#### 3.1. Phonon Spectra

Within the framework of DFPT method, phonon frequencies are computed as second-order derivatives of the total energy with respect to a given perturbation in the form of atomic displacements. The force constants matrix can be obtained by differentiating the Hellmann-Feynman forces on atoms with respect to ionic coordinates. Based on the interatomic force constants, we can obtain the phonon spectra by using Fourier interpolation with specific treatment of the long-range dipole-dipole interaction [

35]. The phonon band structure calculations were performed up to 40 GPa with an interval of 10 GPa. A more careful calculation at 39 GPa was performed for determining the exact pressure value when the phonon became instable.

The obtained ground state phonon dispersion relations in K

_{8}Si

_{46} crystal are shown in

Figure 2a. Clear flat vibrational bands can be easily recognized in the low frequency region around 100 cm

^{−1}. By analysis of partial phonon density of state (PPDOS) as shown in

Figure 3a, we find that these low frequencies are corresponding to a sharp peak located at about 98.8 cm

^{−1} which are originated from the localized vibration of 6

d sites K atoms at Si

_{24} cages. Another somewhat weaker but still clear peak at 128.2 cm

^{−1} also from K(6

d) can be attributed to the asymmetry of the large Si

_{24} cage which takes ellipsoidal shape. A similar phenomenon had also been observed in another type-I silicon-clathrate Na

_{8}Si

_{46} reported by Li et al. [

36]. Moreover, a strong and broad peak centered at 172.1 cm

^{−1} is found to be contributed by the mixing vibrations of K atoms at Si

_{20} cages and the framework Si atoms indicates the intense interaction hybridization between them, this is because the size of Si

_{20} cage is much smaller than Si

_{24} cage which yields a shorter interatomic distance of K-Si. Experimentally, the measured Raman spectrum of K

_{8}Si

_{46} showed noticeable peaks at 94, 119 and 177 cm

^{−1} related to the vibration of K atoms [

9], which is consistent with our results. By inelastic neutron scattering, Mélinon et al. [

37] and Reny et al. [

38] obtained very similar vibrational spectrum of K

_{8}Si

_{46} and found two identified peaks centered at about (100 cm

^{−1}, 170 cm

^{−1}) and (98 ± 2 cm

^{−1}, 161 ± 5 cm

^{−1}) respectively. Our calculated results perfectly reproduced the main characteristics of experimental observation. Under a pressure of 30 GPa, due to the shrinkage of the silicon cages under compress, the PPDOS shows strong mixing of K and Si vibrations. The frequency of K atoms’ motion at both 2

a and 6

d sites became higher, which indicates a further localization of these atoms, as shown in

Figure 3b. However, it is noted that the appearance of massive low frequencies that originate from the framework atoms suggests a collective “soften” of Si-Si bonds which will finally make the host lattice unstable. As the pressure is applied at a value of 40 GPa, it can be seen clearly from

Figure 2c that the frequencies around the M symmetry point become imaginary, which clearly shows the instability of clathrate framework. As illustrated in in

Figure 3c, the PPDOS under 40 GPa also shows a dramatic reduction of frequencies from host lattice as presented in the phonon spectrum. Our calculation results show that a mechanical instability of the silicon framework is believed to be responsible for the pressure-induced volume collapse at about 20 GPa and subsequent amorphization at 32 GPa of K

_{8}Si

_{46} observed experimentally.

However, it is noted that the unstable pressure given in the present work is obviously larger than the experimental observation. This is because our calculations are performed using a perfect K

_{8}Si

_{46} crystal from the view of lattice dynamics without consideration of other possible transition mechanisms (e.g., vacancies formation [

20], local symmetry-breaking [

21] or an electronic topological transition [

14,

15], etc.) associated with the pressure collapse of the clathrate structure. If multi-mechanisms are involved in phase transition, the value of transition pressure would be affected a lot. For instance, the lattice vacancies are actually very likely to be produced in these type-I clathrates, especially in 6

c sites. The experimental observed Cs

_{8}Sn

_{44} was formed from the missing two Sn atoms in the 6

c sites [

39]. Besides, theoretical calculations by Iitaka et al. also showed that 6

c sites lattice vacancies formation under high pressure was indeed energetically preferable. By the model of partially occupied Si sites they explained the transition pressure and change of Raman spectra of both K

