# Layer-dependent thermal and thermoelectric properties of 2D diamond

Samir El Masaoudi, Pascal Puech, Fabrice Piazza, Iann Gerber

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Samir El Masaoudi, Pascal Puech, Fabrice Piazza, Iann Gerber. Layer-dependent thermal and thermoelectric properties of 2D diamond. Journal of Physics: Condensed Matter, 2025, 10.1088/1361-648X/ae0f6d hal-05326345

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Layer-dependent thermal and thermoelectric properties
of 2D diamond

Samir El Masaoudi¹, Pascal Puech², Fabrice Piazza³, and Iann C. Gerber*¹

¹Laboratoire de Physico-Chimie des Nano-objets (LPCNO), UMR5215 CNRS, INSA, 135 avenue de Rangueil, Toulouse, 31077, France

²Centre d'Elaboration des Matériaux et d'Etudes Structurales (CEMES), UPR8011 CNRS, Université de Toulouse, 29 Rue Jeanne Marvig, Toulouse, 31055, France

³Lab. Nanociencia, Pontificia Universidad Católica Madre y Maestra, Autopista Duarte km 1 1/2, Apartado Postal 822, Santiago de los Caballeros, Dominican Republic

September 7, 2025

Abstract

Hydrogenated few layer graphene, also called thin film diamond, two-dimensional
diamond or diamane are promising novel nano-materials with wide and direct band
gap. A theoretical investigation is carried out to study the thermal and thermoelec-
tric properties of hydrogenated graphene-based materials, namely graphane, bilayer
(2LD), and trilayer diamane (3LD). Phonon dispersion relations, calculated within
the harmonic approximation, confirm their dynamical stability. The quasi-harmonic
approximation is then employed to assess the volume-dependent quantities, revealing
positive thermal expansion coefficients for both diamane structures across the entire
temperature range, in contrast to the negative expansion observed in graphane at low
temperatures. The lattice thermal conductivity is obtained by solving the linearized
Boltzmann transport equation using both the single-mode relaxation-time approxima-
tion and the exact iterative scheme which at 300 K, have values of 574, 1314, and
1282 W·m⁻¹·K⁻¹ for graphane, 2LD, and 3LD, respectively. It highlights the influence
of hydrogenation and interlayer bonding on the conductivity. Mode-resolved analysis
indicates that low-frequency acoustic phonons are the primary contributors to heat
conduction, with increased lifetimes in multilayered structures. Thermoelectric prop-
erties are evaluated within the constant relaxation-time approximation, revealing a
significant enhancement of the Seebeck coefficient upon hydrogenation. Among the
studied systems, 2LD displays the best thermoelectric performance, reaching a max-
imum figure of merit (ZT) of approximately 12 × 10⁻³ at room temperature. These
results demonstrate the tunability of thermal transport and thermoelectric efficiency in
hydrogenated graphene structures, underscoring their potential for nanoscale thermal
management and energy conversion applications.

<footnote>*Corresponding author: igerber@insa-toulouse.fr</footnote>
# 1 Introduction

Since the discovery of graphene, 2D materials have been an interest in the world of nanoscience and novel technologies due to their exceptional electronic, mechanical, and thermal properties, which opens the door to numerous innovative applications. The focus of this computational investigation lies on diamanes [1], made of hydrogenated few-layer graphene, where the hydrogen atoms force the interlayer bonding [2] and sp³ hybridization of carbon atoms as shown in Fig. 1(a-b). Sometimes referred to as two-dimensional (2D) or ultra-thin diamond films, they can inherit key properties of diamond, such as its high thermal conductivity [3, 4] and mechanical stiffness [5], while also exhibiting novel characteristics that could arise from their reduced dimensionality. Diamanes were first theorized in 2009 by Chernozatonskii [6] predicting the stability of hydrogenated bilayer graphene (henceforth bilayer or 2LD).

Recently, the synthesis of 2LD was achieved by chemisorbing H atoms on the graphene surface [7], by irradiating with focused electron beams [8] or by compressing a few-layer graphene with high pressure diamond anvil cell [9]. These unique electronic and structural properties make diamanes promising candidates for various applications, including nanoop-tics, nanoelectronics, thermal management, and energy storage [10]. Thermal properties like conductivity or expansion play crucial roles in device performance and stability, for instance. However, the thermal behavior of diamane needs to be thoroughly understood. In this work, we explore the thermal properties of 2LD and 3LD (tri-layer hydrogenated graphene) using first-principles calculations. We compare their behavior with that of graphene, graphane (fully hydrogenated single layer graphene) and bulk diamond. The results are organized as follows: as a first step, density functional perturbation theory (DFPT) is used to determine the different phonon frequencies in the harmonic approximation (HA) along the high-symmetry path of the Brillouin zone. Our calculations reveal that the phonon dispersion of diamanes does not show imaginary frequencies, confirming their dynamical stability. In a second step, quasi-harmonic approximation (QHA) is used, which includes volume changes. It allows to compute the Grüneisen parameter (that indicates the degree of the structure's anharmonicity) and thermal expansion. By fitting the free energy versus volume data and using the Birch-Murnaghan equation of state, it becomes evident that the equilibrium volume increases in the range of 0 to 900 K, showing its volume-dependent character, where diamanes exhibit a more pronounced expansion rate with values that are three and two times higher than bulk diamond for 2LD and 3LD respectively. Lattice thermal conductivity (LTC) is also computed by solving the linearized Boltzmann transport equation (LBTE). We aim to provide a detailed analysis of how hydrogenation and layer stacking affect the thermal properties. Our calculations yield LTC values of 3346, 3128 W·m⁻¹·K⁻¹ for graphene and diamond, which stays consistent with the literature [3, 11, 12, 13, 14, 15] and LTCs of 574, 1314 and 1282 W·m⁻¹·K⁻¹ for graphane, 2LD and 3LD respectively at 300 K. Those results indicate that diamane structures present a high LTC compared to other materials used in technological applications (around 300-400 W·m⁻¹·K⁻¹ for gold and copper [16, 17]) although being lower than graphene and diamond. It highlights the impact of hydrogenation, interlayer C-C bonding and phonon scattering as the number of layers increases. Other authors showed similar LTCs for graphane and 2LD [4, 18, 19, 11], whereas in 3LD systems, LTC have not been studied yet, although Mortazavi et al. [20] have driven an investigation on mechanical, electronic and optical properties of different functionalized
diamanes (H-, F- and Cl-) from two to four layers showing the uniaxial stress-strain response of the different materials depending on the functional group and number of layers.
In the last section we compute the Seebeck coefficient and the electrical conductivity following the methodology in Ref. [21]. We recall that the thermoelectric effect (TE) occurs when a temperature gradient induces an electric potential expressed as $\Delta V = \Sigma\Delta T$ with $\Sigma$ the Seebeck coefficient and $T$ the temperature. The study of the TE aims to quantify its ability to convert this change in temperature into electrical potential. For a material to be used in thermoelectric applications a high Seebeck coefficient is essential, its performance is measured by the dimensionless figure of merit (ZT). We use the LTC to estimate the corresponding ZT figure of merit at room temperature (RT) which are 20, 12 and $10 \times 10^{-3}$ for graphane, 2LD and 3LD respectively. Recently A. H. Reshak [22] computed $\Sigma$ for two configurations of graphane (boat- and chair-like) finding similar results, whereas for 2LD and 3LD our results present, for the first time, the Seebeck coefficient and ZT of these materials, which, although comparable in magnitude, had not been previously reported, highlighting the novelty of this study. These findings will contribute to the growing body of research on 2D materials and their potential for advanced technological applications.

