Independent Channel Method for Lattice Thermal Conductance in Corrugated Graphene Ribbons

Abstract

Graphene’s extraordinary thermal conductivity makes it a compelling material for heat management in microelectronic circuits, lithium-ion batteries, and thermoelectric devices. In this article, we investigate its vibrational modes using a Born–von Karman model that includes first- and second-nearest-neighbor interactions. The resulting phonon dispersion relations agree well with experimental data, including acoustic flexural modes. To analyze phonon transport in mesoscopic graphene ribbons, we use both the Kubo–Greenwood and Landauer formalisms, as well as an independent channel method, which analytically maps zigzag-edged hexagonal ribbons into a set of single and dual chains via a unitary transformation. The resulting lattice thermal conductance spectra exhibit quantized steps that are smoothed in the presence of corrugations. We further explore the effects of temperature-induced rippling and buckling disorders on the phonon transport in graphene ribbons suspended over trenches. The predicted thermal conductance as a function of length and temperature closely matches experimental measurements, demonstrating the effectiveness of the independent channel method for the fully real-space modeling of corrugated graphene ribbons.

1. Introduction

Nowadays, semiconductor-based electronic devices are increasingly fabricated at mesoscopic scales, where thermal management constitutes a critical challenge [1]. Even modest temperature rises can severely degrade a device’s performance. In electrical insulators, heat is primarily carried by phonons, i.e., normal vibrational modes. Among emerging materials, a two-dimensional (2D) allotrope of carbon named graphene has attracted considerable attention due to its exceptional properties, including massless charge carriers [2], unusual phonon dispersion relations [3], and an ultrahigh thermal conductivity [4]. These attributes make graphene a promising candidate for advanced technological applications [5,6].

In general, the phonon transport is less studied than the electronic one, despite its importance in a wide number of physical phenomena, such as heat dissipation [7] and thermoelectricity [8]. Thermal conductivity of graphene as high as 5000 Wm⁻¹ K⁻¹ has been reported using Raman optothermal techniques [9], exceeding those of graphite and diamond [10]. Nevertheless, this exceptional conductivity is highly sensitive to disorder [11]. Moreover, the thermal conductivity of suspended graphene ribbons decreases with ribbon width [12] and increases with sample length for a fixed width [13].

On the theoretical side, phonon transport in graphene has been investigated using diverse approaches. For example, the Boltzmann transport equation was solved within the relaxation-time approximation, combining the effects of anharmonic interactions and edge scattering via Matthiessen’s rule [14]. Molecular dynamics simulations with Nosé–Hoover thermostats reveal almost the same thermal conductivity of graphene nanoribbons along both armchair and zigzag directions [15]. The Lanczos algorithm has also been employed to analyze the influence of edge disorder on phonon transport [16]. However, for mesoscopic graphene ribbons containing multiple structural defects, atomic-scale modeling of phonon scattering requires new strategies and approaches, because its direct calculations carried out on billions of atoms would exceed our current computational capacity.

In this article, we report a unitary transformation that maps zigzag-edged mesoscopic graphene ribbons with correlated ripple and buckling distortions onto a set of independent single and dual channels within the Born–von Karman model, including first- and second-neighbor interactions. The mathematical details of this transformation are provided in Appendix A. We have further developed a new transfer matrix method tailored for dual channels to evaluate the lattice thermal conductance within the Landauer formalism, whose results are verified by the Kubo–Greenwood formula, as shown in Appendix B. The in- and out-of-plane first- and second-neighbor restoring parameters in the Born–von Karman model are determined by fitting the measured phonon dispersion relations of graphene. Finally, the theoretical predictions of thermal conductance, including the thermal contact resistance analyzed in Appendix C, as a function of temperature and ribbon length, are compared against experimental data.

