3.3 Results and discussion
3.3.5 Electronic and thermoelectric properties of ZnGeSb 2 under strain
optimized volume.
1018 1019 Concentration (cm-3) -150
-100 -50 0 50 100 150 200 250 300
S (µV/K)
ZnGeSb2 -p ZnGeSb2 -n ZnGeAs2 -p ZnGeAs2 -n
(a)
1018 1019
Concentration (cm-3) 1018
1020 1022 1024 1026
σ/τ( Ω-1 m-1 s-1 ) ZnGeSb2 -p ZnGeSb2 -n ZnGeAs2 -p ZnGeAs2 -n
(b)
Figure 3.11: Variation of thermopower and electrical conductivity as a function of carrier concen- tration for optimized structure ZnGeSb2 and ZnGeAs2. Solid line for ‘a’ axis, and dashed line for
‘c’ axis
Z Γ Z
-1 -0.5 0 0.5 1
Energy (eV)
EF
(a) (b)
Figure 3.12: Band structure of ZnGeSb2 at -4.6% strain (a) along Γ -Z high symmetric direction, (b) three dimensional representation of the kx, ky projection along Z -Γ -Z direction.
VBM. The schematic of electronic structure over a range of volume is given in Figure 3.14. While analysing the band structure systematically, it is evident that at compressive strain (around 4.6
%), the compound is a normal semiconductor, around optimized volume, the compound possesses massive Dirac state, where it shows highly linearly dispersive band profile, and above 3.2 % tensile strain, the compound shows topological semi-metallic state. This clearly indicate the three states of this compound: normal semiconductor, massive Dirac states and topological semi-metal. In addition to this, tuning of massive Dirac states can be observed by the application of expansive strain. In a similar line we have analysed ZnSnSb2, and we could see that the band gap is closing at 8% tensile strain and remains the same up to 20% tensile strain. A band inversion is observed at the strained state, and the projected band structure of ambient and 8% strained state are given in Figure 3.15.
Further we have concentrated only on ZnGeSb2. In the following paragraph we discuss the transport properties in these states.
The interesting part of the electronic structure of this compound is already discussed, and now one can analyse the thermoelectric properties, and this is summarised in Figure 3.16. In this figure, we have given the variation of transport coefficients as a function of strain at carrier concentration around 5.0×1019 cm−3 at 300 K, and as function of carrier concentrations from 1.0×1018 cm−3 to 1.0×1020cm−3. As we expected from the electronic structure, peculiar behaviour of thermoelectric coefficients are observed. The magnitude of thermopower is found to be decreased for tensile strain and increased for compressive strain, which indicate that at the compressive strained states, the compound has normal semiconducting nature, where the thermopower is high compared to ambi- ent and expanded states are the ones where system has massive Dirac states. The reduction of magnitude of thermopower might be due to the small band mass near the Fermi level. The ther- mopower has a local peak at 2.4% strain, where the electronic topology is changing. The electrical conductivity exhibits a huge increment ( in the order of 1×1025 Ω−1m−1s−1) in the Dirac states compared to normal semiconducting states. Figure 3.16 shows that for negative strains the electrical conductivity is lesser than ambient and expanded states(massive Dirac states). To understand the overall thermoelectric properties, we have given the power factor also (see Figure 3.16), where one can clearly observe that around the optimized volume, there is a huge increment in power factor, with a subsequent decrease, and after 2.4% it starts increasing. Among massive Dirac states, one can clearly see that the strained states which hold the Fermi level far from the Dirac points are more favourable. Similar observations are mentioned in graphene based studies.[212] From the figure it is evident that the magnitude of power factor is almost similar for both holes and electrons, which indicate the possibility of device application in this compound. There is a huge anisotropy in power factor observed in the Dirac states, the ‘c’ axis is more beneficial than the ‘a’ axis for optimized volume. One point we would like to mention here is that all the Dirac states observed in this study were massive Dirac states, where we have observed a gap between two Dirac points, which is bene- ficial for device applications, unlike the massless state in graphene. Recent research in graphene is evolving in a direction where people are trying to open a small gap between the degenerate Dirac point.