Band engineering via biaxial strain for enhanced thermoelectric performance in stannite-type Cu₂ZnSnSe₄

Abstract

The electronic structures of a typical quaternary compound of stannite-type Cu₂ZnSnSe₄ under biaxial strain were investigated by using first-principles calculations, and its p-type thermoelectric properties were calculated on the basis of the semi-classical Boltzmann transport theory. It was found that biaxial strain can be a powerful tool to fine-tune the band structure and thermoelectric properties of stannite-type Cu₂ZnSnSe₄, and the enhancement of thermoelectric properties can be explained from the convergence of the valence bands near the Fermi level. The study offers valuable insight into band engineering via biaxial strain for improving thermoelectric performance of quaternary chalcogenides and similar materials.

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1. Introduction

Thermoelectric materials, which can directly convert waste heat into electricity, are currently receiving significant scientific attention.^1,2 The efficiency of thermoelectric materials is quantified by the dimensionless figure of merit, ZT = S²σT/(κ_e + κ_l), where S, σ, T, κ_e and κ_l are the Seebeck coefficient, electrical conductivity, temperature, electronic thermal conductivity and lattice thermal conductivity, respectively. In order to obtain high ZT, S and σ need to be maximized, while thermal conductivity must be minimized so that the temperature difference can be maintained. Over the last decades, several approaches to reduce the lattice thermal conductivity κ_l have emerged, including solid-solution alloying and nanostructuring,^3,4 and lattice thermal conductivity in these materials has almost approached the amorphous limit.⁵ Consequently, alternative strategy to obtain high ZT value focus on enhancing the power factor S²σ where high Seebeck coefficient and good electrical conductivity are simultaneously required. However, this is not easy because the Seebeck coefficient S and electrical conductivity σ are both related to the electronic states and thus cannot be controlled independently.

The chalcopyrite-like semiconductors with the formula I₂–II–IV–VI₄ (where I = Cu, Ag; II = Zn, Cd; IV = Si, Ge, Sn; and VI = S, Se, Te) have found application in solar-cell absorbers for many years.^6–9 Some crystals of this group, such as Cu₂ZnSnSe₄, Cu₂ZnSnS₄, and Cu₂CdSnSe₄, optimized through indium doping to Sn sites or excess copper doping as substituents at Zn and Cd sites, have been suggested as a new class of wide-band-gap p-type thermoelectric materials.^10–16 These compounds have been derived from binary II₄–VI₄ compounds with four unit cells using the concept of cross substitution (replacement of II₄ by two I, one II and one IV group element, respectively) and maintaining an electron-to-atom ratio of 4.^10,11 It exhibits a naturally distorted structure that can lead to low thermal conductivity.^10,11 Thus, the improvement of ZT of these thermoelectric materials depends on the power factor S²σ. The electronic structure and thermoelectric properties of quaternary compounds I₂–II–IV–VI₄ have been investigated by first-principles calculations,^17,18 and recent studies have demonstrated that the power factor S²σ can be effectively enhanced by the convergence of heavy- and light-hole bands near the Fermi level.^19–24 In this work, we seek to explore convergence of bands via biaxial strain, which has been demonstrated as an effective method to engineer band structure.^25–29 Electronic structures and transport properties of typical quaternary Cu₂ZnSnSe₄ under different biaxial strains will be calculated using first-principles calculations and semi-classical Boltzmann transport theory, and the relationship of their transport behaviors and band convergence induced by biaxial strain will be investigated. It is expected that the present research can offer a useful path to engineer the band structure for improving the thermoelectric performance of quaternary chalcogenides and similar materials.

