###  VII-L Space Assets

The monitoring of UAV can also be performed using sensors in space, such as
radar, imaging, and infrared sensors [263, 264]. The space-based sensors can
provide early warning against several different aerial threats e.g. UAV
swarms. The sensors placed in space have a long-range and wide coverage area
and have added advantage of mobility. However, space sensors need to take into
account the atmospheric, and space effects, and long delays.

###  VII-M Exploring Cost Effective Solutions

To counter malicious UAVs, cost-effective solutions should be explored. For
example, a cost-effective solution is to use simulators. Obtaining data from
real-time measurements is time-consuming and expensive. Moreover, it is
difficult to cover different possible scenarios. On the other hand, simulators
imitating real-world scenarios can be used to generate large amounts of data
in different scenarios at a low cost. The data obtained from simulators can be
used to train the radar systems for DCT in real-time.

##  VIII Concluding Remarks

In this survey paper, we conduct an in-depth analysis of the capabilities,
threats, and challenges associated with UAVs. We offer a comprehensive
overview of three RF-based systems: radars, communication systems, and RF
analyzers for DCT-U. Our survey encompasses fundamental radar processing and
discusses both conventional and modern radar systems for DCT-U. Additionally,
we explore JC&S in active mode and passive radars based on communication
systems for DCT-U. We also discuss various methods for capturing RF signals of
interest emitted by UAVs for DCT-U. The paper addresses the limitations of RF-
based systems for DCT-U and provides insights into future directions based on
lessons learned.

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Thermodynamic Properties of q-deformed massless Dirac fermions in graphene
with Rashba coupling

Rachid Houça***houca.rachid@gmail.com1,2, El Bouâzzaoui
<EMAIL_ADDRESS>Abdelhadi<EMAIL_ADDRESS>Abdellatif Kamal§§§abdellatif.kamal@ensam-casa.ma2,3 and Mohammed El
<EMAIL_ADDRESS>
1Team of Theoretical Physics and High Energy, Department of Physics, Faculty
of Sciences, Ibn Zohr University, Agadir, Morocco,

PO Box 8106, Agadir, Morocco

2Team of Theoretical Physics, Laboratory L.P.M.C., Department of Physics,
Faculty of Sciences, Chouaib Doukkali University, El Jadida, Morocco,

PO Box 20, 24000 El Jadida, Morocco

3Department of Mechanical Engineering, National Higher School of Arts and
Crafts, Hassan II University, Casablanca, Morocco

We study the thermodynamic properties of massless Dirac fermions in graphene
under a uniform magnetic field and Rashba spin-orbit coupling with a
$q-$deformed Heisenberg algebra calculus. The thermodynamic functions such as
the Helmholtz free energy, total energy, entropy and heat capacity are
obtained by using an approach based on the zeta function and Euler-Maclaurain
formula. These functions will be numerically examined for different values of
$\eta={1\over i}\ln(q)$. In particular, the heat capacity in the presence of
deformation, all curves coincide and reach the fixed value $C=6K_{B}$ three
times greater compared to the case of undeformed massless Dirac fermions in
graphene.

PACS numbers: 65.80.Ck, 03.65.-w

Keywords: Graphene, Rashba coupling, zeta function, Euler Maclaurain formula,
partition function, thermodynamic functions, q-deformed algebra.

##  1 Introduction

Graphene, which is an elemental sheet of graphite, consists of a periodic,
two-dimensional arrangement of carbon atoms of monoatomic thickness with a
honeycomb structure. It is the latest member of the carbon allotropic family:
diamond, graphite, $C60$ fullerenes [1] and nanotubes [2]. For the first time
in $2004$, a graphene sheet stable at room temperature was obtained physically
by A. Geim and K. Novoselov [3]. This experiment contradicted the theory that
a graphene sheet was thermodynamically unstable. As this new material
developed by mechanical exfoliation has remarkable and unique properties, they
were awarded the Nobel Prize in Physics in $2010$. Since this discovery,
graphene has been the material most studied by the scientific community for
its new and unique physical properties. Indeed, it has a high superior
electrical mobility [4, 5], an anomalous quantum Hall effect [6], a modulable
band gap [7] and it is a transparent conductor [8] since, in the optical
region, it absorbs only $2.3\%$ of light. It also has good flexibility [9] and
excellent mechanical strength, and its thermal conductivity is ten times
higher than that of copper [10].

