Earth-Abundant Materials for Solar Cells: Cu2-II-IV-VI4 Semiconductors

By Sadao Adachi

Systematically describes the physical and materials properties of copper-based quaternary chalcogenide semiconductor materials, enabling their potential for photovoltaic device applications.

Intended for scientists and engineers, in particular, in the fields of multinary semiconductor physics and a variety of photovoltaic and optoelectronic devices.

• Language: English
• Publisher: Wiley
• Release date: Oct 28, 2015
• ISBN: 9781119052784

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Table of Contents

Cover

Title Page

Preface

Abbreviations and Acronyms

1 Introduction

1.1 Natural Abundance of Elements in the Earth’s Crust

1.2 Solar Radiation Spectrum

1.3 Shockley–Queisser Efficiency Limit

1.4 Fundamental Properties of Photovoltaic Semiconductor Materials

1.5 Solar Cell Device Characteristics

1.6 Prediction of Physical Properties for Complex Material System

References

2 Structural Properties

2.1 Grimm–Sommerfeld Rule

2.2 Crystal Structure and Phase Stability

2.3 Lattice Constant and Related Parameters

2.4 Structural Phase Transition

References

3 Thermal Properties

3.1 Phase Diagram

3.2 Melting Point

3.3 Specific Heat

3.4 Debye Temperature

3.5 Thermal Expansion Coefficient

3.6 Thermal Conductivity

3.7 Thermal Diffusivity

References

4 Elastic, Mechanical, and Lattice Dynamic Properties

4.1 Elastic Constant

4.2 Microhardness

4.3 Lattice Dynamic Properties

References

5 Electronic Energy-Band Structure

5.1 General Remark

5.2 Lowest Indirect and Direct Band-Gap Energies

5.3 Higher-Lying Band-Gap Energy

5.4 External Perturbation Effect on the Band-Gap Energy: Experimental Data

5.5 Effective Mass

5.6 Nanocrystalline Band-Gap Energy

5.7 Heterojunction Band Offset

5.8 Electron Affinity

5.9 Schottky Barrier Height

References

6 Optical Properties

6.1 General Remark

6.2 The Reststrahlen Region

6.3 At or Near the Fundamental Absorption Edge

6.4 The Interband Transition Region

References

7 Carrier Transport Properties

7.1 Electron Transport Properties

7.2 Hole Hall Mobility

7.3 Electrical Resistivity

7.4 Minority-Carrier Transport

7.5 Effect of Grain Boundary

7.6 Proposal: Graded-Absorber Solar Cell Structure

7.7 Proposal: Controlling Transport Properties of Bulk Material by Heat Treatment

References

Appendix A

Appendix B

References

Appendix C

References

Appendix D

References

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 Abundance of elements in the earth’s crust

Table 1.2 Earth abundance per unit atom of raw elementals used for various solar cell applications (in %)

Table 1.3 Crystal structure, lattice constants (a and c), and lowest band-gap energy (Eg) at 300 K for some group IV, III–V, II–VI, and I–III–VI2 semiconductors

Table 1.4 CZTS-related binary and ternary compounds, together with their lattice constants (a, b, c, and aeff) and band-gap energies (Eg) at 300 K

Chapter 02

Table 2.1 Theoretical total energy of Cu2–II–IV–VI4 quaternary semiconductors in different crystal structures relative to that of the kesterite structure (meV/atom)

Table 2.2 Crystal structure, space group, ideal lattice constants (a, b, and c), and effective cubic lattice constant (aeff) of Cu2–II–IV–VI4 semiconductors

Table 2.3 Crystal structure (space group) and lattice constants (a, b, and c) of Cu2Zn–IV–VI4 quaternary semiconductors

Table 2.4 Crystal structure (space group) and lattice constants (a, b, and c) of Cu2Cd–IV–VI4 quaternary semiconductors

Table 2.5 Crystal structure (space group) and lattice constants (a, b, and c) of Cu2Hg–IV–VI4 quaternary semiconductors

Table 2.6 Summary of molecular weight, crystal structure (space group), recommended lattice constants (a, b, c, and β), effective cubic lattice constant (aeff), and crystal density (g) for Cu2–II–IV–VI4 quaternary semiconductors

Table 2.7 Summary of CZTS heteroepitaxy on various substrates

Table 2.8 Crystal structure and lattice constants (a, b, and c) of WZ-type and WZ-derived Cu2–II–IV–VI4 quaternary nanocrystals

Chapter 03

Table 3.1 Melting point Tm of Cu2–II–IV–VI4 quaternary semiconductors

Table 3.2 Summary of melting point Tm (recommended or averaged value), specific heat Cp, Debye temperature θD, thermal expansion coefficient αth, and thermal conductivity κ for Cu2–II–IV–VI4 quaternary semiconductors at 300 K

Table 3.3 Form of the thermal expansion coefficient tensor [α] for Cu2–II–IV–VI4 semiconductors of certain symmetry classes

Table 3.4 Empirical equation, κ(T) = ATn, for the temperature dependence of κ for some Cu2–II–IV–VI4 semiconductors (κ in W/m K)

Chapter 04

Table 4.1 Form of the second-order elastic stiffness [C] (or compliance [S]) tensors for Cu2–II–IV–VI4 semiconductors of certain symmetry classes

Table 4.2 Relation between the elastic stiffness Cij and compliance constants Sij for Cu2–II–IV–VI4 semiconductors crystallizing in the tetragonal stannite ( ) and hexagonal wurtzite (P63mc) structures

Table 4.3 Theoretical elastic stiffness constant for CZTS and CZTSe (in GPa)

Table 4.4 Parameter value describing the relationship between Cij and aeff for 6H-SiC and some WZ-type III–V and II–VI semiconductors (see Figure 4.2)