2. Computational method

First-principles calculations were performed with the Perdew–Burke–Ernzerhof (PBE)³⁰ generalized gradient approximation (GGA) and projector augment wave (PAW)³¹ pseudopotentials, as implemented in Vienna ab initio simulation package (VASP).³² For the geometry relaxation of the primitive unit cell of Cu₂ZnSnSe₄ in stannite structure, a k-point mesh of 13 × 13 × 13 was applied and spin–orbit coupling is not included. The cutoff energy of the plane-wave was set at 400 eV. The energy convergence criterion was chosen to be 10⁻⁶ eV per unit cell, and the forces on all relaxed atoms were less than 0.03 eV Å⁻¹. It is well known that density functional theory (DFT) cannot accurately describe the electronic structure when strongly correlated systems are considered. To correct this problem, we chose on-site Coulomb interactions (DFT + U) method for calculating electronic structure. The effective Coulomb repulsion parameter U was set to be 9 eV based on the published literature of Cu-based ternary semiconductors.³³ The biaxial in-plane strain is defined by

Δa/a₀ × 100% = (a − a₀)/a₀ × 100%,

where a₀ is the optimized lattice constant of the unstrained bulk material, and a is the in-plane lattice constant of the strained state. The out-of-plane lattice parameter c was optimized under the constraint of in-plane lattice parameter a, and the atomic positions were also relaxed.

The Seebeck coefficient S and electronic conductivity over relaxation time σ/τ were obtained using the semi-classical Boltzmann theory in conjunction with rigid band and constant relaxation time approximations. All the calculations of transport properties were implemented in the BoltzTraP package³⁴ which has been successfully predicted the temperature and carrier concentration dependence of transport properties for some thermoelectric materials.^{17,18,35–39} The necessary crystal structures and eigen-energies for BoltzTraP calculation were obtained from ab initio results. It is reasonably to choose DFT + U method for electrical transport calculations of stannite-type Cu₂ZnSnSe₄ (more discussion in the ESI†). In order to get reasonable transport properties, the Brillouin zones of the unit cells were represented by the Monkhorst-Pack special k-point scheme with 31 × 31 × 31 meshes. This provides well-converged transport quantities (see ESI†).

3. Results and discussion

Stannite-type Cu₂ZnSnSe₄ compound with I42̄m space group has the tetrahedral structure, and the atomic arrangement in a unit cell can be analogically derived from the analog of binary II–VI zincblende (ZnSe) compound. The crystal structure of stannite-type Cu₂ZnSnSe₄ is shown in Fig. 1. The calculated lattice parameters of Cu₂ZnSnSe₄ at different strains are listed in Table 1. As we can see from Table 1, the calculated lattice constants of unstrained Cu₂ZnSnSe₄ are a = 0.5747 nm and c = 1.16133 nm, which are compared with the theoretical calculations reported previously (a = 0.577 nm and c = 1.154 nm).¹⁸ The orbital-decomposed band structures of unstrained stannite Cu₂ZnSnSe₄ are presented in Fig. 2. The calculated energy band gap with DFT + U is 0.19 eV. This value is lower than the experimentally estimated energy gaps (1.44 eV),⁴⁰ and this is due to the problems associated with DFT + U used in our calculations, which still cannot entirely describe the exchange–correlation effects of the localized Cu 3d electrons.⁴¹ It can be seen from Fig. 2 that the valence band maximum (VBM) primarily comes from hybridization of the 3d orbitals of Cu and 4p orbitals of Se, whereas the conduction band minimum (CBM) is derived from the Se 4p orbitals and Sn 5s orbitals, and the Zn atoms have no obvious contribution to VBM or CBM. Considered as a stannite structure, Cu₂ZnSnSe₄ can be regarded as composed of tetrahedral (Cu₂Se₄)²⁻ slabs separated by tetrahedral (SnZnSe₄)²⁻ slabs. Analogous to the p-type chalcopyrite compound CuAlSe₂, the (Cu₂Se₄)²⁻ tetrahedral slabs are the electrically conducting units, and the (SnZnSe₄)²⁻ ones are the electrically insulating units.⁴² Following this special structure, the feature in the band structure of stannite-type Cu₂ZnSnSe₄ is also similar to these chalcopyrite semiconductors.⁴² The hybridization of Cu 3d and Se 4p orbitals around the VBM can provide a conduction pathway for charge carriers in the p-type semiconductors, while the weak hybridization of Sn 5s and Se 4p orbitals lies in the CBM, which makes the Sn–Se network an electronically insulating unit.⁴² The combination of electronically conducting and insulating units near the gap edge is responsible for high thermoelectric performance of stannite-type Cu₂ZnSnSe₄.

Fig. 1 Crystal structure of stannite-type Cu₂ZnSnSe₄.