# 2 Calculation Methods

## 2.1 Computational details

All calculations are performed using the Vienna Ab Initio Simulation Package (VASP) [23, 24, 25] along with the general gradient approximation (GGA) for the exchange-correlation energy within the Perdew-Burke-Ernzerhof (PBE) scheme [26]. The core electrons are treated with the plane-augmented wave scheme [27, 28]. We use the perturbation theory (DFPT) to determine phonon and thermal properties, for which a post-processing treatment is done using Phonopy and Phono3py code [29, 30]. A plane-wave cut-off energy of 500 eV is set for the electronic self-consistent convergence. In order to avoid spurious interaction between periodic images in *c* axis, a 20 Å vacuum is introduced above the slab. Structures are optimized with a convergence on Hellmann-Feynman forces set to $10^{-2}$ eV·Å⁻¹. The optimized hexagonal lattice parameters (*a*) for graphane, 2LD and 3LD are found to be 2.54, 2.53 and 2.52 Å respectively. The corresponding interatomic distances between C-C (*d*<sub>C-C</sub>) and C-H (*d*<sub>C-H</sub>) bonds are summarized in Table 1. The hydrogenation of a single graphene layer leads to a wide opening of a direct band gap in the electronic structure at the Γ point [31]. DFT with PBE functional underestimates the band gap energy (*E*<sub>g</sub>) resulting in 3.61 eV, 3.18 eV and 2.85 eV respectively for graphane, 2LD and 3LD. We use the Green function based *GW* approximation to estimate a more accurate value of the band gap [32]. We perform the *GW* calculations using a primitive cell (1 × 1), with a (19 × 19 × 1) k-points sampling of the Brillouin zone (BZ) centered at Γ, a total of 384 bands are included to account for unoccupied states, and the energy cut-off for the response function is set at 150 eV. The fundamental direct gap *E*<sub>g</sub> values estimated with this method are 5.48 eV, 4.8 eV and 4.3 eV for graphane, 2LD and 3LD respectively. They are in agreement with other authors [2, 20, 33]. Results are displayed in Table 1.

The gap tends to decrease with respect to the number of layers. Our first hypothesis
<table><caption>**Table 1:** Optimized lattice parameters, interatomic distances (* for interlayer C-C bonds) and energy band gaps (DFT and GW) for graphane, 2LD and 3LD.</caption><thead><tr><th>Material</th><th>a (Å)</th><th>d<sub>C-C</sub> (Å)</th><th>d<sub>C-H</sub> (Å)</th><th>E<sub>g</sub>(DFT)(eV)</th><th>E<sub>g</sub>(GW) (eV)</th></tr></thead><tbody><tr><td>Graphene</td><td>2.46</td><td>1.42</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Graphane</td><td>2.54</td><td>1.54</td><td>1.11</td><td>3.61</td><td>5.48</td></tr><tr><td>Bi-layer</td><td>2.53</td><td>1.56*-1.54</td><td>1.11</td><td>3.18</td><td>4.80</td></tr><tr><td>Tri-layer</td><td>2.52</td><td>1.55*-1.54</td><td>1.10</td><td>2.85</td><td>4.30</td></tr><tr><td>Diamond</td><td>2.52</td><td>1.54</td><td>—</td><td>4.12</td><td>5.46</td></tr></tbody></table>

was that if we keep adding layer $E_g$ could eventually reach a minimum to then increase and converge towards the diamond value. Based on this assumption we compute $E_g$ bulk diamond obtaining values of 4.12 eV and 5.46 eV with DFT and GW respectively. We then investigate this gap shrinking effect by computing the band gaps from two to eleven layers and found that $E_g$ decreases with each added layer, though it appear to converge around 2.6 eV as shown in Fig. 2 Realizing that $E_g$ does not converge towards the expected 4.12 eV, we change our strategy and compute the band gaps for different diamane configurations grouped in two additional sets: (i) without any hydrogenation, and (ii) with a single side hydrogenation (SSH), again from two to eleven layers. In Fig. 3 we show the electronic band structure of the different configurations sets for three, six and eleven layers. For the SSH model a similar behavior is observed but with a localized state within the band gap. In the non-hydrogenated structures, for two and three layers, a Dirac cone emerges, indicating a graphene-like behavior. For more than three layers, interlayer bonding occurs and the structure begins to resemble diamond, exhibiting an indirect band gap. Within the gap, a few localized states appear, which gradually vanish as the number of layers increases. These mid-gap states are identified as surface states of the non-hydrogenated sides. In the SSH model, the gap tends to converge towards 3 eV approaching the fully hydrogenated limit. In contrast, for the non-hydrogenated structures, a gap opens from four layers onward and gradually decreases until converging to 4.12 eV. Ultimately, the tunability of the band gap in diamanes arises from combined effects of surface termination and interlayer interactions.

To estimate the phonon density of states and dispersion in the HA, computing second-order force constants is mandatory. A (7 × 7) supercell is considered along with (3 × 3 × 1) k-points for the BZ sampling. Since the diamanes are semiconductor, Born effective charges are also required to include non-analytical term correction for the phonon dispersion calculations, by modeling long-range interatomic forces as detailed in [34]. Phonon frequencies are also estimated within the QHA to obtain volume dependent properties. Lattice thermal conductivity and phonon lifetime are computed using a (4 × 4) supercell with the finite displacement method, second and third order force constants being calculated using Phonopy along with Phono3py module. With its constants, thermal conductivity is computed solving the linearized Boltzmann transport equation (BTE) [35].
## 2.2 Harmonic approximation

