Ab initio lattice dynamical studies of silicon clathrate frameworks and their negative thermal expansion

The thermal and lattice dynamical properties of seven silicon clathrate framework structures are investigated with ab initio density functional methods (frameworks I, II, IV, V, VII, VIII, and H). The negative thermal expansion (NTE) phenomenon is investigated by means of quasiharmonic approximation and applying it to equal time displacement correlation functions. The thermal properties of the studied clathrate frameworks, excluding the VII framework, resemble those of the crystalline silicon diamond structure. The clathrate framework VII was found to have an anomalous NTE temperature range up to 300 K and it is suitable for further studies of the mechanisms of NTE. Investigation of the displacement correlation functions revealed that in NTE, the volume derivatives of the mean square displacements and mean square relative displacements of atoms behave similarly to the vibrational entropy volume derivatives and consequently to the coefficients of thermal expansion as a function of temperature. All studied clathrate frameworks, excluding the VII framework, possess a phonon band gap or even two in the case of framework V.

Figure 1

Schematic figures of the silicon clathrate structures included in this work. The vertices of the polyhedral cages present silicon atoms. Crystallographic unit cell edges are drawn in black. For a more comprehensive description of the structural characteristics, see [36].

Figure 2

Phonon dispersion relations along high symmetry paths in the first Brillouin zone for (a) $d$-Si (experimental values indicated by diamonds [48]) and (b) for structure VII.

Figure 3

Phonon dispersion relations along high symmetry paths in the first Brillouin zone for structures I, II, IV, V, VIII, and H.

Figure 4

Heat capacities at constant volume for studied structures. Only structure VII is slightly different from the other structures (indicated with a dash-dotted line).

Figure 5

Heat capacity at constant volume as a function of temperature and frequency in QHA. Every separate line represents the value of Eq. (8) for a particular phonon frequency as a function of the temperature.

Figure 6

Grüneisen parameters for the studied structures, each dot represents one mode at a particular $\mathbf{q}$ point.

Figure 7

Top: Linear CTE for the studied cubic structures. The experimental values for $d$-Si are indicated with circles [18]. Bottom: Volumetric CTE for the studied hexagonal structures, $d$-Si is included as a reference.

Figure 8

$\Delta S/\Delta V$ for each phonon mode in the $d$-Si, II, and VIII structures as a function of the temperature. The zero level is at the point where the lines representing the phonon modes coincide at $T=0$ K. The acoustic modes are indicated with dash-dotted lines.

Figure 9

Displacement autocorrelation functions for $d$-Si and the clathrate framework II [Eq. (12)]. Each line represents the displacement autocorrelation function for one atom in the primitive unit cell.

Figure 10

The primitive unit cell of the clathrate framework II shown in (a) [100], (b) [111], and (c) general direction. Unit cell edges are drawn in black.

Figure 11

Parallel and perpendicular correlation functions for $d$-Si within same unit cell, i.e., for the nearest neighbors.

Figure 12

Parallel and perpendicular MSRD within the same unit cell for structure II. (a) Normalized parallel MSRD. (b) Parallel MSRD. (c) Perpendicular MSRD. Different lines represent different atoms $\kappa$, the second atom ${\kappa }^{\prime }=2$ in all cases (see Fig. 10).

Figure 13

MSD-VD and MSRD-VD values for $d$-Si calculated from Eqs. (25), (27), and (28). The MSRD-VD values are for the nearest neighbors. The acoustic modes are indicated with a dash-dotted line.

Figure 14

MSD-VD for three different atoms in structure II calculated from Eq. (25). Acoustic modes are indicated with a dash-dotted line.