Theory of anharmonic phonons in two-dimensional crystals

Anharmonic effects in an atomic monolayer thin crystal with honeycomb lattice structure are investigated by analytical and numerical lattice dynamical methods. Starting from a semiempirical model for anharmonic couplings of third and fourth orders, we study the in-plane and out-of-plane (flexural) mode components of the generalized wave vector dependent Grüneisen parameters, the thermal tension and the thermal expansion coefficients as a function of temperature and crystal size. From the resonances of the displacement-displacement correlation functions, we obtain the renormalization and decay rate of in-plane and flexural phonons as a function of temperature, wave vector, and crystal size in the classical and in the quantum regime. Quantitative results are presented for graphene. There, we find that the transition temperature $T_{\alpha }$ from negative to positive thermal expansion is lowered with smaller system size. Renormalization of the flexural mode has the opposite effect and leads to values of $T_{\alpha }\approx 300\mathrm{K}$ for systems of macroscopic size. Extensive numerical analysis throughout the Brillouin zone explores various decay and scattering channels. The relative importance of normal and umklapp processes is investigated. The work is complementary to crystalline membrane theory and computational studies of anharmonic effects in two-dimensional crystals.

Figure 1

Phonon modes of graphene along the high symmetry crystallographic direction $\Gamma K\text{−}M\Gamma$ (Ref. [35]).

Figure 2

Schematic plot of $A_1$ and $B_1$ atoms belonging to the unit cell with its corresponding first-nearest neighbors.

Figure 3

Thermal tension coefficient ${\beta }_T$ for system sizes $l={10}^{2}a$ (circles) and ${10}^{4}a$ (squares). The (a) out-of-plane ${\beta }_T\left(\text{Z}\right)$ and (b) in-plane ${\beta }_T(\perp )$ components are given in units of dyn ${\mathrm{cm}}^{-1}{\mathrm{K}}^{-1}$. Notice in the inset the size dependence at low $T$ of ${\beta }_T(\perp )$.

Figure 4

Thermal expansion coefficient ${\alpha }_T$ for $l={10}^{2}a$ and ${10}^{4}a$ sample sizes. Units ${\mathrm{K}}^{-1}$.

Figure 5

Linewidths for $\lambda =\text{L}$ (circles) and T (squares) at different temperatures as indicated. The 2D crystal size is $l={10}^{4}a$.

Figure 6

(a) Quantity $c_{\text{Z}}^2={\Sigma }^{\prime }(\vec{q},\mathrm{Z})/q^2$ as a function of $T$, units ${\mathrm{cm}}^2{\mathrm{s}}^{-2}$. (b) Thermal tension ${\beta }_T$(Z) and (c) thermal expansion ${\alpha }_T$ evaluated with renormalized flexural mode frequency $\Omega (\vec{q},\mathrm{Z})$ (filled squares) as a function of $T$. ${\beta }_T$(Z) is given in units of dyn ${\mathrm{cm}}^{-1}{\mathrm{K}}^{-1}$ and ${\alpha }_T$ in ${\mathrm{K}}^{-1}$. The 2D crystal size is $l={10}^{4}a$. Compare with Fig. 4.

Figure 7

System size $l_{\alpha }$ as a function of temperature $T_{\alpha }$ for change from negative to positive thermal expansion. Discrete points are calculated self-consistently with renormalization of flexural mode.

Figure 8

Generalized Grüneisen parameters $\gamma (\vec{q},\lambda )$ for (a) ZA, (b) TA, and (c) LA acoustic phonon modes of graphene. ${\gamma }_x(\vec{q},\lambda )$ (${\gamma }_y(\vec{q},\lambda )$), where $x$ ($y$) refer to strain ${\epsilon }_{xx}$ (${\epsilon }_{yy}$), contribution is indicated by a dashed-red (dotted-blue) line.

Figure 9

Temperature dependence of the thermal expansion coefficient ${\alpha }_T$ of graphene. Units ${\mathrm{K}}^{-1}$. Sample size $l=80a$.

Figure 10

Temperature dependence of $\Gamma (\vec{q},\lambda )$ for (a) ZA, (b) TA, and (c) LA acoustic phonon modes of graphene. Units ${\mathrm{cm}}^{-1}$.

Figure 11

Phonon linewidths $\Gamma (\vec{q},\lambda )$ for $T=1$ (left) and 300 K (right). The logarithmic scale for the $y$ axis is used to allow comparison of normal and umklapp contributions. Units ${\mathrm{cm}}^{-1}$.

Figure 12

Contribution of the different scattering channels to the phonon linewidths at $T=1$ K (left) and 300 K (right). The total linewidth (a) is composed of normal (b) and umklapp (c) processes. Units ${\mathrm{cm}}^{-1}$.

Figure 13

Contribution of the different scattering channels to the phonon linewidths at 1 K (left panels) and 300 K (right panels) in the long-wavelength regime. Units ${\mathrm{cm}}^{-1}$ $[q_x$ ($x$ axis) is given in units of ${\AA }^{-1}]$.

Figure 14

Contour plots of the contribution of selected scattering channels inside the BZ for the TA and LA acoustic phonon modes of graphene at ${\vec{q}}_A,{\vec{q}}_B$, and ${\vec{q}}_C$ (i.e., the middle points of $\Gamma \text{−}K\text{−}M\text{−}\Gamma$). Units ${\mathrm{cm}}^{-1}$ $[q_x$ ($x$ axis) and $q_y$ ($y$ axis) are given in units of ${\AA }^{-1}]$.

Figure 15

Temperature dependence of the third- (left panel) and fourth-order (right panel) band shifts for the (a) ZA, (b) TA, and (c) LA acoustic phonon modes of graphene. Units ${\mathrm{cm}}^{-1}$.

Figure 16

Discrete mesh with 225 $\vec{q}$ points inside the irreducible part (IP) of the 1BZ (left). Comparison of meshes in the vicinity of the $\Gamma$ point (right).

Figure 17

Comparison of results for the linewidth (left panels) and phonon shift (right panels) for different number of $q$ points in the mesh along the BZ.

Figure 18

Effects of the auxiliar variables $\eta$ and $\epsilon$ on the evaluation of the principal part and the delta function in the calculation of the phonon shift (right panels) and linewidths (left panels).

Figure 19

Contour plots of the contribution of selected scattering channels along the BZ for the TA and LA acoustic phonon modes of graphene at ${\vec{q}}_A,{\vec{q}}_B$, and ${\vec{q}}_C$ (i.e., the middle points of $\Gamma \text{−}K\text{−}M\text{−}\Gamma$). Units ${\mathrm{cm}}^{-1}$ $[q_x$ ($x$ axis) and $q_y$ ($y$ axis) are given in units of ${\AA }^{-1}]$.