_{8}Si

_{46} and Ba

_{8}Si

_{46} [

20]. Moreover, our recent work showed that 6

c sites lattice vacancies increased the compressibility of clathrate greatly while guest atoms vacancies hardly had any influence on this property [

19]. Experimentally, by performing high quality in situ high-pressure angle-dispersive X-ray powder diffraction measurements, Li et al. found a highly disordered Si framework from analysis of the obtained anomalously large Si thermal parameters [

13]. Also, present lattice dynamics calculation for K

_{8}Si

_{46} shows that the Si-Si bond “softens” under high pressure which again provides theoretical possibility for the formation of lattice vacancies. If one considers this mechanism, the volume collapse pressure of K

_{8}Si

_{46} is believed to be reduced. Thus, in view of these results, guest K atoms displacement induced phonon instability from earlier ab initio phonon band structure calculations [

8] is indeed more likely to be caused by the disadvantages of the finite displacement method in treating these large cell calthrate compound, especially under high pressure. Our calculated results based on DFTP method obviously gives a more convincing and clear physical picture for the instability of K

_{8}Si

_{46} under high pressure.

#### 3.2. Thermal Properties from Quasi-Harmonic Approximation

The results of calculated phonon spectra and phonon density of state can be used to compute the thermodynamic properties using the quasi-harmonic approximation (QHA) [

40]. In the QHA, the phonon Helmholtz free energy is given by:

where

k_{B} is the Boltzmann constant.

ħ is Planck’s constant and

f(

ω) is the phonon density of states (PDOS). Through a series calculation of PDOS of K

_{8}Si

_{46} with different volumes, the volume dependence of Helmholtz free energy

F_{vib}(

V,

T) can be obtained. Then the vibrational contribution to the entropy, the specific heat at constant volume and isothermal bulk modulus can be derived by:

The Grüneisen parameters can be computed by the volume derivative of (−

TS):

Then, the volume coefficient of thermal expansion and constant pressure heat capacity (

C_{p}) follows:

From QHA calculation, zero-point energy

F_{vib} (

T = 0) of K

_{8}Si

_{46} compound is determined to be 2.688 eV. Moreover, the calculated variation of volume thermal expansion coefficient

α_{V} with temperature under different pressures are illustrated in

Figure 4 from which we can find that

α_{V} increases rapidly with temperature below about 200 K and pressure imposes a strong restraint on the lattice expansion. At room temperature and zero pressure, the

α_{V} is predicted as 6.26 × 10

^{−5} K

^{−1}, corresponding to a linear thermal expansion coefficient

α_{L} as 2.09 × 10

^{−5} K

^{−1} which is very close to the experimental value of Na

_{8}Si

_{46} (about 2.0 × 10

^{−5} K

^{−1}) given by Qiu et al. [

41]. In addition, another important thermodynamic quantity of Grüneisen parameters which is difficult to determine experimentally can also be predicted by QHA method. The obtained temperature dependencies of Grüneisen parameters under pressure of 0, 10, 20 and 30 GPa are presented in

Figure 5. It can be found that the Grüneisen parameters become almost constant in relation to the varied temperature under high pressure. At ambient conditions, the Grüneisen parameter is found to be 2.47. For congener compound Na

_{8}Si

_{46} under same condition, this value was reported as 2.68 [

41].

