Analytic Electrically Charged Black Holes in $F(R)$-ModMax Theory

Motivated by a new model of nonlinear electrodynamics known as Modified Maxwell (ModMax) theory, an exact analytical solution for black holes is obtained by coupling ModMax nonlinear electrodynamics and $F(R)$ gravity. Then, the effects of the system's parameters ($F(R)$-ModMax gravity parameters) on the event horizons are analyzed. The obtained black holes thermodynamic properties in the $F(R)$-ModMax theory are investigated by extracting their thermodynamic quantities such as Hawking temperature, electric charge, electric potential, entropy, and also total mass. The first law of thermodynamics for the system under study is evaluated. Next, by considering these black holes, the impact of various parameters on both the local stability and global stability are investigated by examining the heat capacity and the Helmholtz free energy, respectively. Finally, the thermodynamic geometry of the black hole in $F(R)$-ModMax gravity is investigated by applying the thermodynamic metric (the HPEM metric).

Nonlinear electrodynamics (NED) can describe, for example; i) the virtual electron-positron pairs' self-interaction [31][32][33].ii) the NED field modifies the gravitational redshift around super-strong magnetized compact objects [34,35], iii) removing the singularities due to both the black hole (which is known as the regular black hole [36][37][38][39]) and Big Bang [40][41][42].Notably, the regular black hole is an object whose spacetime has the horizon, but there is no curvature singularity.In this regard, regular black holes with multi-horizon have been studied in general relativity, F (R) gravity, and Gauss-Bonnet gravity in the presence of NED (see Ref. [39], for more details).iv) NED can explain the radiation propagation within particular substances [43][44][45][46].In addition, the NED field's effects affect pulsars and higher-magnetized neutron stars [47,48].In this regard, Born and Infeld [49] introduced the first model of NED in 1934.This particular NED model removes several of the problems that are encountered in Maxwell's theory, including the elimination of the electric field's singularity at the center of point particles.Power-Maxwell NED theory (PM NED) is also an interesting NED theory, whose Lagrange function is an arbitrary power of Maxwell's Lagrange function [50][51][52][53].The PM NED remains invariant when subjected to the conformal transformations g µν → Ω 2 g µν and A µ → A µ (here, A µ and g µν , respectively, represent the electrical gauge potential and the metric tensor).In the framework of PM NED, it was highlighted that point particles may possess a finite electric field at their center and charges could potentially have a finite self-energy [54,55].The Maxwell electrodynamics is particularly noteworthy for its duality and scale invariance.In this regard, in 2020, Bandos, Lechner, Sorokin, and Townsend introduced a modified Maxwell model (where is known as the ModMax) of nonlinear duality-invariant conformal electrodynamics.In essence, the ModMax model of NED exhibits both the duality and also conformal symmetries, similar to Maxwell's theory [56].
Examining the black holes' exciting features in any gravity can provide important information about the theory from both theoretical and observational perspectives.On the other hand, in the gravitational collapse scenario, all forms of matter, including charged particles, are assimilated by the black hole.Hence, it becomes crucial to examine the interplay between the black hole, the linear and nonlinear electromagnetic fields, and the impact of these fields on the geometry in the context of F (R) gravity.In this regard, various black holes in the F (R) gravity framework with (or without) matter field were extracted in Refs. .Furthermore, the field equations of F (R) gravity pose a challenge as they are intricate 4th order differential equations, making it arduous to derive analytical solutions for black holes, especially in the presence of matter field such as NED theories.Based on the specified characteristics of F (R) gravity and the ModMax-NED theory (which includes the same symmetries as Maxwell's theory, i.e. electromagnetic duality and conformal invariance), we are interested in extracting analytical solutions for black holes in this theory.
By connecting the geometrical quantities such as surface gravity to temperature and horizon area to entropy, the black hole as a thermodynamic system is studied by Bekenstein and Hawking [85,86].Next, for non-rotating and uncharged black holes, dM = T dS was established as the first law of black hole thermodynamics [86,87].In this regard, studying the thermodynamical properties of black holes, especially phase transition has assimilated much interest .