Table 4.5 Elastic constants in the tetragonal stannite-type crystal approximation and related properties of CZTS, Cu2ZnSn(SxSe1−x)4, and CZTSe at 300 K

Table 4.6 Expression for the reciprocal of Young’s modulus in the direction of the unit vector l = (l1, l2, l3) in the various crystal systems of Cu2–II–IV–VI4 semiconductors

Table 4.7 Expression for the linear compressibility in the direction of the unit vector l = (l1, l2, l3) in the various crystal systems of Cu2–II–IV–VI4 semiconductors

Table 4.8 Functional expression for sound velocity propagating parallel (‖ c) and perpendicular to the c-axis (⊥ c) in a tetragonal stannite-type lattice of Cu2–II–IV–VI4 semiconductors

Table 4.9 Microhardness H of Cu2–II–IV–VI4 semiconductors at 300 K

Table 4.10 Raman frequency for CZTS at 300 K (in cm−1)

Table 4.11 Raman frequency for some possible inclusion materials in CZTS at 300 K

Table 4.12 Raman frequency for CZTSe in the stannite (kesterite) lattice at 300 K (in cm−1)

Table 4.13 Raman frequency for some possible inclusion materials in CZTSe at 300 K

Table 4.14 Raman frequency for some t-Cu2–II–IV–VI4 semiconductors at 300 K (in cm−1)

Table 4.15 Raman frequency obtained from single-crystalline CZTS at 300 K

Table 4.16 Raman frequency obtained from single-crystalline Cu2Zn–IV–Se4 (IV = Ge, Sn) quaternary semiconductors in the kesterite-type structure at 300 K (in cm−1)

Table 4.17 Raman frequency for single-crystalline Cu2ZnSi–VI4 (VI = S, Se) quaternary semiconductors in the WZ-stannite structure at 300 K (in cm−1)

Table 4.18 Raman frequency for single-crystalline o-Cu2ZnGeS4 quaternary semiconductor at 300 K (in cm−1)

Table 4.19 Raman frequency for polycrystalline kesterite CZTS measured at 300 K, together with that obtained from theoretical calculation (in cm−1)

Table 4.20 Far-IR phonon frequency for some Cu2–II–IV–VI4 quaternary semiconductors at 300 K

Chapter 05

Table 5.1 Lowest indirect and direct band-gap energies Eg for Cu2Zn–IV–VI4 quaternary semiconductors at 300 K

Table 5.2 Lowest indirect and direct band-gap energies Eg for Cu2Cd–IV–VI4 quaternary semiconductors at 300 K

Table 5.3 Lowest indirect and direct band-gap energies Eg for Cu2Hg–IV–VI4 quaternary semiconductors at 300 K

Table 5.4 Summary of the lowest indirect and direct band-gap energies Eg for Cu2–II–IV–VI4 quaternary semiconductors at 300 K

Table 5.5 Lowest indirect and direct band-gap energies (Eg) for some possible inclusion materials in CZTS and CZTSe at 300 K

Table 5.6 CP energy for some Cu2–II–IV–VI4 semiconductors determined by SE measurements at 300 K (in eV)

Table 5.7 Empirical equation (Eq. 5.6) for the lowest indirect and direct band-gap energy variations with temperature T for some Cu2–II–IV–VI4 quaternary semiconductors. The temperature coefficient (∂Eg/∂T) corresponds to that at 300 K

Table 5.8 Γ-valley electron effective masses, (⊥c), (||c), and , for CZTS and CZTSe, together with those for some wurtzite III–V and II–VI semiconductors (in m0). The CZTS and CZTSe values were obtained by theoretical calculation

Table 5.9 Theoretical Γ-valley hole effective masses, (⊥c) and (||c), for CZTS and CZTSe (in m0)

Table 5.10 Γ-valley DOS ( ) and conductivity hole effective masses ( ) for kesterite-type CZTS and stannite-type CZTSe with i = A, B, or C (in m0)

Table 5.11 Lowest direct band-gap energy Eg for some Cu2–II–IV–VI4 quaternary nanocrystals. All these data were determined from the optical absorption measurements at 300 K

Table 5.12 Experimentally determined band offsets (ΔEc and ΔEv) and offset ratio (ΔEc:ΔEv) in the II–VI (window)/II–VI (buffer) heterojunction system used for Cu2–II–IV–VI4 solar cell devices

Table 5.13 Experimentally determined band offsets (ΔEc and ΔEv) and offset ratio (ΔEc:ΔEv) in the CdS/CZTS, CdS/CZTSe, and some buffer/CZTS heterojunction systems

Chapter 06

Table 6.1 First-order Sellmeier parameter used in the calculation of n(λ) (Eq. 6.83; λ in µm) for Cu2–II–IV–VI4 quaternary semiconductors at 300 K, together with ε∞ values estimated from Equation 6.94

Table 6.2 MDF CP parameter used in the calculation of ε(E) for Cu2ZnGeS4 quaternary

Chapter 07

Table 7.1 Temperature and mass dependences of the electron mobility for different scattering mechanisms in semiconductors

Table 7.2 Hole mobility μh and resistivity ρ of Cu2Zn–IV–VI4 quaternary semiconductors at 300 K

Table 7.3 Hole mobility μh and resistivity ρ of Cu2Cd–IV–VI4 quaternary semiconductors at 300 K

Table 7.4 Hole mobility μh and resistivity ρ of Cu2Hg–IV–VI4 quaternary semiconductors at 300 K

Table 7.5 Minority-electron lifetime (τe), diffusion length (Le), and power conversion efficiency (η) for p-type CZTS at 300 K