Table 1 The optimized lattice constants of stannite-type Cu₂ZnSnSe₄ under different strains

Strain (%)	a (Å)	c (Å)	dCu–Se (Å)
−1	5.6873	11.7241	2.4360
0	5.7447	11.6133	2.4439
1	5.8022	11.4800	2.4509
2	5.8596	11.3317	2.4558
3	5.9170	11.1673	2.4599
4	5.9745	11.0291	2.4668

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Fig. 2 The orbital-decomposed band structure of stannite Cu₂ZnSnSe₄: (a) the 3d-orbital of Cu (blue), (b) the 5s-orbital of Sn (olive), and (c) the 4p-orbital of Se (red). The Fermi level is set to zero. Note that the upper part of the valence bands are mainly composed of Cu 3d and Se 4p orbitals.

Experimental works to date have found that Cu₂ZnSnSe₄ tends to form p-type semiconductors, and thus we mainly focus on discussion in VBM. To further illustrate the features of the electronic structures of stannite-type Cu₂ZnSnSe₄, the valence bands near the Fermi level are shown in Fig. 3(a). Stannite-type Cu₂ZnSnSe₄ is a compound possessing tetragonal structure with distorted tetrahedra. The crystal field splitting in such a crystal structure leads to the triply degenerate valence band Γ_15v into a non-degenerate band Γ_4v and a doubly degenerate band Γ_5v.^21,43 As we can see from Fig. 3(a), the non-degenerate band Γ_4v belongs to light band while the degenerate band Γ_5v is heavy-hole band. The combination of heavy-hole and light-hole bands near the VBM has been reported in the literature to be responsible for good thermoelectric performance of p-type semiconductors.⁴⁴ The crystal field splitting energy, Δ_CF = E_max(Γ_5v) − E_max(Γ_4v), is positive when Γ_5v lies above Γ_4v and negative when Γ_5v lies below Γ_4v. The calculated crystal field splitting energy Δ_CF of unstrained stannite Cu₂ZnSnSe₄ shows a value of 0.13 eV, which is comparable to the reported value.⁴⁵ It is proposed that thermoelectric performance of a material can be assessed by quality factor B, which is proportional to N_v/m*κ_l, where N_v is the number of degenerated valleys for the band, m* is electron effective mass, and κ_L is lattice thermal conductivity.²⁴ In order to obtain high thermoelectric performance, one has to increase the number of band valleys (N_v) while keep the effective mass m* and thermal conductivity κ_l suppressed. The effective mass m*is determined by the curvature of the band dispersion. As can be seen from Fig. S4 (see ESI†), the curvature of the band dispersion appears to have almost no change under strains, suggesting that the effective mass does not change with strain. As for these chalcopyrite like semiconductors, these values of κ_L are usually very low,^10–16 suggesting that the thermoelectric performance of these compounds can be improved from enhancement of N_v which is determined by the energy-splitting parameter Δ_CF.^21,24 Due to the lattice distortion under applied strain, the energy-splitting parameter can be substantially modified, and it will cause the convergence of the light and heavy-hole bands near the Fermi level. Fig. 3(b) shows the energy-splitting parameters Δ_CF of stannite-type Cu₂ZnSnSe₄ at each given value of strain. As shown in Fig. 3(b), the energy-splitting parameters Δ_CF can be tuned by applied biaxial strain, indicating the convergence of the bands at Γ point can be fine-tuned by the presence of strain, offering the possibility for enhancing the thermoelectric performance of this compound. Especially, the energy-splitting parameters Δ_CF reaches nearly zero with applied 2% tensile strain, and it is expected that the thermoelectric properties of Cu₂ZnSnSe₄ can be enhanced near this strain.

Fig. 3 (a) Calculated valence-band structure of stannite-type Cu₂ZnSnSe₄ near the Fermi level. Γ_4v is a nondegenerate band and Γ_5v is a doubly degenerate band. Δ_CF is the energy split at the top of Γ_4v and Γ_5v bands. (b) Energy-splitting parameters Δ_CF of stannite Cu₂ZnSnSe₄ at each given value of strain.

