Hydrogenation effects on the thermal and magnetic properties of mono- and bilayer graphene

Abstract

In the present study, the nearest-neighbor tight-binding model has been employed to calculate the density of states (DOS), electronic heat capacity (EHC), and Pauli magnetic susceptibility (PMS) of hydrogenated systems, namely monolayer graphone and graphane, bilayer graphone–graphene, and bilayer graphane–graphene. Then, the results have been compared with that of monolayer and simple bilayer graphene. It was found that the behaviors of hydrogenated systems differ from those of monolayer and bilayer graphene near the Fermi Level. Also, monolayer graphone and bilayer graphone–graphene exhibit a high peak near the Fermi level. Graphane monolayer, on the other hand, has no states around the Fermi level. Furthermore, bilayer graphane–graphone, similar to graphene, is a semimetal. Also, Schottky anomaly peaks in the EHC curves and crossovers in the PMS curves can be observed, which have divided the domain into two regions of low and high temperature. Compared to hydrogenated systems, the Schottky anomaly in graphene monolayer and bilayer graphene occurred at lower temperatures, while the PMS of monolayer graphone and bilayer graphone–graphene were faster than other systems in reaching the crossover. From the theoretical standpoint, these phenomena are due to the proportional relation of the PMS and EHC with the DOS.

Appendix

Fig. 5

a Structure of graphene with the primitive vectors ${{\varvec{a}}}_{1}$ and ${{\varvec{a}}}_{2}$. Dashed lines show the BLUC. b Schematic depiction of the BLUC of simple bilayer graphene

1.1 Graphene

The lattice structure of graphene monolayer is shown in Fig. 5a. As observed, the BLUC of graphene includes two nonequivalent atoms. Therefore, the TB Hamiltonian of this structure is represented by a $2\times 2$ matrix:

$$\begin{aligned} {\varvec{{{\mathcal {H}}}_k}}=-\left( \begin{array}{cc} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] \\ t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} 0\\ \end{array} \right) . \end{aligned}$$

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1.2 Graphane

Fig. 6

a Top view of graphane lattice structure with the primitive vectors ${{\varvec{a}}}_{1}$ and ${{\varvec{a}}}_{2}$. Dashed lines show the BLUC. b Schematic representation of the BLUC of bilayer graphone–graphene

The lattice structure of graphane monolayer is illustrated in Fig. 6a. According to this figure, the BLUC of graphane contains four nonequivalent atoms. Therefore, the TB Hamiltonian of this structure is given by a $4\times 4$ matrix:

$$\begin{aligned} {\varvec{{{\mathcal {H}}}_k}}=-\left( \begin{array}{cccc} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} t^{\prime } \\ t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} 0 &{} t^{\prime } &{} 0 \\ 0 &{} t^{\prime } &{} \varepsilon ^{\mathrm {H}} &{} 0 \\ t^{\prime } &{} 0 &{} 0 &{} \varepsilon ^{\mathrm {H}} \\ \end{array} \right) . \end{aligned}$$

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1.3 Bilayer graphene

According to Fig. 5b, the BLUC of the simple bilayer graphene (AA-stacked) includes four nonequivalent atoms. Therefore, the TB Hamiltonian of the bilayer graphene would be shown by a $4\times 4$ matrix:

$$\begin{aligned} {\varvec{{{\mathcal {H}}}_k}}=-\left( \begin{array}{ccccccc} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} t_\bot \\ t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} 0 &{} t_\bot &{} 0 \\ 0 &{} t_\bot &{} 0 &{} t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] \\ t_\bot &{} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 \\ \end{array} \right) , \end{aligned}$$

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in which $t_\bot =t_0/7$ is the inter-layer hopping parameter between the carbon atoms [32].

1.4 Bilayer graphone–graphene

As it is shown in Fig. 1b, the BLUC of bilayer graphone–graphene (AA-stacked) includes five nonequivalent atoms. Therefore, the TB Hamiltonian of this structure is represented by a $5\times 5$ matrix:

$$\begin{aligned} {\varvec{{{\mathcal {H}}}_k}}=-\left( \begin{array}{ccccc} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} t_\bot &{} 0 \\ t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} 0 &{} t^{\prime }_\bot &{} 0 &{} 0\\ 0 &{} t^{\prime }_\bot &{} 0 &{} t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} t^{\prime }\\ t_\bot &{} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} 0\\ 0 &{} 0 &{} t^{\prime } &{} 0 &{} \varepsilon ^{\mathrm {H}}\\ \end{array} \right) , \end{aligned}$$

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where $t_\bot =t_0/7$ is inter-layer hopping parameter between pristine carbon atoms [32], and $t^{\prime }_\bot =3t_0/14$ refers to the inter-layer hopping parameter between the carbon atoms in graphene layer and the carbon atoms bonded to the hydrogen atoms in the graphone layer.

1.5 Bilayer graphane–graphene

According to Fig. 6b, since the BLUC of bilayer graphane–graphene (AA-stacked) includes six nonequivalent atoms, its TB Hamiltonian is given by a $6\times 6$ matrix:

$$\begin{aligned} {\varvec{{{\mathcal {H}}}_k}}=-\left( \begin{array}{cccccc} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} t^{\prime }_\bot &{} 0 &{} 0 \\ t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} 0 &{} t^{'}_\bot &{} 0 &{} 0 &{} 0 \\ 0 &{} t^{\prime }_\bot &{} 0 &{} t_0\left[ 1+2f^{*}({{\varvec{k}}})\right] &{} t^{\prime } &{} 0 \\ t^{\prime }_\bot &{} 0 &{} t_0\left[ 1+ 2f({{\varvec{k}}})\right] &{} 0 &{} 0 &{} t^{\prime }\\ 0 &{} 0 &{} t^{\prime } &{} 0 &{} \varepsilon ^{\mathrm {H}} &{} 0\\ 0 &{} 0 &{} &{} t^{\prime } &{} 0 &{} \varepsilon ^{\mathrm {H}}\\ \end{array} \right) . \end{aligned}$$

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