The more general framework of the $q-$deformation theory for a real parameter
$q$ has found great success and has attracted considerable attention from
physicists and mathematicians. The interesting physical application was
started by the introduction of the $q-$deformed harmonic oscillator by
Biedenharn [11] and Macfarlane [12] in $1989$. Quantum mechanics can be
considered as a deformation (the deformation parameter is $\hbar$) of
classical mechanics and relativistic mechanics is another deformation (with
$1\over c$ as the deformation parameter) of classical mechanics. In the same
sense, quantum mechanics can be seen as the limit of a more general theory
depending on one or more deformation parameters.

The study of the dynamic behavior of systems is a central question in physics
and mathematics. These systems provide fundamental and general results which
have found major applications not only in physics, but also in all other
branches of science, as well as in technology; however, the harmonic
oscillator is the simplest and most fundamental theoretical model of
mechanical and electrical oscillatory phenomena. The definitions and main
properties of independent and time dependent harmonic oscillators and damped
harmonic oscillators have been studied by several authors [13, 14].

It was lately shown that the regular standard thermodynamics of Boltzmann-
Gibb’s statistics are no longer suitable for studying all physical systems,
including the attitude of complex systems controlled by the Tsallis non-
extensive statistics [15] and non-equilibrium statistics of the q-deformed
superstatistics [16, 17]. The concept of superstatistics was first developed
by Wilk and Wlodarczyk [18] before Beck and Cohen [19] latter reworded the
theory.

Motivated by the work done on thermodynamic proprieties under magnetic field
and Rashba coupling [23] we will generalize this last work by introducing the
notion of the $q$-deformed harmonic oscillator and see the influence of the
parameter $q$ on the various thermodynamic quantities. For this, we consider a
massless Dirac fermions in monolayer graphene with the magnetic field applied
perpendicular to the graphene layer. Through investment of The $q$-deformed
algebra of the quantum oscillator defined by q-deformed Heisenberg algebra we
express our Hamiltonian in terms of creation and annihilation operators to
obtain the solutions of the energy spectrum, this last will be used to
determine the partition function which will help us to calculate and plot
numerically the different thermodynamic functions in order to make
conclusions. The present paper is organized as follows. In section 2, we give
an overview on q-deformed Heisenberg algebra which serves to determine,
explicitly, the exact eigenvalues in terms of $q$-deformed parameter. In
section 3, we will look for the partition function which will be the key to
determine the different thermodynamic functions such as the free energy of
Helmholtz, internal energy, entropy and heat capacity. section 4, will be
devoted to the numerical results and discussions as well as comparison with
literature. We conclude our results in the final section.

##  2 Theoretical model

###  2.1 q-deformed quantum theory

The q-deformed algebra of the quantum oscillator is defined by q-deformed
Heisenberg algebra in terms of creation and annihilation operators
$a^{\dagger}$ and $a$, respectively, and number operator $N$ by [20, 21, 22]

$[a,a]=[a^{\dagger},a^{\dagger}]=0,\quad[a,a^{\dagger}]_{q}=aa^{\dagger}-q^{-1}a^{\dagger}a=q^{N},\quad[N,a^{\dagger}]=a^{\dagger},\quad[N,a]=-a$
(1)

where deformation parameter $q$ is real and the observed value of q has to
satisfy the non-additivity property

$[x+y]\neq[x]+[y]$ (2)

In addition, the operators obey the relations

$[N]=a^{\dagger}a,\quad[N+1]=aa^{\dagger}$ (3)

The $q-$Fock space spanned by orthornormalized eigenstates $|n\rangle$ is
constructed according to

$|n\rangle={(a^{\dagger})^{n}\over\sqrt{[n]!}}|0\rangle,\quad a|0\rangle=0$
(4)

Both $q-$factorial and $q-$numbers are defined, respectively, by

$[n]!=[n][n-1][n-2]\cdots[1],\quad[n]={q^{n}-q^{-n}\over q-q^{-1}}$ (5)