2. The Model

To investigate graphene’s vibrational modes, we use the Born–von Karman model, in which the interaction potentials $V_{m,n}^{(1)}$ and $V_{m,n}^{(2)}$, respectively, for first- and second-neighboring atoms m and n, with displacement vectors $u_m$ and $u_n$ from their equilibrium positions, are given by [17]

$$\left\{\begin{array}{l}{V}_{m,n}^{(1)}={\displaystyle \frac{{\gamma }_{m,n}-{\alpha }_{m,n}}{2}}{\left[(u_m-u_n)\cdot \hat{z}\hspace{0.17em}\right]}^{\hspace{0.17em}2}+\hspace{0.33em}{\displaystyle \frac{{\alpha }_{m,n}}{2}}\hspace{0.33em}{\left|\hspace{0.17em}{u}_m-u_n\right|}^2\\ V_{m,n}^{(2)}={\displaystyle \frac{{{\gamma }^{\prime }}_{m,n}-{{\alpha }^{\prime }}_{m,n}}{2}}{\left[(u_m-u_n)\cdot \hat{z}\hspace{0.17em}\right]}^{\hspace{0.17em}2}+\hspace{0.33em}{\displaystyle \frac{{{\alpha }^{\prime }}_{m,n}}{2}}\hspace{0.33em}{\left|\hspace{0.17em}{u}_m-u_n\right|}^2\end{array}\right.,$$

(1)

where ${\alpha }_{m,n}$ (${{\alpha }^{\prime }}_{m,n}$) and ${\gamma }_{m,n}$ (${{\gamma }^{\prime }}_{m,n}$) denote the in-plane and out-of-plane restoring force constants for first (second) neighbors, respectively. In this article, we align the zigzag direction of graphene with the X-axis and include second-neighbor interactions only for bonds parallel to this axis, which capture the acoustic flexural modes (ZA) in the phonon dispersion relations of pristine graphene.

Using a two-atom unit cell (see Figure 1a), the dynamical matrix ${\varphi }_{\mu ,\nu }^{(s)}(m,n)\hspace{0.17em}=\hspace{0.17em}{\partial }^2{V}_{m,n}^{(s)}/\partial u_{m,\mu }\partial u_{n,\hspace{0.17em}\nu }$, with $s\in \hspace{0.17em}\{1,2\}$ indexing first- and second-neighbor interactions, can be written in the k-space as

$$\Phi (k)=\left[\begin{array}{cc}3\hspace{0.17em}{\varphi }^{(1)}+[2-g_2(k)]\hspace{0.17em}{\varphi }^{(2)}& -g_1(k)\hspace{0.17em}{\varphi }^{(1)}\\ -g_1^{\ast }(k)\hspace{0.17em}{\varphi }^{(1)}& 3\hspace{0.17em}{\varphi }^{(1)}+[2-g_2(k)]\hspace{0.17em}{\varphi }^{(2)}\end{array}\right],$$

(2)

where $g_1(k)=\exp (i{k}_y{a}_{\mathrm{L}}/\sqrt{3})+2\exp [-i{k}_y{a}_{\mathrm{L}}/(2\sqrt{3})]\cos \hspace{0.17em}(k_x{a}_{\mathrm{L}}/2)$, $g_2(k)=2\cos \hspace{0.17em}(k_x{a}_{\mathrm{L}})$,

$${\varphi }^{(1)}=\left[\begin{array}{ccc}\alpha & 0& 0\\ 0& \alpha & 0\\ 0& 0& \gamma \end{array}\right] \mathrm{and}\hspace{0.17em}\hspace{0.17em}{\varphi }^{(2)}=\left[\begin{array}{ccc}{\alpha }^{\prime }& 0& 0\\ 0& {\alpha }^{\prime }& 0\\ 0& 0& {\gamma }^{\prime }\end{array}\right],$$

(3)

being $a_{\mathrm{L}}=0.246 \mathrm{nm}$, the lattice constant of pristine graphene. The phonon dispersion relations $\mathsf{\omega }(k)$ follow from the secular equation $\left|\Phi (k)-M{\mathsf{\omega }}^2(k)I\hspace{0.17em}\right|=0$, where M is the mass of carbon atoms, and I denotes the identity matrix.

High-symmetry points of the first Brillouin zone, namely $\Gamma \hspace{0.17em}=(0,0)$, $\mathrm{K}\hspace{0.17em}=(1\hspace{0.17em},\hspace{0.17em}0)4\pi /(3a_{\mathrm{L}})$, and $\mathrm{M}=(\sqrt{3},\hspace{0.33em}1)\pi /(\sqrt{3}{a}_{\mathrm{L}})$, are illustrated in Figure 1b. Theoretical phonon dispersion relations ${\mathsf{\omega }}_{\mathrm{T}}\left(k\right)$ (solid lines), obtained from the Born–von Karman model (Equation (1)) with the non-linear constraint for flexural modes $(\partial {\mathsf{\omega }}_{\mathrm{ZA}}^2/\partial k_x)(\Gamma \hspace{0.17em})=0$ along the Γ-Κ direction, i.e., ${\gamma }^{\prime }=-\gamma /4$, are plotted in Figure 1c, in comparison with the experimental data obtained from the inelastic X-ray (circles) [18], Raman (triangles) [19], and neutron (rhombuses) [20] scatterings, as well as the infrared absorption (IR) (stars) [21] and the electron energy loss spectroscopy (EELS) (squares) [22,23,24].