There exist multiple approaches to peruse the phase transition of black holes, such as examining their heat capacity and utilizing geometrical thermodynamics (GTD).The study of the heat capacity reveals two distinctive points which are known as bound and phase transition points.i) the bound point determines when the numerator of heat capacity (or temperature) is zero.At this point, the sign of the temperature changes, and we can separate the non-physical and physical black holes.ii) the heat capacity divergences determine the phase transition points.Furthermore, the heat capacity's positivity (negativity) ensures the system's thermal (in)stability.So, the study of the heat capacity gives us information about the thermal stability, the phase transition, and physical bound points of the system [112][113][114].
GTD is an alternative approach used to evaluate the phase transition of black holes.GTD involves constructing a thermodynamic metric using thermodynamic potentials, such as entropy or internal energy, and their derivatives with respect to the system's extensive parameters.The divergences of the Ricci scalar of this thermodynamic metric provide insights into the phase transition points.Various thermodynamic metrics, including those proposed by Weinhold [115,116], Ruppeiner [117][118][119], Quevedo [120,121], and Hendi-Panahiyan-EslamPanah-Momennia (referred to as HPEM's metric) [122][123][124][125], have been introduced to investigate this phenomenon.However, it has been observed the Ricci scalar's singularities in the Weinhold and Ruppeiner metrics do not align with the singularities of heat capacity, making them unsuitable for explaining the thermodynamic properties of different black holes.This issue arises from the lack of Legendre invariance in these metrics.To address this problem, Quevedo introduced a new type of thermodynamic metric, known as Quevedo's metric, which is invariant under Legendre transformations [120,121].Other thermodynamic metrics have also been proposed in the literature [126][127][128][129], but they have their own limitations.In this context, Hendi-Panahiyan-EslamPanah-Momennia developed a new metric, HPEM's metric, which overcomes the shortcomings of previous thermodynamic metrics and effectively distinguishes between phase transition and bound points [122][123][124][125]. Therefore, we consider HPEM's metric to study the bound and phase transitions points, and also stability conditions for the obtained black holes in F (R)−ModMax theory.
This paper follows the following outline.The subsequent section presents an introduction to the field equations in F (R) gravity in the presence of ModMax NED.The electric black hole solution in F (R)-ModMax gravity will be derived and the influence of the parameters on these black holes will be evaluated in Section III.Section IV will cover the thermodynamic quantities and an examination of the first law of thermodynamics.The forthcoming section will investigate the impact of various parameters on both local and global stability through the utilization of heat capacity and Helmholtz free energy.This analysis will be carried out in Section V. Section VI will focus on the study of the phase transition and the physical limitation points for the extracted black holes within the framework of GTD using HPEM's metric.The final section will be dedicated to concluding remarks.

II. FIELD EQUATIONS IN F(R)-MODMAX THEORY
In this study, we investigate the coupling of the ModMax field (as the source of matter) with F (R) gravity.In four-dimensional spacetime, the action of F (R)-ModMax theory is given by where F (R) = R + f (R), in which R and f (R), respectively, are scalar curvature and a function of scalar curvature.
In this paper, we consider the Newtonian gravitational constant and the speed of light equal to 1, i.e., G = c = 1.The second term in the above action is devoted to the ModMax Lagrangian (L), which is defined [56,130] where γ is known as the parameter of ModMax theory.γ is a dimensionless parameter.Also, S, and P are, respectively, a true scalar, and a pseudoscalar, which are defined in the following forms and where A µ is the gauge potential) is the electromagnetic tensor).In addition, F equals to F µν F µν , where F µν = 1 2 ǫ ρλ µν F ρλ .This nonlinear electromagnetic theory reduces to Maxwell's theory, when γ = 0.Moreover, g = det(g µν ) is the determinant of metric tensor g µν , in the action (1).
In this work, we are interested in considering the electrically charged case of the ModMax theory.In other words, we want to obtain electrical charged black hole solutions by coupling F (R) theory and the ModMax nonlinear electrodynamics theory.Therefore, we have to consider P = 0 in the above equations.For this purpose, we are able to extract the equations of motion of F (R)-ModMax theory of gravity, which lead to where f R = df (R) dR .Also, T µν defines as the energy-momentum tensor, which is given by and E µν in Eq. ( 5), is defined as where L S = ∂L ∂S .So, the ModMax field equation (Eq.( 5)) for the electrically charged case reduces to We consider a static spherically symmetric spacetime as in which g(r) defines as the metric function that we must find.