In a periodic crystalline structure, atoms oscillate around the equilibrium position. We can define the crystal potential energy as a function of the displacement of the atoms. This potential can be written as an expanded Taylor series as follows:

$$ 
\begin{aligned}
V = \Phi_o &+ \sum_{lk} \sum_{\alpha} \Phi_{\alpha}(lk) u_{\alpha}(lk) \\
&+ \frac{1}{2} \sum_{ll'kk'} \sum_{\alpha\beta} \Phi_{\alpha\beta}(lk, l'k') u_{\alpha}(lk) u_{\beta}(l'k') \\
&+ \frac{1}{3!} \sum_{ll'l''kk'k''} \sum_{\alpha\beta\gamma} \Phi_{\alpha\beta\gamma}(lk, l'k', l''k'') u_{\alpha}(lk) u_{\beta}(l'k') u_{\gamma}(l''k'') \\
&+ \dots,
\end{aligned} 
\quad (1) $$

where V is the crystal potential, $\Phi_o$, $\Phi_\alpha$, $\Phi_{\alpha\beta}$, $\Phi_{\alpha\beta\gamma}$ the zero-th, first-, second- and third-order force constants respectively, $lk$ the atomic position and $u_\alpha = r(lk) - r_{eq}(lk)$ denotes the displacement of atom $\alpha$ at $r(lk)$ with respect to his equilibrium position at $r_{eq}(lk)$. Since the atoms are in equilibrium at a minimum the first-order term is assumed to be $\Phi_\alpha(lk) = 0$. The second-order term corresponds to the Hessian matrix. Eigenvalues must be either positive or equal to zero for the system being at a stable minimum. Within the HA scheme we are interested in the second-order term and thus ignore the higher order terms. The force constants are computed taking the second derivative of the potential V with respect to the displacements of atoms $\alpha$ and $\beta$ following the equation:

$$ \Phi_{\alpha\beta}(lk, l'k') = \frac{\partial^2 V}{\partial u_{\alpha}(lk) \partial u_{\beta}(l'k')} = -\frac{\partial F_{\beta}}{\partial u_{\alpha}(lk)}, \quad (2) $$

where $F_\beta$ are the forces for a given set of Kohn-Sham orbitals [36]. Then the density of states of phonon can be determined with the equation:

$$ DOS(\omega) = \frac{1}{N} \sum_{qj} \delta(\omega - \omega_{qj}), \quad (3) $$

with N the number of unit cells in the crystal, $\omega$ the phonon frequency, $q$ and $j$ the wave vector and band index respectively. The energy of the phonon system is given by statistical mechanics taking into account the Bose-Einstein distribution as

$$ E_{phonon} = \sum_{qj} \hbar\omega_{qj} \left[ \frac{1}{2} + \frac{1}{e^{\hbar\omega_{qj}/k_B T} - 1} \right]. \quad (4) $$

Here $k_B$ and $\hbar$ are the Boltzmann and reduced Planck constants respectively and T the temperature. The HA is applied to both diamane systems (Fig. 1(a-b)) and graphane in order to see the influence of the number of layers on such properties.

Using the calculated phonon frequencies and thermodynamic relations, we compute the Helmholtz free energy ($F$), the heat capacity at constant volume, and the entropy as functions of temperature [29, 37]. With the following equations:
$$F = \frac{1}{2} \sum_{qj} \hbar\omega_{qj} + k_B T \sum_{qj} \ln \left[ 1 - \exp \left( -\frac{\hbar\omega_{qj}}{k_B T} \right) \right], \quad (5)$$

$$C_v = \sum_{qj} k_B \left( \frac{\hbar\omega_{qj}}{k_B T} \right)^2 \frac{\exp\left(\frac{\hbar\omega_{qj}}{k_B T}\right)}{\left[\exp\left(\frac{\hbar\omega_{qj}}{k_B T}\right) - 1\right]^2}, \quad (6)$$

$$S = \frac{1}{2T} \sum_{qj} \hbar\omega_{qj} \coth \left( \frac{\hbar\omega_{qj}}{2k_B T} \right) - k_B \sum_{qj} \ln \left[ 2 \sinh \left( \frac{\hbar\omega_{qj}}{2k_B T} \right) \right]. \quad (7)$$

## 2.3 Quasi-Harmonic approximation

Although the harmonic approximations provides a reasonable starting point, it cannot capture the anharmonic effects necessary to describe volume-dependent properties such as volume change and thermal expansion. The QHA approximates these effects by applying the HA at different volumes, as phonon properties will vary accordingly. Phonon frequencies are computed for different unit cell volumes with an expansion and contraction of ±5%.

The mode-Grüneisen parameter $\gamma_{qj}$ is computed for both of diamane nanosheet structures, this is a measure of the anharmonicity of the crystal [38], and is defined as follows:

$$\gamma_{qj} = - \frac{\gamma_{eq}^{0K}}{2\omega_{qj}^2} \sum_j e_{qj} \frac{\partial D_q}{\partial V} e_{qj}^*, \quad (8)$$

with $D_q$ the dynamical matrix for the wave vector $\vec{q}$, $\gamma_{eq}^{0K}$ the equilibrium volume at 0 K and $\vec{e}_{qj}$ the eigenvector for phonon band $j$. The temperature dependent Grüneisen parameter is obtained with the following equation:

$$\gamma(T) = \frac{\sum_{qj} \gamma_{qj} C_v^{qj}}{C_v}. \quad (9)$$

As temperature increases, most materials expand in volume. This thermal expansion $\alpha(T)$ provides useful information about their crystalline structure and is linked to the mode Grüneisen parameter. It can be determined using the equation:

$$\alpha(T) = \frac{C_v(T)\gamma(T)}{\mathcal{V}(T)B(T)}, \quad (10)$$

$B(T)$ is the bulk modulus and $\mathcal{V}(T)$ the volume of the studied material. In the case of diamond, this volume is simply that of the unit cell. However, for 2D materials, one must carefully define it. In this work, we set the thickness as the distance between the two hydrogen atoms located at the highest and lowest layers along the vacuum direction, plus twice the van der Waals H radius (one for each side), as illustrated in Fig. 1c.
## 2.4 Lattice thermal conductivity

In order to obtain the temperature dependent lattice thermal conductivity $\kappa_L(T)$ of a system, an accurate description of the anharmonicity of the crystalline structure is needed. The main contribution to LTC is the phonon scattering, thus it is necessary to account for the phonon-phonon interactions. The third order forces in Eq. (1) are computed using the finite displacement method which consists on small displacements on triplets of atoms, corresponding to $\alpha, \beta, \gamma$, in the Eq. (1). By combining HA calculations for each displacement, we obtain the atomic forces from which we compute the force constants and collision matrices. The last step is to solve the linearized Boltzmann transport equation (LBTE) as $\kappa_L(T) \propto \frac{2k_B T^2}{\mathcal{V}}$, where $\kappa_L$ is the lattice thermal conductivity, $k_B$ Boltzmann constant, $T$ temperature and $\mathcal{V}$ the volume of the structure as for the thermal expansion expression. For 2D structures, the equation is solved with a q-points sampling mesh over the first BZ of $(45 \times 45 \times 1)$ q-points, finding well-converged results, whereas for diamond a neat convergence is found with a smaller sampling of $(30 \times 30 \times 30)$ q-points mesh.

The LBTE is solved both within the single-mode relaxation time (SMRT) approximation and using the exact solution proposed by Chaput, as described in [39]. The SMRT approach is the most popular among researchers because of its simplicity and low computational cost [40]. However, this model assumes that all scattering processes contribute evenly to the thermal conductivity. This assumption becomes problematic when normal scattering processes dominate over umklapp processes, leading to an inaccurate estimation of the thermal conductivity, as discussed by Nika et al. [41].

