Graphene as a two-dimensional (2D) honeycomb lattice with sp² hybridized carbon atoms discovered in 2004 and guaranteed its future applications both theoretically and experimentally.^[1] Owing to its novel properties as a weird material sit is applicable in nanoelectronics and optoelectronics.^[1–8] In graphene, conductance electrons are from the tangential π and π* bands at the limit of zero doping, i.e., at Dirac points of first Brillouin zone. S, p_x, and p_y orbitals are responsible for strong covalent bonding (in-plane bonding) while p_z orbitals lead to the weak van der Waals bonding (out-of-plane bonding). Gapless nature of graphene limits the direct utilization of graphene in nanoelectronic and nanophotonic devices.^[9,10] Thus, opening or tuning a band gap is useful for technological application of graphene, especially in highly motivated new graphene-based devices. Hydrogenation of graphene has been realized and predicted not only theoretically but also experimentally in 2007 and 2009^[11,12] to induce a finite band gap. It has been found that in new 2D material, hydrogenated graphene which is called graphane with sp³ hybridization, carbon atoms react with hydrogen (H) atoms and consequently create a band gap about 3.5 eV. (In Fig. 1, two kinds of monolayer, chair-like (CL) and table-like (TL) graphene are presented). Furthermore, in graphene, p_z orbitals of carbon atoms are saturated with hydrogen atoms. On the other hand, bilayer graphene has attracted a great deal of attentions recently.^[13–15] In bilayer graphene, energy dispersion diagram is parabolic and massive chiral fermions are responsible for the low-energy excitations unlike in monolayer graphene in which Dirac fermions are responsible for the linear energy dispersion diagram. Also, in this paper we have considered AA-stacked (simple (S)) and AB-stacked (Bernal (B)) bilayer graphene, which are hydrogenated on top surface of upper layer and bottom of lower layer with different sublattices (as shown in Fig. 2). For the case of AA-stacking, an A (B) atom in the upper layer is stacked directly above A (B) atom in the lower layer while for AB-stacking, an A (B) atom in the upper layer is stacked above the center of honeycomb lattice in the lower layer (producing different configurations).

Fig. 1.

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	Fig. 1. (color online) Schematic representation of hydrogenated monolayer graphene (a) chair-like, (b) table-like. Dashed lines illustrate the Bravais lattice unit cell and primitive vectors are denoted by a1 and a2. Also a0 is the inter-atomic distance.

Fig. 2.

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	Fig. 2. (color online) Schematic illustration of hydrogenated bilayer graphene and their top views for case (a) AA-stacked and (b) AB-stacked.

Recently, several works have been done to compare the thermal properties (including thermal conductivity and temperature dependence of EHC) of low-dimensional systems with macroscale ones.^[16–19] Balandin in 2011 found that graphene is the best thermal conductor in nature.^[17] Furthermore, with changing in orbitals hybridization of carbon atoms, electronic properties of materials can be changed and therefore affect other properties. Hybridization from sp² to sp³ (from graphene to graphane) leads to the changing of the dynamics of lattice charge carriers. It is necessary to mention that the thermal properties of a material have merit to find applications in thermal management and thermoelectric.^[20,21] Since all nanoelectronic applications are closely related to the thermal properties, the investigation of thermodynamic properties is important. The EHC of a system defines as the ratio of the amount of heat which is used by the carriers (here, Dirac fermions) to the rising in temperature of the system. Gharekhanlou et al.^[22] reported that graphane-based materials can be used as bipolar transistor and have introduced a 2D p–n junction based on graphane.^[23] Savini et al.^[24] used a kind of p-doped graphane to fabricate a prototype high-T_c electron–phonon superconductor.

