Low bending rigidity and large Young's modulus drive strong flexural phonon renormalization in two-dimensional monolayers

Abstract

Many intriguing phenomena such as the wave-like hydrodynamic heat flow, the logarithmic divergence of electrical resistivity at low temperatures and microscale kirigami are driven by flexural acoustic (ZA) phonons in two-dimensional (2D) materials. Yet, a definitive first-principles description of their dispersion, with explicit consideration of the crystal anharmonicity and the stability of large 2D monolayers against thermal fluctuations, is lacking in the literature. Using first-principles calculations, we show that the bending rigidity ($κ$) controls the anharmonic renormalization of the ZA phonons throughout the Brillouin zone in 2D monolayers, with stronger renormalization in low-$κ$ materials like germanene and weaker effects in high-$κ$ materials like molybdenum disulphide. Furthermore, the ZA phonons at long wavelengths undergo an additional renormalization to stabilize the flat phase of the 2D monolayers against thermal fluctuations, which is modulated by the competing influence of the bending rigidity and the in-plane Young's modulus in all materials. The resulting renormalized ZA phonon dispersions are qualitatively and quantitatively different from those commonly used by the first-principles community, thus motivating a re-examination of the ZA phonon-driven unconventional thermal and electronic phenomena in 2D as well as lower-dimensional systems. Our work provides new insights into the role of nanoscale crystal anharmonicity and macroscale elasticity in shaping the vibrational properties of 2D materials and will inform novel engineering applications that are exclusive to low dimensions such as kirigami, with materials beyond graphene.

Figures (6)