In order to investigate how strain affects the electronic states of p-type Cu₂ZnSnSe₄, the partial charge density of the conduction bands near Fermi energy at different strains are shown in Fig. 4. The partial charge density distribution can directly shows the real space distribution of the corresponding electronic states.³¹ As show in Fig. 4(a), the distribution of partial charge density shows an obvious antibonding characteristic between Cu and Se states, which forms an antibonding conductive network [–Cu–Se–Cu–] for hole transport in Cu₂ZnSnSe₄. Such characteristic of conductive network agrees with some other Cu-based semiconductors.^36,39,42 It can be seen from Fig. 4(a) that there are spherical-like electrons around Se atoms in the three-dimensional charge density distributions. These electronic states of spherical-like shape are composed of ns² around Se atoms, which represents the localization of these core electrons of Se atoms. As we can see from Fig. 4, the spherical-like electron distribution around Se atoms become more localized with increased strain, and this is because the Cu–Se bond length increases with the increase of tensile strain. As Se electrons become more localized under strain, the antibonding Cu–Se becomes less hybridized, and it will lead to electrical conductivity decrease with increased tensile strain.

Fig. 4 Contour plots of the partial charge density in the upper portion of the valence bands (−1–0 eV) of stannite-type Cu₂ZnSnSe₄ on the Cu–Se–Cu plane under typical strains: (a) no strain, (b) 2%, and (c) 4%. The Fermi levels are all set to zero. The unit of charge density is e Å⁻³.

In order to further understand the electronic structure of stannite-type Cu₂ZnSnSe₄ under strains, the projected density of states (PDOS) at three typical strains in the energy interval between −7 eV and 5 eV are shown in Fig. 5. From the PDOS of unstrained Cu₂ZnSnSe₄, it confirms again that the bottom of the conduction band primarily comes from the hybridization between Sn 5s and Se 4p orbitals, and the top of the valence band is mainly due to the hybridization between Cu 3d and Se 4p orbitals in the Cu–Se antibonding states. As we can see from Fig. 5, the obvious change of PDOS of Cu₂ZnSnSe₄ under strains lies in that the energy positions of these peaks of Se 4p orbitals near VBM become sharper, suggesting that Se 4p orbitals become more localized and the hybridization between Cu 3d and Se 4p orbitals decreases. The result is consistent with the analysis of the electronic states of stannite-type Cu₂ZnSnSe₄ under strain. As the Cu–Se antibonding states become less hybridized under strain, the electrical conductivity of p-type Cu₂ZnSnSe₄ decreases with increased tensile strain.

Fig. 5 Calculated projected density of states for stannite-type Cu₂ZnSnSe₄ under different strains. (a) no strain, (b) 2% strain (c) 4% strain. The Fermi levels are set to zero.

The total density of states (DOS) of stannite-type Cu₂ZnSnSe₄ near Fermi level under typical strains are shown in Fig. 6. It is well known that a rapid change of the DOS near the VBM is a good indicator of large Seebeck coefficient.^44,46 Therefore, the slope of the DOS near the band gap plays an important role in determining the transport properties of p-type Cu₂ZnSnSe₄. As shown in Fig. 6, the total DOS under 2% strain had the largest slope under three conditions, indicating that the value of Seebeck coefficient of Cu₂ZnSnSe₄ is the biggest under 2% tensile strain.

Fig. 6 Total density of states of stannite-type Cu₂ZnSnSe₄ under typical strains with the zero energy set to the valence-band maximum.

Electrical transport properties of Cu₂ZnSnSe₄ were calculated by combining Boltzmann transport theory under constant relaxation time approximation and electronic band structures. The Seebeck coefficient S is directly fixed by the electronic structure, and it is independent of relaxation time τ. The calculated temperature dependence of Seebeck coefficients S for the unstrained Cu₂ZnSnSe₄ at different doping levels are shown in Fig. 7. Here, we only calculate Seebeck coefficients for the 10% and 15% p-type doping of Cu₂ZnSnSe₄ since these doping concentrations have been studied experimentally. The carrier concentration of 15% doped Cu₂ZnSnSe₄ is adopted to experimentally measured 6 × 10²⁰ cm⁻³ which comes from ref. 10, and the carrier concentration of 10% p-type doped Cu₂ZnSnSe₄ is set to be 4.3 × 10²⁰ cm⁻³, which is slightly lower than the theoretical carrier concentrations (5.2 × 10²⁰ cm⁻³). As seen in Fig. 7, the Seebeck coefficient increases with increased temperature in the whole temperature range, as most of the thermoelectric materials.^18,36,39 At the same temperature, the Seebeck coefficients of Cu₂ZnSnSe₄ decrease as the doping level increases. The computed Seebeck coefficients of doped Cu₂ZnSnSe₄ agree well with those observed in experiments,¹⁰ and it indicates that our calculations of transport properties are reliable.