Here, we are about to compare the partial and total DOS and EHC of these systems. We will present the corresponding results based on Green’s function calculations. Monolayer graphene with full hydrogenation and fully hydrogenated bilayer graphene are both exhibit semiconducting characteristics similarly. As long as tight-binding Harrison model describes the dynamics of charge carriers. The outline of this paper is as follows. Section 2 describes the Hamiltonian and related calculation details. In Section 3, we will show the total and specially orbital DOS and EHC. In Section 4 the numerical results are explained and finally, Section 5 is the summary of the present paper.

The Harrison model describes the low-energy dynamics of Dirac fermions in the honeycomb lattice for all graphene-type configurations. As shown in Fig. 1, each unit cell includes 4 atoms: two carbons and two hydrogens for both chair-like and table-like structures. In each configuration, 2s, 2p_x, 2p_y, and 2p_z orbitals of each carbon atom and 1s orbital of each hydrogen atom participate in the sp³ hybridization, so the Hamiltonian can be written as a 10 × 10 matrix in the following form

Other elements are H_H1H1(k) = −2.4, H_H1B(k) = 0_1×4 where 0 uses for zero matrix. Also H_H1H2(k) = 0 for CL while it is −2.4 for TL, H_BB(k) = H_AA(k) and

The primitive unit cell vectors of honeycomb lattice in Fig. 1 are given by

By substituting Eqs. (1) and (2) into Eq. (5), Green’s function matrices can be found. The obtained Green’s functions in Eq. (5) has been used to find DOS and EHC. The details of calculations in terms of Green’s functions are presented in the next section.

The summation of imaginary part of Green’s function elements results the DOS.^[32] By engaging Eqs. (1), (2), and (5) and setting i ω_n → E + i0⁺ as a numerical calculation in which 0⁺ is a very small real number, the total and partial density of states can be eventuated

The study of C(T) in the monolayer and bilayer graphene lattice constitutes the main aim in this work. In the next section, the results are presented.

In this paper, we have obtained the total and partial electronic density of states and electronic heat capacity of monolayer and bilayer graphene in the presence of hydrogenation. We have implemented a tight-binding Harrison Hamiltonian model including local energy terms to describe the dynamics of massless Dirac and massive chiral fermions. We normalized all energies over the intra-layer hopping energy, eV.^[25–27] In this regard, for full contribution of s and {p_x, p_y, p_z} orbitals as well as for separately contribution of them, we have plotted DOS of hydrogenated monolayer and bilayer graphene over the whole bandwidth in Fig. 3.

Fig. 3.

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	Fig. 3. (color online) (a) The total and (b)–(f) orbital electronic densities of states for all kinds of monolayer and bilayer graphane.

Figure 3(a) shows that the monolayer graphane is a semiconductor with the band gap around 3.49 and 0.55 for chair-like and table-like structures, respectively in qualitative agreement with Refs. [34]–[36]. On the other hand, for AA-stacked and AB-stacked bilayer graphane, the band gaps are 2.58 and 2.89, respectively in agreement with Ref. [37]. Although, we have listed the band gap values of all orbitals in Table 1 for the better understanding of panels (a)–(f). What the reader confronts here may be different with the reported values in Table 1. We know that the band gap is the corresponding value to D(E) = 0 and this cannot be seen clearly in Fig. 3. For this reason, we have calculated the value of band gaps separately as presented in Table from which we are witness that in all the cases, the total DOS and contribution of p_z orbital behave similarly near the Fermi level, E = 0 (columns 1 and 5 of numbers). In graphene, the s, p_x, and p_y orbitals are responsible for the strong covalent bonding (in-plane bonding) while p_z orbitals lead to the weak van der Waals bonding (out-of-plane bonding). On the other hand, we know that in graphene the tangential π and π^* bands at Dirac points of the first Brillouin zone are form of the conductance electrons which are the main response for all interesting properties of graphene. So, total results should be close to p_z orbitals.

Table 1.

Table 1.

