Introduction

Copper–zinc–tin–chalcogenides-based quaternary semiconductors have received wide attention as potential solar cell absorber materials. An ideal solar energy absorber should have a band gap around 1.5 eV and a high optical absorption coefficient for the visible light. Based on this fact, the band gaps and optical properties of the copper-based systems were extensively studied both experimentally [1–3] and theoretically [4, 5]. The band gap energy (E g) of Cu2ZnSnS4 (CZTS) and Cu2ZnSnSe4 (CZTSe) are around 1.0–1.5 eV, which is suitable for photovoltaic applications. Moreover, knowledge of the optical properties, such as the dielectric function and the optical absorption coefficient, is required to analyse optical measurements as well as to optimize the solar cell devices. Various research groups reported the electronic structure calculations [4–8]. But new approach, such as TB-mBJ, is introduced recently. Overall, TB-mBJ is a decent alternative for GW or hybrid functional, achieving a comparable accuracy at a far lower cost. The combination of TB-mBJ exchange and generalise gradient approximation (GGA) correlation produced accurate semiconductor gaps for numerous semiconductors and insulators [9–13]. In this paper, the full potential linearized augmented plane wave (FP-LAPW) method with the TB-mBJ potential has been used to investigate the electronic and optical properties of the copper-based solar cell materials.

Method of calculation

As a computational tool, for Cu2ZnSnX4 (X = S, Se), we have employed the full potential augmented plane wave method (APW + lo) based on density functional theory (DFT) using Vienna package WIEN2K code (FP-LAPW) [14]. We have used the GGA within the Perdew–Burk–Ezrenhof (PBE) parameterization [15] and TB-mBJ [9–11] potential. A plane wave expansion has been done with K max × R max equal to 7 and l max = 10 are kept constant throughout the calculations. The charge density is Fourier expanded up to G max = 12(Ry)1/2. A 2 × 2 × 2 division for k-point sampling is used, and the tetrahedral method [16] has been employed for the Brillouin zone integrations. The Zn (3d 10 4s 2), Cu (3p 6 4s 2 3d 9), Sn (4d 10 5s 2 5p 2), S (3s 2 3p 4), and Se (3d 10 4s 2 4p 4) orbital are treated as valence states. The muffin tin radii for Zn, Cu, Sn, S, and Se are chosen to be 2.25, 2.22, 2.48, 1.97, and 1.97 a.u., respectively, for stannite structure and for kesterite structure, the radii are 2.22, 2.34, 2.41, 1.91, and 1.91 a.u., respectively. A number of iterations (40) are dedicated to accomplish self consistency. Less than 0.00001Ry of the total energy difference is used as per formula unit for succeeding iterations.

The frequency-dependent complex dielectric function ε(ω) = ε 1(ω) + iε 2(ω) is known to describe the optical response of the medium at all phonon energies E = ℏω. The imaginary part of the ε 2(ω) in the long wavelength limit has been obtained directly from the electronic structure calculation, using the relation [17],

$$ \varepsilon_{2} (\omega ) = \left( {\frac{{4\pi^{2} e^{2} }}{{m^{2} \omega^{2} }}} \right)\sum\limits_{i,j} {\int {\left\langle {i\left| M \right|j} \right\rangle^{2} \,f_{i} (1 - f_{i} )\delta (E_{f} - E_{i} - \omega )d^{3} k} } , $$
(1)

where M is the dipole matrix, i and j are the initial and final states, respectively, f i is the Fermi distribution function for the ith state, and E i is the energy of electron in the ith state. The real part of the dielectric function [ε 1(ω)] can be extracted using the Kramers–Kronig relation [18]

$$ \varepsilon_{1} (\omega ) = 1 + \frac{2}{\pi }p\int\limits_{0}^{\infty } {\frac{{\omega^{\prime } \varepsilon_{2} (\omega^{\prime } )d\omega^{\prime } }}{{\omega^{\prime 2} - \omega^{2} }}} , $$
(2)

where p stands for the principal value of the integral. The knowledge of both the real and imaginary parts of the dielectric tensor allows the calculation of the important optical properties, such as absorption coefficient, electron energy loss function, and real part of optical conductivity.

The absorption coefficient, α(ω) and real part of the optical conductivity, Re[σ(ω)] are calculated using the relation as follows [19–21]:

$$ \alpha (\omega ) = \sqrt 2 \omega \left[ {\sqrt {\varepsilon_{1}^{2} (\omega ) + \varepsilon_{2}^{2} (\omega ) - \varepsilon_{1} (\omega )} } \right]^{1/2} , $$
(3)
$$ \text{Re} [\sigma (\omega )] = \frac{{\omega \varepsilon_{2} }}{4\pi }. $$
(4)

Results and discussion

Structural properties

The crystalline and optical properties of CZTS and CZTSe have been discussed and compared with available literature results [1, 5]. The I2–II–IV–VI4 quaternary semiconductors have two fundamental crystal structures: one is the kesterite structure (KS) [space group \( I\bar{4} \)], the other is the stannite structure (SS) [space group \( I\bar{4}2m \)], which are shown in Fig. 1. In all of the structures, Se or S (group VI) atom is surrounded by two Cu (group I) atoms, one Sn (group IV) atom and one Zn (group II) atom, therefore, the octet rule is obeyed.

