and a 34$\times$34$\times$1 dense electronic
grid were sufficient to achieve a good convergence of the total energy value.
A $17\text{\,}\mathrm{\text{\r{A}}}$ vacuum buffer layer was added along the
non-periodic $z$ direction to avoid undesirable interactions of the unit cell
with its images. The electronic transport properties were obtained using the
BOLTZTRAP code28 with an electronic k-mesh of 60$\times$60$\times$1.

To improve the electronic bandgap description, which is underestimated in the
PBE framework due to self-interaction errors,29, 30 we used the hybrid
exchange-correlation functional proposed by Heyd–Scuseria–Ernzerhof
(HSE06).31, 32, 33 The Fock exchange was treated using the adaptive compressed
exchange (ACE) method 34 with a k-mesh of 6$\times$6$\times$1\. Due to the
heaviness of the calculation, we kept a 12$\times$12$\times$1 electronic
k-mesh to deal with the PBE part of the Hamiltonian. Also, we used the same
cutoff energy of the PBE simulation.

To analyze the structural stability of the tetragraphene (TG) monolayer, a
phonon dispersion spectra over a 6$\times$6$\times$1 phonon (q) grid was also
computed using a density function perturbation theory approach (DFPT),35, 36
implemented in QE. A 24$\times$24$\times$1 q-grid was used to obtain well-
converged phonon density of states (PhDOS). Using the QERAMAN code 37
interfaced with QE, first-order resonant Raman spectra at various laser
energies were also estimated.

The elastic constants of the TG monolayer were determined using the
THERMO$\\_$PW code,38, 39, which is also interfaced with QE. The code
generally applies strain along several directions of the material and fits
several strain-stress curves to extract the corresponding elastic constants.
For the case of the 2D TG, a total of twelve relaxation steps and an
electronic 34$\times$ 34$\times$1 grid produced converged values.

To probe the excitonic effects, we used a maximally localized Wannier
functions tight binding (MLWF-TB) Hamiltonian to deal with quasi-particle
effects and optical properties, as implemented in the WanTiBEXOS code.40 The
MLWF-TB Hamiltonian was obtained at the HSE06 level from the Wannier90
package.41 The Wannierization procedure was performed considering the $s$ and
$p$ orbitals of the carbon atoms. The Bethe–Salpeter equation (BSE) 42, 43 was
solved using a Coulomb truncated 2D potential (V2DT) 44 and a k-mesh of
33$\times$33$\times$1\. For the BSE solution, the highest three valence bands
and the lowest two conduction bands were taken into account. A smearing of
$0.05\text{\,}\mathrm{eV}$ was utilized to compute the absorption coefficient
at the BSE and independent particle approximation (IPA) levels.

##  3 Results

###  3.1 Structural Properties

Figure 1:  (a) Top and side views of a TG monolayer. The red parallelogram
indicates the primitive unit cell. (b) The first Brillouin zone and its
respective high symmetry k-points.

The top and side views of the TG monolayer are shown in Fig 1(a), and the
corresponding Brillouin zone (BZ) with the high symmetry points are displayed
in Fig 1(b). The primitive unit cell contains six C atoms and the structure
belongs to the $Cmm2$ space group, with a lattice constant of
$3.79\text{\,}\mathrm{\text{\r{A}}}$ and an angle of
$73.19\text{\,}\mathrm{\SIUnitSymbolDegree}$ between the in-plane lattice
vectors. Unlike graphene, the TG monolayer is not planar and has a buckling
distance of $0.58\text{\,}\mathrm{\text{\r{A}}}$ between two out-of-plane
carbon atoms.

Figure 2: A variable cell relaxation simulation shows that a 1OT carbon
structure is unstable, transitioning to the more energy-favorable TG
structure.