In general, the equations governing F (R) gravity with a nonlinear matter field (Eq.( 4)) are intricate.Therefore, deriving a precise analytical solution is a challenging task.To resolve this problem, one can consider the traceless energy-momentum tensor for the nonlinear matter field (like the ModMax field), one can extract an exact analytical solution from F (R) gravity coupled to a nonlinear matter field.So, to get the solution for a black hole with constant curvature in F (R) theory of gravity coupled to the ModMax field, it is necessary for the trace of the stress-energy tensor T µν to be zero [131,132].Assuming the constant scalar curvature R = R 0 = constant, then the trace of the equation (4) turns to where We can solve the equation (10) to obtain R 0 which leads to By replacing Eq. (11) within Eq. ( 4), the F (R)-ModMax theory's equations of motion can be found in the following format It is notable that, the equation of motion in F (R)-ModMax theory (12) reduces to GR-ModMad theory of graviry when f R0 = 0.
To obtain a radial electric field, we take the following form for the gauge potential By utilizing the provided gauge potential and equations ( 8) and ( 9), we are able to derive the subsequent differential equation.
where, respectively, the prime and double prime represent the first and second derivatives of r.The solution of the equation ( 14) yields where q represents an integration constant that is associated with the electric charge.
We are now able to obtain precise analytical solutions for the metric function g (r) by taking into account the metric (9), the derived h(r), and the field equations (12).Consequently, we derive the subsequent set of differential equations in which eq tt , eq rr , eq θθ and eq ϕϕ , respectively, are the components of tt, rr, θθ and ϕϕ of field equations (12).By utilizing the aforementioned differential equations, we can obtain a precise solution for the constant scalar curvature (R = R 0 = constant).After careful consideration and performing several calculations, the metric function can be expressed in the subsequent form where m 0 is an integration constant.It is noteworthy that this constant of integration is connected to the black hole's geometric mass.Furthermore, any of the field equations ( 12) is satisfied by the obtained solution (18).We should limit ourselves to f R0 = −1 in order to have physical solutions.In addition, we can see the effect of ModMax's theory in the fourth term of the solution (18), and the effect of F (R) gravity both in the third and fourth terms of the metric function (18).Notably, Reissner-Nordström-(A)dS black hole is covered by considering f R0 = 0, R 0 = 4Λ and γ = 0, i.e., Here, we study the Kretschmann scalar (R αβγδ R αβγδ ) in order to find the singularity of spacetime.Indeed, this quantity gives us information about the existence of the singularity in spacetime.For this purpose, we calculate the Kretschmann scalar of the spacetime (9), which is and by replacing the metric function (18) within Eq. ( 20), we have The obtained Kretschmann scalar includes three important points, which are i) It diverges at r = 0, i.e., lim there exists a curvature singularity situated at the coordinate r = 0.
ii) The Kretschmann scalar is finite for r = 0.
iii) The effect of ModMax's parameter appears in the Kretschmann scalar.Although, the divergence of the electrical field is removed by considering γ −→ ∞, this limit cannot remove the curvature singularity at r = 0.In other words, in the limit γ −→ ∞, we have lim r→0 R αβγδ R αβγδ −→ ∞.
The asymptotical behavior of the Kretschmann scalar and the metric function are given by lim where indicate that the spacetime will be asymptotically (A)dS, when we consider R 0 = 4Λ, and Λ > 0 (Λ < 0).It is notable that, the asymptotical behavior is independent of γ.In other words, the parameter of ModMax does not affect the asymptotical behavior of the spacetime.In this context, our objective is to determine the real roots of the acquired metric function (18) because these roots can give us information about the horizons (inner and outer horizons) of the solution.The objects known as black holes possess a curvature singularity situated at r = 0, which is concealed by a minimum of one event horizon.Notably, we can have black holes without the event horizon, which is known as the naked singularities.