Fig. 1
figure 1

(Color online) The crystal structures of CZTS_KS and CZTS_SS, respectively

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The calculated total energies within GGA as function of the volume were used for determination of theoretical lattice constant and bulk modulus. Equilibrium lattice constant, bulk modulus and its pressure derivative are calculated by fitting the calculated total energy to the Murnaghan’s equation of state given in equation

$$ E(V) = E_{0} + \left[ {\frac{{(V_{0} /V)^{{B_{0}^{\prime } }} }}{{B_{0}^{\prime } - 1}} + 1} \right] - \frac{{B_{0} V_{0} }}{{B_{0}^{\prime } - 1}}, $$
(5)

where E 0 is the minimum energy at T = 0 K, B 0 is the bulk modulus at the equilibrium volume, and B 0′ is pressure derivative of the bulk modulus at the equilibrium volume. The equilibrium volume is given by the corresponding total energy minimum as shown in Fig. 2a–d. The volume corresponding to the lowest energy is used in the determination of equilibrium lattice constant. The calculated values of lattice constant and bulk modulus are presented in Table 1. It is clearly seen that the structural stability of these compounds is more stable in kesterite than stannite structure. We have also optimized the positional parameter and the c/a ratio for all the compounds; it is found that the c/a ratio is very close to 2. The bulk modulus is slightly higher in KS when compared to SS for CZTS; whereas the bulk modulus is greater in SS when compared to KS for CZTSe. The calculated lattice constants for both KS and SS are in good agreement with the experimental values.

Fig. 2
figure 2

(Color online) a–d represent the equilibrium volume with corresponding total energy minimum of CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively

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Table 1 Structural properties of tetragonal body-centered structure: equilibrium lattice parameter (a.u.), c/a ratio, and anion position are given in fractions of the lattice vectors, bulk modulus (GPa), GGA band gap (eV), and TB-mBJ band gap (eV)
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Density of states (DOS)

Figure 3a–d shows calculated partial and total density of states for all four CZTS and CZTSe. Overall, they have rather similar energy distribution of the eigenstates. The VB–DOS contains mainly a hybridization of Cu-d and anion (S and Se)-p, although there are also contributions from Zn-spd, Cu-s, and Sn-p in the lower energy region of the DOS (i.e., −6.5 to −2 eV). The CB-DOS consists mainly of cation-s anion-p hybridization. In the range between −2.5 and −4 eV below the Fermi level, Sn-p shows highest contribution. Within the range −1.28 to 1.5 eV around E F, Zn-s shows major contribution. On the other hand, in conduction band side anion, (S and Se)-p shows a peak in between 1.2 and 1.7 eV. So, the band gap is formed between the hybridized “p” like states of anion in the valence band and the antibonding states in the conduction band of Zn-s like states [1, 21].

Fig. 3
figure 3

(Color online) a–d represent the calculated partial density of states and total density of states with the TB-mBJ approximation for CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively

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By analyzing the partial density of these compounds, it is found that the s–p interaction modifies the gap of these materials. The anion-p like states are pushed up and Zn-s like states are pushed down. The bands above 2.5 eV mainly consists of the hybridised bands of Zn “s” like, Sn “p” like and anion “p” like bands. The energy gap is about 1.5 eV. The higher CBs do not contribute to the optical absorption in the low-energy regime. The calculated band gap values are reported in Table 1. Moreover, our calculated band gap values are in good agreement with earlier theoretical predictions and experimental investigations [1, 22–26].

Optical properties

The optical properties of the CZTX (X = S, Se) can be described by means of complex dielectric function ε(ω). The dielectric function describes the linear response of the system to an electromagnetic radiation, which is related to the interaction of photons with the electrons. The magnitude of optical properties depends on the polarization of photons. For the tetragonal unit cell structure, we calculate two independent components of dielectric constant ε(ω). One perpendicular component and the other parallel component, which arises due to polarization along the z direction. All the other optical constants, such as refractive index n(ω), the extinction coefficient k(ω), the optical reflectivity r(ω), and absorption coefficient α(ω), are then computed from the components of the dielectric function.

Figures 4 and 5a–d show the real and imaginary parts of the dielectric function. The real part of the dielectric function, ε 1(ω), represents the dispersion of the incident photons by the materials, while imaginary part of the dielectric function, ε 2(ω) is related to the energy absorbed by the material. The values of the static dielectric constant for a field perpendicular or parallel to the tetragonal c-axis are listed in Table 2. The dielectric constant ε 1(0) is obtained from the zero-frequency limit of ε 1(ω) and it corresponds to the electronic part of the static dielectric constant of the material. It is a parameter of fundamental importance in many aspects of materials properties.

