The structural and electronic properties of Cu(In_1−xB_x)Se₂ as a new photovoltaic material

Abstract

The newly synthesized solar cell absorber material Cu(In_1−xB_x)Se₂ based on a solvothermal method with a special solvent is investigated using density functional theory. Our results show that Cu(In_1−xB_x)Se₂ has an extremely high formation enthalpy which indicates that it is hard to synthesize using a traditional evaporation method. The band gap of Cu(In_1−xB_x)Se₂ increases with increasing B content with a larger bowing parameter than that of Cu(In_1−xGa_x)Se₂. The band alignment of CuInSe₂ and CuBSe₂ is of type II. And the upshift of the valence band maximum is much smaller than that of the conduction band maximum for CuBSe₂, which indicates that pure CuBSe₂ can easily be p-type doped but that n-type doping could be more difficult in the alloy with a high B content. Therefore, we expect that the high efficiency Cu(In_1−xGa_x)Se₂ solar cell material has a low B content.

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I. Introduction

Ternary CuInSe₂ possesses a zinc-blend derived chalcopyrite crystal structure and an excellent optical absorption coefficient, and is widely studied as an absorber material in solar cells. The band gap of CuInSe₂ is about 1.04 eV, which is smaller than the optimal single-junction value (1.5 eV) according to the Shockley–Queisser model.¹ However, the band gap of CuInSe₂ can be tuned over a large range through alloying CuInSe₂ with its sister semiconductors Cu–III–VI₂ (III = Al, Ga, VI = S and Se), since these compounds can form various solid solutions. Among these solid solutions, one outstanding material is Cu(In_1−xGa_x)Se₂ (CIGS) which possesses the world record of optical to electrical efficiency in thin film solar cells.² Changing the Ga content can tune the band gap from 1.04 to 1.68 eV. However, Ga is an expensive raw material, which limits the large scale production of CIGS solar cells. Therefore, it is necessary to explore alternative alloys to CuGaSe₂ which can tune the band gap of CuInSe₂ to close to 1.5 eV and also have the excellent optical properties of CuInSe₂.

Apart from Al and Ga, B also lies in the same column as In in the periodic table. However, there is seldom research about the Cu(In_1−xB_x)Se₂ alloy, since, until recently, the Cu(In_1−xB_x)Se₂ alloy was synthesized through a solvothermal based method.³ In this paper, the authors also created Cu(In_1−xB_x)Se₂ solar cell devices with an efficiency of 2.34%. A theoretical understanding of CuBSe₂ as well as the Cu(In_1−xB_x)Se₂ alloy is still lacking. DFT calculations enable us to predict the electronic structure properties of both CuBSe₂ and the Cu(In_1−xB_x)Se₂ alloy. We can expect that the band gap of CuBSe₂ will be larger than that of CuGaSe₂ and CuAlSe₂, and a low concentration of B rather than Ga will tune the band gap of CuInSe₂ to the ideal value of 1.5 eV. In this study, the miscibility and the compositional dependence of the band gap change as well as band alignments are determined using first-principles calculations.

The electronic structure and the compositional dependence of the physical properties of Cu(In_1−xB_x)Se₂ are studied based on the special quasi-random structure (SQS). We find out that Cu(In_1−xB_x)Se₂ has an extremely high formation enthalpy, which indicates that Cu(In_1−xB_x)Se₂ is hard to synthesize using a traditional evaporation method. The band gap of Cu(In_1−xB_x)Se₂ increases with increasing B content and the band gap bowing parameter is larger than that of Cu(In_1−xGa_x)Se₂ because of the larger band gap of CuBSe₂ than CuGaSe₂. The band alignment of CuInSe₂ and CuBSe₂ shows that the band offset mainly results from the conduction band maximum (CBM) upshift of CuBSe₂, which indicates that n-type doping could be more difficult in the alloy with a high B content. Therefore, the high efficiency Cu(In_1−xB_x)Se₂ solar cells have a low B content compared with that of Cu(In_1−xGa_x)Se₂.

II. Computational details

The electronic structure and total energy calculations are performed using density functional theory as implemented in the plane wave VASP code.⁴ The generalized gradient approximation of the Perdew–Burke–Ernzerhof (PBE-GGA) exchange–correlation functional is used.^5,6 The interaction between the core electrons and the valence electrons is included using the standard frozen-core projector augmented-wave PAW potentials.^7,8 The cutoff energy for the plane wave basis is set to 300 eV. For the Brillouin zone integration, we used the 4 × 4 × 4 k-point mesh for 64-atom cells.