To find the roots, it is better to solve the metric function.Here, the metric function is a fourth-order function of r, and it is not easy to get an exact solution.Therefore, the metric function is plotted against the variable r in Fig. 1 to obtain these roots.As shown in Fig. 1, for R 0 > 0 (or dS case if we consider R 0 = 4Λ), we encounter three different cases.In the first case, there are three roots, which are an inner root, an event horizon, and an outer root (cosmological horizon), respectively.In the second case, there are two roots (extreme case and cosmological horizon).In the third case, one root (cosmological horizon) exists.For R 0 < 0 (or AdS case if we consider R 0 = 4Λ), the solution may have three different cases which are: i) two roots (an inner horizon and an event horizon).ii) one root (extreme case).iii) naked singularity.By modifying certain parameters, it is possible to achieve an event horizon that encompasses the singularity located at r = 0.The results validate that the solution acquired in Eq. ( 18) may be associated with the black hole solution in F (R)-ModMax theory.
FIG. 1: The function g(r) is plotted against r for various parameter values, with the left panels corresponding to R 0 = 1 and the right panels corresponding to R 0 = −1.
Now, we have the opportunity to examine the impact of the parameters in F (R)-ModMax theory on the event horizon.Notably, we want to evaluate the effects of the electrical charge (q), the parameter of ModMax (γ), F (R)'s parameters (f R0 , and R 0 ) on this kind of black hole.Our results are: i) The effect of the electrical charge indicates that by increasing q, the number of roots decreases.In other words, the higher charged black hole in F (R)-ModMax theory does not have an event horizon, and we encounter a naked singularity (see Fig. 2a).
ii) The effect of the ModMax theory reveals that the number of roots and the radius of the event horizon increase by increasing γ.Indeed, a black hole with large γ, has two roots (see Fig. 2b).
iii) In Fig. 2c, we can see the effect of f R0 on the obtained black holes in F (R)-ModMax theory.This figure indicates that by increasing f R0 , the number of roots and radius of black holes increase.
iv) The effect of R 0 appears in Fig. 2d.The behavior of black holes under this parameter is the same as the electrical charge.In other words, by increasing |R 0 |, the number of roots and the radius of the black hole decrease.FIG.2: The function g(r) versus r is plotted for various parameter values.

IV. THERMODYNAMICS
Here, we compute the conserved and thermodynamic quantities of the electrical charged black holes within the F (R)-ModMax theory.Next, we will examine the first law of thermodynamics.
The initial step involves the computation of the Hawking temperature for the black holes.The Hawking temperature can be determined using the following equation where κ is the superficial gravity of these black holes, which is given by in which r + and χ = ∂ t are the radius of the events horizon and the Killing vector, respectively.Before obtaining the Hawking temperature, we have to derive an expression for the mass (m 0 ) using the event horizon radius (r + ), R 0 , and the charge (q), resulting in where we extracted m 0 by equating g(r) = 0.
By applying the extracted metric function (18), and by substituting the mass (26) in Eq. ( 25), we get the superficial gravity as Now, we have reached a stage where we can acquire the Hawking temperature.For this purpose, we replace Eq. ( 27) into Eq.( 24), which leads to As one can see, the Hawking temperature of black holes in F (R)-ModMax theory is dependent on the electrical charge (q), the parameters of F (R) gravity (f R0 , and R 0 ), as well as ModMax's parameter (γ).
In classical thermodynamics, the positive (negative) values of temperature are interpreted as (non-)physical solutions, i.e., the roots of temperature separate physical solutions from non-physical ones.Therefore, the roots of temperature determines bound points.To find the bound points (or real roots of temperature), we solve the Hawking temperature.Our analysis reveals that there is only one real root for the Hawking temperature, which is given by where indicates that there is no real root for the Hawking temperature when γ → ∞ (see the dashed line in Fig. 3a, for more details).
We follow our study to evaluate the behavior of the high energy and asymptotic limits of the temperature.In high energy limit of the obtained temperature is given by which reveals that the parameter of ModMax theory plays an important role in this limit of the temperature.In other words, the temperature is always negative for small black holes by considering the small value of γ.But, for γ → ∞, the temperature goes to positive infinity (see four diagrams in Fig. 3a).
On the other hand, the asymptotic limit of the temperature only depends on R 0 , i.e.
which is dependent on one of the parameters of F (R) gravity.The asymptotic limit of the temperature is always positive (negative) when R 0 < 0 (R 0 > 0).To confirm the obtained results of the various behaviors (high energy and asymptotic limit) for the Hawking temperature and study other effects of the mentioned parameters such as q, and f R0 , we plot the Hawking temperature versus r + in Fig. 3.
