Comparative study of phonon spectrum and thermal expansion of graphene, silicene, germanene, and blue phosphorene

Based on first-principles calculations using density functional theory, we study the vibrational properties and thermal expansion of monoatomic two-dimensional honeycomb lattices: graphene, silicene, germanene, and blue phosphorene. We focus on the similarities and differences of their properties, and try to understand them from their lattice structures. We illustrate that, from graphene to blue phosphorene, a phonon band gap develops due to large buckling-induced mixing of the in-plane and out-of-plane phonon modes. This mixing also influences their thermal properties. Using quasiharmonic approximation, we find that all of them show negative thermal expansion at room temperature.

Figure 1

Structures of (a)–(d) graphene, silicene, germanene, and blue phosphorene and (e) the corresponding first Brillouin zone. (a) and (c): top view. (b) and (d): side view. Graphene is flat (a)–(b), while all others are buckled (c)–(d). The lattice constants are $a=2.47,3.87,4.06,3.28\AA$, and $\mathrm{\Delta }=0,0.45,0.69,1.24\AA$, for graphene, silicene, germanene, and blue phosphorene, respectively.

Figure 2

Phonon dispersion and density of states of (a) graphene, (b) silicene, (c) germanene, and (d) blue phosphorene. We use the following color code: LA (blue), TA (Green), ZA (Red), ZO (violet), TO (light blue), LO (black).

Figure 3

(Left) Energy per unit cell as a function of relative lattice constant $a/a_0$ within the range of $-0.005\le a/a_0\le 0.005$, with $a_0$ the equilibrium lattice constant. (Right) Temperature dependence of bulk modulus.

Figure 4

Grüneisen parameters of (a) graphene, (b) silicene, (c) germanene, and (d) blue phosphorene along the high symmetric lines within the Brillouin zone. Inset: the Grüneisen parameters of ZA mode from $\mathrm{\Gamma }$ to $M$ in full range. We use the same color code as that in Fig. 2.

Figure 5

Fitting (green dashed line) of the total free energy (electronic plus phononic) to the third order Birch-Murnaghan equation of state at representative temperatures for (a) graphene, (b) silicene, (c) germanene, and (d) blue phosphorene.

Figure 6

Thermal expansion coefficient as a function of temperature. (Left) Fitting the third-order Birch-Murnaghan equation of state. The strain applied are $\pm 0.5\%$. (Right) Calculated from the Grüneisen method.

Figure 7

Thermal expansion coefficients of blue phosphorene for different strains from $\pm 0.5\%–\pm 2\%$. This shows that the results depend sensitively on the strains applied. Inset: The phonon dispersion near $\mathrm{\Gamma }$ point at different strain applied. For larger strain, a large fraction of ZA modes goes negative, indicating the failure of QHA. A small strain ($\pm 0.5\%$) is needed in the calculation to minimize this effect.