Table 7.6 Minority-electron lifetime (τe), diffusion length (Le), and power conversion efficiency (η) for p-type CZTSe at 300 K

Table 7.7 Minority-electron lifetime (τe), diffusion length (Le), and power conversion efficiency (η) for p-type Cu2–II–IV–VI4 alloy semiconductors at 300 K

Appendix A

Table A.1 Structural properties of CZTS and CZTSe at 300 K

Table A.2 Thermal properties of CZTS and CZTSe at 300 K

Table A.3 Elastic and mechanical properties of CZTS and CZTSe at 300 K

Table A.4 Raman frequency for CZTS and CZTSe in the kesterite (stannite) lattice at 300 K (in cm−1)

Table A.5 CP energy in CZTS and CZTSe at 300 K

Table A.6 Theoretically obtained Γ-valley electron and hole effective masses in kesterite-type CZTS and stannite-type CZTSe (in m0)

Table A.7 Band offsets (ΔEc and ΔEv) and offset ratio (ΔEc:ΔEv) in the CZTS- and CZTSe-based heterojunctions to be used for solar cell applications

Table A.8 Electron affinity and Schottky barrier height in CZTS and CZTSe

Table A.9 Static and high-frequency dielectric constants for CZTS and CZTSe at 300 K

Table A.10 Exciton parameter for CZTS and CZTSe

Table A.11 Optical constant at the specific wavelength of CZTS and CZTSe at 300 K

Table A.12 Carrier transport properties of CZTS and CZTSe at 300 K

Appendix B

Table B.1 Structural properties of CdS and ZnO at 300 K

Table B.2 Thermal properties of CdS and ZnO at 300 K

Table B.3 Elastic and mechanical properties of CdS and ZnO at 300 K

Table B.4 Raman frequency for CdS and ZnO at 300 K (in cm−1)

Table B.5 Mode Grüneisen parameter for the long-wavelength phonons in CdS and ZnO

Table B.6 Long-wavelength phonon deformation potential for w-CdS (in cm−1)

Table B.7 Piezoelectric stress and strain constants and Fröhlich coupling constant for w-CdS and ZnO

Table B.8 CP energy in c-CdS, w-CdS, and ZnO at 300 K (in eV)

Table B.9 Γ-valley electron effective mass in c-CdS, w-CdS, and ZnO (in m0)

Table B.10 Luttinger VB parameter for c-CdS, w-CdS, and ZnO

Table B.11 Deformation potential for c-CdS, w-CdS, and ZnO (in eV)

Table B.12 Electron affinity and Schottky barrier height in c-CdS, w-CdS, and ZnO

Table B.13 Static and high-frequency dielectric constants for c-CdS, w-CdS, and ZnO at 300 K

Table B.14 Exciton parameter for w-CdS and ZnO

Table B.15 Optical constant at the specific wavelength of c-CdS, w-CdS, and ZnO at 300 K

Table B.16 Carrier transport properties of c-CdS, w-CdS, and ZnO at 300 K

Appendix C

Table C.1 Optical constants of Cu2ZnSiSe4 at 300 K

Table C.2 Optical constants of Cu2ZnGeS4 at 300 K

Table C.3 Optical constants of Cu2ZnGeSe4 at 300 K

Table C.4 Optical constants of CZTS at 300 K

Table C.5 Optical constants of CZTSe at 300 K

Appendix D

Table D.1 Optical constants of c-CdS at 300 K

Table D.2 Optical constants of w-CdS for E⊥c (ordinary ray) at 300 K

Table D.3 Optical constants of w-CdS for E‖c (extraordinary ray) at 300 K

Table D.4 Optical constants of ZnO for E⊥c (ordinary ray) at 300 K

Table D.5 Optical constants of ZnO for E‖c (extraordinary ray) at 300 K

List of Illustrations

Chapter 01

Figure 1.1 Abundance of elements in the earth’s crust

Figure 1.2 Solar radiation spectra at both the top of the earth’s atmosphere (AM0) and sea level (AM1.5)

Figure 1.3 SQ efficiency limit together with conversion efficiencies obtained from various semiconductor solar cells and dye-sensitized TiO2 solar cell. The shaded region indicates an optimal Eg range (~1.0–1.7 eV) in the SQ efficiency limit. The solid and open circles show the results obtained from one pn-junction and multijunction solar cells, respectively. CIGS = Cu(Ga,In)Se2

Figure 1.4 (a) Band-gap energy Eg versus lattice constant a for a number of group IV, III–V, and II–VI semiconductors crystallizing in the cubic structure at 300 K [4]. The shaded region indicates an optimal Eg range (~1.0–1.7 eV) in the SQ efficiency limit (see Figure 1.3). (b) Spectral irradiance (SI) for sunlight at AM1.5

Figure 1.5 (a) Band-gap energy Eg versus effective lattice constant aeff for a number of I–III–VI2 semiconductors crystallizing in the chalcopyrite structure at 300 K. The Eg versus aeff data for CZTS and CZTSe quaternaries are also shown by the open circles. The shaded region indicates an optimal Eg range (~1.0–1.7 eV) in the SQ efficiency limit (see Figure 1.3). (b) Spectral irradiance (SI) for sunlight at AM1.5

Figure 1.6 Room-temperature optical absorption spectra α(E) for a number of semiconductors which are important or to be used for photovoltaic device applications: (a) group IV, (b) III–V, (c) II–VI, and (d) I–III–VI2 semiconductors [5, 6]. The α(E) spectra for anisotropic semiconductors correspond to those for the ordinary ray (E⊥c)

Figure 1.7 Spectral efficiency for some different kinds of solar cells along with the spectral irradiance at AM1.5