The resulting force constants are

$$\gamma \approx 104.1 \mathrm{N}/\mathrm{m}, \mathrm{with} \alpha =3\gamma , {\alpha }^{\prime }=\gamma /4 \mathrm{and} {\gamma }^{\prime }=-\gamma /4\hspace{0.17em},$$

(4)

obtained by minimizing the root-mean-square deviation given by

$$\sigma ={\left\{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{s=1}^N{\left[{\mathsf{\omega }}_{\mathrm{T}}\left(k_s\right)-{\mathsf{\omega }}_{\mathrm{E}}\left(k_s\right)\right]}^2}\right\}}^{1/2},$$

(5)

where $N=709$ is the number of experimental data ${\mathsf{\omega }}_{\mathrm{E}}\left(k_s\right)$ of Refs. [18,19,20,21,22,23,24], shown as open symbols in Figure 1c, and they are compared with ${\mathsf{\omega }}_{\mathrm{T}}\left(k_s\right)$ of the same color.

The analytical phonon dispersion relations, illustrated in Figure 1c, are given by

$$\left\{\begin{array}{c}{\mathsf{\omega }}_{\mathrm{XY},\hspace{0.17em}\Omega }^2(k)={\displaystyle \frac{\alpha \left[3\hspace{0.17em}\pm \hspace{0.17em}|g_1(k)|\right]\hspace{0.33em}+{\alpha }^{\prime }\left[2-g_2(k)\right]}{M}}\\ {\mathsf{\omega }}_{\mathrm{Z},\hspace{0.17em}\Omega }^2(k)={\displaystyle \frac{\gamma \left[3\hspace{0.17em}\pm \hspace{0.17em}|g_1(k)|\right]\hspace{0.33em}+{\gamma }^{\prime }\left[2-g_2(k)\right]}{M}}\end{array}\right.,$$

(6)

where $\Omega \in \{\hspace{0.17em}\mathrm{O}, \mathrm{A}\}$. The plus (minus) sign in Equation (6) corresponds to the optical (O) [acoustic (A)] branches for both the in-plane (XY) and out-of-plane (Z) vibrational modes. As seen in Figure 1c, the theoretical dispersion relations agree well with the experimental data, capturing the essential features of the measured graphene’s phonon spectra, producing a standard deviation of $\sigma \approx 15.7$ THz.

Now, let us consider a suspended graphene ribbon of length ${\mathcal{L}}_{\mathrm{G}}$ over a trench, clamped by two metallic contacts deposited on a crystalline silicon substrate, following the experimental setup in Ref. [13]. High-temperature annealing at 573 K, followed by cooling, leads to a thermal expansion mismatch between graphene and its supporting structure that produces ripples and buckling [25] in the suspended ribbon, as schematically shown in Figure 2. These thermally induced corrugations perturb the atomic arrangement, reducing the lattice thermal conductivity. Although graphene’s ripple textures on platinum supports have been experimentally observed [26,27], their quantitative impact on phonon transport remains largely unexplored.

Figure 2 shows a suspended graphene ribbon connected to the left and right graphene leads anchored on two platinum blocks with top surface area ${\mathcal{L}}_{\mathrm{Pt}}^2$, separated by a crystalline silicon slab of length ${\mathcal{L}}_{\mathrm{Si}}$, as illustrated in Figure 2c. The thermal expansion of a material is given by [28,29]

$${\mathcal{L}}_f={\mathcal{L}}_i\exp \left[{\displaystyle {\int }_{T_i}^{T_f}\lambda (T)\hspace{0.17em}dT}\right],$$

(7)

where $\lambda (T)$ is the temperature-dependent coefficient of thermal expansion; $T_i$ and $T_f$ are the initial and final temperatures; and ${\mathcal{L}}_i$ and ${\mathcal{L}}_f$ are the corresponding initial and final system length. Experimental data for graphene ${\lambda }_{\mathrm{G}}(T)$, platinum ${\lambda }_{\mathrm{Pt}}(T)$, and crystalline silicon ${\lambda }_{\mathrm{Si}}(T)$, are taken from Refs. [29,30,31] to determine the thermal expansion or contraction of each material.