For the study of diamane's LTCs, a detailed analysis of the computational parameters is carried out, as an abnormally low thermal conductivity was obtained when all atomic interactions were included in the calculation of the third-order force constants, compared to higher values reported in the literature [18, 4]. Therefore, the vacuum distance, the number of k-points used for BZ sampling, the supercell's size, and the cut-off distance limiting the range of atom pairs considered in the third-order force constants are varied. This analysis is motivated by the fact that certain parameters can significantly affect the final results of the calculations. Three vacuum spacings are tested: 20 Å, 25 Å, and 30 Å; k-point grids of $(1 \times 1 \times 1)$ and $(5 \times 5 \times 1)$ is used; and the cut-off distance is varied between 2 Å and 5 Å. Test simulations are performed for 2LD, calculated at 300 K.

## 2.5 Thermoelectric coefficients and the ZT figure of merit

A deeper understanding on the TE properties is carried out by including electron-phonon scattering. To compute electronic related properties a finer sampling of the BZ is needed and therefore we use a denser k-point grid of $(41 \times 41 \times 1)$. The electron-phonon interaction is computed with BoltzTrap2 code [42]. The code uses a constant relaxation-time approximation to evaluate the TE coefficient such as electric conductivity, the electronic contribution to the thermal conductivity relative to the relaxation-time, $\frac{\sigma(T\mu)}{\tau}$ and $\frac{\kappa_e}{\tau}$ respectively. The Seebeck coefficient $\Sigma$ and the ZT figure of merit defined as:

$$ZT = \frac{\sigma \Sigma^2 T}{\kappa_L + \kappa_e}, \qquad (11)$$
as in Ref. [43]. The estimated values for $τ$ lie between 10 fs and 100 ps, since $τ$ is not directly computed.

# 3 Results and discussion

## Harmonic Approximation

Phonon dispersion gives us valuable information on the dynamical stability of diamane nanosheets. Results are collected and the phonon band structure of 2LD and 3LD are plotted in Fig. 4, for comparison. It is necessary to emphasize that on the phonon dispersion, around $3000 \text{ cm}^{-1}$, few frequencies appear due to the presence of highly vibrating C-H bonds which are not depicted for readability reasons. H atoms are here to saturate dangling bonds and obtain well-defined sp³ structure mainly affecting the surface-localized vibrations (C-H stretch/bend) and the outermost out-of-plane modes. Its impact on in-plane acoustic heat carriers diminishes rapidly with thickness because the surface-atom fraction scales as inverse of the number of layers. Consequently, as number of layers increases the phonon spectrum and mode heat currents become dominated by the diamond-like carbon core, and the computed properties are expected to converge toward bulk diamond when increasing in the number of layers. Moreover, three acoustic modes at the $Γ$ point are observed due to the low-dimensionality of the structures, the longitudinal mode (LA), the transverse mode (TA), and the flexural out-of-plane mode (ZA). Phonon frequencies range from 0 to nearly $1300 \text{ cm}^{-1}$ showing high density states for the highest frequency modes.

The data is plotted in Fig. 5 for temperatures below 3000 K insets show the behavior below 300 K. As expected, the free energy $F$ decreases as the temperature grows. Showing that $F$ is higher for 2D materials than for bulk diamond, with values between 20 and $23 \text{ kJ} \cdot \text{mol}^{-1}$ compared to ~18 kJ·mol⁻¹. $F$ remains constant with negligible variations below 200 K, beyond that it slowly decreases. Free energies of different diamanes become negative around 1500 K whereas for diamond it is at 1600 K.

From Fig. 5b at temperatures below 200 K heat capacity $C_v$ of all materials follows a power-law behavior. Bulk diamond exhibits the expected Debye's $T^3$ dependence typical of three-dimensional solids, while the 2D structures, such as graphane and diamanes, shows a lower power-law exponent, closer to $T^2$, as predicted for two-dimensional systems. This distinction arises because, although each atom vibrates in all three spatial directions, the reduced dimensionality of 2D materials limits the phonon density of states at low energies, leading to the $T^2$ dependence. To better illustrate this behavior, the inset of Fig. 5b is plotted on a $log-log$ scale, the slope of the curves at low temperature visually confirm the theoretical exponents: a slope of 3 for bulk diamond and a slope closer to 2 for the 2D materials. The inset also highlights that, at low temperatures, graphane exhibits the highest $C_v$, requiring more energy to increase its temperature. In contrast, for the multi-layered structures, $C_v$ remains lower until around 250 K. Beyond this temperature, the trend reverses, and bulk diamond shows the highest heat capacity, also being the first to reach the Dulong-Petit limit ($3R \sim 25 \text{ kJ mol}^{-1}$), which all materials approach at high temperatures. This behavior reflects the influence of dimensionality.

The entropy $S$ follows a similar behavior, increasing with temperature for all materials
Fig. 5c. At low temperatures, *S* is also low as the system is in a more ordered state, it slowly increases following an exponential shape until reaching RT. At higher temperatures a slow and almost linear increase is observed. Within that temperature range, materials with fewer layers exhibit higher entropy at elevated temperature. Similar to heat capacity, the entropy of bulk diamond increases faster, being the highest among them.

2D materials behavior is consistent with what is expected, since a higher number of layers and degrees of freedom yields to a greater disorder whereas for bulk diamond the stability of the structure along with the high number of atoms will constraint the degrees of freedom. By analyzing the figures, we notice a consistent trend: adding layers cause their properties to converge towards those of bulk diamond.

## Quasi-Harmonic Approximation

By constraining (expanding and contracting) the unit cell volume, we obtain the phonon frequency dependence on volume, which is useful for capturing anharmonic effects. Fig. 6 shows the variation of *F* as function of the volume for temperatures going from 0 K to 900 K with a 100 K increment, results are fitted with the Birch-Murnaghan equation of state. We can observe a shift of the volume at the minimum energy point with the increase on temperature. That allows to determine the change on the equilibrium volume with respect to the temperatures and thus the Grüneisen parameter and thermal expansion.

Fig. 7 shows the change in $\gamma(T)$ against temperature, for diamond it converges to 0.9 which agrees well with the literature [44, 45]. For 2LD and 3LD, $\gamma(T)$ converges to 0.8 at high temperatures, whereas for graphane the limit is at 0.6 evidencing the anharmonic behavior of the phonon vibrational modes.

Thermal expansion is computed and plotted against temperatures ranging from 0 to 1000 K. In Fig. 8, 2LD and 3LD are compared with graphane and diamond. At first sight we observe two different profiles, for bulk diamond it is nearly zero until reaching RT then slowly increases to converge at high temperatures, this behavior is in agreement with different sources [46, 47, 48].