The valence band offset of ΔE_v(CuInSe₂/CuBSe₂) can be defined as:^10,11

Here,

where and are the core level to valence band maximum energy separations for CuInSe₂ and CuBSe₂, respectively. ΔE_C,C′ is the core level binding energy difference between CuInSe₂ and CuBSe₂, which is derived from the calculation of the 1 × 1 (001) oriented (CuInSe₂)_n/(CuBSe₂)_n superlattice. We have tested that n = 2 can keep the core level of the innermost layer on each side of the superlattice bulk-like. In our calculations, all the structure parameters of the superlattice are fully relaxed. The conduction band offset ΔE_c is defined as:

ΔE_c = ΔE_g + ΔE_v (4)

where ΔE_g is the experimental band gap difference between CuInSe₂ and CuBSe₂.

Several methods have been developed to investigate random alloys in the past, for example, virtual crystal approximation (VCA)¹² and cluster expansion (CE).^13–15 However, VCA is non-structural and CE is time consuming. The SQS method is an alternative approach to investigate a random alloy, and constructs a finite supercell to best match the correlation functions of a standard random alloy.^16,17 Therefore, the SQS is a finite supercell, which includes the microstructure and can be adopted for first-principles calculations. The SQS approach is widely used in the study of solid solution alloys with great success.^18–20

In this work, we use the SQS to construct the Cu(In_1−xB_x)Se₂ 64-atom supercell random alloy with the lattice vectors:

a⃑₁ = (2, 0, 0)a, a⃑₂ = (0, 2, 0)a, a⃑₃ = (0, 0, 2η)a (5)

where η = c/2a, and a and c are the short and long lattice constants, respectively, of the tetragonal chalcopyrite cell. The atomic positions for the SQS at x = 0.25 and x = 0.5 are listed in Table 1, and their structural correlation functions [capital Pi, Greek, macron]k,m as well as ideal random alloy correlation functions are listed in Table 2, which shows that the constructed SQSs in this paper are reasonably good.

Table 1 The atomic coordinates and the occupation of the SQS for In and B in Cu(In_1−xB_x)Se₂ at the concentrations x = 0.25 and x = 0.5 used in our work. Only the mixed sublattice coordinates are shown for clarity

Type	x = 0.25			Type	x = 0.5
	Coordinates				Coordinates
B	0.00	0.25	0.75	B	0.00	0.25	0.75
B	0.50	0.75	0.75	B	0.50	0.25	0.75
B	0.25	0.25	0.00	B	0.50	0.75	0.75
B	0.25	0.75	0.00	B	0.75	0.50	0.25
In	0.00	0.75	0.75	B	0.25	0.25	0.00
In	0.50	0.25	0.75	B	0.25	0.75	0.00
In	0.25	0.00	0.25	B	0.75	0.25	0.00
In	0.25	0.50	0.25	B	0.00	0.50	0.50
In	0.75	0.00	0.25	In	0.00	0.75	0.75
In	0.75	0.50	0.25	In	0.25	0.00	0.25
In	0.75	0.25	0.00	In	0.25	0.50	0.25
In	0.75	0.75	0.00	In	0.75	0.00	0.25
In	0.00	0.00	0.50	In	0.75	0.75	0.00
In	0.00	0.50	0.50	In	0.00	0.00	0.50
In	0.50	0.00	0.50	In	0.50	0.00	0.50
In	0.50	0.50	0.50	In	0.50	0.50	0.50

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Table 2 Atomic correlation functions [capital Pi, Greek, macron]k,m of the SQS used in our calculation at concentrations x = 0.25 and 0.5, and compared with the ideal values (2x − 1)^k of the random alloy

	[capital Pi, Greek, macron]2,1	[capital Pi, Greek, macron]2,2	[capital Pi, Greek, macron]2,3	[capital Pi, Greek, macron]2,4	[capital Pi, Greek, macron]3,1	[capital Pi, Greek, macron]4,1
x = 0.25
SQS	1/4	1/4	1/4	0	0	0
Random	1/4	1/4	1/4	1/4	−1/8	1/16
x = 0.5
SQS	0	0	−1/8	−1/4	−1/4	−1/16
Random	0	0	0	0	0	0