[213] This shows the significance of the investigated material. Overall, the maximum power factor observed at the optimized state is several orders of magnitude higher than the power factor observed for the normal semiconducting state, and also the magnitude of power factor observed in the Dirac state is higher than the power factor of well known thermoelectric materials.[214] In figure 3.16(f), we have compared the power factor of our investigated compound at it’s massive Dirac
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.13: Band structure of ZnGeSb2 (a) band structure at 1.6% strain along Z -Γ -Z direction, (b) band structure at 2.4% strain along Z-Γ -Z direction, (c) band structure at 2.6 % strain along Z -Γ -Z direction,(d) band structure at 3.2% strain along X -Γ -X direction, Calculateds,pprojected band structure for different strains (black circle represent ’s’ bands, red circle represent ’p’ bands):
e) -4.6% strain f) 2.4% strain g)2.8% strain h) 3.2% strain
(a)
Figure 3.14: The schematic of electronic structure as a function of strain
Z Γ N X P Γ -2
-1 0 1 2
Energy(eV)
Γ6
Γ8 EF
(a)
Z Γ N X P Γ -2
-1 0 1 2
Energy(eV)
Γ6Γ8 EF
(b)
Figure 3.15: Projected band structure for ZnSnSb2 at a) ambient and b) 8% hydrostatic strain, (black circle represent ’s’ bands, red circle represent ’p’ bands)
state with other materials. The magnitude of thermal conductivity also plays crucial role in the performance of thermoelectric materials. The investigated phonon dispersion and calculated Debye temperature gave strong evidence of low lattice thermal conductivity. The magnitude of thermal conductivity of chalcopyrite family is previously investigated[180], and they have reported that the prototype structure ZnGeP2 and ZnGeAs2 have lattice thermal conductivity of 158 mW(cmK)−1 and 117 mW(cmK)−1 respectively. As we know that the magnitude of lattice thermal conductivity will reduce if one replace As with Sb, the magnitude of lattice thermal conductivity of ZnGeSb2
can be expected to be lesser than that of ZnGeAs2. To estimate the range of power factor (S2σ), which is independent of relaxation time, we have assumed the relaxation time as 1×10−14 s and 1×10−15 s , and the calculated electrical conductivity (σ) and power factor (S2σ) are given in Table 3.5. We have observed high power factor due to the ultra high conductivity. The conductivity values observed for both the relaxation time are very high compared to other established materials, and the comparison is given in Table 3.5. In addition to this, conductivity value is higher than the reported conductivity of the Dirac semi-metal Cd3As2[215]. In Figure 3.17, we have summarised the TE properties of ZnGeSb2 in schematic way. In the era of thermoelectric materials, we have observed the enhancement of power factor (S2σ) due to some external effect like pressure or in low dimensional materials. Here in the present study, the bulk optimized structure itself showed a huge power factor.
To predict the exact figure of merit, the knowledge of the relaxation time is needed. The relaxation time of graphene is predicted to be in the order of 1×10−15.[216] If one can experimentally verify the lattice thermal conductivity and relaxation time of ZnGeSb2, it will turn out to be an excellent thermoelectric material.
-4 -2 0 2 Strain (in % ) -200
-100 0 100 200
p = 1020 cm-3 n = 1020 cm-3
S (µV/K)
(a) (b)
(c) (d)
(e) (f)
Figure 3.16: (a,b,c) Variation of thermopower, electrical conductivity and power factor as a function of strain. Solid line for ‘a’ axis, and dashed line for ‘c’ axis, (d,e) Variation of power factor as a function of concentration, f) Comparison of power factor with other materials PbTe[214], solid line for p -type and dashed line for n- type.
Figure 3.17: Schematic of phase change and comparison with well established compounds, PbTe[214], Cd3As2[215], YPtBi[221]