Figure 1.8 (a) Optical absorption spectra α(E) for w-CdS and ZnO [5, 6], (b) normalized EQE of CZTS and CZTSSe solar cells, and (c) normalized EQE of CZTSe and CZTSSe solar cells. The spectral irradiance at AM1.5 is shown in (b) and (c). The experimental photovoltaic characteristics of CZTS, CZTSSe, and CZTSe solar cells are taken from Ennaoui et al. [9], Woo et al. [10], and Repins et al. [11], respectively

Figure 1.9 (a) A roughened surface and (b) its equivalent EM layer thickness d on a substrate of the original material

Chapter 02

Figure 2.1 3D perspective view of (a) kesterite-type and (b) stannite-type CZTS structures

Figure 2.2 (a) WZ-kesterite-type and (b) WZ-stannite-type Cu2–II–IV–VI4 semiconductors

Figure 2.3 XRD patterns for (a) LT- and (d) HT-Cu2ZnGeS4 crystals taken from the ASTM cards (JCPDS 01-074-8334 and 00-026-0572, respectively). They crystallize in the stannite (LT-Cu2ZnGeS4) and WZ-stannite structures (HT-Cu2ZnGeS4). The ASTM card images are also shown in (b) for t-Cu2GeS3 (00-041-1035), (c) for ZB-ZnS (c- or β-ZnS; 00-005-0566), (e) for m-Cu2GeS3 (01-088-0827), and (f) for WZ-ZnS (h- or α-ZnS; 00-036-1450)

Figure 2.4 (a) Experimental XRD trace for t-CZTS at 300 K [19]. The corresponding quasibinary ASTM card images are shown in (b) for t-Cu2SnS3 (01-089-4714) and in (c) for ZB-ZnS (00-005-0566)

Figure 2.5 (a) Experimental XRD trace for t-CZTSe at 300 K [20]. The corresponding quasibinary ASTM card images are shown in (b) for c-Cu2SnSe3 (01-089-2879) and in (c) for ZnSe (00-037-1463)

Figure 2.6 Summary of easily or normally grown crystal structure for Cu2–II–IV–VI4 quaternary semiconductors

Figure 2.7 (a) Lattice constants a and c versus Cu composition x in Cu-poor kesterite-type Cu2(1–x)ZnSnSe4 quaternary at 300 K. (b) Lattice constant ratio c/a and effective cubic lattice constant aeff, calculated from (a, c) in (a), versus x in Cu-poor kesterite-type Cu2(1−x)ZnSnSe4 quaternary. The solid lines in (a) and (b) show the linear least-squares fit results

Figure 2.8 Lattice constants a, b, and c versus molecular weight M for some t-Cu2–II–IV–VI4 (solid circles) and o-Cu2–II–IV–VI4 semiconductors (open circles) at 300 K. The solid lines show the least-squares fit results given by Equations 2.5 and 2.6

Figure 2.9 Lattice constant ratio c/a versus molecular weight M for t-Cu2–II–IV–VI4 semiconductors at 300 K, together with those for some typical WZ semiconductors. Because Cu2–II–IV–VI4 has two formula units in the unit cell, the M values for the WZ semiconductors are multiplied by four. The c/a = 2.0 and 1.633 lines correspond to those for the ideal tetragonal and WZ lattices, respectively

Figure 2.10 (a) Effective cubic lattice constant aeff and (b) X-ray crystal density g versus molecular weight M for some t-Cu2–II–IV–VI4 (solid circles) and o-Cu2–II–IV–VI4 semiconductors (open circles) at 300 K

Figure 2.11 (a) Lattice constants a and c versus Zn composition x for Cu2ZnxCd1−xSnS4 alloy at 300 K. (b) Lattice constant ratio c/a and effective cubic lattice constant aeff, calculated from (a, c) in (a), versus x for Cu2ZnxCd1−xSnS4 alloy. The solid lines in (a) and (b) show the linear least-squares fit results (see Eqs. 2.9 and 2.10)

Figure 2.12 (a) Lattice constants a and c versus Ge composition x for Cu2ZnGexSn1−xSe4 alloy at 300 K. (b) Lattice constant ratio c/a and effective cubic lattice constant aeff, calculated from (a, c) in (a), versus x for Cu2ZnGexSn1−xSe4 alloy. The solid lines in (a) and (b) show the linear least-squares fit results given by Equations 2.11–2.13

Figure 2.13 Lattice constants a, b, and c for Cu2CdGexSn1−xS4 alloy plotted against Ge composition x at 300 K. The stannite and WZ-stannite structures can be grown in the x regions of ~0–0.12 and ~0.9–1.0, respectively, whereas the two-phase alloys are synthesized at x ~ 0.12 − 0.9. The effective cubic lattice constant aeff versus x data for Cu2CdGexSn1−xS4 alloy are also plotted in (d). The solid lines in (a–c) represent the Vegard’s linear relations. The solid line in (d) is also calculated using Equation 2.14

Figure 2.14 Lattice constants a, b, and c versus x for t-Cu2ZnGe(SxSe1−x)4 (solid circles) and o-Cu2ZnGe(SxSe1−x)4 alloys (open circles) at 300 K. The experimental data are taken from Doverspike et al. [34]. The solid lines show the least-squares fit results given by Equations 2.15 and 2.16. The open triangles represent the values obtained using the extrapolation scheme (x → 0)

Figure 2.15 Effective cubic lattice constant aeff versus x for t-Cu2ZnGe(SxSe1−x)4 (solid circles) and o-Cu2ZnGe(SxSe1−x)4 alloys (open circles) at 300 K. The solid line shows the Vegard’s law relation given by Equation 2.17