Table 1. The total and orbital energy gaps of hydrogenated monolayer and bilayer graphene in terms of . . Structure Total sC sH Chair-like monolayer graphane 1.15 1.09 1.06 1.0 1.26 1.77 Table-like monolayer graphane 0.86 0.94 1.86 1.0 0.77 0.94 AA-stacked bilayer graphane 0.18 0.25 0.45 0.89 0.23 1.53 AB-stacked bilayer graphane 0.95 0.55 0.52 0.89 0.87 1.6

Table 1. The total and orbital energy gaps of hydrogenated monolayer and bilayer graphene in terms of . .

Figure 4 shows the temperature dependence of partial and total EHC of monolayer chair-like, table-like, bilayer AA-stacked, and AB-stacked graphane. It is illustrating that there is a crossover, renowned as Schottky anomaly^[33] in all curves due to the Van-Hove singularities in DOS when the thermal energy reaches the energy gap between the subbands which parts the temperature dependence of the EHC into two temperature regions, low and high temperatures. The Schottky anomaly as an interesting effect can be explained in terms of the changing in the entropy of a system. As we know, at zero temperature only the lowest energy level is occupied and the entropy is equal to zero. In this regard, there is a very little probability of transition to a higher energy level however as the temperature increases, the entropy increases too monotonously and therefore the probability of transition goes up. As soon as the temperature closes to the difference between the energy levels in the system, a broad peak appears which is corresponding to a large change in the entropy for a small change in temperature. At high temperatures all of levels are occupied, so there is again a little change in the entropy for small changes in temperature and thus a lower specific heat capacity.^[38] Furthermore, it is well known that the EHC of semiconductors at low temperature can be written as C(T) ∝ e^−Δ/T.^[33,39] According to this formula, when Δ decreases, the C(T) increases at low temperature region as shown in Fig. 4 while decreases at high temperature region. In this region, less value of energy is needed for the temperature increasing of the system rather than for the low temperature region. Comparatively, C(T) reaches almost linearly to this anomalous peak in lower temperatures and bears more decay after that. Here, the low temperature defines as T < T_Shottky and high temperature as T > T_Shottky, in which T_Shottky is the critical temperature point (about 4.55 eV) for C(T)_max. Moreover, at high temperatures all curves go towards a common value because the temperature effects plays a dominant role rather than the other phenomena. Another point is in connection with the compared electronic heat capacity of these four structures. As can be seen easily, the maximum value for the total EHC among these structures is related to the bilayer AA-stacked graphene while its minimum is for the monolayer table-like graphene. Also this is valid for other orbital contributions except for the p_y orbitals which is the same for both monolayer and bilayer structures.

Fig. 4.

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	Fig. 4. (color online) (a) The total and (b)–(f) orbital electronic heat capacity for all kinds of monolayer and bilayer graphane.

Surprisingly, the behavior of p_y orbitals for two monolayer and two bilayer structures are the same. As we know, each C atom has s, p_x, p_y, and p_z orbitals with symmetric forms. By hydrogen doping, symmetry breaking occurs and since we have selected the x and y axes, respectively as horizontal and vertical vectors, the p_x orbitals from A atoms that are overlapped with that of B atoms are affected less than p_y orbitals because p_y orbitals have more accessible free space on the honeycomb lattice with this selection x–y coordinates. If we choose the y–x coordinates in our calculations, it is obvious that the p_x orbitals are affected more than p_y orbitals. For this reason, doped hydrogen does not change p_y orbitals interestingly and have the same contributions.

In summary, we have obtained qualitative results of electronic density of states and electronic heat capacity of the hydrogenated monolayer chair-like, table-like, bilayer AA-stacked, and AB-stacked graphene. The contribution of different orbitals on the heat capacity over a wide temperature range is compared. The Green’s function approach gives findings in Harrison model. It is concluded that the maximum and minimum band gap is for monolayer chair-like and bilayer AA-stacked graphene while the maximum and minimum Schottky anomaly is for bilayer AA-stacked graphene and monolayer table-like graphane, respectively.