Calculation of Mode Grüneisen Parameters Made Simple

A novel method to calculate mode Grüneisen parameters of a material from first principles is presented. This method overcomes the difficulties and limitations of existing approaches, based on the calculation of either third-order force constants or phonon frequencies at different volumes. Our method requires the calculation of phonon frequencies of a material at only the volume of interest, it is based on the second-order differentiation of a corrected stress tensor with respect to normal mode coordinates, and it yields simultaneously all the components of the mode Grüneisen parameters tensor. In this work, after discussing conceptual and technical aspects, the method is applied to silicon, aluminum, scandium fluoride, and a metallic alloy. These calculations show that our method is straightforward and it is suited to be applied to the broad class of materials prone to exhibit structural instabilities, or presenting anisotropy, or chemical and/or structural disorder.

Figure 1

Values of a mode Grüneisen parameter obtained from Eq. (13) using increasing values of ${\zeta }_k=\sqrt{2}{q}_k{\omega }_k$. Inset: Values of the “corrected” pressure times $V$ (in atomic units of energy) versus ${\zeta }_k$. The colored disks indicate that a mode parameter can be calculated by carrying out a single total energy calculation (red disk) with ions fixed in a configuration accommodating displacements resulting from a normal mode coordinate equal to $q_k$. These results were obtained by considering a large supercell of fcc Al (inset) described by using a classical energy scheme.

Figure 2

Mode Grüneisen parameters of fcc Al computed by using classical interatomic potentials and a cubic supercell containing 864 atoms. Light blue disks show results obtained by using the conventional method based on numerical differentiation of phonon frequencies computed at different volumes, whereas blue circles show parameters computed by using our method. The absolute average difference between these two sets of data is 0.06.

Figure 3

Mode Grüneisen parameters of Si calculated by using a periodic DFT approach and a cubic supercell containing 216 atoms. Dark yellow disks are used to show parameters obtained by using the conventional method, whereas red circles show results obtained by using our method. The absolute average difference between the two sets of values is 0.05. Inset: Average Grüneisen parameter versus temperature calculated by using Eq. (2).

Figure 4

Thermodynamic Grüneisen parameters tensor of a hcp Mg-Li alloy versus temperature derived from Eq. (2) and generalized mode parameters (shown in the inset) calculated by using our method. Aqua, dark green, and blue colored symbols and solid lines show the $xx$, $yy$, and $zz$ components of the mode and thermodynamic Grüneisen parameters tensors, respectively. Light green colors are used to show the $yz$, $xz$, and $xy$ components of the symmetric tensors.