Figure 2.16 (a) Lattice constants a and c versus S composition x for t-Cu2ZnGe(SxSe1−x)4 alloy at 300 K. The experimental data are taken from Heinrich et al. [35]. (b) Lattice constant ratio c/a and effective cubic lattice constant aeff, calculated from (a, c) in (a), versus x for t-Cu2ZnGe(SxSe1−x)4 alloy. The solid lines in (a) and (b) show the linear least-squares fit results (see Eqs. 2.18 and 2.19)

Figure 2.17 Plots of sin (2θ) versus x data for (a) (112), (b) (220)/(204), and (c) (312)/(116) diffraction peaks observed in Cu2ZnSn(SxSe1−x)4 alloy at 300 K. The experimental data are taken from Gao et al. [36]. The solid lines show the linear least-squares fit results

Figure 2.18 Lattice constants a and c versus x for kesterite-type Cu2ZnSn(SxSe1−x)4 pentanary alloy at 300 K. The experimental data are taken from Todorov et al. [25], Gao et al. [36], Ionkin et al. [37], Levcenco et al. [38], and Nagaoka et al. [39]. The solid lines show the least-squares fit results given by Equation 2.22

Figure 2.19 (a) Lattice constant ratio c/a and (b) effective cubic lattice constant aeff versus x for kesterite-type Cu2ZnSn(SxSe1−x)4 pentanary alloy at 300 K. The lattice parameters used for these calculations are taken from Figure 2.18 (Todorov et al. [25], Gao et al. [36], Ionkin et al. [37], Levcenco et al. [38], and Nagaoka et al. [39]). The solid line in (b) shows the result calculated using Equation 2.23

Figure 2.20 Lattice constants a and c for t-Cu2CdGeSe4 and a, b, and c for WZ-stannite-type Cu2CdGe(SxSe1−x)4 plotted against alloy composition x at 300 K. The WZ-stannite-type solid solutions were formed at x ≥ 0.11, whereas the stannite-type solid solutions with the participation of WZ-stannite crystals were formed in the lower x region of 0.02–0.1. The extent of the stannite-type solid solutions was observed only in the limited region of x ≤ 0.02

Figure 2.21 Temperature dependences of a and c for t-Cu2ZnGeS4 and Cu2ZnGeSe4 quaternaries at T = 300–670 K. The experimental data are taken from Zeier et al. [41]. The solid lines in (a) and (b) are a guide to the eye and show the trend toward slight increase in a and c with increasing T

Figure 2.22 (a) Schematic representation for a tetragonal epitaxial layer–cubic substrate system, where a0 is the lattice parameter for the cubic substrate and a and c are the lattice parameters for the tetragonal epitaxial layer. (b) Projections of the cubic or tetragonal lattice atoms (e.g., CZTS) and sapphire atoms (Al and O) on the (111) and (0001) crystallographic planes, respectively. The lattice mismatch is assumed to be zero

Figure 2.23 (a) WZ-type and (b) ZB-type crystal structures of Cu2–II–IV–VI4 semiconductor. The Cu, group II, and group IV cations can be randomly situated in their lattice sites

Figure 2.24 (a) Experimental and (b) simulated XRD patterns for WZ-type CZTS nanocrystals at 300 K. The ASTM card images for m-Cu2SnS3 (01-070-6338) and WZ-ZnS (00-036-1450) are also shown in (c) and (d), respectively

Figure 2.25 (a) XRD pattern for CZTSe nanocrystals at 300 K. The simulated XRD traces for WZ-kesterite-, WZ-, kesterite-, and ZB-CZTSe are also shown in (b–e), respectively.

Figure 2.26 (a) Lattice parameters a and c plotted against x for WZ-Cu2ZnxCd1−xSnS4 nanorods at 300 K. The experimental data were measured using XRD (open circles) and EDX techniques (solid circles). The c/a versus x data obtained from (a) are plotted in (b). The solid lines in (a) and (b) represent least-squares fit results represented by Equations 2.30 and 2.31, respectively. The dashed line in (b) corresponds to the ideal WZ value (c/a = 1.633)

Figure 2.27 (a–c) Lattice constants, a, b, and c, and (d) effective cubic lattice constant aeff versus x for WZ-derived Cu2ZnSn(SxSe1−x)4 pentanary alloy (solid circles) at 300 K. The solid lines show the least-squares fit results given by Equations 2.32 and 2.33. The kesterite aeff values taken from Cao et al. [57] and Li et al. [91] are also shown in (d) by the open and inverse open triangles, respectively

Figure 2.28 Lattice parameters a and c/2 versus temperature T for kesterite-type CZTS quaternary. Note that these data exhibit structural phase transition at high temperatures (~866–883°C) from tetragonal (kesterite) to cubic (ZB) structure

Chapter 03

Figure 3.1 Phase diagram of Cu2GeS3–ZnS quasibinary system. The open circles (liquidus), solid circles (solidus), and solid triangles (subliquidus and subsolidus) represent the experimental DTA data. The melting points of Cu2GeS3 (1220 K) [6] and ZnS (2196 K) [7] are shown by the gray circles. The hexagonal (WZ) to cubic (ZB) phase-transition temperature at 1293 K for ZnS is also shown by the inverse open triangle. a, L (liquid) + α-ZnS (WZ); b, α-ZnS; c, β-ZnS (ZB); d, L + α-Cu2ZnGeS4 (WZ-stannite); e, α-ZnS + α-Cu2ZnGeS4; f, Cu2GeS3 + α-Cu2ZnGeS4; g, β-ZnS + α-Cu2ZnGeS4; h, Cu2GeS3 + β-Cu2ZnGeS4 (stannite); i, β-ZnS + β-Cu2ZnGeS4.