Considering an initial temperature $T_i=573 \mathrm{K}$, at which the device has no lattice mismatch as a consequence of annealing [25], and a final measurement temperature $T_f\hspace{0.17em}<T_i$, two types of lattice-mismatch strain arise. The first occurs at the graphene–platinum interface and is described by

$${\epsilon }_{\mathrm{G}/\mathrm{Pt}}(T_f)=[{\mathcal{L}}_{\mathrm{G}}^{\mathrm{Lead}}(T_f)-{\mathcal{L}}_{\mathrm{Pt}}(T_f)]/{\mathcal{L}}_i^{\mathrm{Lead}},$$

(8)

with ${\mathcal{L}}_i^{\mathrm{Lead}}={\mathcal{L}}_{\mathrm{G}}^{\mathrm{Lead}}(T_i)={\mathcal{L}}_{\mathrm{Pt}}(T_i)$. This mismatch induces ripples along both the X- and Y-directions within the graphene leads. The second mismatch strain stems from the opposite signs of the thermal expansion coefficients, ${\lambda }_{\mathrm{G}}(T)<0$ and ${\lambda }_{\mathrm{Si}}(T)>0$, giving

$${\epsilon }_{\mathrm{G}/\mathrm{Si}}(T_f)=[{\mathcal{L}}_{\mathrm{G}}(T_f)-{\mathcal{L}}_{\mathrm{Si}}(T_f)]/{\mathcal{L}}_i,$$

(9)

where ${\mathcal{L}}_i={\mathcal{L}}_{\mathrm{G}}(T_i)={\mathcal{L}}_{\mathrm{Si}}(T_i)$. The strain in Equation (9) produces a buckling profile in the suspended ribbon along the X-direction, accompanied by a Y-direction rippling penetration from the left and right leads, as illustrated in Figure 2a–c. These combined corrugations are considered in this article.

Let us consider a zigzag-edged graphene ribbon that contains L (even number) transverse armchair lines and W (odd number) atoms in each line, and exhibits rippling and buckling angles ${\theta }_l$ along the X-direction (see Figure 2a) and rippling angles ${\phi }_l$ along the Y-direction (see Figure 2b’’), as well as a rippling penetration as shown in Figure 2b. Figure 2d illustrates the mapping process of this corrugated graphene ribbon into independent single and dual channels through a unitary transformation $\Xi$, which block-diagonalizes the dynamical matrix Φ of the considered ribbon given by

$$\Phi =\left[\begin{array}{ccccccc}\ddots & \ddots & \ddots & \ddots & \ddots & ⋮& ⋰\\ \ddots & A_{l-2}& B_{l-2}& C_{l-2}& 0& 0& \dots \\ \ddots & B_{l-2}& A_{l-1}& B_{l-1}& C_{l-1}& 0& \ddots \\ \ddots & C_{l-2}& B_{l-1}& A_l& B_l& C_l& \ddots \\ \ddots & 0& C_{l-1}& B_l& A_{l+1}& B_{l+1}& \ddots \\ \dots & 0& 0& C_l& B_{l+1}& A_{l+2}& \ddots \\ ⋰& ⋮& \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right],$$

(10)

whose submatrices are $B_l=I_W\otimes {\Theta }_l$, $C_l=I_W\otimes {{\Theta }^{\prime }}_l$,

$$A_{2n-1}=\left[\begin{array}{cccc}{\epsilon }_{2n-1}^{-}& 0& \dots & 0\\ 0& a_{2n-1}& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& a_{2n-1}\end{array}\right], \mathrm{and} A_{2n}=\left[\begin{array}{cccc}{a}_{2n}& 0& \dots & 0\\ 0& \ddots & \ddots & ⋮\\ ⋮& \ddots & a_{2n}& 0\\ 0& \dots & 0& {\epsilon }_{2n}^{-}\end{array}\right],$$