We can see the effects of different parameters on the Hawking temperature in Fig. 3.There are two extrema points that belong to the maximum and minimum values of the temperature.Actually, by increasing the radius of the black hole, the temperature reaches a maximum value (the first extremum) and then decreases to a minimum value (the second extremum).After the second extremum point, the temperature increases.
Our analysis of the effects of γ, q, f R0 , and R 0 on r +T =0 (root of the temperature) reveal that: i) r +T =0 decreases by increasing the parameter of ModMax theory, and finally, for the very large value of γ, we have no root, as we expected from the equation (29).See Fig. 3a, for more details.
ii) The effect of the electrical charge on the root of the temperature shows that by increasing q, r +T =0 increases (see Fig. 3b).
iii) Fig. 3c indicates that r +T =0 decreases by increasing f R0 .iv) The root of the temperature is not very sensitive to changes of R 0 (see Fig. 3d).But the asymptotic limit of the temperature depends on completely this parameter, as we expected it from Eq. (31).
Our results reveal that the small black holes (i.e., r + < r +T =0 ), cannot be physical because the temperature is negative in this area, except for the large value of γ.In other words, small black holes are physical, provided the parameter of ModMax theory has a large value.We can get the electric charge of the black hole in F (R)-ModMax theory, by using the Gauss law, which leads to One can find the electric potential at the event horizon (U ) in the following form where F tr = qe −γ r 2 .We can apply a modification of the area law in the F (R) theory of gravity [133], to extract the entropy of black holes, which yields in which A is the horizon area as Now, we can obtain the entropy of ModMax-black holes in F (R) gravity by replacing Eq. ( 35) within Eq. (34), which leads to The Ashtekar-Magnon-Das (AMD) approach enables us to determine the total mass of black holes in the F (R)-ModMax theory [134,135], in the following form Here, we expand the total mass by substituting the mass (26) within the equation ( 37), and we get where indicates that the total mass is dependent on the parameters of the electrical charge, the parameters of F (R) gravity, as well as ModMax's parameter.
In high energy limit of the total mass is given by which depends on q and γ.The total mass of small black holes is always positive for finite values of γ.Also, M of small black holes is zero (i.e., lim r+→0 M = 0) when γ → ∞, see the dashed line in Fig. 4a.
The asymptotic limit of M is obtained which depends on the parameters of F (R) gravity (f R0 , and R 0 ).Also, the asymptotic limit of M is always positive (negative) when R 0 < 0 (R 0 > 0).To see the effects of γ, q, f R0 , and R 0 on the total mass of black holes, we plot Fig. 4. Our analysis states that for a finite value of γ, there is an extreme point (a minimum point) of the total mass.Indeed, M decreases by increasing r + , and reaches a minimum value.After this extreme point, the total mass increases by increasing the radius of black holes.Notably, there is no extreme point for the total mass when we consider the large value of γ (see Fig. 4a, for more details).In this case, the total mass is an increasing function of r + .
In summary, our examination of the impacts of different factors on the total mass are as follows: i) The extreme point depends on the parameter of ModMax theory.Considering a finite value of ModMax's parameter, by increasing γ, the extreme point decreases and the minimum point is located at a smaller radius (see Fig. 4a).
ii) The effect of electrical charge on the total mass presented in Fig. 4b.The results indicate that by increasing q, the extreme point increases, and it is located at a large radius.
iii) The minimum of total mass increases by increasing f R0 .In addition, for the large values of f R0 , the extreme point is located in a smaller radius (see Fig. 4c).iv) We can see the effect of R 0 on the total mass in Fig. 4d.The extreme point is not very sensitive to the parameter of R 0 .As one can see in Eq. ( 40) and Fig. 4d, the asymptotic limit of M is sensitive to R 0 .Now, we can evaluate the first law of thermodynamics.In other words, the obtained conserved and thermodynamics quantities in Eqs. ( 28), ( 32), ( 33), (36), and (38), satisfy the first law of thermodynamics in the following format where T = ∂M ∂S Q , and U = ∂M ∂Q S , respectively, are in agreement with the obtained relations in Eqs. ( 28) and (33).