Figure 3.2 Phase diagram of Cu2GeSe3–ZnSe quasibinary system. The open circles (liquidus), solid circles (solidus), and solid triangles (subliquidus and subsolidus) represent the experimental DTA data. The melting points of Cu2GeSe3 (1033 K) [6] and ZnSe (1793 K) [7] are shown by the gray circles. a, L (liquid) + ZnSe; b, L + α-Cu2ZnGeSe4 (HT modification); c, L + β-Cu2ZnGeSe4 (stannite); d, L + Cu2GeSe3; e, α-Cu2ZnGeSe4; f, α-Cu2ZnGeSe4 + β-Cu2ZnGeSe4; g, β-Cu2ZnGeSe4; h, ZnSe + α-Cu2ZnGeSe4; i, Cu2GeSe3 + β-Cu2ZnGeSe4; j, ZnSe + β-Cu2ZnGeSe4.

Figure 3.3 Phase diagram of Cu2GeTe3–ZnTe quasibinary system. The open and solid circles correspond to the cooling and heating data, respectively. The melting points of Cu2GeTe3 (777 K) [6] and ZnTe (1568 K) [7] are shown by the gray circles. a, L (liquid) + ZnTe; b, ZnTe + α-Cu2ZnGeTe4 (HT modification); c, L + α-Cu2ZnGeTe4; d, Cu2GeTe3 + β-Cu2ZnGeTe4 (stannite); e, ZnTe + β-Cu2ZnGeTe4.

Figure 3.4 Phase diagram of Cu2SnS3–ZnS quasibinary system. This phase diagram corresponds to a triangulating section in the Cu2S–SnS2–ZnS system. The melting points of CZTS (1253 K), Cu2SnS3 (1128 K) [6], and ZnS (2196 K) [7] are shown by the solid (CZTS) and gray circles (Cu2SnS3 and ZnS). The hexagonal (WZ) to cubic (ZB) phase-transition temperature at 1293 K for ZnS is also shown by the inverse open triangle. a, L (liquid) + α-ZnS (WZ); b, α-ZnS; c, α-ZnS + β-ZnS (ZB); d, β-ZnS; e, L + β-ZnS; f, L + CZTS; g, β-ZnS + CZTS; h, Cu2SnS3 + CZTS.

Figure 3.5 Phase diagram of Cu2SnSe3–SnSe2–ZnSe system in the polythermal SnSe2–Cu2ZnSnSe4 section with A = 50 mol% Cu2Se + 50 mol% ZnSe. The melting points of SnSe2 (948 K) [12] and CZTSe (1074 K) are shown by the gray circles. a, L (liquid) + β-ZnSe solid solution; b, L + γ-SnSe2 solid solution; c, L + β-ZnSe solid solution + γ-SnSe2 solid solution; d, L + β-ZnSe solid solution + HT-CZTSe; e, γ-SnSe2 solid solution + HT-CZTSe; f, HT-CZTSe; g, LT-CZTSe (tetragonal) + HT-CZTSe; h, LT-CZTSe; i, γ-SnSe2 solid solution; j, γ-SnSe2 solid solution + LT-CZTSe.

Figure 3.6 Phase diagram of Cu2SnS3–CdS system. The melting points of Cu2SnS3 (1128 K) [6] and CdS (1748 K) [7] are shown by the gray circles. a, liquid (L) + β solid solution; b, β solid solution; c, L + α solid solution; d, L + Cu2CdSnS4; e, α solid solution; f, α solid solution + Cu2CdSnS4; g, β solid solution + Cu2CdSnS4.

Figure 3.7 Phase diagram of Cu2SnSe3–CdSe system. The melting points of Cu2SnSe3 (970 K) [6] and CdSe (1531 K) [7] are shown by the gray circles. a, liquid (L) + β solid solution; b, L + Cu2CdSnSe4; c, L + α solid solution; d, α solid solution; e, α solid solution + Cu2CdSnSe4; f, β solid solution + Cu2CdSnSe4; g, β solid solution.

Figure 3.8 Phase diagram of Cu2SnTe3–CdTe system. The melting points of Cu2SnTe3 (680 K) [6] and CdTe (1365 K) [7] are shown by the gray circles. a, liquid (L) + β solid solution; b, Cu2SnTe3 incongruent character region; c, L + Cu2CdSnTe4; d, α solid solution; e, α solid solution + Cu2CdSnTe4; f, β solid solution + Cu2CdSnT4; g, β solid solution.

Figure 3.9 Phase diagram of Cu2GeS3–HgS system. The melting points of Cu2GeS3 (1220 K) [6] and HgS (1093 K) [12] are shown by the gray circles.

Figure 3.10 Phase diagram of Cu2GeSe3–HgSe system. The melting points of Cu2GeSe3 (1033 K) [6] and HgSe (1072 K) [7] are shown by the gray circles. a, L (liquid) + Cu2GeSe3; b, L + Cu2HgGeSe4; c, L + HgSe; d, Cu2HgGeSe4; e, HgSe + Cu2HgGeSe4; f, Cu2HgGeSe4 + Cu2Hg3GeSe6; g, Cu2Hg3GeSe6; h, HgSe + Cu2Hg3GeSe6; i, HgSe; j, Cu2GeSe3 + Cu2HgGeSe4; k, HgSe + Cu2HgGeSe4.

Figure 3.11 Phase diagram of Cu2GeTe3–HgTe system. The melting points of Cu2GeTe3 (777 K) [6] and HgTe (943 K) [7] are shown by the gray circles. a, L (liquid) + Cu2−xTe/GeTe/Cu2GeTe3; b, L + Cu2GeTe3; c, Cu2GeTe3; d, L + Cu2HgGeTe4; e, Cu2HgGeTe4; f, L + HgTe; g, HgTe; h, Cu2GeTe3 + Cu2HgGeTe4; i, HgTe + Cu2HgGeTe4.