(11)

with $3W\times 3W$ elements each. Here, $I_W$ is the $W\times W$ identity matrix, ⊗ is the Kronecker product, and ${\Theta }_l$ and ${{\Theta }^{\prime }}_l$ are 3 × 3 matrices given by Equations (A22) and (A23), respectively. In Equation (11),

$${\epsilon }_l^{-}=-\hspace{0.17em}{\Theta }_{l-1}-{\Theta }_l-\hspace{0.17em}{{\Theta }^{\prime }}_{l-2}-\hspace{0.17em}{{\Theta }^{\prime }}_l \mathrm{and} a_l=\left[\begin{array}{cc}{\overline{\epsilon }}_l& {\phi }_l\\ {\phi }_l& {\overline{\epsilon }}_l\end{array}\right],$$

(12)

where ${\overline{\epsilon }}_l={\epsilon }_l^{-}-{\phi }_l$ with ${\phi }_l$ given by Equation (A25). The unitary transformation is given by

$$\Xi =\left[\begin{array}{ccccc}\ddots & \ddots & \ddots & ⋮& ⋰\\ \ddots & U_{l-1}^{3\mathrm{D}}& 0& 0& \dots \\ \ddots & 0& U_l^{3\mathrm{D}}& 0& \ddots \\ \dots & 0& 0& U_{l+1}^{3\mathrm{D}}& \ddots \\ ⋰& ⋮& \ddots & \ddots & \ddots \end{array}\right],$$

(13)

whose submatrices $U_l^{3\mathrm{D}}$ with $3W\times 3W$ elements can be written for $l=2n-1$,

$$U_{2n-1}^{3\mathrm{D}}={\displaystyle \frac{1}{\sqrt{W}}}\left[\begin{array}{cccc}1& -\sqrt{2}y& \dots & -\sqrt{2}y\\ z^T& R_{-}(1,1)& \dots & R_{-}(1,N)\\ ⋮& ⋮& \ddots & ⋮\\ z^T& R_{-}(N,1)& \dots & R_{-}(N,N)\end{array}\right]\otimes I_3,$$

(14)

and for $l=2n$,

$$U_{2n}^{3\mathrm{D}}={\displaystyle \frac{1}{\sqrt{W}}}\left[\begin{array}{cccc}{z}^T& R_{+}(N,1)& \dots & R_{+}(N,N)\\ ⋮& ⋮& \ddots & ⋮\\ z^T& R_{+}(1,1)& \dots & R_{+}(1,N)\\ 1& \sqrt{2}y& \dots & \sqrt{2}y\end{array}\right]\otimes I_3,$$

(15)

where $N=(W-1)/2$ is the number of dual channels, $y=(0,1)$, $z=(1,1)$, and

$$R_{\pm }(m,j)=\pm \sqrt{2}\left[\begin{array}{cc}\sin [2(2j-1)m\pi /W]& \cos [2(2j-1)m\pi /W]\\ -\sin [2(2j-1)m\pi /W]& \cos [2(2j-1)m\pi /W]\end{array}\right],$$

(16)

being $m=\hspace{0.17em}1,\dots ,N$ and $j=\hspace{0.17em}1,\dots ,N$. Applying $\Xi$ to Φ, i.e., ${\Xi }^{†}\Phi \hspace{0.17em}\Xi$, and after a reordering permutation (see, for example, Equation (A70)), we obtain a block-diagonal matrix given by

$$\stackrel{⌢}{\Phi }=\left[\begin{array}{cccc}{\mathit{S}}_0^{3\mathrm{D}}& 0& \dots & 0\\ 0& {\mathit{D}}_1^{3\mathrm{D}}& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& {\mathit{D}}_N^{3\mathrm{D}}\end{array}\right],$$

(17)

where ${\mathit{S}}_0^{3\mathrm{D}}$ and ${\mathit{D}}_j^{3\mathrm{D}}$ are matrices given by Equation (A34), corresponding to the single-channel and the j-th dual channel, respectively.