V. THERMAL STABILITY
In order to investigate the thermal stability of a black hole as a thermodynamic system, our focus lies on examining the impact of parameters within the F (R)-ModMax theory.This analysis will be conducted through the utilization of both heat capacity and Helmholtz free energy.FIG.4: The total mass M versus r + for different parameters.

A. Heat Capacity
In the domain of the canonical ensemble, the local stability of a thermodynamic system can be assessed through the utilization of heat capacity.The heat capacity, whether positive or negative, serves as an indicator of the system's thermal stability.It is worth noting that a positive heat capacity corresponds to thermal stability.Therefore, we examine the local stability of black holes in the F (R)-ModMax theory by employing the heat capacity.
The heat capacity is defined in the following form to obtain the heat capacity, we re-write the total mass and the Hawking temperature of the black hole (38) in terms of the electrical charge (32), and the entropy (36), which lead to Now, we can get the heat capacity by considering Eqs. ( 43) and ( 44) within Eq. ( 42), and after some calculation, we have In the realm of black holes, the root of heat capacity (C Q = T = 0) is considered as a border line between nonphysical (T < 0), and physical (T > 0) black holes.Hereafter, we name the root of heat capacity (or the root of temperature) as a physical limitation point.In simpler terms, the heat capacity undergoes a sign change at the point of physical limitation.Furthermore, it has been suggested that the divergences in heat capacity serve as crucial indicators of phase transition critical points in black holes.Consequently, the heat capacity can be utilized to identify both the phase transition critical and physical limitation points in black holes.So, we determine these points by solving the following relations The physical limitation point is obtained by utilizing Equation ( 44) and solving it with respect to entropy, resulting in The above relation imposes a constraint on R 0 .Indeed, to have the real root, it is necessary to adhere to the condition In addition, the root of the obtained temperature (47), depends on the parameters of γ, Q, f R0 , and R 0 .It is worth mentioning that in the previous section, we deliberated on the impact of these parameters on the temperature's root.
To evaluate the phase transition critical points, we have to solve the relation which impose the constraint R 0 ≥ −(1+fR 0 ) 3Q 2 e −γ , in order to have the real divergent point(s).Our analysis from Eq. ( 48) reveals that the heat capacity may have three different cases: The first case: there is one divergence point when γ → ∞.Indeed, the real and positive divergence point is located The second case: by considering R 0 = −(1+fR 0 ) 3Q 2 e −γ , the divergence points reduce from two points to one point.In other words, we encounter with one divergence point at The third case: for finite value of γ, and R 0 ≥ −(1+fR 0 ) 3Q 2 e −γ , there are two divergence points which are located in S div1 , and S div2 .Now we are in a position to study the local stability by using the obtained temperature and heat capacity.Our findings are given with more details in Fig. 5 and Table.I.
Our study of the temperature and the heat capacity, simultaneously, reveals some information about the physical and local stability of the black holes in F (R)-ModMax theory.We mention the which are: i) There are two physical and stable areas.These areas are located at S root < S < S div1 , and S > S div2 .Indeed, the temperature and the heat capacity are positive in these areas.This result also indicates that the black holes in the ranges S < S root , and S div1 < S < S div2 , cannot be physical and stable objects, whereas the medium black holes (S root < S < S div1 ), and large black holes (S > S div2 ) satisfy the local stability and are physical and stable objects.See Fig. 5, and Table.I, for more details.

TABLE I:
The local stability of the black holes in F (R)-MadMax theory for finite value of γ, and R 0 < 0.
T > 0 CQ > 0 local stability and physical area ii) By increasing the parameter of ModMax theory, S root , and S div1 decrease but S div2 increases.In other words, although the physical area increases, but the local stability for black holes in the ranges S root < S < S div1 , and S > S div2 decreases by increasing γ (see two panels in Fig. 5a).In addition, in the limit γ → ∞, there is only one divergence point, i.e., S div2 , which states that the large black holes (i.e., S > S div2 ) can satisfy physical and local stability, simultaneously.
iii) We can see the effect of the electrical charge on physical and local stability in two panels of Fig. 5b.The results show that by increasing Q, S root , and S div1 increase but S div2 decreases.It means that the local stability areas (i.e., S root < S < S div1 , and S > S div2 ) increase by increasing Q.In other words, the higher electrically charged black holes have large physical and stability areas, simultaneously.Notably, by comparing Fig. 5a and Fig. 5b, together, we can see that the electrical charge acts the opposite of ModMax's parameter.iv) Our findings in Fig. 5c, indicate that S root , and S div1 are not very sensitive to changes of f R0 .However, S div2 increases by increasing f R0 .It means that the local stability area of large black holes in F (R)-ModMax theory decreases.
v) S root and S div1 change by varying |R 0 |, but these changes are less than S div2 .In other words, by increasing |R 0 |, the second divergence point (S div2 ) decreases, which leads to increasing the local stability area.