Figure 3.12 Maps of phase composition for (a) Cu2GeS3–Cu2SnS3–CdS and (b) Cu2GeS3–Cu2GeSe3–3CdS–3CdSe systems at 670 K.

Figure 3.13 Phase diagram of Cu2CdGeS4–Cu2CdSnS4 pentanary alloy system. a, L (liquid) + Cu2Cd3GeS6; b, L + CdS; c, L + CdS + Cu2Cd3GeS6; d, L + CdS + Cu2CdSnS4; e, L + Cu2CdGeS4 + Cu2Cd3GeS6; f, L + Cu2CdSnS4 + Cu2Cd3GeS6; g, L + Cu2CdSnS4; h, Cu2CdGeS4; i, Cu2CdSnS4; j, Cu2CdGeS4 + Cu2CdSnS4.

Figure 3.14 Phase diagram of Cu2CdGeS4–Cu2CdGeSe4 alloy system. The solid line from T ~ 1100 to 1300 K shows the solidus curve obtained by Marushko et al. [26] that can be approximated by Equation 3.1. The heavy dashed line indicates a value of T = 670 K used in preparing this alloy system.

Figure 3.15 Melting point Tm versus (a) molecular weight M and (b) effective cubic lattice constant aeff for some Cu2–II–IV–VI4 semiconductors. The solid lines in (a) and (b) show the least-squares fit results represented by Equations 3.2 and 3.3, respectively

Figure 3.16 Theoretical specific heat Cp versus temperature T for CZTS and CZTSe. The theoretical Cp data are taken from He and Shen [30] for CZTS and He and Shen [31] for CZTSe. The experimental synthetic diamond, GaAs, and CdTe data [7] are also plotted. The light dashed line shows the validity of Debye’s T³ law. Note that the specific heat values for CZTSe were plotted after dividing the original data by eight. The Dulong–Petit’s value of 3R is also indicated by the horizontal arrow

Figure 3.17 Specific heat Cp at constant pressure versus temperature T for single-crystalline CZTS. The experimental data are taken from Nagaoka et al. [32]. Note that the experimental specific heat values were plotted after dividing the original data of Nagaoka et al. by eight. The dashed line at 0 ≤ T < 25 K represents the validity of Debye’s T³ law. The Cp values at T > 25 K calculated using Equation 3.8 are also shown by the solid line. The Dulong–Petit’s value of 3R is indicated by the horizontal arrow

Figure 3.18 Debye temperature θD at 300 K versus effective cubic lattice constant aeff for some group IV, III–V, and II–VI semiconductors. The solid line represents the least-squares fit result given by Equation 3.11. The experimental θD value of ~302 K for CZTS and estimated θD value of ~320 K using Equation 3.11 for CZTSe are shown by the black and gray circles, respectively

Figure 3.19 Linear thermal expansion coefficients αth for a and c of kesterite-type CZTS (t-CZTS). The αth values were calculated from the lattice constants (a and c) versus T data at T ~ 300–1000°C measured by Schorr and Gonzalez-Aviles [36]. Note that CZTS exhibits the structural phase transition from the tetragonal structure to the cubic ZB-type structure (c-CZTS) at T ~ 866–883°C (see details in Figure 2.28). A negative thermal expansion behavior can be observed in the limited cubic structure region

Figure 3.20 Thermal conductivity κ at 300 K versus molecular weight M for some Cu2–II–IV–VI4 quaternary and III–V binary semiconductors. The solid lines show the least-squares fit results represented by Equations 3.16 and 3.17

Figure 3.21 Thermal conductivity κ and electrical resistivity ρ versus temperature T for t-Cu2ZnGeS4. The experimental data are taken from Heinrich et al. [51]. An insulator-to-metal (I-to-M) phase transition is observed to occur at T ~ 550 K. The solid lines show the results calculated using Equation 3.19 (see details in Table 3.4)

Figure 3.22 Thermal conductivity κ and electrical resistivity ρ versus temperature T for Cu2ZnGeSe4. The experimental data are taken from Zeier et al. [40] (κ: solid circles) and Heinrich et al. [51] (ρ: open circles). An insulator-to-metal (I-to-M) phase transition is observed to occur at T ~ 450 K. The solid lines show the results calculated using Equation 3.19 (see details in Table 3.4)

Figure 3.23 Thermal conductivity κ versus temperature T for CZTS (solid circles) and CZTSe (open circles). The experimental data are taken from Liu et al. [42]. The solid lines show the results calculated using Equation 3.19 with n = −1.7 for CZTS and −0.9 for CZTSe (see also Table 3.4)

Figure 3.24 Thermal conductivity κ and electrical resistivity ρ versus temperature T for CZTS. The experimental data are taken from Yang et al. [43]. The solid line shows the result calculated using Equation 3.19 (see details in Table 3.4). The dashed line also gives the relation of ρ(T) ∝ Tn with n = −3.2

Figure 3.25 Thermal conductivity κ and electrical resistivity ρ versus temperature T for CZTSe. The experimental data are taken from Dong et al. [44]. The solid line shows the result calculated using Equation 3.19 (see details in Table 3.4)

Figure 3.26 Thermal conductivity κ versus temperature T for Cu2CdSnSe4 quaternary. The experimental data are taken from Liu et al. [46] (open circles) and Fan et al. [47] (solid circles). The light and heavy solid lines show the results calculated using Equation 3.19 with n = −1.3 and −1.0, respectively (see also Table 3.4)