For $n\in\mathbb{N}$ with $[0]!=1$. The eigenvalues of the $q-$deformed one-
dimensional harmonic oscillator are

$E_{n}={\hbar\omega\over 2}\left([n]+[n+1]\right)$ (6)

Considering the definition of basic number given in (5), and making
$q=e^{i\eta}$, the eigenvalues become

$E_{n}={\hbar\omega\over 2}{\sin[\eta(n+{1\over 2})]\over\sin[{\eta\over 2}]}$
(7)

###  2.2 Eigenvalue problem

We consider a massless Dirac fermions in monolayer graphene with the magnetic
field applied perpendicular to the graphene layer. Low energy quasiparticles
in graphene with Rashba spin orbit coupling (RSOC) interaction can be well
described by the Dirac-type Hamiltonian

$H=v_{F}\left(\eta\sigma_{x}\pi_{x}+\sigma_{y}\pi_{y}\right)+\lambda_{R}\left(\eta\sigma_{x}s_{y}-\sigma_{y}s_{x}\right)$
(8)

where the conjugate momentum $\pi_{x}$ and $\pi_{y}$ can be written in
symmetric gauge $\vec{A}=\frac{B}{2}(-y,x)$ as

$\pi_{x}=p_{x}-\frac{eB}{2}y,\qquad\pi_{y}=p_{y}+\frac{eB}{2}x.$ (9)

Where $B$ and $v_{F}=10^{6}m/s$ are respectively the uniform magnetic field
and the Fermi velocity, the parameter $\eta=\pm 1$ labels the valley degrees
of freedom, $\sigma=\left(\sigma_{x},\sigma_{y}\right)$ are the Pauli matrices
of pseudospin operator on $A(B)$ lattice cites. The present system presents
the intrinsic spin orbit coupling (SOC), but its value is very weak compared
to the RSOC [23], for this we have neglected it because it will not influence
on the physical properties of the studied system.

Fixing a certain intra-Landau-level quantum number, we denote by
$|r_{A,B},n,\sigma\rangle={(a^{\dagger})^{n}\over\sqrt{[n]!}}|r_{A,B},n,\sigma\rangle$
a state in the nth Landau level with spin direction
$\sigma\in\\{\uparrow,\downarrow\\}$ , and all other eigenstates are of the
form
$|\Psi\rangle=(|r_{A},n,\uparrow\rangle,|r_{B},n-1,\downarrow\rangle,|r_{B},n,\uparrow\rangle,|r_{A},n-1,\downarrow\rangle)^{t}$.
The Hamiltonian (8) around a single Dirac point $(\eta=+1)$ with these
considerations is given by

$H=\left(\begin{array}[]{cccc}0&0&v_{F}\left(\pi_{x}-i\pi_{y}\right)&0\\\
0&0&0&v_{F}\left(\pi_{x}+i\pi_{y}\right)\\\
v_{F}\left(\pi_{x}+i\pi_{y}\right)&0&0&-2i\lambda_{R}\\\
0&v_{F}\left(\pi_{x}-i\pi_{y}\right)&2i\lambda_{R}&0\\\ \end{array}\right).$
(10)

To diagonalize the Hamiltonian (10), it is convenient to introduce the usual
bosonic operators in terms of the conjugate momentum

$a=\frac{\ell_{B}}{\sqrt{2}\hbar}\left(\pi_{x}-i\pi_{y}\right),\qquad
a^{\dagger}=\frac{\ell_{B}}{\sqrt{2}\hbar}\left(\pi_{x}+i\pi_{y}\right)$ (11)

which verify the commutation relation $[a,a^{\dagger}]=q^{N}$,
$\ell_{B}=\sqrt{\frac{\hbar}{eB}}$ is the magnetic length. Express (10) in
terms of $a$ and $a^{\dagger}$ to obtain