At low temperatures, graphane shows a negative $\alpha(T)$, thus a thermal contraction, a behavior that has already been reported [49]. Meaning that its volume slightly decreases until reaching a minimum at 100 K, beyond that point contraction turns into expansion increasing the volume faster and eventually converging at very high temperatures making it very sensitive to temperature changes. This behavior of graphane in the low temperature range is shared by graphene as explained in [50]. Acoustic modes, unlike optical modes, are more sensitive at low temperatures and can be excited with ease. It is the dominant contribution to $\alpha(T)$ in this temperature range whereas optical modes contribute much more for higher temperatures. Analyzing the mode Grüneisen parameter of each mode we found out that the transversal acoustic ZA mode is negative for graphane as shown in Fig. 9 the same way as for graphene in accordance with [50] leading to thermal contraction at low temperatures. On the other hand, diamanes have a positive $\alpha(T)$ in the whole temperature range. Negative thermal expansion arises from very low dimensionality and has its origin in the ZA phonon branch, by adding layers we introduce interlayer interactions that stiffen the out-of-plane modes, we have computed the Grüneisen parameter for each phonon mode for 2LD and 3LD, confirming that there is no or very little negative contribution from the ZA branch as shown in Fig. 10,
and thus, having a positive thermal expansion. At temperatures above 300 K, both 2LD and 3LD show a slower thermal expansion rate compared to graphane. However, 2LD expands faster than 3LD. Both structures tend to a plateau, exhibiting more stable behavior at higher temperatures.

Among the studied materials, graphane shows the highest sensitivity to thermal expansion, with negative values at very low temperatures and exponential growth at high temperatures. 2LD and 3LD maintain positive thermal expansion throughout the entire range, with the bilayer expanding more rapidly than the trilayer. Diamond is the most stable material in terms of thermal expansion, showing relatively low values and an almost constant behavior over the entire temperature range. In summary, 2D materials reach higher values for $\alpha(T)$ which is lowered by the number of layers, it is reasonable to expect that if we keep adding layers, $\alpha(T)$ will eventually converge toward that of diamond.

## Lattice Thermal Conductivity

LTC is first computed for diamond and graphene to ensure the reliability of the code and the accuracy of the method, comparing our results with previous works. [15, 14, 18, 13]. For graphene we define a thickness of 3 Å using the same criterion as for thermal expansion explained in section 2.3.

As shown in Fig. 11 graphene and diamond present similar and very high thermal conductivity. In literature values reported are typically between 2000 and 4000 W·m⁻¹·K⁻¹ at 300 K (see Table 3 for comparison with the different sources). In our calculations graphene and diamond reach a LTC of 3346 and 3128 W·m⁻¹·K⁻¹ respectively. Since we are dealing with extremely thin structures, its out-of-plane component is not trivial to define so the focus of interest here is the in-plane thermal conductivity for 2D materials, and because of its isotropicity $\kappa_{xx} = \kappa_{yy}$, idem for diamond where $\kappa_{xx} = \kappa_{yy} = \kappa_{zz}$.

<table><caption>**Table 2:** Lattice thermal conductivity (LTC) values for bilayer diamane with different vacuum distances, k-point grids, supercell sizes, and cut-off distances at 300K</caption><thead><tr><th></th><th colspan="3">20</th><th colspan="3">25</th><th colspan="3">30</th></tr></thead><tbody><tr><th scope="row">Vacuum (Å)</th><td colspan="3"></td><td colspan="3"></td><td colspan="3"></td></tr><tr><th scope="row">k-points</th><td colspan="3">1×1×1</td><td colspan="3">5×5×1</td><td colspan="3">1×1×1</td></tr><tr><th scope="row">Supercell</th><td colspan="3">4×4</td><td colspan="3">4×4</td><td colspan="3">4×4</td></tr><tr><th scope="row">Cut-off (Å)</th><td>2</td><td>3</td><td>4</td><td>3</td><td>3</td><td>3</td><td>4</td><td>5</td><td>3×7</td></tr><tr><th scope="row">LTC (W·m⁻¹·K⁻¹)</th><td>1486.2</td><td>1306.1</td><td>79.4</td><td>1457.9</td><td>1487.5</td><td>1314.4</td><td>22.2</td><td>10.2</td><td>1497.9</td><td>1091.2</td></tr></tbody></table>

The results reported in Table 2 show a dramatic drop in the thermal conductivity values when the cut-off distance exceeds 4 Å. The number of k-points and the vacuum size do not appear to have any significant relevance on the final result whereas the supercell-size seem to have some influence when dealing with high cut-off distances. However, the cut-off distance has a notable impact, especially when too many near-neighbors are considered. One hypothesis is that including atoms located too far apart from each other, and thus do not interact significantly, may introduce some numerical noise into the third-order force constants, thereby leading to a significant reduction in the computed LTC. Moreover, the
$\kappa_L$ value reported for a (7 × 7) supercell appears to decrease when a 4 Å cutoff distance is applied, although not as drastically as in the (4 × 4) supercell. This suggests that the supercell size may also play a role in the accuracy of third-order force constant calculations and should be considered carefully. The computed $\kappa_L$ can vary enormously depending not only on those parameters but also with the lattice parameter which can shift the group velocities/phase space and thus the LTC. The works of Zhu et al. [51] show how a difference in a with our calculation (2.49 versus 2.53 Å respectively) and a different way of evaluating the third order constants can lead to different results for $\kappa_L$. Therefore the third order force constants for each structure are calculated by avoiding the use of a cut-off distance larger than 3 Å along with a (4 × 4) supercell, a vacuum spacing of 30 Å along the c axis and a k-point sampling mesh of (1 × 1 × 1). Using the same method, we obtain the LTCs of graphane, 2LD and 3LD which at 300 K have values of 574, 1314 and 1282 W·m⁻¹·K⁻¹ respectively. In Fig. 12, $\kappa_L$ are computed with direct solution of LBTE and with SMRT approximation showing very different values according to the method used. SMRT shows significantly lower values than the direct solution of LBTE, they are reported and compared to literature in Table 3. To compare both computational approaches we determine the ratio SMRT/LBTE for each material. Results shows a significant drop of 82, 56, 55, 32 and 35% on $\kappa_L$ for graphene, graphane, 2LD, 3LD and diamond respectively. The classification of three-phonon scattering processes is based on momentum conservation principles within the first Brillouin zone. Normal processes satisfy the momentum conservation condition $q_1 + q_2 = q_3$, where all wavevectors remain within the first Brillouin zone boundaries. In contrast, umklapp processes require a reciprocal lattice vector G to satisfy momentum conservation: $q_1 + q_2 = q_3 + G$, effectively transferring momentum to the crystal lattice. The three-phonon phase space P counts the number of $(\lambda, \lambda', \lambda'')$ triplets that satisfy energy and crystal-momentum conservation. We resolve it into normal and umklapp channels,

$$P_N^{\pm}(\lambda) = \frac{1}{N} \sum_{\lambda', \lambda''} \delta(\omega_{\lambda} \pm \omega_{\lambda'} - \omega_{\lambda''}) \delta_{q_1 \pm q_2 - q_3, 0}, \quad (12)$$

$$P_U^{\pm}(\lambda) = \frac{1}{N} \sum_{\lambda', \lambda''} \sum_{G \neq 0} \delta(\omega_{\lambda} \pm \omega_{\lambda'} - \omega_{\lambda''}) \delta_{q_1 \pm q_2 - q_3, G}, \quad (13)$$