Another new 2D structure with a unit cell containing six atoms is the recently
predicted 1OT structure 45 of transition metal dichalcogenides (TMDs), namely
PdS2, PdSe2, and PdSSe, whose excitonic properties have been recently
investigated.46 Its BZ is a tilted irregular hexagon, a signature of the
monoclinic spacegroup (Fig. 2). We tested the carbon version of this topology
using a variable cell relaxation simulation. We observed that the C-1OT system
is unstable and undergoes a phase transition to the TG structure (Fig. 2).

The cohesive energy (Ecoh) of tetragraphene is determined by the values of its
total energy and the energy of each carbon atom in the unit cell. In eV/atom:

$E_{\rm{coh}}=\frac{E_{TG}-6E_{C}}{6}\leavevmode\nobreak\ ,$

where $E_{TG}$ is the total energy, and EC is the energy of an isolated C
atom. We obtained the value of
$-8.36\text{\,}\mathrm{eV}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$,
indicating that the structure is energetically stable. This value is slightly
larger than that of graphene
($-9.23\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
eV/atom) 47 and graphyne
($-8.582\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
),48 but closer to the one for penta-graphene
($-8.324\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$).49

Figure 3:  2D polar representation of (a) Young’s modulus (N/m), (b) Shear
modulus (N/m), and (c) Poisson’s ratio in the $xy$ plane for the TG monolayer.

###  3.2 Elastic constants and mechanical stability

The response of any material to the deformations under which it is subjected
depends on several parameters, including its elastic constants. These
constants are also important to estimate its mechanical stability. For a 2D
system, in specific, the in-plane related elastic constants are the most
important, namely Cij where {i,j=1,2,6}. For the TG monolayer, we obtained
C11=$158.1\text{\,}\mathrm{GPa}$, C22=$155.1\text{\,}\mathrm{GPa}$,
C12=$9.4\text{\,}\mathrm{GPa}$, and C66=$69.4\text{\,}\mathrm{GPa}$. Since
C11, C22, C66 > 0 and C12 < C11 and C22, the material can be considered
structurally stable. To obtain vacuum-independent results, the values have to
be multiplied by 0.1$c$, where $c$ is the distance along the non-periodic $z$
direction. We further obtained
C11=$287.1\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ N/m,
C22=$281.7\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$,
C12=$17.1\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$, and
C66=$126.0\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$, which are near to the
values reported in the work of Wei et al. .21

To verify TG’s structural anisotropy, orientation-dependent parameters should
be determined. In particular, we evaluated Young’s modulus (Y), the shear
modulus (G), and the Poisson ratio ($\nu$) as a function of an in-plane angle
$\theta$ (0$\leq\theta\leq 2\pi$), and the different compliance’s
Sij=C${}_{ij}^{-1}$ {i,j=1,2,6}50, 51:

$\displaystyle\frac{1}{{\rm{Y}}(\theta)}$ $\displaystyle=$ $\displaystyle
S_{11}\cos^{4}{\theta}+S_{22}\sin^{4}{\theta}$ $\displaystyle+$ $\displaystyle
2S_{16}\cos^{3}{\theta}\sin{\theta}$ $\displaystyle+$ $\displaystyle
2S_{26}\cos{\theta}\sin^{3}{\theta}$ $\displaystyle+$
$\displaystyle(2S_{12}+S_{66})\cos^{2}{\theta}\sin^{2}{\theta}.$

$\nu(\theta)=-{\rm{Y}}(\theta)(A+B\psi).$ (2)

$\frac{1}{4{\rm{G}}(\theta)}=C+D\cos{(4\phi+2)}.$ (3)

$A=\Big{[}\Big{(}S_{11}+S_{22}-S_{66}\Big{)}/2+3S_{12}\Big{]}/4,$ (4)

$B=\frac{\sqrt{\Big{(}S_{26}-S_{16}\Big{)}^{2}+\Big{(}S_{12}-\frac{\Big{(}S_{11}+S_{22}-S_{66}\Big{)}}{2}\Big{)}^{2}}}{4},$
(5)

$C=\frac{S_{11}+S_{22}-2S_{12}+S_{66}}{8},$ (6)

