Properties at ambient conditions - Results and discussions

5.3 Results and discussions

5.3.1 Properties at ambient conditions

The structure of the studied compounds is given in Figure 5.1, and these compounds crystallize in orthorhombic structure with space groupPnma in the form CaSrX (X: Si, Ge, Sn, Pb). Computed ground state properties of all the studied materials are reported in Table 5.1, which indicate the compatibility between the present values and other reported theory and experimental values. In order to examine the electronic structure properties in detail, we have calculated the band structure with several exchange correlation functionals such as GGA and TB-mBJ, together with inclusion of spin orbit coupling. From the analysis, we have observed that spin orbit effect is negligible in the studied compounds. Band gaps of all the compounds are reported in Table 5.1, and all the compounds show semiconducting nature. The band structure calculated along different high symmetry directions for all the investigated compounds are presented in Figure 5.2. For CaSrSi, we have represented the band structure by combining two cases, with spin-orbit coupling and without spin-orbit coupling (see red dot and black solid lines in Figure 5.2(a)), and from the figure, it is evident that spin-orbit effect is very minimum in this compound. The band profile of all the compounds is almost similar, and band gap is found to decrease from CaSrSi to CaSrPb. A direct band gap is observed at Γ high symmetry point. Along all crystallographic directions, ‘a’,‘b’, ‘c’, the bands are observed to be dispersive in the valence band, with dispersion being little higher along ‘b’ axis. A similar scenario can be observed in conduction band. A flat band is observed along R-S high symmetry direction in valence band in all the compounds. A thorough investigation of the band structure of these compounds reveal a highly linearised Dirac like band profile along Γ - Y direction in conduction band. Figure 5.3 represents the projected band structure, which confirms the domination of Si-p states near valence band and a competing Si-p and Sr-d states in the conduction band. Figure 5.4 presents the total density of states of all the compounds and partial density of states of CaSrSi, which also indicate similar contributions from Si-p and Sr-d states near Fermi level. The calculated elastic

Z Γ Y T R S U Γ X -4

-3 -2 -1 0 1 2 3 4

Energy(eV)

CaSrSi

E_F

(a)

Z Γ Y T R S U Γ X -4

-3 -2 -1 0 1 2 3 4

Energy(eV)

CaSrGe

E_F

(b)

Z Γ Y T R S U Γ X

-4 -3 -2 -1 0 1 2 3 4

Energy(eV)

CaSrSn

E_F

(c)

Z Γ Y T R S U Γ X -4

-3 -2 -1 0 1 2 3 4

Energy(eV)

CaSrPb

E_F

(d)

Figure 5.2: Band structure of investigated compounds, a) CaSrSi with and without inclusion of spin orbit coupling, solid lines represent band structure with spin orbit coupling and red dots represent the band structure without spin-orbit coupling, b) CaSrGe, c) CaSrSn d) CaSrPb with spin orbit coupling

Y Γ Y -1

-0.5 0 0.5 1

Energy(eV)

CaSrSi

E_F

(a)

Figure 5.3: Projected band structure of CaSrSi, black circle represents Si-p states and red circle represents Sr-d states

constants of all the investigated compound are presented in Table 5.2. The born stability criteria is found to be satisfied for all the compounds, confirming the mechanical stability of these compounds.

The elastic constants are comparable with iso-structural materials. In Table 5.2 we have reported the bulk modulus and Debye temperature of all the investigated compounds. The magnitude of Debye temperature is observed to decrease from Si to Pb, and the range of the same indicate the probability of low thermal conductivity. Further the phonon dispersion plot of CaSrSi and CaSrGe are presented in Figure 5.5. The less dispersed phonon modes indicate the probability of lesser thermal conductivity in these materials. The range of frequency, where the acoustic and optical phonon modes interact is found to be decreased from CaSrSi to CaSrGe, and the corresponding frequencies are 70 cm⁻¹ to 50 cm⁻¹, implying the low thermal conductivity in these compounds, which is appropriate for TE applications. A unique band profile along Γ -Y direction encourage us to examine the electronic structure in detail, and we have analysied the topological behavior of these studied compounds. All the investigated compounds are invariant under time reversal symmetry, and the topological character of the same can be connected with Z2topological invariant.

Moreover, the system possesses inversion symmetry and the corresponding space group also fall into non-symmorphic crystalline group. As the consequence of the coexistence of time reversal symmetry and inversion symmetry, Z2 topological invariant can be connected to the parity of bands in each TRIM point (time reversal invariant momentum) in the Brillouin zone. Here we have to consider the case of three dimensional system, in which the system possess 8 TRIM points, which are (0,0,0), (0.5,0,0), (0.0, 0.5,0.0), (0.0, 0.0, 0.5), (0.5, 0.5, 0.0), (0.5, 0.0, 0.5), (0.0, 0.5, 0.5), and (0.5, 0.5, 0.5). For this purpose we have adapted the method suggested by Liang Fu and Charles L. Kane [277]. The equation for Z2 topological invariant is, (-1)^ν =Q

γi

-3 -2 -1 0 1 2 3 Energy (eV)

0 10 20 30 40

DOS (States/eV)

CaSrSi CaSrGe CaSrSn CaSrPb

(a)

0 10 20 30

40 Total

Ca-Total Sr-Total Si -Total

0 0.5 1

1.5 _{Ca -s}

Ca -p Ca -d

0 0

0.5 1 1.5

DOS (States/eV)