$H=\left(\begin{matrix}0&0&\frac{\sqrt{2}\hbar v_{F}}{\ell_{B}}a&0\\\
0&0&0&\frac{\sqrt{2}\hbar v_{F}}{\ell_{B}}a^{\dagger}\\\ \frac{\sqrt{2}\hbar
v_{F}}{\ell_{B}}a^{\dagger}&0&0&-2i\lambda_{R}\\\ 0&\frac{\sqrt{2}\hbar
v_{F}}{\ell_{B}}a&2i\lambda_{R}&0\\\ \end{matrix}\right).$ (12)

To find the solution of the energy spectrum we act the Hamiltonian on the
state $|\Psi\rangle$ leading to the eigenvalue equation

$\left(\begin{matrix}-E&0&\frac{\sqrt{2}\hbar v_{F}}{\ell_{B}}a&0\\\
0&-E&0&\frac{\sqrt{2}\hbar v_{F}}{\ell_{B}}a^{\dagger}\\\ \frac{\sqrt{2}\hbar
v_{F}}{\ell_{B}}a^{\dagger}&0&-E&-2i\lambda_{R}\\\ 0&\frac{\sqrt{2}\hbar
v_{F}}{\ell_{B}}a&2i\lambda_{R}&-E\\\
\end{matrix}\right)\left(\begin{matrix}|r_{A},n,\uparrow\rangle\\\
|r_{B},n-1,\downarrow\rangle\\\ |r_{B},n,\uparrow\rangle\\\
|r_{A},n-1,\downarrow\rangle\\\ \end{matrix}\right)=\left(\begin{matrix}0\\\
0\\\ 0\\\ 0\\\ \end{matrix}\right)$ (13)

giving the following system of equations

$\displaystyle-E|r_{A},n,\uparrow\rangle+\frac{\sqrt{2}\hbar
v}{\ell_{B}}a|r_{B},n,\uparrow\rangle=0$ (14)
$\displaystyle-E|r_{B},n-1,\downarrow\rangle+\frac{\sqrt{2}\hbar
v}{\ell_{B}}a^{\dagger}|r_{A},n-1,\downarrow\rangle=0$ (15)
$\displaystyle\frac{\sqrt{2}\hbar
v}{\ell_{B}}a^{\dagger}|r_{A},n,\uparrow\rangle-E|r_{B},n,\uparrow\rangle-2i\lambda_{R}|r_{A},n-1,\downarrow\rangle=0$
(16) $\displaystyle\frac{\sqrt{2}\hbar
v}{\ell_{B}}a|r_{B},n-1,\downarrow\rangle+2i\lambda_{R}|r_{B},n,\uparrow\rangle)-E|r_{A},n-1,\downarrow\rangle=0.$
(17)

These can be solved to obtain a second order equation for the eigenvalues

$E^{2}\pm 2\lambda_{R}E-+\left(\hbar\omega_{D}\right)^{2}[n]=0,\qquad
n=0,1,2\cdots$ (18)

where $\omega_{D}=v_{F}\sqrt{\frac{2eB}{\hbar}}$ is the Dirac constant. The
following solutions of the last equations are the form

$\displaystyle
E_{1,n}^{\pm}=-\lambda_{R}\pm\sqrt{\left(\hbar\omega_{D}\right)^{2}[n]+\lambda_{R}^{2}}$
(20) $\displaystyle
E_{2,n}^{\pm}=\lambda_{R}\pm\sqrt{\left(\hbar\omega_{D}\right)^{2}[n]+\lambda_{R}^{2}}$

We note that the preceding energies depend to the Rashba coupling parameter
$\lambda_{R}$ and the $q-$deformation parameter, now using the equation (5) to
find

$\displaystyle
E_{1,n}^{\pm}=\lambda_{R}\left(-1\pm\sqrt{1+\left({\hbar\omega_{D}\over\lambda_{R}}\right)^{2}{\sin(n\eta)\over\sin(\eta)}}\right)$
(21) $\displaystyle
E_{2,n}^{\pm}=\lambda_{R}\left(1\pm\sqrt{1+\left({\hbar\omega_{D}\over\lambda_{R}}\right)^{2}{\sin(n\eta)\over\sin(\eta)}}\right)$
(22)

Figure 1: (Color online) Eigenvalue $E$ versus $n$ for different values of
$q-$deformed parameters, $\eta=0,0.2,0.4,0.6$, respectively for the values of
the magnetic field and Rashba coupling parameter $B\sim 10^{-3}$ and
$\lambda_{R}=0.014meV$ [23].