Fig. 7 Calculated Seebeck coefficient S of the unstrained Cu₂ZnSnSe₄ as a function of temperature. The data represented by dots are experimental measurements from ref. 10.

The results for thermoelectric transport properties of stannite-type Cu₂ZnSnSe₄ at each given value of strain are shown in Fig. 8. As we can see from Fig. 8(a), the Seebeck coefficients increase at first and then descend with increased tensile strain with a peak appears under 2% strain, while the value of Seebeck coefficient is the lowest under −1% compressive strain. The trend clearly demonstrates that the peak Seebeck coefficients are achieved around Δ_CF = 0, and this is consistent with the previous analysis of the band convergence under strain. In Fig. 8(b), the electrical conductivity of Cu₂ZnSnSe₄ decreases almost linearly with increased tensile strain, and the value is the highest under −1% compressive strain. For Cu₂ZnSnSe₄ strained in tension the Cu–Se bond length increases with the increase of tensile strain, and it will weaken the Cu–Se antibonding. As a result, the electrical conductivity decreases with increased tensile strain, and this result also agrees well with the partial charge density of Cu₂ZnSnSe₄ under tensile strain discussed above. Combining the Seebeck coefficient with electrical conductivity, the results for power factor under strain are shown in Fig. 8(c). Under tensile strain, the enhancement of Seebeck coefficient makes it possible to compensate the reduction of electrical conductivity, resulting in an enhancement of power factor of p-type Cu₂ZnSnSe₄ under 1% and 2% strains. Meanwhile, we note from Fig. 3(a) that energy-splitting parameters Δ_CF become negative under 3% and 4% strains, suggesting that the over-converged band structure is not good for enhancement of thermoelectric performance. The influence of strain on the transport properties of typical p-type Cu₂ZnSnSe₄ offers a general strategy for band convergence to improve the thermoelectric performance of the available I₂–II–IV–VI₄ thermoelectric materials.

Fig. 8 Calculated thermoelectric transport properties of p-type stannite Cu₂ZnSnSe₄ at each given value of strain at 700 K with the hole-concentration p = 6 × 10²⁰ cm⁻³. (a) Seebeck coefficient. (b) The electrical conductivity divided by relaxation time. (c) p-type power factors divided by relaxation time. The solid lines are provided as guides to eyes.

4. Conclusions

In summary, the band structures of strained and unstrained stannite-type Cu₂ZnSnSe₄ have been investigated using first-principles calculations. The transport properties of p-type Cu₂ZnSnSe₄ under different strains have been estimated based on semi-classical Boltzmann transport theory. It is found that the energy-splitting parameters Δ_CF of Cu₂ZnSnSe₄ can be tuned by applied strain, indicating that the convergence of the bands at Γ point can be induced by the presence of strain. Results suggest that band convergence induced by biaxial strain leads to enhancement of power factor of stannite-type Cu₂ZnSnSe₄, which offers an alternative route for improving the thermoelectric performance of this compound. Although this work only investigates Cu₂ZnSnSe₄ as a representative material, the mechanism of band convergence via biaxial strain is applicable to other quaternary semiconductors I₂–II–IV–VI₄ and other similar materials.

Acknowledgements

The work was carried out at Shenzhen Key Laboratory of Nanobiomechanics. We acknowledge the support by National Natural Science Foundation of China (11172255 and 11325420), National Basic Research Program of China (973 Program, 2015CB755500), Scientific Research Fund of Hunan Provincial Education Department (14C0461), and SIAT Innovation Program for Excellent Young Researchers (201405).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra00477b

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