Assessment on lattice thermal properties of two-dimensional honeycomb structures: Graphene, $h$-BN, $h$-MoS${}_2$, and $h$-MoSe${}_2$

The linear thermal expansion coefficients of two-dimensional honeycomb structures graphene, $h$-BN, $h$-MoS${}_2$, and $h$-MoSe${}_2$ are systematically studied by using first-principles based quasiharmonic approximation. This approach is first tested on diamond crystal and excellent agreement with the available experimental data is achieved. Our simulations show that the linear thermal expansion coefficients of graphene and $h$-BN are more negative than that of their multilayered counterparts graphite and white graphite. In addition, there is a remarkable distinction between the coefficients of these two materials in particular at low temperatures. Contrary to graphene and $h$-BN, lattice thermal expansion coefficient of MoS${}_2$ and MoSe${}_2$ are always positive, and the values are comparable with those predicted for diamond.

Figure 1

LTEC for diamond as a function of temperature. Our QHA-LDA (black solid line) and QHA-GGA (blue dashed line) results are compared with the ab initio calculations (Ref. [24], green dash dot dot line) and experimental measurements (Ref. [45], black solid circles). The inset shows the results up to 300 K. Here, we also compare our results with very recent experiments (red solid squares and green solid diamonds) reported in Ref. [46].

Figure 2

LTHE for graphene as a function of temperature. Our QHA-LDA-VASP (black solid line), QHA-GGA-(red dash dot dot line), are compared with the ab initio calculations (Ref. [24], green dash dot line) and experimental measurements for graphene (Ref. [30], orange solid circles, Ref. [29], red solid up triangles, Ref. [31] green solid down triangles), and graphite (Ref. [47] blue solid diamonds). Inset shows the calculated mode Grüneisen parameters for graphene.

Figure 3

LTEC for single layer $h$-BN (QHA-GGA black solid line, QHA-LDA red dashed line) and graphene (QHA-GGA green dotted line, QHA-LDA blue dash dot dot line) as a function of temperature. Our results are compared with the experimental measurements for hexagonal BN (Ref. [48], red solid squares, Ref. [49], orange solid circles, and Ref. [50], blue up-triangles). Inset shows the calculated mode Grüneisen parameters of single layer $h$-BN (black solid line) in comparison with graphene (green dashed line).

Figure 4

Calculated phonon dispersions of (a) graphene, (b) $h$-BN, (c) $h$-MoS${}_2$, and (d) MoSe${}_2$ structures along high symmetry lines of the Brillouin zone.

Figure 5

LTEC for single layer $h$-MoS${}_2$ (QHA-GGA black solid line, QHA-LDA red dashed line) and $h$-MoSe${}_2$ (QHA-GGA blue dash dot line, QHA-LDA green dash dot dot line) crystals as a function of temperature. Inset shows the calculated mode Grüneisen parameters of $h$-MoS${}_2$ (black solid line) and $h$-MoSe${}_2$ (green dashed line).