Sr -s Sr -p Sr-d

-3 -2 -1 0 1 2 3

Energy (eV) 0

0.5 1

1.5 _{Si -s}

Si -p

(b)

Figure 5.4: a) Total density of states of all the compounds, b)Total and partial density of states of CaSrSi

, whereγi are parities of each band. Computed Z2 topological invariants of all the compounds are represented in Table 5.3. Due to the semiconducting nature of the studied compounds, we have considered all the occupied bands for the calculations, and we have found that at Γ point, parity is -1 and at all the other points the value is +1, and lead to a non zero value for Z2 parameter. A strong topological character has been observed in all the investigated compounds. Further we have calculated the surface states for all the compounds and presented in figure 5.6. Well linearized Dirac cone like surface bands is observed around Γ high symmetry point, which again confirm the strong topological nature in this compound. In the next paragraph we try to understand the TE properties of these compounds.

Thermopower, electrical conductivity and power-factor as a function of carrier concentrations at different temperatures are studied for all the compounds. Figure 5.7 to 5.10 represent the TE properties of CaSrSi, CaSrGe, CaSrSn and CaSrPb as a function of both hole and electron concentrations at different temperatures. The difference in band dispersion along different crystallographic directions are reflected in thermopower. Around 300 K, thermopower along ’a’-axis has secured the highest value than ’c’-axis value, which is followed by ’b’-axis value. When we move to higher temperature, the difference between ’a’-axis and ’c’-axis values are reducing, and ’b’-axis value remains the lowest.

Around 700 K onwards, bi-polar conduction has been observed for low concentration range. Coming to electron doping, the thermopower values are more isotropic and bipolar conduction is observed similar to that of holes. The magnitude of thermopower for electron doping is lesser than the same for hole doping. Figure 5.7 (c,d) present the electrical conductivity for CaSrSi for both holes and electrons at different temperatures. At around 300 K, conductivity along ’b’-axis is found to be more, and ’c’ -axis has the lowest value. Up to 700 K the trend is same, and around 900 K at low concentrations, the conductivity is found to be invariant as a function of carrier concentration, and for higher concentration it varies linearly. In the case of electron doping, for all temperatures the b-axis value is higher, and other two axes values are found to be almost same. Figure 5.8 (c,d), 5.9 (c,d) and 5.10 (c,d) reported the electrical conductivity values for CaSrGe, CaSrSn and CaSrPb

Γ Z T Y S 0

50 100 150 200 250 300

Frequency (cm-1 )

(a)

Γ Z T Y S

0 50 100 150 200 250

Frequency (cm-1 )

(b)

Γ Z T Y S

0 50 100 150 200

Frequency (cm-1 )

(c)

Γ Z T Y S

0 50 100 150 200

Frequency (cm-1 )

(d)

Figure 5.5: a) Calculated phonon dispersion plot for all the investigated compounds a)CaSrSi, b)CaSrGe, c)CaSrSn, d)CaSrPb

respectively. Power factor of CaSrSi is presented in Figure 5.7 (e,f). For hole doping, significant difference is observed between each crystallographic directions, and ’b’-axis is found to have highest value. But for electron doping, we could see little anisotropic character, where b -axis value is higher than ’a’ and ’c’ axes. Since all the investigated compounds show bipolar conduction at higher temperatures, we have presented the magnitude of thermopower over a temperature range of 100 K to 500 K around the carrier concentration of 10¹⁸ cm⁻³ for all the compounds in Figure 5.11. From the figure it is quite evident that, all the investigated compounds posses appreciable value of thermopower over the temperature range of 100 K to 500 K. Electrical conductivity and power-factor of other compounds are also found to show similar behaviour as that of CaSrSi, as seen from figures 5.8 to 5.9.

Table 5.2: Calculated Elastic constants of CaSrX (X: Si, Ge, Sn, Pb)

Parameter CaSrSi CaSrGe CaSrSn CaSrPb

C¹¹(GPa) 68.72 64.58 57.17 53.98

C²²(GPa) 77.23 74.64 66.35 61.99

C³³(GPa) 73.52 69.37 62.19 59.15

C44(GPa) 35.23 34.02 29.69 28.27

C55(GPa) 28.98 27.89 23.54 22.06

C66(GPa) 34.11 32.58 28.10 26.33

C12(GPa) 19.85 19.01 17.52 16.94

C13(GPa) 19.24 17.83 15.44 14.39

C²³(GPa) 22.53 20.23 19.16 17.38

DebyeT(K) 340.79 295.44 253.82 212.41

Table 5.3: Calculated Z2 topological invariants of CaSrX (X: Si, Ge, Sn, Pb) at ambient

CaSrSi CaSrGe CaSrSn CaSrPb

ν⁰ 1 1 1 1

ν1 0 0 0 0

ν2 1 1 1 1

ν3 0 0 0 0

As we mentioned earlier, the figure of merit is an important parameter for thermoelectric materials. The magnitude of power-factor for all the compounds at 300 K around the concentration range

(a) (b)

Figure 5.6: a) Calculated surface band structure for all the compounds along (110) surface a)CaSrSi, b)CaSrGe, c)CaSrSn, d)CaSrPb

10¹⁸ 10¹⁹ 10²⁰ 10²¹ Hole Concentration (cm^-3 ) 0

200 400 600

S (µV/K)

a-300K b-300K c-300K a-500K b-500K

(a)

10¹⁸ 10¹⁹ 10²⁰ 10²¹

Electron Concentration (cm^-3 ) -600

-400 -200 0

S (µV/K)