These similarities in thermdynamic properties of type I clathrates doped with same group elements have also been reported by many experimental works, for example, the thermal expansion coefficients of Ba(Sr)

_{8}Ga

_{16}Ge

_{30} and Sr

_{8}Ga

_{16
}Ge

_{30} [

42] were just found to be almost identical to each other, and so were Rb

_{8}Sn

_{44}□

_{2} and Cs

_{8}Sn

_{44}□

_{2} (□ means lattice vacancy) [

43]. Consequently, in general, the thermal expansion coefficient of intermetallic clathrates is assumed to mainly depend on the bonding of the framework atoms. The nature of the guest atoms just make a small contribution due to their weaker ionic bonding to the host structure. However, in the case of Ba

_{8}Si

_{46}, which has been attracting extensive attention due to the discovery of its superconductivity, the measured

α_{L} at room temperature was found to be only 1.2 × 10

^{−5} K

^{−1}. Besides, a drop of

α_{L} of Ba

_{8}Si

_{46} occurred at the superconducting transition temperature and the frequency-dependent Grüneisen parameter of Ba

_{8}Si

_{46} indicated strong anharmonicity of the lattice vibrations for low energy mode with a value of up to 8.6 while higher energy modes are much less anharmonic with a value somewhat below 2 [

44]. These novel results of Ba

_{8}Si

_{46} are quite different from those of Na(K)

_{8}Si

_{46} which reveals the non-negligible role of different hybridization between the guest and host atoms in studying the vibrational properties for doped clathrate compound. Moreover, the vacancies which are very likely to be formed in clathrate compounds can also significantly influence the system vibrational properties. Their presence can decrease the average bonding strength among host atoms and induce the displacement of guest atoms from their ideal positions. Thus, the anharmonicity of system is enhanced which would yeild an increment of the thermal expansion coefficient and Grüneisen parameters at zero pressure that had been identified by the experimental work for type-I Ge-based clathrates [

45]. However, this effect became more unintelligible under high pressure because the formation of vacancies can also increase the compressbility of the clathrate which would lead the volume collapse under compress. So, further theoretical simulation with considering the formation of vacancies in hosts framework is expected to explore the effects of these vacancies on the vibrational properties of clathrate system under high pressure.

Because the experimental specific heat was conducted at constant pressure namely

C_{p}, in

Figure 6, the resulting dependence of

C_{p} on temperature calculated from QHA method at zero pressure is illustrated, the experimental data up to room temperature from Stefanoski have also been plotted together for a comparison [

46]. It can be found that they agree with each other excellently (for instance, the specific heat at room temperature determined by present work is 1151.5 J·mol

^{−1}·K

^{−1}, experimental measured value is 1159.6 J·mol

^{−1}·K

^{−1}) which indicates the validity of present lattice dynamic simulation within DFTP. At low temperatures, the temperature dependence of specific heat presents a similar behavior to thermal expansion coefficient. When the temperature is higher than about 400 K, specific heat gradually approaches the Dulong-Petit limit, i.e., 3

nN_{A}k_{B} (about 1397.2 J·mol

^{−1}·K

^{−1}) which is followed by all solid at high temperatures. In our former study [

47], we found the specific heat of Na

_{8}Si

_{46} predicted by quasi-harmonic Debye model was underestimated obviously at low temperature which revealed the limitation of the Debye model in dealing with these doped clathrate compounds. The experimental heat capacities of Na

_{8}Si

_{46} were finally reproduced by treating the special “rattle” modes of captured Na atoms in cages as Einstein oscillators. In the present work, a full phonon calculation within DFPT method can give a rather exact description of vibrational properties of these clathrate compounds which avoids a discriminatory treatment of the host lattice and encapsulated atoms. The specific heat of K

_{8}Si

_{46} under pressure of 30 GPa is also presented in

Figure 6, from which it can be found that pressure can decrease the specific heat considerably due to the suppression of lattice vibration.

The comparison of the calculated specific heat to that predicted by the Debye model leads to the concept of the temperature dependent Debye temperature. The obtained Debye temperature at the high temperature limit is 550 K, which is consistent with the reported experimental result of 577 K [

48]. In addition, the thermal conductivity

ĸ_{L} contributed by the lattice can be estimated by the Debye equation [

49]:

where

C is the volumetric heat capacity,

λ is the mean free path of phonons, assumed as the average distance between the guest atoms,

V_{m} is the velocity of sound which can be derived from Debye temperature [

50]. In this way, the lattice thermal conductivity is given as 1.64 W m

^{−1}·K

^{−1} which is comparable to other reported room temperature thermal conductivities of type-I silicon based clathrate compound [

48].