Electronic band structure Ej (k) of CuInSe2 (upper panels), CuIn0.5Ga0.5Se2 (middle panels), and CuGaSe2 (lower panels) along four directions. The energies are referred to the VBM (dashed lines). Spin-orbit coupling is included, however, the notation of the energy bands (j = v1, v2, v3, and c1) refers to a spin-independent band indexing where c1 represents the lowest CB and v1 represents the topmost VB; these VBs are highlighted with colors in the online version. The solid lines show the full-potential results from Refs. 18 and 19, the circles are the results of the full band parameterization from Ref. 15, and the dotted lines represent the parabolic band approximation. Notice that the parabolic bands describe the twouppermost VBs poorly in the directions (100), (110), and (112).
Close-up of Fig. 1, demonstrating the strong non-parabolicity of the three uppermost VBs. Our parameterized energy bands Ej (k) consider the average of the two spinor states with σ = ↓ and ↑, even though there is a relatively large split of the spin-up- and spin-down-like bands in the (100)-direction. This average approximation is justified by = , which overall make the material spin-independent. Therefore, the notation of the energy bands is j = v1, v2, and v3 refers to a spin-independent band indexing.
Constant energy surfaces Sj (E) for the two uppermost VBs in CuIn0.5Ga0.5Se2 for the energies E = 1 meV (left column) and E = 200 meV (right column). The energy surfaces demonstrate that the VBs have ellipsoidal shapes in the vicinity of the Γ-point, but that they are very non-parabolic and anisotropic away from the Γ-point. In the supplementary information (Ref. 22), we present Sj (E) for the CB, as well as corresponding results for CuInSe2 and CuGaSe2.
Total DOS of the VBs (left panels) and of the CB (right panels) for CuInSe2, CuIn0.5Ga0.5Se2, and CuGaSe2. The solid lines show the full band parameterization (fbp), and the dashed lines represent the parabolic band approximation (pba). The energies refer to the VBM. Notice the different scales in the figures for the VB and the CB. The results demonstrates that the non-parabolicity of the bands strongly affect the DOS dispersions.
Carrier concentration p or n as functions of the quasi-Fermi energy EF * of the VBs EF , v * and of the CB EF , c *. Left column shows the results for large energy scale up to 0.5 eV, and right column displays a close-up for small Fermi energies. In the figure, |ΔE| is the positive energy difference Ev 1(0) − EF , v * for the VBs and EF , c * − Ec 1(0) for the CB. The carrier concentrations consider external band filling in intrinsic materials at T = 0 K. The results demonstrate that the parabolic band approximation strongly underestimates the band filling of both the VBs and CB.
The DOS mass of the VBs and the CB in CuInSe2, CuIn0.5Ga0.5Se2, and CuGaSe2. The upper (lower) panel shows in a wider (narrower) energy region. This energy-dependent mass generates accurate quasi-Fermi energy EF , v * and EF , c * as function of the carrier concentration; see Eq. (3). |ΔE| is the energy difference Ev 1(0) − EF , v * for the VBs and EF , c * − Ec 1(0) for the CB; cf. Fig. 5.
(a) Band-gap energy Eg and Fermi energy EF for 1 ≤ T ≤ 900 K of intrinsic CuInSe2, CuIn0.5Ga0.5Se2, and CuGaSe2, determined from the full band parameterization. In this figure, we also present the Fermi distribution f(E) of CuIn0.5Ga0.5Se2 for T = 300 K and 600 K. (b) Intrinsic carrier concentration as function of temperature. For comparison, the theoretical result for GaAs and Si using the parabolic band approximation is given.
Fermi level as function of the temperature 20 ≤ T ≤ 600 of p-type CuInSe2, CuIn0.5Ga0.5Se2, and CuGaSe2 for the effective doping concentration NA = 1013, 1014, 1015, … , and 1019 acceptors/cm3. The energy scale EF p − EF describes the Fermi energy with respect to the intrinsic EF ; see Fig. 7. Dashed lines represent the VBM with respect to the intrinsic Fermi level. Solid and dotted lines represents the full band parameterization and the parabolic band approximation, respectively.
Free carrier concentration as function of the temperature in p-type CuInSe2, CuIn0.5Ga0.5Se2, and CuGaSe2 for the effective doping concentration NA = 1013, 1014, 1015,…, and 1019 acceptors/cm3. Solid and dotted lines represents the full band parameterization and the parabolic band approximation, respectively. The full band description of the energy dispersion is important for high doping concentrations and/or in the intrinsic region for cross-gap excitations at high temperatures.
Γ-point energy gaps Eg , Fermi energies EF with respect to theVB maximum, and free carrier concentrations n = p in intrinsic CuIn1− x Ga x Se2 (x = 0, 0.5, and 1) at temperature T = 300 K. Corresponding Fermi energies EF p and hole concentrations p are presented in p-type CuIn1− x Ga x Se2 with effective acceptor concentrations of NA = 1017 and 1019 cm−3. At T ≈ 0 K, the band gap is Eg = 1.04, 1.33, and 1.67 eV for x = 0, 0.5, and 1, respectively.
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