$D=\frac{\sqrt{\Big{(}S_{26}-S_{16}\Big{)}^{2}+\frac{\Big{(}S_{66}+2S_{12}-S_{11}-S_{22}\Big{)}^{2}}{4}}}{4}.$
(7)

$\psi$ and $\phi$ are defined in a way that:

$\tan{\psi}=\frac{S_{26}-S_{16}}{S_{12}-\frac{\Big{(}S_{11}+S_{22}-S_{66}\Big{)}}{2}}$
(8)

$\tan{\phi}=\frac{2\Big{(}S_{16}-S_{26}\Big{)}}{S_{66}+2S_{12}-S_{11}-S_{22}}\leavevmode\nobreak\
.$ (9)

From Fig. 3, we realize that both Y and G are almost isotropic, with maximum
values of $286.0\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ and
$133.7\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$, respectively. However, the
polar graph of the Poisson’s ratio ($\nu$) shows an anisotropic behavior
reaching a maximum value of 0.09 for $\theta$=45o, 135o, 225o, and 315o.
Having $\nu$ < 0.5 satisfied further evidences the structural stability of the
TG monolayer.

###  3.3 Electronic Properties

Figure 4:  (Left panel) Tetragraphene monolayer electronic band structure at
the PBE (blue curves) and HSE06 (red curves) levels. (Right panel) Projected
Density of States (PDOS) onto the carbon’s $s$ and $p$ orbitals. The Fermi
level value is set at $0\text{\,}\mathrm{eV}$.

The electronic band structure in the left panel of Fig. 4 shows that the TG
monolayer has a PBE(HSE06) direct band gap of
$1.63\text{\,}\mathrm{eV}$($2.66\text{\,}\mathrm{eV}$) localized at the high
symmetry $\Gamma$ point. The projected density of states (PDOS) at the PBE
level, shown in the right panel of Fig. 4, demonstrates that the top valence
band (TVB) has a majority contribution from the $p_{z}$ orbital followed by a
contribution from the $s$ orbital. However, the main contribution of the
bottom conduction band (BCB) comes from the $p_{x}$ orbital, but also with
contribution from the $s$ and $p_{y}$ orbitals. These features differ from
those observed in graphene, where the $p_{z}$ orbitals majorly compose these
states. This difference can be attributed to the buckling in the tetragraphene
case, which additionally has a $\sigma$-bond character in the carbon-carbon
interactions, in contrast to the $\pi$ electrons contributing to the BCB and
TVB in graphene.

###  3.4 Vibrational properties and Raman spectra

The phonon dispersion of the TG monolayer along the $\Gamma$-C-Y-V-$\Gamma$-Y
symmetry path is displayed in Fig. 5. Since the structure has six carbon atoms
in its primitive unit cell, we expect eighteen vibrational modes, three of
which are acoustic and the other fifteen optical. No negative phonon
frequencies were observed along the selected path, consistent with a
structurally stable structure. The flexural mode (ZA), the lowest phonon
branch, follows a parabolic dispersion in the vicinity of the $\Gamma$ point,
a signature of 2D materials, 52, 53, 54, 55, 56, 57 including graphene.58
Conversely, the longitudinal (LA) and transverse (TA) acoustic modes show a
linear dispersion behavior.

Two interesting facts can be inferred from the phonon dispersion branches.
Firstly, an optical-optical phonon bandgap of $\sim$
$296\text{\,}{\mathrm{cm}}^{-1}$ can be observed between the low and high
optical modes. The ability to tune such a bandgap by applying strain or
stacking up more layers can be an essential feature in applications involving
selective frequency transmission. Secondly, the highest four optical modes are
flatter (low dispersion) than the others, translating into sharp peaks in the
material’s phonon DOS (PhDOS). Consequently, such modes have a small group
velocity and, thus, do not significantly affect the thermal phonon transport
properties.

Figure 5: Phonon dispersion (left panel) of the TG monolayer and its phonon
DOS (right panel). Figure 6: Optical phonon modes, at the $\Gamma$ point, of a
TG monolayer. The arrows indicate the directions of the atoms’ displacements.