B. Helmholtz Free Energy
The global stability of a thermodynamic system is determined by the negative value of the Helmholtz free energy in the canonical ensemble.In order to examine the global stability of black holes in F (R)-ModMax theory, our objective is to assess the Helmholtz free energy.It is worth mentioning that, in the usual thermodynamics, the Helmholtz free energy is typically defined as that in relation to the black holes (U = M ), it turns to the following relation Using Eqs. ( 43) and ( 44), we can get the Helmholtz free energy, which yields To study the global stability of black holes, we have to determine the negative value of the Helmholtz free energy.To achieve this objective, we can determine the roots of the Helmholtz free energy (51) by solving the equation F (T, Q) = 0, which leads to which indicates that there is only one real root of the Helmholtz free energy if we respect the constraint R 0 ≤ 1+fR 0 Q 2 e −γ .Notably, for R 0 < 0, this constraint is automatically satisfied.
In order to observe the impact of different parameters on the global stability regions of black holes, we graph the Helmholtz free energy against S in four sections of Figure 6.Our results are: i) The global stability area is located at S > S F =0 .Indeed, there are two different areas, before and after of the root of Helmholtz free energy (S < S F =0 , and S > S F =0 ).The Helmholtz free energy is positive in the range S < S F =0 , which indicates that the black holes cannot satisfy the global stability.In other words, the small black hole in F (R)-ModMax theory is not a global stable system.Conversely, the Helmholtz free energy exhibits a negative value within the range of S > S F =0 .So, the large black holes are global stable systems (see all panels in Fig. 6).
ii) Our analysis of Fig. 6a and Fig. 6d, indicate that by increasing γ and |R 0 |, the root of the Helmholtz free energy (S F =0 ) decreases, which leads to increasing the global stability area.
iii) By increasing Q and f R0 , the global stability area decreases because S F =0 increases (see Fig. 6b and Fig. 6c, for more details).

FIG. The Helmholtz free energy F versus S for different values of the parameters.
As a result, we found that the large black holes in F (R)-ModMax theory can satisfy the local and global stability conditions, simultaneously.This result is extracted by comparing the areas of the local and global stabilities in Figs. 5, and 6.
Previous research has examined the limitations of Ruppeiner, Weinhold, Fisher-Rau, and Quevedo metrics in accurately assessing the thermal phases of certain types of black holes (see Refs. [123,124,144] for further information).Consequently, we employ the HPEM's metric to explore the thermal phases of electrically charged black holes in the F (R)-ModMax theory.
The HPEM's metric is introduced as [122] where . After some calculation, we can find the numerator and denominator of the Ricci scalar of HPEM's metric in the following forms in which Our findings, in Fig. 7, reveal that the divergence points of the Ricci scalar of HPEM's metric coincides completely with both the phase transition critical and the physical limitation points of the heat capacity.So, all the thermodynamic criteria are included in the divergences of the Ricci scalar of HPEM's metric.In addition, the divergences of the Ricci scalar of HPEM's metric different before and after the physical limitation points with the phase transition critical points.Indeed.The sign of the Ricci scalar of HPEM's metric changes before and after divergence, which is related to the physical limitation point.Nevertheless, the signs of the Ricci scalar exhibit identical characteristics in the vicinity of the critical points of the phase transition.These divergences are known as Λ divergences.So, by adopting this methodology, we can differentiate between physical constraints and critical points of phase transitions.FIG.7: Ricci scalar of HPEM's metric R HP EM , the heat capacity C Q , and the temperature T versus S for different values of the parameters.