where $\delta_{q_1 \pm q_2 - q_3, 0}$ and $\delta_{q_1 \pm q_2 - q_3, G}$ are Kronecker deltas. In 2D materials the geometry severely limits umklapp solutions at low $\omega$, yielding a strongly reduced $P_U^{\pm}(\omega)$. After computing three-phonon scattering rates $\Gamma_{N/U}(\lambda)$, we find $\Gamma_N \gg \Gamma_U$ for low-frequency acoustic modes, which confirms this kinematic picture represented in Fig. (13 and 14). Hence, the SMRT underestimates $\kappa_L$ because it treats dominant normal processes as resistive, whereas the full LBTE does not. These findings highlight the predominance of normal scattering over umklapp processes in 2D materials, as previously discussed [41]. This effect becomes more pronounced as the number of layers decreases. Thermal conductivity follows an exponential decay with the increase of temperature for all materials. As shown in Fig. 12 hydrogenation of graphene leads to a drastic reduction in thermal conductivity of approximately 70–80%, a behavior already reported by Pei et al [52]. In contrast, diamanes exhibit a sharper and more abrupt reduction in thermal conductivity with increasing temperature. Furthermore, 2LD and 3LD, present similar behavior both with very high conductivities at low temperature. At higher temperatures their values tend to converge and remain below 300, 400
and 500 W·m⁻¹·K⁻¹ for graphane, 2LD and 3LD respectively. However diamanes present remarkably high thermal conductivity at RT when compared to other common materials. To our knowledge, there are no direct, published measurements of the in-plane LTC for H-terminated diamanes (2LD/3LD). Existing experimental work has focused on graphene (with widely quoted RT values ~ 1300 – 5000 W m⁻¹ K⁻¹ depending on sample and/or size) and on its functionalization, which strongly suppresses heat transport, consistent with the sp² → sp³ conversion [53]. Diamane synthesis has been verified (Raman signatures of sp³ carbon in hydrogenated bilayer graphene), but thermal transport data are not yet available [7].
<table><caption>**Table 3:** Calculated LTC values at 300 K for graphene, graphane, bi- and tri-layer diamane, and diamond, along with literature reported data, the methods used, and their respective references. All values are reported in W·m⁻¹·K⁻¹.</caption><thead><tr><th>Material</th><th>Reported values</th><th>Method</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="12">Graphene</td><td>3346</td><td>Direct solution to LBTE</td><td>This Work</td></tr><tr><td>609</td><td>SMRT</td><td>This Work</td></tr><tr><td>2700</td><td>ShengBTE including 3-phonon scattering (3ph)</td><td>[13]</td></tr><tr><td>1300</td><td>ShengBTE including 4-phonon scattering (4ph)</td><td>[13]</td></tr><tr><td>3200</td><td>Optimized Tersoff, Gauss-Seidel</td><td>[14]</td></tr><tr><td>800</td><td>SMRT</td><td>[14]</td></tr><tr><td>4000</td><td>Callaway model</td><td>[11]</td></tr><tr><td>400</td><td>SMRT</td><td>[11]</td></tr><tr><td>3636</td><td>ShengBTE</td><td>[18]</td></tr><tr><td>4840 – 5300</td><td>Measurement on suspended single layer graphene</td><td>[53]</td></tr><tr><td rowspan="4">Graphane</td><td>574</td><td>Direct solution to LBTE</td><td>This Work</td></tr><tr><td>258</td><td>SMRT</td><td>This Work</td></tr><tr><td>450–500</td><td>SMRT</td><td>[11]</td></tr><tr><td>1900</td><td>Callaway model</td><td>[11]</td></tr><tr><td rowspan="6">Bi-layer</td><td>1314</td><td>Direct solution to LBTE</td><td>This Work</td></tr><tr><td>589</td><td>SMRT</td><td>This Work</td></tr><tr><td>2034</td><td>ShengBTE</td><td>[4]</td></tr><tr><td>1145</td><td>ShengBTE</td><td>[18]</td></tr><tr><td>1360</td><td>ShengBTE</td><td>[19]</td></tr><tr><td>1282</td><td>Direct solution to LBTE</td><td>This Work</td></tr><tr><td>873</td><td>SMRT</td><td>This Work</td></tr><tr><td rowspan="10">Diamond</td><td>3128</td><td>Direct solution to LBTE</td><td>This Work</td></tr><tr><td>2021</td><td>SMRT</td><td>This Work</td></tr><tr><td>2300</td><td>ShengBTE (3ph)</td><td>[12]</td></tr><tr><td>2000</td><td>ShengBTE (3ph+4ph)</td><td>[12]</td></tr><tr><td>2700–3000</td><td>Conjugate gradient method</td><td>[15]</td></tr><tr><td>3320</td><td>Measurement on 0.07% <sup>13</sup>C enriched diamond</td><td>[3]</td></tr><tr><td>2600</td><td>Measurement on 0.5% <sup>13</sup>C enriched diamond</td><td>[3]</td></tr><tr><td>2230</td><td>Measurement on 1.0% <sup>13</sup>C natural abundance diamond</td><td>[3]</td></tr><tr><td>2190</td><td>Measurement on Natural diamond</td><td>[3]</td></tr></tbody></table>

The analysis of phonon mode-resolved properties for graphane, 2LD and 3LD reveals some differences in the phononic behavior of each material. As shown in Figs. 15 to 17, the main contributions to thermal conductivity arise from low-frequency phonon modes, particularly in the acoustic range. This behavior is closely related to two key factors: the phonon relaxation times ($\tau_{\lambda}$) and the group velocities ($|v_{\lambda}|$).

For both diamanes, acoustic modes exhibit relatively long lifetimes of the order of 10 to almost $10^3$ ps showing a clear difference in $\tau_{\lambda}$ between acoustic and optic modes, modes at higher frequencies have lower lifetime. As the number of layers decreases, from 3L diamane
to graphane, a progressive reduction in both $\tau_{\lambda}$ and $\bar{\kappa}_{\lambda}$ is observed, more evident in the low-frequency range. This trend can be attributed to enhanced phonon scattering processes induced by increasing structural complexity and interlayer interactions.

Regarding group velocities, although the differences among the three materials are less pronounced, a slight decrease is noticeable with the addition of more layers for the in-plane TA acoustic modes which goes from $12.5 \text{ km} \cdot \text{s}^{-1}$ for graphane to $\sim 11.5 \text{ km} \cdot \text{s}^{-1}$ for both diamanes, whereas for the out-of-plane ZA acoustic mode the velocity increases for frequencies between $100$ and $400 \text{ cm}^{-1}$ when adding more layers with a maximum on $|\vec{v}_{\lambda}|$ of 8, 9 and $10 \text{ km} \cdot \text{s}^{-1}$ for graphane, 2LD and 3LD respectively, resulting in a higher thermal conductivity.

Altogether, these findings explain why diamanes exhibit higher thermal conductivity compared to graphane structures, highlighting that thermal transport efficiency depends not only on individual phonon properties but also on structural characteristics that modulate scattering and phonon coherence.