Fig. 6 exhibits the fifteen optical modes of the TG monolayer. Optical modes 3
and 4 are characterized by almost perfect perpendicular vibrations of four of
the six carbon atoms, while all atoms vibrate perpendicularly in the seventh
mode. Nevertheless, most of the remaining modes have their carbon atoms
vibrating within the plane of the structure, with small motions along the
perpendicular direction, except for modes 5, 6, and 9.

Not all phonon modes are Raman (R) or Infrared active (IR). Fig. 7 displays
the calculated Raman spectra for linear and circular polarizations and at
different laser energy values (Elaser), ranging from the far IR to the
ultraviolet (UV) regime. Three peaks occur at
$688.46\text{\,}{\mathrm{cm}}^{-1}$, $1007.6\text{\,}{\mathrm{cm}}^{-1}$, and
$1700.58\text{\,}{\mathrm{cm}}^{-1}$. We realize that the first and third
peaks remain intense enough for the different laser energies. Yet, for values
larger than or equal to $2.33\text{\,}\mathrm{eV}$ and as Elaser increases,
the second peak’s intensity fades away, becoming negligible compared to the
others. The first peak could be attributed to the fifth mode in Fig. 6, while
the second peak could be ascribed to the ninth mode. The third peak
corresponds to the fifteenth optical mode.

Naively, we might consider that all modes with zero intensity in the IR are
Raman active. To ensure that only the modes mentioned above are Raman active,
we calculated the intensity of the IR modes (Fig. 8). Our calculations reveal
that optical modes 3, 7, 12, and 14 are IR-active. We thus speculate that the
remaining eight modes are silent; they are neither IR nor Raman active.

Figure 7: Raman spectra with laser excitation energies in the infrared,
visible, and ultraviolet regimes for linearly (XX, XY) and circularly
($\sigma$+$\sigma$+, $\sigma$+$\sigma$-) polarized light. “I” refers to the
intensity of the peaks in arbitrary units. Figure 8: The IR spectrum of the TG
monolayer and its corresponding vibrational modes.

###  3.5 Thermoelectric Properties

In Fig. 9-(a), we present the electrical conductivity tensor $\sigma^{B}$
calculated using the Boltztrap code without carrier excess, with a null
chemical potential ($\mu=0$), as a function of temperature, considering the X
and Y directions within the 2D plane. The electrical conductivity is almost
zero for temperatures below $600\text{\,}\mathrm{K}$. This behavior is
characteristic of semiconductors, as electrons would acquire sufficient energy
to overcome the energy gap barrier only above a specific temperature threshold
. We also observe that the electrical conductivity along both directions
within the plane is nearly identical, with a slight difference at higher
temperatures. Hence, as far as electrical transport is concerned, the system
is essentially isotropic.

Figure 9: (a) Electrical conductivity tensors as a function of temperature,
considering the X and Y directions within the 2D plane. (b) The Arrhenius plot
required to determine the relaxation time, as described in reference 59.

Fig. 9-(b) demonstrates the Arrhenius plot considering the electrical
conductivity presented in Fig. 9-(a). From the plot’s slope, we have estimated
that tetragraphene’s bandgap is approximately $1.7\text{\,}\mathrm{eV}$, a
value very close to that deduced from the electronic band structure.
Constructing the Arrhenius plot enables us to determine the linear
coefficient, discussed in the work of Tromer et al. 59, needed to estimate the
electronic relaxation time $\tau$ as:

$\tau(T)=\frac{\Gamma(T_{0})}{\Gamma(T)}\frac{ln(\Gamma(T_{0}))}{ln(\Gamma^{Ref}(T_{0}))}\tau^{Ref.}(T_{0}),$
(10)

where $\tau^{Ref}$ is a previously calculated reference value for nitrogen-
doped graphene at the reference temperature $T_{0}=350$ $\mathrm{K}$. We will
use the values $\tau^{Ref.}(350)=14.3$ $\mathrm{fs}$ and,
$\Gamma^{Ref.}(350)=11.997$, from Ref. 59. The quantity $\Gamma$ is an
auxiliary function given by:

$\Gamma(T)=ln\bigg{[}\frac{\sigma^{B}(T)}{\beta}\bigg{]}.$ (11)

Once we have confirmed the system is isotropic, we will consider the average
quantities along the 2D plane, where
$\sigma=\frac{(\sigma^{B}_{XX}+\sigma^{B}_{YY})}{2}$ and
$\beta=\beta_{XX}=\beta_{YY}$ is an estimated parameter derived from the
BoltzTraP code. Fig. 10 shows the electronic relaxation time ($\tau$) as a
function of temperature (T), using the scheme discussed previously. We note
that $\tau(\rm{T})$ presents an exponential decay behavior, as expected for 2D
systems.

Figure 10: Electronic relaxation time ($\tau$) as a function of temperature of
the TG monolayer. Figure 11: Lattice thermal conductivity $\kappa_{L}$ of the
TG monolayer as a function of temperature.

Another critical parameter required for analyzing the thermoelectric
properties of a particular system is the lattice thermal conductivity
($\kappa_{L}$). Using the method outlined in Ref.60, we estimate the value for
tetragraphene to be approximately $\kappa_{L}(300)=118$
$\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$. This value is significantly lower
than that of graphene ($3084.6$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$),
graphenylene ($600.0$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$), and
T-graphene ($800.0$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$), but
comparable to the thermal conductivity of the biphenylene network ($208.0$
$\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$) 59.

To determine the values of $\kappa_{L}$ at higher temperatures, we assume that
its temperature dependence follows a power-law relationship with an exponent
equal to $-1$, a characteristic of 2D systems.61 We will ,therefore, use a
straightforward heuristic approach by defining the function as $118\times$
($300$/T), with T ranging from $300$ up to $800$. The values of $\kappa_{L}$
for different temperatures are depicted in Fig. 11.

Figure 12: Thermoelectric coefficients as a function of positive (electron
doping) and negative (hole doping) values of the chemical potential at
different temperatures.

Fig. 12 depicts the various thermoelectric properties as a function of a
varying chemical potential $\mu$ (Fermi level at zero) at
$350\text{\,}\mathrm{K}$, $550\text{\,}\mathrm{K}$, and
$800\text{\,}\mathrm{K}$. Negative chemical potential values correspond to
hole doping, while positive values denote electron doping. It is important to
note that the electrical conductivity $\sigma$, the electronic contribution to
thermal conductivity $\kappa^{e}$, and consequently, the power factor $PF$
($S^{2}\times\sigma$) have been adjusted by the relaxation time values.

In Fig. 12-(a), we observe a notable null region near the Fermi level for the
electrical conductivity. This region corresponds to the energy gap of the
material, where electrons lack sufficient energy to transition from the
valence band to the conduction band. Only for slightly higher chemical
potential values does the electrical conductivity exhibit non-zero values due
to the formation of excitons. Additionally, the material displays more
efficient hole transport than electron transport, maximizing $\sigma$ in the
case of holes. Furthermore, the electrical conductivity decreases with
increasing temperature, which is attributed to the fact that at higher
temperatures, the number of charge carriers increases, affecting the mobility
and reducing the electrical conductivity.

The Seebeck coefficient $S$ is presented in Fig. 12-(b), which physically
represents the magnitude of the thermoelectric voltage generated in response
to a given temperature. Notably, the figure shows a nearly symmetrical
behavior with respect to $\mu=0$. Moreover, the highest contribution to the
Seebeck coefficient occurs in regions around the Fermi level.

The contribution of electrons/holes to the material’s thermal electronic
conductivity is shown in Fig. 12-(c). Compared to the lattice thermal
conductivity shown in Fig. 11, we observe that the contribution of
electrons/holes is at least one order of magnitude lower. Hence, phonons play
a substantially significant role in the material’s thermal conductivity. To
maximize the material’s thermoelectric efficiency, exerting control over
phonons is paramount.