In graphene, the total thermal conductance ($K_{\mathrm{G}}$) is the sum of the electronic ($K_{\mathrm{G}}^{\mathrm{el}}$) and phononic $(K_{\mathrm{G}}^{\mathrm{ph}})$ contributions, i.e., $K_{\mathrm{G}}=\hspace{0.17em}{K}_{\mathrm{G}}^{\mathrm{el}}\hspace{0.17em}+K_{\mathrm{G}}^{\mathrm{ph}}$. Reported values indicate $K_{\mathrm{G}}^{\mathrm{el}}(40\hspace{0.17em}\mathrm{K})\hspace{0.17em}\approx \hspace{0.17em}{10}^{-10}\mathrm{W}\hspace{0.17em}{\mathrm{K}}^{-1}$, $K_{\mathrm{G}}^{\mathrm{el}}(300\hspace{0.17em}\mathrm{K})\hspace{0.17em}\approx \hspace{0.17em}{10}^{-9}\mathrm{W}\hspace{0.17em}{\mathrm{K}}^{-1}$, and $K_{\mathrm{G}}^{\mathrm{ph}}(30\hspace{0.17em}\mathrm{K}-300\hspace{0.17em}\mathrm{K})\approx {10}^{-7}\mathrm{W}\hspace{0.17em}{\mathrm{K}}^{-1}$ [13,32,33]; thus, $K_{\mathrm{G}}^{\mathrm{ph}}\gg K_{\mathrm{G}}^{\mathrm{el}}$. In particular, the phonon transport in graphene ribbons can be studied by means of the Landauer formalism [33] as

$$K_{\mathrm{G}}\cong K_{\mathrm{G}}^{\mathrm{ph}}={\displaystyle \frac{{\hslash }^2}{2\pi k_{\mathrm{B}}{T}^2}}{\displaystyle \underset{0}{\overset{\infty }{\int }}d\mathsf{\omega }\frac{{\mathsf{\omega }}^2\exp (\hslash \mathsf{\omega }/k_{\mathrm{B}}T)}{{[\exp (\hslash \mathsf{\omega }/k_{\mathrm{B}}T)-1]}^2}}\hspace{0.17em}\mathcal{T}(\mathsf{\omega }),$$

(18)

where $\hslash$ and $k_{\mathrm{B}}$ are, respectively, the reduced Planck and Boltzmann constants, T is the temperature, and $\mathcal{T}(\mathsf{\omega })$ is the phonon transmittance at angular frequency ω.

Using the independent channels in Equation (17) and the transfer matrix method for dual channels developed in Appendix B, the phonon transmittance $\mathcal{T}(\mathsf{\omega })$ of a corrugated graphene ribbon, decomposed as a single channel ($j=0$) and N dual channels (see Figure 2d), can be written as

$$\mathcal{T}(\mathsf{\omega })={\displaystyle {\sum }_{j=0}^N{\mathcal{T}}^{(j)}(\mathsf{\omega })},$$

(19)

where ${\mathcal{T}}^{(j)}(\mathsf{\omega })$ is the phonon transmittance of the j-th channel given by Equation (A73).

3. Results

In Figure 3, the phonon transmittance is shown for two graphene ribbons with the same length of 0.9 µm, but different widths: (a) 11 atoms and (b) 7043 atoms. The inset (b’) illustrates a magnified view of Figure 3b around $\mathsf{\omega }=280 \mathrm{THz}$, where quantized values of $T(\mathsf{\omega })$ are observed for the pristine case. Hence, the smoothness of curves in Figure 3b is a wider ribbon effect.

In Figure 3, the phonon transmittances in pristine and corrugated graphene ribbons are, respectively, denoted by gray and magenta lines, whose graphene–platinum thermal mismatch is ${\epsilon }_{\mathrm{G}/\mathrm{Pt}}=0.00348$, obtained by assuming a flat ribbon at the annealing temperature $T_i=573 \mathrm{K}$ and a corrugated ribbon at $T=300 \mathrm{K}$, which yields ripple angles ${\theta }_{\mathrm{lead}}\hspace{0.17em}\approx 6.76°$ and ${\phi }_{\mathrm{lead}}\approx 5.86°$ in both graphene leads (see Equations (A15) and (A17)). A second mismatch, arising from the different thermal expansion coefficients of graphene and crystalline silicon, gives ${\epsilon }_{\mathrm{G}/\mathrm{Si}}=0.00185$. It generates a catenary-type buckling profile (see Equation (A16)) with curvature radius ${\mu }^{-1}=4.33\hspace{0.17em}\mathsf{\mu }\mathrm{m}$ at the middle of the suspended graphene ribbon, as illustrated in Figure 2a.