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III. Results and discussion

In the chalcopyrite CuInSe₂, each Se anion coordinates two Cu and two In, while each cation (Cu or In) is tetrahedrally coordinated by four Se atoms. The computed and experimental lattice parameters^21,22 and the anion displacement for CuInSe₂ and CuBSe₂ are given in Table 3. For CuInSe₂ our theoretical structure parameters are in good agreement with the experimental ones. For CuBSe₂, there is no experimental data about its structure and electronic structure. To see the electronic structure of CuBSe₂ clearly, we plotted the total and partial density of states (DOS) for the 16-atom CuBSe₂ as shown in Fig. 1. From the partial DOS, we can see that the upper valence band is derived mainly from the hybridization of the Se 4p and Cu 3d states, similar to those of CuInSe₂ and CuGaSe₂.²² The low conduction band derives mainly from the hybridization of the Se 4s, 4p, 3d and B 2s, 2p states. The calculated band gaps for CuBSe₂ and CuInSe₂ are 1.506 eV and 0.02 eV, respectively, which are smaller compared with the experimental values since the GGA functional cannot well describe the hybridization of the Cu 3d states and Se 4p states and therefore GGA investigations usually underestimate the band gap of semiconductors. However, the general chemical trend of the band gap variation is reproduced in the GGA calculations, i.e., the band gap of CuBSe₂ is larger than that of CuInSe₂. Much work has shown that the hybrid functional (HSE06 functional) describes the localized Cu 3d orbitals more correctly than (semi)local-density functionals^23–25 by substituting part of the short range PBE-GGA exchange energy with the short range Hartree–Fock (HF) exchange energy. We also use the HSE06 functional to evaluate the band gaps of CuInSe₂ and CuBSe₂, which yield the values of 1.0 eV and 3.14 eV.

Table 3 The comparison of the crystal constants a, η = c/2a, and u and the band gaps from the experimental values and GGA optimized results for both CuInSe₂ and CuBSe₂

	CuInSe2	CuBSe2
	Exp./GGA	GGA
a (Å)	5.782/5.865	5.337
η	1.004/1.006	0.911
u	0.224/0.219	0.202
Gap (eV)	1.04/0.02	1.506

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Fig. 1 The total and partial density of states of CuBSe₂.

The formation enthalpy describes the miscibility of the alloys. For CuIn_1−xB_xSe₂ the formation enthalpy is defined as:

ΔH(x) = E(x) − (1 − x) E_CuInSe2 − xE_CuBSe2 (6)

where E_CuInSe2 and E_CuBSe2 represent the total energy of pure CuInSe₂ and CuBSe₂, and E(x) is the total energy of the alloy at a composition x. The relationship between the alloy formation enthalpy and its composition x often obeys a quadratic function, i.e.,

ΔH(x) = (1 − x)ΔH(0) + xΔH(1) + Ωx(1 − x) (7)

where Ω is the interaction parameter indicating the solubility of the alloy. The calculated formation enthalpies of the Cu(In_1−xB_x)Se₂ alloy at five different compositions, namely x = 0, 0.25, 0.5, 0.75 and 1, are shown in Fig. 2 with blue triangles. The positive formation enthalpies indicate that the alloy prefers phase segregation into CuInSe₂ and CuBSe₂ at zero temperature, and mixing In and B cations costs additional energy to form the random alloy. By using eqn (7) to fit the calculated formation enthalpies, we get the interaction parameter Ω = 927 meV per atom. This value is much larger than that of Cu(In_1−xGa_x)Se₂ (11 meV per atom).²⁶ The extremely large interaction parameter suggests that thousands of Kelvin are needed to transform the two compounds into an alloy, which is mainly caused by the large chemical mismatch between the In and B atoms and the inert nature of the B element. As a result, Cu(In_1−xB_x)Se₂ can hardly be synthesized using a traditional evaporation method. This is also the reason why, until recently, the Cu(In_1−xB_x)Se₂ alloy was only synthesized using a solvothermal-based method with a special solvent. The band gap of CuBSe₂ is larger than that of CuGaSe₂, so less B than Ga is needed to tune CuInSe₂ to the optimized band gap value.

Fig. 2 The calculated formation enthalpy for the Cu(In_1−xB_x)Se₂ alloy as a function of the alloy composition x. The fitting curve according to eqn (7) is also given with the interaction parameter Ω.

Since the Cu(In_1−xB_x)Se₂ alloy is already synthesized, we will show the band gap changes depending on the composition. Usually, the dependence of the band gap on the composition can be described using the following equation for random semiconductor alloys:

E_g(x) = (1 − x)E_g(CuInSe₂) + xE_g(CuBSe₂) − bx(1 − x) (8)

where E_g denotes the band gap, and b is the bowing parameter. The band gaps for the Cu(In_1−xB_x)Se₂ alloy at x = 0, 0.25, 0.5, 0.75 and 1 are calculated using the PBE functional and the fitted bowing parameter b ∼1.68 eV is obtained. It is true that the PBE functional often underestimates the band gaps of semiconductors, but the calculated bowing parameters are accurate because the bowing parameters depend on the band gap difference, so the band gap errors induced by the PBE functional are systematically canceled in the calculation. The results shown in Fig. 3 are the shifted band gaps using the scissor operator, so that the band gap at x = 0 is equal to the experimental value of about 1.04 eV and the band gap at x = 1 is equal to the HSE06 band gap value of about 3.14 eV. After correction, the band gaps at different compositions can be compared directly with the experimental values.