A critical parameter in the context of thermoelectric materials and devices,
which significantly influences their performance, is the power factor
($PF=S^{2}\times\sigma$), presented in Fig. 12-(d). For the TG monolayer, we
observe that the power factor reveals a peak efficiency when the chemical
potential approaches approximately $\pm 1$$\mathrm{V}$ due to the distinct
contributions from holes and electrons. Notably, the maximum power factor
occurs at different temperatures for holes and electrons, with
$350\text{\,}\mathrm{K}$ and $800\text{\,}\mathrm{K}$ being the optimal
temperatures. These findings indicate the critical influence of the chemical
potential and temperature on thermoelectric performance, highlighting the
importance of carefully considering these factors in designing and optimizing
thermoelectric materials and devices.

The power factor provides valuable insights into a material’s suitability for
thermoelectric applications by highlighting its ability to convert thermal
gradients into electrical power. However, it is important to note that while a
high $PF$ is desirable, it does not provide a complete picture of a material’s
overall thermoelectric efficiency. Other factors, such as thermal conductivity
and device design, must also be considered to fully determine the performance
of thermoelectric systems.

The quantity that comprehensively determines the thermoelectric conversion
efficiency is known as the figure of merit, denoted as ZT, and is evaluated as
follows:

$ZT=\frac{S^{2}\sigma}{(\kappa^{e}+\kappa_{L})}T.$ (12)

Note that $ZT$ is the ratio of the power factor quantity and the thermal
conductivity multiplied by the temperature parameter $T$.

Figure 13: Figure of merit ZT as a function of the chemical potential for
different temperatures $T$.

Finally, Fig. 13 portrays the variation of ZT with the chemical potential at
three disparate temperatures for the TG monolayer. Akin to the case of the PF,
the peak values are also located at chemical potential values of approximately
$\pm 1$. These values are highly temperature-dependent and tend to increase
with increasing temperature. This behavior is due to the substantial decrease
in the thermal conductivity of the lattice $\kappa_{L}$ with increasing
temperature, as observed in Fig. 11. The maximum efficiency for holes ($0.04$)
is nearly twice that obtained for electrons ($0.02$) at $T=800$ $\mathrm{K}$.
At lower $T$ values, the peak intensities are similar. Still, those associated
with holes are always slightly larger.

###  3.6 Excitonic and Optical properties

Figure 14: The excitonic band structure (DFT-HSE06-BSE) of the Tetragraphene
monolayer.

As extensively reported, excitonic effects tend to be significant in 2D
materials 43, 62, 63 considering the description of the optical absorption
spectrum. From the TG’s excitonic band structure in Fig. 14, we can observe
that despite the direct nature of the electronic band structure, we have an
indirect exciton ground state of $2.30\text{\,}\mathrm{eV}$, between the
$\Gamma$ and Y symmetry points, resulting in an excitonic binding energy of
$0.36\text{\,}\mathrm{eV}$.

This nontrivial, indirect excitonic behavior can be explained by the electron-
hole pair orbital characteristics combined with the Coulomb interaction that
binds both, which also depends on their momentum. An analogous behavior, where
the bandgap is direct and the excitonic ground state is indirect, has been
reported in the literature.64, 65 The direct exciton ground state is
bright,i.e, optically active, with an energy of $2.41\text{\,}\mathrm{eV}$.
The indirect nature of the fundamental excitonic ground state enables phonon-
assisted optical transitions with photon excitation energy lower than the
optical band gap of $2.41\text{\,}\mathrm{eV}$. However, since the difference
between the direct and indirect excitonic ground states is low ($\approx$
$0.09\text{\,}\mathrm{eV}$), other optical band gap energy variations are not
expected for phonon-assisted transitions.