In Figure (1), we present the eigenvalues of the $q-$deformed massless Dirac
fermions in graphene, thus, the energy levels shows that when there is no
deformation the energy is quantified and has a parabolic form and symmetrical
compared to the quantization axis $n$, but for a given deformation we notice
that the parabolic form tends towards an ellipse, on the other hand there is
an appearance of a second quantification via the periodicity of ellipses.

Now if we consider very small deformation and neglect all terms proportional
to $\eta^{4}$, we have

$\displaystyle E_{1,n}^{\pm}=\epsilon^{\pm}_{1,n}\pm\Delta\epsilon(n)$
$\displaystyle E_{2,n}^{\pm}=\epsilon^{\pm}_{2,n}\pm\Delta\epsilon(n)$

where $\varrho$, $\epsilon^{\pm}_{1,n}$, $\epsilon^{\pm}_{2,n}$ and
$\Delta\epsilon(n)$ are defined by

$\displaystyle\epsilon^{\pm}_{1,n}=\lambda_{R}\left[-1\pm\sqrt{1+\varrho^{2}n}\right]$
(24)
$\displaystyle\epsilon^{\pm}_{2,n}=\lambda_{R}\left[1\pm\sqrt{1+\varrho^{2}n}\right]$
(25)
$\displaystyle\Delta\epsilon_{n}=-\frac{\lambda_{R}\varrho^{2}}{12}\frac{n^{3}}{\sqrt{1+\varrho^{2}n}}\eta^{2}$
(26) $\displaystyle\varrho={\hbar\omega_{D}\over\lambda_{R}}$ (27)

The term $\Delta\epsilon_{n}$ is the correction on the energy when the
deformation exists, without deformation i.e. $q\rightarrow 1$
($\eta\rightarrow 0$) the last energies are reduced to the expressions already
found in [23].

##  3 Thermodynamic quantities

We will study the thermodynamic properties of massless Dirac fermions in
graphene with Rashba coupling in contact with a thermal reservoir at finite
temperature. For simplicity, we assume that only fermions with positive energy
$(E>0)$ are regarded to constitute the thermodynamic ensemble [23]. We start
by evaluating

$\mathbb{Z}={\rm Tr}e^{-\beta H}=\sum_{n=0}^{+\infty}\left(e^{-\beta
E_{1,n}^{+}}+e^{-\beta E_{2,n}^{+}}\right)$ (28)

where $\beta=\frac{1}{k_{B}T}$, $k_{B}$ is the Boltzmann constant and $T$ is
the equilibrium temperature. Using (2.2-28), we show that $\mathbb{Z}$ takes
the form

$\displaystyle\mathbb{Z}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}e^{-\beta\left(\epsilon^{+}_{1,n}+\Delta\epsilon_{n}\right)}+e^{-\beta\left(\epsilon^{+}_{2,n}+\Delta\epsilon_{n}\right)}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}e^{\beta\Delta\epsilon_{n}}e^{-\beta\left(\epsilon^{+}_{1,n}+\epsilon^{+}_{2,n}\right)}$

noting here that the term $e^{-\beta\Delta\epsilon_{n}}$ is very small than
$1$ then the development limit around $0$ give

$\displaystyle\mathbb{Z}$ $\displaystyle\simeq$
$\displaystyle\sum_{n=0}^{+\infty}\left(1+\beta\Delta\epsilon_{n}\right)e^{-\beta\left(\epsilon^{+}_{1,n}+\epsilon^{+}_{2,n}\right)}$
(30) $\displaystyle\mathbb{Z}$ $\displaystyle\simeq$
$\displaystyle\mathbb{Z}_{0}+\mathbb{Z}_{1}$

with the partitions functions of no deformed system $\mathbb{Z}_{0}$ and the
correction partition function $\mathbb{Z}_{1}$ have as expressions