To compare with measured thermal conductance, the theoretical prediction must include thermal contact resistance $R_{\mathrm{c}}(T)$ at the two interfaces between the graphene leads and the platinum electrodes, as discussed in Appendix C (see Equation (A84)). Accordingly, the predicted thermal conductance is

$$K(T,{\mathcal{L}}_{\mathrm{G}})={[\Delta R_{\mathrm{G}}^{\mathrm{corrugated}}(T,{\mathcal{L}}_{\mathrm{G}})+R_{\mathrm{G}}^{\mathrm{ballistic}}(T)+R_{\mathrm{c}}(T)]}^{-1},$$

(20)

where $R_{\mathrm{G}}^{\mathrm{ballistic}}(T)$ is the ballistic thermal resistance given in Equation (A81), and $\Delta R_{\mathrm{G}}^{\mathrm{corrugated}}(T,{\mathcal{L}}_{\mathrm{G}})=R_{\mathrm{G}}^{\mathrm{corrugated}}(T,{\mathcal{L}}_{\mathrm{G}})-R_{\mathrm{G}}^{\mathrm{flat}}(T,{\mathcal{L}}_{\mathrm{G}})$ accounts for the difference between the corrugated thermal resistance $R_{\mathrm{G}}^{\mathrm{corrugated}}={(K_{\mathrm{G}}^{\mathrm{corrugated}})}^{-1}$ and that of a flat ribbon $R_{\mathrm{G}}^{\mathrm{flat}}={(K_{\mathrm{G}}^{\mathrm{flat}})}^{-1}$, both obtained within the Born–von Karman model using Equation (18).

In Figure 4, the predicted lattice thermal conductance (red lines) obtained from Equation (20) is compared with experimental data of Ref. [13] (blue spheres).

In Figure 4, the predicted lattice thermal conductance is obtained from Equation (20) for suspended graphene ribbons of width $\mathcal{W}=1.5 \mathrm{\mu }\mathrm{m}$ and a buckling profile described by Equation (A16). The ribbon is connected to two corrugated graphene leads, whose rippling angles—generated by the thermal mismatch at the graphene–platinum contact—are ${\theta }_{\mathrm{lead}}\hspace{0.17em}\in [6.76°,9.04°]$ and ${\phi }_{\mathrm{lead}}\in [5.86°,7.83°]$ when the temperature decreases from 300 K to 40 K. The conductance is calculated using the independent channel method of Appendix A, together with the dual-channel transfer matrix method of Appendix B, and includes the thermal contact resistance from Appendix C. In these calculations, both the buckling profile and the ripple angles depend on the temperature, assuming a flat ribbon at 573 K immediately after annealing. We further consider the penetration of Y-oriented ripples with a depth of $L_d=500$ transverse lines from each lead into the suspended graphene ribbon, as illustrated in Figure 2b.

Finally, observe in Figure 4 an excellent agreement between the theoretical prediction (red lines) and the experimental data (blue spheres) from Ref. [13] over a broad range of temperatures and graphene ribbon lengths.

4. Conclusions

In this article, we study the normal vibrational modes and lattice thermal conductance of zigzag-edged and corrugated graphene ribbons within the Born–von Karman model. This study has been carried out by means of a new independent channel method, which maps the suspended graphene ribbon with rippling and buckling into a set of decoupled single and dual channels, using an analytical unitary transformation without introducing additional approximations (see Appendix A). We further extended the transfer matrix formalism to treat dual channels with interactions between next-nearest neighbors (see Appendix B). The in- and out-of-plane restoring force constants are determined by fitting the experimental phonon dispersion relations of pristine graphene.

To compare with experimental results, we additionally consider the thermal contact resistance and mismatches induced by differential thermal expansion, since graphene has a negative thermal expansion coefficient, while crystalline silicon and platinum have predominantly positive coefficients. In the measured device, the graphene ribbon is suspended between two platinum contacts mounted on a crystalline silicon block. The resulting expansion mismatch generates the buckled profile of the suspended section and produces ripples in the graphene leads along both the X- and Y-directions, as shown in Figure 2c.

The calculated lattice thermal conductance of corrugated graphene ribbons aligns excellently with the experimental data of Ref. [13], validating the independent channel method for mesoscopic ribbons containing over 4 × 10⁸ atoms. Beyond replicating experiments, the combination of independent channel and real-space renormalization methods establishes a robust, efficient, and scalable framework for studying the excitation dynamics in aperiodic solids [34]. Consequently, this study not only deepens our fundamental understanding of heat transport in two-dimensional materials but also paves the way for rationally designing next-generation, graphene-based thermoelectric and electronic devices, where precise thermal management is critical. Nevertheless, the present study did not include ribbon-edge reconstructions, as successfully treated in the electron transport on a graphene transistor [35]. This extension is currently under development.