Figure 15: TG monolayer optical properties: (a) absorption coefficient, (b)
refractive index, (c) reflectivity; at BSE (solid curves) and IPA (dashed
curves) levels, considering linear $\hat{x}$ (blue curves) and $\hat{y}$ (red
curves) polarizations.

The optical linear response, at IPA and BSE approximations, represented by the
absorption coefficient, refractive index, and reflectivity, are outlined in
Fig. 15 (a), (b), and (c), respectively. From Fig. 15(a), we can observe that
the quasi-particle effects not only result in a redshift of the spectrum but
also change the absorption intensity of the peaks, which are smaller at the
BSE level. The system shows a small optical anisotropy independent of BSE or
IPA levels, resulting in a different peak absorption intensity for $\hat{x}$
and $\hat{y}$ light polarizations. Furthermore, the optical bandgap from BSE
($2.41\text{\,}\mathrm{eV}$) and IPA ($2.66\text{\,}\mathrm{eV}$) calculations
remain the same.

Fig. 15(b) shows that at $0\text{\,}\mathrm{eV}$, the refractive index at the
IPA and BSE levels is approximately $1.47$. The quasi-particle effects seem to
enhance (reduce) the refractive index for optical excitations lower (higher)
than $4.0\text{\,}\mathrm{eV}$. A similar comportment is shown in the
reflectivity (Fig. 15(b)), which achieves a maximum value of $0.07$ ($0.09$)
at the BSE (IPA) level, meaning that less than $10\text{\,}\mathrm{\char
37\relax}$ of the light is reflected off this material.

##  4 Conclusions

In this work, we have carried out a complete investigation into the
mechanical, thermoelectric, and excitonic properties of a tetragraphene
monolayer using DFT simulations and semiempirical methods. This unique
material is characterized by a quasi-2D structure with a combination of
squared and hexagonal rings built in a buckled shape. Our findings identify
the tetragraphene monolayer as a direct bandgap semiconductor with a bandgap
value of about $2.66\text{\,}\mathrm{eV}$ at the HSE06 level. It also exhibits
an indirect excitonic ground state with an energy of
$2.30\text{\,}\mathrm{eV}$, leading to an excitonic binding energy of
$0.36\text{\,}\mathrm{eV}$. Furthermore, the lattice thermal conductivity of
tetragraphene is approximately $\kappa_{L}(300)=118$
$\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$. This value is significantly lower
than that of graphene ($3084.6$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$),
graphenylene ($600.0$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$), and
T-Graphene ($800.0$ $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$), but
comparable to the thermal conductivity of the biphenylene network ($208.0$
$\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{K}$). Monolayer tetragraphene is
mechanically stable, possessing an ultralow Poisson ratio of 0.09. Regarding
the thermoelectric properties, our results show that the maximum figure of
merit (efficiency) occurs for holes at an energy of $-1\text{\,}\mathrm{eV}$
below the Fermi level. The same pattern is preserved at higher temperatures
but with larger values of ZT reaching 0.04 at T=$800\text{\,}\mathrm{K}$ in
the case of hole doping.

The authors are grateful for the computational resources provided by Centro
Nacional de Processamento de Alto Desempenho in São Paulo-CENAPAD-SP (proj
634, 897, and 909) and the Lobo Carneiro HPC (NACAD) at the Federal University
of Rio de Janeiro (UFRJ) (proj 133 and 135). A.C.D. also thanks the financial
support from National Council for Scientific and Technological Development
(CNPq, grant number $408144/2022-0$) and Federal District Research Support
Foundation (FAPDF, grants number $00193-00001817/2023-43$ and
$00193-00002073/2023-84$). L.A.R.J. acknowledges the financial support from
FAP-DF grants $00193.00001808/2022-71$ and $00193-00001857/2023-95$ and FAPDF-
PRONEM grant $00193.00001247/2021-20$, and CNPq grant $350176/2022-1$. RMT and
DSG acknowledge support from CNPq and the Center for Computational Engineering
and Sciences at Unicamp, FAPESP/CEPID Grant ($2013/08293-7$).

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