In summary, the comparative analysis of graphane, 2LD and 3LD underscores how subtle structural variations influence phonon transport properties. The relatively long lifetimes of acoustic modes, coupled with notable group velocities, favor efficient heat conduction in diamanes. As the number of layers increases, enhanced phonon scattering and interlayer interactions slightly reduce phonon lifetimes but still support higher thermal conductivities than in graphane. These results highlight that both intrinsic phonon dynamics and external structural factors play critical roles in optimizing thermal transport behavior.

## Thermoelectric Properties

As for LTC, it is essential to validate the method by comparing results with the known data of graphene and diamond. In Fig. 18 we plot the Seebeck coefficient and the scaled electrical conductivity $\frac{\sigma}{\tau}$ which are very close to values estimated by [54, 55, 56, 57]. It has to be noted that obtaining electron-phonon relaxation times directly from VASP is not a trivial task. Accurate calculations require very dense k- and q-point meshes, explicit evaluation of electron-phonon matrix elements, and a careful treatment of long-range polar interactions in two-dimensional systems, which makes the procedure computationally demanding [58]. For this reason, and in line with reported values for graphene and diamond where typical electron-phonon scattering times lie in the 10 fs-100 ps range [43, 59], we adopt this interval as a reasonable estimate for $\tau$ in our transport calculations.

In graphene, ultrafast optical and transport studies have shown carrier relaxation times governed by electron-phonon scattering up to the order of a few tens of picoseconds [59, 60, 61]. In diamond, time-resolved spectroscopy has revealed relaxation times up to 7 ps of picoseconds, depending on temperature and excitation energy [62, 63]. Fig. 19 shows the figure of merit ZT at 300K as a function of energy for graphene and diamond. In the case of graphene, the ZT profile exhibits a series of sharp peaks symmetrically distributed around the Fermi level, characteristic of its linear band dispersion and high carrier mobility. ZT increases with longer relaxation-times. However the values remain low, below $1.5 \times 10^{-3}$, reflecting graphene's intrinsically high electronic thermal conductivity. Diamond in the other hand presents a different behavior. The figure of merit is negligible within the wide band gap region, but it increases significantly near the conduction and valence band edges. Notably,
the values of ZT exceed $35 \times 10^{-3}$ for the longest relaxation time considered suggesting potential for thermoelectric performance under heavy doping or defect engineering [64].

The Seebeck coefficient for graphane, 2LD and 3LD is shown in Fig. 20, as observed Σ increases when graphene is fully hydrogenated varying weakly for 2LD and 3LD, in contrast there is a decrease in $\frac{\sigma}{\tau}$, specially at high temperatures, a behavior that is closer to diamond. In [22] the author computed Σ for two different configurations of graphane obtaining similar results.

Fig. 21 shows the figure of merit ZT at 300K as function of energy for (a) graphane, (b) 2LD and (c) 3LD. For all systems ZT increases with the relaxation time $\tau$ and presents sharp peaks close to the band edges.

In the case of graphane, two clear peaks appear at $E - E_F \approx -0.2$ and 3.2 eV, reaching a maximum value of ZT $\approx 20 \times 10^{-3}$ for $\tau = 10$ ps. Compared to graphene, the values are significantly higher due to the reduced electronic thermal conductivity. This maximum value is gradually reduced by the addition of new layers, hence for 2LD and 3LD ZT reaches values of $12 \times 10^{-3}$ and $10 \times 10^{-3}$. The reduction is related to enhanced $\kappa_e$ as more layers are added.

These results suggest that the optimal configuration in terms of thermoelectric performance under the constant $\tau$ approximation is the 2LD structure, which provides a balance between high Seebeck coefficient and suppressed electronic thermal conductivity.

# 4 Conclusion

The investigation of thermal properties of different graphene-related materials such as graphane, 2LD and 3LD shows how the different magnitudes evolve through temperature variation. Using first-principle calculations we describe how phonon frequencies calculated within the harmonic approximation are useful to extract different thermal properties. Such properties show a thermal behavior that is typically found in 2D materials where the number of layers have an important influence over it.

In this paper we show that diamanes are dynamically stable materials since no imaginary frequency is observed in the phonon dispersion curve. Free energy decreases over the temperature range whereas heat capacity and entropy increases. Thermal properties dependence on the number of layers is shown as it converges towards those of bulk diamond when this number grows. This last statement is also proved by going beyond the harmonic approximation, with variations in volume of the unit cell from which the thermal expansion is computed. This expansion remains positive for the diamanes across the temperature range studied, unlike the thermal contraction observed in graphane at low temperatures which is due to the high contribution of ZA acoustic mode on the Grüneisen parameter. The thermal expansion values in diamanes decrease as the number of layers increases, indicating that as the material thickness grows, its thermal behavior becomes more similar to that of three-dimensional structures like diamond.

When the phonon-phonon interactions are included and the third order force constant is computed, we were able to determine the LTC using two different approaches. The SMRT [40] which is computationally less resource demanding gives inaccurate results for most of 2D structures the reason for it is that normal and umklapp phonon scattering processes are
not taken into account properly and are considered to contribute evenly while usually in 2D materials normal scattering is the dominant process. On the other hand the direct solution of Boltzmann transport equation computed with Chaput's method [39] gives more reliable results which shown in agreement with other authors and with other methods. It shows that thermal conductivity of diamanes is considerably higher than the one of graphane and with values that are almost the half of graphene and diamond at 300 K. Indeed, going from one layer (graphane) to multilayer (diamanes) increases the LTC, with similar values when considering either a two or a three layered diamane in contrast to thermal expansion which tends to converge to that of diamond with the increasing number of layers. This behavior is consistent when observing how the different phonon modes contribute to the LTC.

Regarding the thermoelectric properties, the Seebeck coefficient increases significantly when graphene is fully hydrogenated, as in the case of graphane, and remains relatively stable when increasing the number of layers in diamane. This indicates that the band structure changes induced by hydrogenation play a major role in enhancing S, while interlayer coupling has a more moderate impact. The figure of merit ZT also improves substantially with respect to pristine graphene. In graphane, the presence of a band gap and reduced thermal conductivity lead to higher values of ZT. Adding layers leads to a decrease on the ZT due to the increase in electronic thermal conductivity $\kappa_e$.