Fig. 4
figure 4

(Color online) a–d represent the real (ε 1) parts of dielectric functions versus photon energy (eV). Solid lines show the TB-mBJ results, and dashed lines the GGA ones

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Fig. 5
figure 5

(Color online) a–d represent the imaginary (ε 2) parts of dielectric functions versus photon energy (eV) for CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively. Solid lines show the TB-mBJ results, and dashed lines the GGA ones

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Table 2 Scissor correction in eV and real part of dielectric functions ɛ 0(ω) at zero frequency using both PBE (GGA) and TB-mBJ potentials
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From the application point of view, it is very interesting to determine the transition corresponding to each peak in the spectrum. The peak in the ε 2(ω) spectrum distributes in a sequence similar to that in density of states. The main difference of dielectric functions between TB-mBJ and GGA becomes noticeable in the energy range from 0 to 5 eV. Transitions from valence bands at top to conduction bands at the lower state contribute to the optical spectra’s major part [21]. The dramatically increasing point of the mBJ imaginary part ε 2 is shifted to higher photon energy with respect to that of the GGA ε 2. The rightward shift of the TB-mBJ ε 2 results between 2 and 5 eV with respect to the GGA ones reflects the fact that the TB-mBJ energy gap is substantially improved over the GGA one. The zero-frequency real part ε 1 of static dielectric functions and scissor operator are reported in Table 2, in comparison with earlier calculated value [1, 5, 27].

Furthermore, the dielectric functions of all four components have a strong ε 2(ω) peak in the energy range between 3.6 and 4.3 eV. This peak arises from the response at \( \overrightarrow {k} \)-states always from the Γ-point, with a strong contribution from the flat dispersion of the top most VB and of the lowest CB. The two kesterite compounds have more pronounced peaks around 1.6–2.0 eV, whereas the two stannite compounds show much stronger anisotropy of their response peaks. In the kesterite compounds, the upper VBs are energetically close together at 0–0.5 eV at the BZ edges, whereas the upper VBs in the stannite compounds have more split with energy in the region of about 0.5–1.0 eV. It is the symmetry of these VBs in combination with their energy dispersions that cause the anisotropy [1, 5].

The calculated real part of the optical conductivity (frequency dependent) using GGA and TB-mBJ potentials is shown in Fig. 6a–d. The material’s conductivity increases with an increase of energy from 3.3 to 9.03 eV. Here, the conductivity for both the structures of CZTS and CZTSe has the highest value. Afterwards, it decreases with the energy; higher energy implies that the material does not interact with incident photon.

Fig. 6
figure 6

(Colar online) a–d represent the calculated real part of the optical conductivity spectra of CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively

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The absorption band edges of CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS with scissor correction using GGA and TB-mBJ are shown in Fig. 7. The band gap energies (E g) of CZTS and CZTSe were calculated from the Tauc’s plot assuming α 2 ∝ (hν − E g ), where α is the absorption coefficient and hν is the photon energy (h is the Planck’s constant and υ is the frequency). It is found that the theoretical absorption spectra of CZTS_KS and CZTSe_KS agree well with the available experimental spectra [1, 22, 28, 29] in their shape. The estimated band gaps using TB-mBJ including scissor correction are 1.58, 1.63, 1.34, and 1.55 eV for CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively, (Fig. 7). All these values are quite close to the experimental value for a single-junction solar cell [28, 30–33]. The absorption coefficient was higher than 104 cm−1 in the visible region. Actually, the absorption coefficient, α(ω) of CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS represent the linear optical response from the valence bands to the lowest conducting bands. All three components have comparable absorption, although with different photon energies for the onset to absorption (i.e., the band gap energy). The absorption starts in the energy of fundamental gap. In the energy range of 1.1–1.8 eV, the absorption peaks are due to the transition of electron in the hybridized states of Cu-d and anion-p to empty states in the CB. The value of α(ω) increases as the energy increases to its higher value. Therefore, it is worthy to investigate extensively CZTS and CZTSe for the sake of obtaining high-efficient materials.

Fig. 7
figure 7

(Color online) a–d represent the calculated (αhυ)2 versus energy gap (eV) of CZTS_KS, CZTS_SS, CZTSe_KS, and CZTSe_SS, respectively

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Conclusion

In summary, we have used TB-mBJ exchange potential implemented in WIEN2K (FP-LAPW) method to investigate the electronic and optical properties of Cu2ZnSnX4 (X = S, Se). The calculated results show that the energy gaps are substantially improved by TB-mBJ over GGA values toward experimental results and also we found that the compounds are direct band gap semiconductors. The behavior of optical dielectric functions (both real and imaginary parts) and optical conductivity and absorption coefficients as functions of photon energy agree well with the experimental and other literature values. The present study reveals that these excellent improvements are achieved because TB-mBJ potential describes accurately the hybridization between the Zn-s and anion (S and Se)-p states.