• Figure 1: Origin of strong anharmonic renormalization of ZA phonons and elastic constants in germanene.(a). Long wavelength (small-$q$) behavior of $\omega/q^2$ for ZA phonons in different 2D monolayers calculated using the bare harmonic IFCs. The ZA phonons in all 2D monolayers have a quadratic dispersion in the small-$q$ limit, with the curvature $\left[2\omega/q^2\right]$ being the smallest for germanene and the largest for MoS$_2$. (b). Bar plots of the relative changes in the $\kappa_0\left(T\right)$ and $Y^{\text{2D}}_0\left(T\right)$ of four 2D monolayers caused by the anharmonic renormalization of the bare harmonic IFCs using the SCAP framework. Germanene exhibits the strongest renormalization of the elastic constants due to the small curvature of its ZA phonons at small $q$, resulting in their high thermal occupation that drive large thermal excursions of atoms in the out-of-plane direction, as shown in (c), thus resulting in strong anharmonic renormalization of the harmonic IFCs. The contrasting relative magnitudes of renormalization of the $\kappa_0\left(T\right)$ and the $Y^{\text{2D}}_0\left(T\right)$ in all materials is caused by the much stronger anharmonic renormalization of the appropriately-scaled cross-plane harmonic IFCs $\left[a^2\Xi_{zz}/\kappa_0 = a^2\left(\psi_{zz} - \chi_{zz}\right)/\kappa_0\right]$ compared to that of their in-plane counterparts $\left[\Xi_{xy}/Y^{\text{2D}}_0\right]$, as shown e.g., for germanene at 300 K in (d).
• Figure 2: Temperature-dependent acoustic phonon dispersions in germanene along high symmetry directions, along with the bare (i.e., $T \sim 0$ K) dispersions. We observe a stronger anharmonic renormalization of ZA phonons relative to that of LA and TA phonons, from the BZ center [$\Gamma$-point] extending till about mid-way to the BZ edge, driven by the small $\kappa_0$ of germanene.
• Figure 3: Renormalization of the elastic constants of 2D monolayers with system size at 300 K.(a) Bending rigidity, $\kappa$, of 2D monolayers as a function of the sample size ($L$), showing a universal scaling of $\kappa\left(L\right)\sim L^{0.82}$ for samples larger than a micron in all 2D monolayers while approaching their respective $\kappa_0\left(T\right)$ for nanoscale samples. The calculated $\kappa$'s agree well with the available measurements in the literature for graphene that span several orders of magnitude from $\sim 1$ eV for a supported sample that mimics a nanoscale system to $\sim 10^3$ eV for suspended samples with dimensions exceeding 10 $\mu$m. For the suspended samples, the system size is chosen as a geometric mean of the reported lengths and widths in Ref. blees_graphene_2015. (b) Dimensionless vacuum polarization [$b_0\left(T\right)I\left(k\right)$] as a function of the wave vector $k$ for different 2D materials, which causes the transition from the small-$L$ [$b_0\left(T\right)I\left(k\right) \ll 1$] to large-$L$ [$b_0\left(T\right)I\left(k\right) \gg 1$] trends in the elastic constants through the SCSA equations. The quantity $I\left(k\right)$ is plotted as an inset for reference. (c) Two-dimensional Young's modulus, $Y^{\text{2D}}$, of 2D monolayers as a function of $L$, showing a weaker universal scaling relative to that of $\kappa$$\left[Y^{\text{2D}}\left(L\right)\sim L^{-0.36}\right]$ for samples larger than a micron in all 2D monolayers, while approaching their respective $Y^{\text{2D}}_0\left(T\right)$ for nanoscale samples. The scaling exponents of $\kappa$$\left[\eta_\kappa\right]$ and $Y^{\text{2D}}$$\left[\eta_Y\right]$ satisfy the Ward identity for the 2D rotation group given by $\eta_Y = 2\eta_\kappa - 2$.
• Figure 4: Temperature dependence of the critical wave vector ($q_{\text{critical}}$) for the renormalization of the elastic constants and its physical origin.(a) The material-dependent factor $\frac{Y^{\text{2D}}_0\left(T\right)}{2\kappa_0\left(T\right)^2}$ at different temperatures for four 2D monolayers. (b) Temperature-dependence of $q_{\text{critical}}$ for graphene and germanene. The strong temperature-dependence of $\frac{Y^{\text{2D}}_0\left(T\right)}{2\kappa_0\left(T\right)^2}$ in germanene weakens the temperature-dependence of $q_{\text{critical}}$, resulting in a $\sim T^{0.15}$ scaling as opposed to a $\sim T^{0.4}$ scaling observed for graphene. Our predicted $q_{\text{critical}}$ for graphene at 300 K is consistent with that predicted using atomistic calculations bowick_non-hookean_2017.
• Figure 5: Renormalization of the ZA phonon dispersion in 2D monolayers.(a) The ratio $\omega/q^2$ plotted as a function of $q$ for graphene in the temperature range of $\left[\text{T}_{min}, \text{T}_{max}\right]$ = $\left[100\ \text{K}, 600\ \text{K}\right]$. The solid black line is the result from the DFPT implementation in Quantum ESPRESSO, and the dashed yellow line is the result after the Born-Huang rotational invariance and stress-free equilibrium conditions are enforced on the DFPT result from Quantum ESPRESSO, representing the phonon modes obtained from the bare harmonic IFCs (i.e., $T \sim 0$ K). The colored contours are the results after the application of the SCAP renormalization described in this work. The curves plotted with the bright color-map represent the result of the SCAP implementation that are computed from the renormalized harmonic IFCs at different temperatures, while preserving the rotational invariance and stress-free equilibrium conditions. The curves plotted with the dark color-map are the results of the SCSA framework applied to the temperature-dependent phonons and the elastic constants from SCAP, thus capturing the departure from the quadratic behavior of the ZA phonon dispersion at small $q$. (b) Same as in (a) for hBN in the temperature range of $\left[\text{T}_{min}, \text{T}_{max}\right]$ = $\left[100\ \text{K}, 1000\ \text{K}\right]$. (c) Same as in (a) for germanene in the temperature range of $\left[\text{T}_{min}, \text{T}_{max}\right]$ = $\left[100\ \text{K}, 750\ \text{K}\right]$.