Among the structures studied, 2LD exhibits the most favorable balance between a high Seebeck coefficient and moderate electronic thermal conductivity under constant $\tau$, making it the most promising candidate for thermoelectric optimization. While materials with intrinsically high lattice thermal conductivity may seem poor thermoelectrics, studying their performance is still meaningful. The Seebeck coefficient and electrical conductivity provide direct insight into the electronic structure and potential power factor, which can be competitive under proper doping or gating [65, 66, 52, 67]. The high in-plane conductivity combined with a favorable power factor could also enable integration in nanoscale cooling or energy-harvesting devices [68]. $\kappa_L$ is not a fixed quantity and can be suppressed through strain, isotopic engineering, and functionalization as it has been shown in diamond and graphene-based systems [69, 3]. Meaning that further enhancements on the thermoelectric performance could be achieved by external stimuli that simultaneously affect the electronic and phononic subsystems. Electrostatic gating or light chemical doping allows the Fermi level to be tuned near the band edges, enhancing the Seebeck coefficient and power factor as demonstrated in graphene/hBN devices [68, 70]. Similarly, strain engineering has been shown to substantially reduce the lattice thermal conductivity in sp³-derived carbon systems while modifying the band dispersion [4, 71]. Controlled functionalization or isotopic disorder can further suppress phonon transport without strongly perturbing the electronic structure [52, 3, 69]. Taken together, these strategies suggest that although pristine 2LD exhibits high $\kappa_L$, it remains possible to achieve significant improvements in ZT by combining carrier tuning with phonon engineering. Therefore, assessing the thermoelectric baseline of 2LD provides a valuable reference for future optimization.
Acknowledgments

This research was funded by ANR in the GLADIATOR Project ANR-21-CE09-0003 and partially funded by the NanoX ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir" for the research program NV-DIAM and by the "Fondo Nacional de Innovación y Desarrollo Científico y Tecnológico (FONDOCyT)", Dominican Republic, grant 2022-1A1-095, and by the U.S. Office of Naval Research Global and the Southern Office of Aerospace Research & Development, grant N62909-24-1-2000. We thank the "Calcul en Midi-Pyrénées" initiative CALMIP through project p0812, as well as TGCC, IDRIS and CINES under the allocation 2024-A0160906649 made by GENCI for computer time.
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Top and side views of diamanes: (a) bi-layer, (b) tri-layer, and (c) the effective thickness of the different structures. From top to bottom: graphane, bi-layer, and tri-layer diamane. Carbon and hydrogen atoms are represented in brown and pink, respectively.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Evolution of $E_g$(DFT) in function of the number of layers in (a) H-diamane (b) SSH-diamane and (c) non-hydrogenated diamane.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

(a)

<figure><img src="image_5.png" /></figure>

(b)

<figure><img src="image_6.png" /></figure>

(c)

<figure><img src="image_7.png" /></figure>

(d)

<figure><img src="image_8.png" /></figure>

(e)

<figure><img src="image_9.png" /></figure>

(f)

<figure><img src="image_10.png" /></figure>

(g)

<figure><img src="image_11.png" /></figure>

(h)

<figure><img src="image_12.png" /></figure>

(i)

**Figure 3:** Electronic band structure evolution with respect of the number of layers and hydrogen coverage. In the first row (a-c) for H-diamane, the second row (d-f) depicts SSH-diamane and the last row (g-i) is for non-H diamanes, from left to right the number of layers increase starting with 3, 6 and 11 layers for the last column.
<figure><img src="image_13.png" /><figcaption>(b) Bi-layer diamane</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c) Tri-layer diamane</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 4:** Phonon dispersion along the high-symmetry path Γ → M → K → Γ for (a) graphane, (b) bi-layer diamane, and (c) tri-layer diamane.</figcaption></figure>
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 5:** Thermal property evolution with temperature ranging from 0 to 3000 K for graphane, bi-layer diamane, tri-layer diamane, and diamond. Panels show: (a) Free energy, (b) Heat capacity, and (c) Entropy. Insets highlight the behavior below 300 K.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 6:** Free energy vs volume for temperatures going from 0 K top curve to 900K bottom curve. Red line links the minimum energy point for each temperature.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**Figure 7:** Average mode-Grüneisen parameter for graphane, diamanes and diamond.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 8:** Thermal expansion of graphane, bi- and tri-layer diamane and diamond.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 9:** (Top) Graphane mode Grüneisen parameter $\gamma_{qj}$ for phonon modes. (Bottom) Phonon dispersion of graphane through the high symmetry path $\Gamma \rightarrow M \rightarrow K \rightarrow \Gamma$. Labels correspond to the acoustic longitudinal, transversal and flexural modes, LA, TA and ZA respectively. Coloured lines from the top figure correspond to the same phonon mode on bottom figure.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 10:** (Top) mode Grüneisen parameter $\gamma_{qj}$ for phonon modes. (Bottom) Phonon band dispersion through the high symmetry path $\Gamma \rightarrow M \rightarrow K \rightarrow \Gamma$ for (a) 2LD and (b) 3LD. Coloured lines from the top figure correspond to the same phonon mode on bottom figure.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Figure 11:** Lattice thermal conductivity of graphene and diamond for temperature ranging from 150 to 800 K, solid line is $κ$ computed with exact solution and dotted lines with SMRT.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Figure 12:** Lattice thermal conductivity of graphane and bi- and tri-layer diamane for a temperature range between 150 and 800 K, solid line is $κ$ computed with exact solution and dotted lines with SMRT.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>**Figure 13:** From left to right: (a, d) three-phonon phase space at T = 300 K, (b, e) Normal vs Umklapp processes and (c, f) Phase Space vs Temperature for graphene (top) and diamond (bottom).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Figure 14:** From left to right: (a, d, g) three-phonon phase space at T = 300 K, (b, e, h) Normal vs Umklapp processes and (c, f, i) Phase Space vs Temperature for graphane (top), 2LD (middle) and 3LD (bottom).</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Figure 15:** Phonon mode-resolved properties of graphane, from top to bottom; thermal conductivity $\bar{\kappa}_{\lambda}$, phonon life-time $\tau_{\lambda}$ and group velocity $|v_{\lambda}|$ at 300 K. Color gradient indicate phonon modes going from acoustic (blue) to optic (red)</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>**Figure 16:** Phonon mode-resolved properties of 2L-diamane, from top to bottom; thermal conductivity $\bar{\kappa}_{\lambda}$, phonon life-time $\tau_{\lambda}$ and group velocity $|v_{\lambda}|$ at 300 K. Color gradient indicate phonon modes going from acoustic (blue) to optic (red)</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>**Figure 17:** Phonon mode-resolved properties of 3L-diamane, from top to bottom; thermal conductivity $\bar{\kappa}_{\lambda}$, phonon life-time $\tau_{\lambda}$ and group velocity $|v_{\lambda}|$ at 300 K. Color gradient indicate phonon modes going from acoustic (blue) to optic (red)</figcaption></figure>
<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /><figcaption>**Figure 18:** Seebeck coefficient and scaled electrical conductivity as function of carrier concentration for different temperatures of (a) graphene and (b) diamond.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>(a) Graphene</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(b) Diamond</figcaption></figure>

**Figure 19:** Figure of merit ZT at 300 K as function of energy of (a) graphene and (b) diamond.
<figure><img src="image_35.png" /><figcaption>(b) Bi-layer diamane</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>(c) Tri-layer diamane</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>**Figure 20:** Seebeck coefficient and scaled electrical conductivity as function of carried concentration for different temperatures of (a) graphane and (b) 2L and (c) 3L diamane.</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>(b) Bi-layer diamane</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>(c) Tri-layer diamane</figcaption></figure>

<figure><img src="image_40.png" /></figure>

**Figure 21:** Figure of merit ZT at 300 K as function of energy of (a) graphane and (b) 2L and (c) 3L diamane.
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