Fig. 3 The band gap of the Cu(In_1−xB_x)Se₂ alloy as a function of the composition x. The band gap shifted using the scissor operator so the band gap at x = 0 agrees with the experimental value and at x = 1 is equal to the HSE06 calculated value.

From Fig. 3 we can see the monotonically increasing nature of the band gap of the Cu(In_1−xB_x)Se₂ alloy with the composition parameter x. The bowing parameter of Cu(In_1−xB_x)Se₂ is larger than that of the Cu(In_1−xGa_x)Se₂ alloy (0.1–0.3 eV)^9,10, which is caused by the larger size and chemical mismatch between In and B compared to In and Ga. However, the absolute bowing parameter value is large, indicating that the Cu(In_1−xB_x)Se₂ alloy has poor tolerance to the chemical and size difference of the mixed cations. Because the band gap of CuBSe₂ is larger than that of CuInSe₂, only 30% of B is needed to alloy into CuInSe₂ to tune the Cu(In_1−xB_x)Se₂ band gap to about 1.5 eV. For comparison, 30% Ga can tune the band gap of CuIn_1−xGa_xSe₂ to a smaller value of about 1.1 eV.

To understand how the band gap increases with increasing B composition in CuIn_1−xB_xSe₂, we calculate the band offsets of CuInSe₂ and CuBSe₂. The valence band maximum (VBM) offsets of the two compounds are small (the offset is less than 0.02 eV), but the CBM shifts up significantly from CuInSe₂ to CuBSe₂, as shown in Fig. 4. As a result, the CBM offset is 2.1 eV between CuInSe₂ and CuBSe₂. The small valence band offset is related to the states of the VBM of these compounds which primarily consists of the Se 4p and Cu 3d orbitals; therefore, the substitution of In with Ga or B does not change the VBM states and energies significantly. However, the CBM mainly consists of the antibonding state of the hybridization between the group III (In, B) s and p orbitals and Se 4s, 4p and 3d orbitals. Because the In–Se bond length is larger than the B–Se bond length, the repulsion between In and Se is smaller than between B and Se; as a result, the antibonding CBM states of CuBSe₂ are pushed to a higher energy level than those of CuInSe₂. Similarly, the CBM energy of CuGaSe₂ is lower than that of CuBSe₂.

Fig. 4 The band alignments of CuInSe₂, CuGaSe₂ and CuBSe₂ in eV.

The calculated band alignments in Fig. 4 show that the band alignment is of type II. The band gap of CuGaSe₂ and CuBSe₂ increasing with the Ga and B composition is mainly contributed by the conduction band upshift, with a much smaller contribution from the valence band downshift. With this finding, the electrical conductivity of these alloys can be predicted. According to the doping limit rule,²⁷ a material is difficult to be n-type doped if the CBM energy is high and difficult to be p-type doped if the VBM energy is low. The experiments and theoretical calculations have shown that the CuIn_1−xGa_xSe₂ samples have an intrinsic p-type conductivity but are hard to be n-type doped with high Ga concentrations, due to the compensation of acceptor defects.^28–30 The small valence band offset indicates that the Cu(In_1−xB_x)Se₂ alloy has an intrinsic p-type conductivity, and the much higher CBM of CuBSe₂ compared to that of CuInSe₂ suggests that n-type doping could be more difficult in a Cu(In_1−xB_x)Se₂ alloy as the band gap increases.

IV. Conclusion

Using first-principles calculations, we have investigated the electronic properties of CuBSe₂ as well as the Cu(In_1−xB_x)Se₂ random alloy as a function of the composition x. Our results show that Cu(In_1−xB_x)Se₂ has an extremely high formation enthalpy which indicates that it is hard to synthesize using a traditional evaporation method. The band gap increases monotonically with increasing B content, and the bowing parameter is larger for Cu(In_1−xB_x)Se₂ than that for Cu(In_1−xGa_x)Se₂ due to the larger size and chemical mismatch between In–B and In–Ga. The band gap increases with B content, which primarily comes from the large conduction band upshift. This indicates that Cu(In_1−xB_x)Se₂ shows an intrinsic p-type conductivity similar to CuInSe₂ and their n-type doping could be difficult as the B content increases. Based on these results, we predict that high efficiency Cu(In_1−xB_x)Se₂ solar cells should have low B content.

Acknowledgements

This work was supported by the Hunan Provincial joint fund for young talent training project under Contract No. 2015JJ6022 and the National Nature Science Foundation of China under Contract No. 11472103.

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Footnote

† PACS numbers: 71.20.Nr, 71.23.k, 73.61.Le.

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