VII. CONCLUSIONS
In this work, we have exacted analytical solutions in F (R)-ModMax theory of gravity.Then, we calculated the Kretschmann scalar of the obtained analytical solutions in order to find an essential singularity.Our results indicated there was a curvature singularity at r = 0.In order to determine the roots, a graphical representation of the metric function has been plotted against the variable r in Figure 1.Our findings revealed that for R 0 > 0, there were three different roots, an inner root, an event horizon, and a cosmological horizon.If R 0 < 0 and certain parameters are adjusted, the solution could exhibit an inner horizon and an event horizon (two roots), a single root (in the extreme case), or a naked singularity.Our analysis indicated that we could have an event horizon that covered the singularity at r = 0 by adjusting the parameters.So, the obtained solution could be related to the black hole solution within the F (R)-ModMax theory.
The thermodynamic quantities of charged black holes within the F (R)-ModMax theory were calculated, and the validity of the first law of thermodynamics was checked.Our analysis indicated that the obtained conserved and thermodynamic quantities of these black holes (such as the Hawking temperature, the electrical charge, the electrical potential, entropy, and the total mass) satisfied the first law of thermodynamics.Furthermore, an assessment was conducted to analyze the impact of different factors on both the Hawking temperature and the total mass.Our results revealed there was one real root for the Hawking temperature, which is dependent on different parameters.It is notable that, this root was removed in the limit γ → ∞.By considering R 0 < 0 and different values of parameters, the temperature was positive for large black holes.In addition, there were two extremum points for the Hawking temperature (except in the limit γ → ∞).The total mass was always positive and there was one extremum point that depended on different parameters.Notably, there was no extremum point for the total mass when γ → ∞.
We have studied the heat capacity of the charged black holes in F (R)-ModMax theory to investigate the local stability.We found two critical points (or phase transition points), and one physical limitation point for the heat capacity.These points depended on the parameters of the system.Our findings indicated that there were two local stability and physical areas, simultaneously.These areas are located at S root < S < S div1 , and S > S div2 .In other words, the electrical charged black holes in F (R)-ModMax theory of gravity could be stable and physical objects, simultaneously, when are located in the ranges S root < S < S div1 (or medium black holes) and S > S div2 (or large black holes).
We have examined the Helmholtz free energy (F ) to assess the global stability of the acquired black holes.We found that there was one real root of the Helmholtz free energy (S F =0 ), which depended on the parameters of F (R) gravity (i.e., R 0 , and f R0 ), the electrical charge Q, and ModMax's parameter (γ).The global (in)stability area was located at S > S F =0 (S < S F =0 ).In other words, there were two different areas, before and after the root of Helmholtz free energy (S < S F =0 , and S > S F =0 ).The Helmholtz free energy was positive in the range S < S F =0 , which indicated that the black holes could not satisfy the global stability.Conversely, the Helmholtz free energy exhibited a negative value within the range of S > S F =0 .Therefore, the large black holes were global stable systems.
In the last section, we have studied geometrical thermodynamics by using HPEM's metric for the obtained electrical charge black holes in F (R)-ModMax theory.In Fig. 7, it was observed that the Ricci scalar of HPEM's metric exhibited divergence points that aligned with both phase transition critical and the physical limitation points of the heat capacity.Furthermore, the divergences of the Ricci scalar before and after the physical limitation points were distinct from those at the phase transition critical points.Indeed, the alteration in the Ricci scalar sign of HPEM's metric occurred before and after divergence, indicating a connection to the physical limitation point.Nevertheless, the Ricci scalar signs remained consistent around the critical points of phase transition.Consequently, the utilization of HPEM's metric allowed us to differentiate between phase transition critical and physical limitation points.
Given the importance of the NED field in various aspects of physics, the study of modified gravitational theories such as the F (R) theory with NED fields or F (R) models inspired by NED (which can describe some phenomena observed in astrophysics and cosmology) is very interesting.For example, Born-Infeld-F (R) gravity [149,150] has shown some interesting properties in cosmology and black holes.Born-Infeld-F (R) gravity opened new ways to answer some questions of gravitational dynamics at low energies.This theory efficiently explains the structure of stars without the need to reconsider the convenient approximation of a perfect fluid.We can therefore extend our model by a Lagrangian reform (similar to the Born-Infeld-F (R) theory) and explore cosmological and astrophysical applications in the future.

FIG. 3 :
FIG. 3: The Hawking temperature T versus r + for different values of the parameters.