$\displaystyle\mathbb{Z}_{0}=\sum_{n=0}^{+\infty}e^{-\beta\left(\epsilon^{+}_{1,n}+\epsilon^{+}_{2,n}\right)}$
(31)
$\displaystyle\mathbb{Z}_{1}=\sum_{n=0}^{+\infty}\beta\Delta\epsilon_{n}e^{-\beta\left(\epsilon^{+}_{1,n}+\epsilon^{+}_{2,n}\right)}$

the partition function $\mathbb{Z}_{0}$ for no deformed system is already
calculated in [23] and it has the following expression

$\mathbb{Z}_{0}=\left[{2\over\varrho^{2}}\left(\tau^{2}-1\right)+1\right]\cosh{1\over\tau}.$
(32)

Where $\tau={k_{B}T\over\lambda_{R}}$ is the reduced temperature. Indeed, the
second term can be evaluated by using the Euler-Maclaurin formula; starting by
the equation

$\mathbb{Z}_{1}=\frac{\varrho^{2}\eta^{2}}{6\tau}\cosh{1\over\tau}\sum_{n=0}^{+\infty}\frac{n^{3}}{\sqrt{1+\varrho^{2}n}}e^{-{1\over\tau}\sqrt{1+\varrho^{2}n}}$
(33)

to solve the last sum it’s convenient to approximate integrals by finite sums,
or conversely to evaluate finite sums and infinite series using integrals,
indeed we put

$f(x)=\frac{x^{3}}{\sqrt{1+\varrho^{2}x}}e^{-{1\over\tau}\sqrt{1+\varrho^{2}x}}$
(34)

using the Euler–Maclaurin formula

$\sum_{x=0}^{+\infty}f(x)={1\over
2}f(0)+\int_{0}^{+\infty}f(x)-\sum_{p=1}^{+\infty}{B_{2p}\over
2p!}f^{(2p-1)}(0)$ (35)

$B_{2p}$ are the Bernoulli numbers, and $f^{(2p-1)}$ is the derivative of
order $(2p-1)$. Up to $p=1$, the values $f(0)$ and $f^{(1)}(0)$ are nulls,
then with the straightforward calculation the final form of $\mathbb{Z}_{1}$
have the form

$\mathbb{Z}_{1}={16\eta^{2}\over\varrho^{6}}\left(15\tau^{6}+15\tau^{5}+6\tau^{4}+\tau^{3}\right)e^{-{1\over\tau}}\cosh{1\over\tau}$
(36)

Finally, the compact final form of the q-deformed partition function of the
system

$\mathbb{Z}=\left({2\over\varrho^{2}}\left(\tau^{2}-1\right)+1+{16\eta^{2}\over\varrho^{6}}\left(15\tau^{6}+15\tau^{5}+6\tau^{4}+\tau^{3}\right)e^{-{1\over\tau}}\right)\cosh{1\over\tau}$
(37)

Since we have inferred the partition function of our framework, we would now
be able to determine all related thermodynamic quantities. The determination
of all thermal properties, such as the Helmholtz free energy $F$, internal
energy $U$, heat capacity $C$ and entropy $S$, can be obtained through the
expression of the partition function $\mathbb{Z}$ by using the following
relations [23]:

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\lambda_{R}\tau\ln\mathbb{Z}$ (38) $\displaystyle U$
$\displaystyle=$
$\displaystyle\lambda_{R}\tau^{2}\frac{\partial\ln\mathbb{Z}}{\partial\tau}$
$\displaystyle\frac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle-{1\over\lambda_{R}}\frac{\partial F}{\partial\tau}$
$\displaystyle\frac{C}{k_{B}}$ $\displaystyle=$
$\displaystyle{1\over\lambda_{R}}\frac{\partial U}{\partial\tau}.$

Then, we will numerically investigate the above thermodynamic functions to
underline the conduct of our framework. This will be finished by giving a few
plots under reasonable conditions and making various discussions.

##  4 Numerical Results and discussions

To make a reference to reality of graphene, we restrict our study to the low-
energy regime, which may be reached by fixing an appropriates values of the
Rashba coupling parameter $\lambda_{R}$ and the external magnetic field $B$.
Indeed, for $B\simeq 10^{-3}T$ and $\lambda_{R}=0.014meV$. The thermodynamic
functions versus the reduced temperature $\tau$ for the fixed values of
$\eta=0,0.2,0.4,0.6,0.8,0.9$.

Figure 2: (Color online) Thermodynamic functions of $q-$deformed Dirac
fermions in graphene with Rashba coupling versus the reduced temperature
$\tau$ for different values of the $q-$deformed parameter
$\eta=0.0,0.2,0.4,0.6,0.8,0.9$, respectively for the values of the magnetic
field and Rashba coupling parameter $B\sim 10^{-3}T$ and
$\lambda_{R}=0.014meV$ [23].

It is clearly seen that the common remark between the four curves is the
$\eta$-deformed parameter does not influence on the thermodynamic properties
of the system in the low temperature regime. In figure (2.a) The free energy
$F$ decreases gradually with increasing of temperature at a given
$\eta$-deformed parameter and decreases with $\eta$ at a given temperature. In
figure (2.b) we observe that at high temperature our system follows Joule’s
first law in both cases, with and without deformation, thus in the case where
the $\eta-$deformed parameter is not zero the internal energy in this regime
is asymptotic to $U=6\lambda_{R}\tau$, but when $\eta=0$ the internal energy
become asymptotic to $U=2\lambda_{R}\tau$, then we observe that for two cases
the internal energy in high temperature regime depends only on the reduced
temperature $\tau$, then we conclude for two cases the kinetic energy of
translation of molecules is the unique form of energy of $N$ atoms contained
in a volume $V$ of the system. In figure (2.c) there are two remarks to report
in low temperature in particular for $0<{S\over k_{B}}<1.7$ the entropy is
negative which can be explained by the less disorder of the system [23], in
the case where ${S\over k_{B}}>1.7$ the entropy increases when
$\eta$-parameter increases. For the tree curves at the top (a,b,c) we deduce
that the parameter $\eta$-parameter plays the same role of the doping of the
graphene, however when $\eta$ increase the thermodynamic properties such as
entropy, internal energy increases with $\eta$-parameter in the same way when
we dope pure graphene with boron atoms $B$ or nitrogen $N$ and vice versa
[24], and for the free energy of Helmholtz, it decrease when $\eta$ decrease
similarly when the concentration of the doped atoms in graphene decrease. What
is remarkable is in figure (2.d) we observe that without q-deformation our
system at high temperature obeys to the Dulong-Petit law, but when the
q-deformation is introduced, the heat capacity passes through a maximum in low
temperature regime, that is, the point where the temperature changes very
little as energy is supplied to the system, most of the energy is used to
excite the carbon atoms of the ground state in the excited state, rather than
increasing the kinetic energy of the system, that on the one hand, on the
other hand at high temperature the heat capacity coincide and reach the fixed
value $C=6K_{B}$ three times greater compared to the case of no deformed
massless Dirac fermions in graphene which can be explained by the increasing
of degree of freedom of the system due to the introduction of the
$\eta$-deformed parameter.

##  5 Conclusion

In this paper, after a brief insight on the notion of the q-deformed harmonic
oscillator, we have studied the thermodynamic properties of Dirac fermions in
graphene in this deformation formalism, we have found the eigenvalues of the
considered system via q-deformed annihilations and creations operators. It was
shown that the eigenvalues of our system are more general than in the case
where there is no deformation, and especially we tested them in the limiting
case $\eta=0$ where the ordinary results were well recovered. The eigenvalues
are used together with a method based on the zeta function and Euler-
Maclaurain formula to determine the partition function according to the
q-deformed parameter. Therefore the thermodynamic functions, such as the
Helmholtz free energy, total energy, entropy and heat capacity, were obtained
in terms of the q-deformed parameter.

Subsequently, some cases were studied related to the $q$-deformed parameter.
Indeed, we numerically analyzed the plotted curves which allowed us to make
important remarks on the influence of deformation on the thermodynamic
properties of our system. We also found a similarity between the doping
concentration and the q-deformed parameter for the graphene system [24].
Finally, it was shown that the Dulong-Petit law is no longer verified when the
q-deformed harmonic oscillator notion is introduced where the heat capacity at
high temperature tends to a constant value $C=6k_{B}$ three times greater in
comparison with the Dirac fermions in graphene [23].

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