Figure 3.27 Thermal conductivity κ and electrical conductivity σ versus composition x for Cu-rich Cu2+xCd1−xSnSe4 quaternary. The experimental data are taken from Liu et al. [46]. The heavy solid line shows the result calculated using Equation 3.20. The light solid line is drawn through the experimental points as a guide to the eye

Figure 3.28 Thermal conductivity κ and electrical resistivity ρ versus temperature T for Cu2CdSnTe4. The experimental data are taken from Dong et al. [49]. The solid lines show the results calculated using Equation 3.19 (see details in Table 3.4). The dashed line gives the functional relation of κ(T) ∝ T

Figure 3.29 Thermal conductivity κ versus temperature T for Cu2CdSnTe4, together with those for GaAs and CdTe. The experimental data are taken for Cu2CdSnTe4 from Dong et al. [49], for GaAs from Adachi [52], and for CdTe from Adachi [53]. The solid lines show the results calculated using Equation 3.19 with A = 2.40 × 10³ and n = −1.1 for Cu2CdSnTe4, A = 7.50 × 10⁴ and n = −1.3 for GaAs, and A = 2.20 × 10⁴ and n = −1.4 for CdTe

Figure 3.30 Plots of Lorenz number L versus T for Cu2CdSnTe4. The L values are obtained from the κ and ρ data in Figure 3.28 using Equation 3.18. The theoretical Lorenz number of 2.45 × 10−8 WΩ/K² is also shown by the dashed line

Figure 3.31 (a) Thermal resistivity W versus x for CuxAu1−x alloy at 300 K. The solid circles show the experimental data obtained from ordered CuAu and Cu3Au alloys, whereas the open circles represent the experimental data for disordered alloys. Note that this metallic alloy system shows an ordered phase at x = 0.5 (CuAu = L10 (tetragonal)) and x = 0.75 (Cu3Au = L12 (cubic)). The solid and dashed lines represent the results calculated using Equation 3.30 with CCu–Au = 0.0815 m K/W and with and without properly taking into account the alloy ordering effect at x = 0.5 and 0.75 (CuAu and Cu3Au), respectively. (b) Thermal resistivity W versus x for SixGe1−x alloy at 300 K. The solid line shows the result calculated using Equation 3.30 with WSi = 1.67 × 10−2 m K/W, WGe = 6.4 × 10−3 m K/W, and CSi–Ge = 0.5 m K/W.

Figure 3.32 Thermal resistivity W versus x for t-Cu2ZnGe(SxSe1−x)4 pentanary alloy at 300 K. The experimental data are taken from Heinrich et al. [51]. The solid line represents the result calculated using Equation 3.31 with CS–Se = 0.65 m K/W

Figure 3.33 Bowing parameter CA–B versus Γ determined for some III–V ternary alloys at 300 K. The solid line shows the least-squares fit result given by Equation 3.34. The CS–Se value of ~0.65 m K/W obtained in Figure 3.32 is also shown by the solid circle

Figure 3.34 Thermal resistivity W versus x for Cu2ZnSn(SxSe1−x)4 at 300 K. The solid line is calculated using Equation 3.31 with W(CZTS) = 0.212 m K/W, W(CZTSe) = 0.235 m K/W, and CCZTS–CZTSe = CS–Se = 0.30 m K/W, whereas the dashed line is obtained by introducing CS–Se = 0.65 m K/W into Equation 3.31. The CZTS (x = 1.0) and CZTSe (x = 0) values are shown by the solid circles (W = κ−1; Table 3.2)

Chapter 04

Figure 4.1 Elastic stiffness constant Cij versus effective cubic lattice constant aeff for some group IV, III–V, and II–VI semiconductors with cubic and hexagonal (WZ) structures. The solid lines represent the least-squares fit results given by Equation 4.5. The aeff values for CZTS and CZTS are indicated by the vertical dashed lines

Figure 4.2 Elastic stiffness constant Cij versus effective cubic lattice constant aeff for 6H-SiC and some WZ-type III–V and II–VI semiconductors. The solid lines show the least-squares fit results given by Equation 4.5. The fit-determined Aij and Bij values are listed in Table 4.4. The aeff values for CZTS and CZTS are indicated by the vertical dashed lines

Figure 4.3 Spherically averaged longitudinal and transverse sound velocities vl and vt in t-Cu2ZnGe(SxSe1−x)4 pentanary alloy as a function of alloy composition x at 300 K.

Figure 4.4 Spherically averaged longitudinal and transverse sound velocities (vl and vt) and average sound velocity ( ) in Cu2ZnSn(SxSe1−x)4 as a function of alloy composition x, calculated using Equations 4.23 and 4.24, respectively. The endpoint (quaternary) and pentanary values are also summarized in Table 4.5

Figure 4.5 Microhardness H versus molecular weight M for some Cu2–II–IV–VI4 quaternary semiconductors. The solid line shows the least-squares fit result given by Equation 4.25

Figure 4.6 Microhardness H versus effective cubic lattice constant aeff for some Cu2–II–IV–VI4 quaternary semiconductors. The solid line shows the least-squares fit result given by Equation 4.26

Figure 4.7 The first Brillouin zone for the tetragonal lattice

Figure 4.8 (a) Phonon dispersion and (b) phonon DOS (P-DOS) curves along the principal symmetry directions of CZTS. The straight solid and dashed lines give the dispersion relations for the LA and TA phonons, respectively. The room-temperature Raman spectrum IR for CZTS measured by Altosaar et al. [21] is also shown in (c)

Figure 4.9 (a) Phonon dispersion and (b) phonon DOS (P-DOS) curves along the principal symmetry directions of CZTSe. The straight solid and dashed lines give the dispersion relations for the LA and TA phonons, respectively. The room-temperature Raman spectrum IR for CZTSe measured by Altosaar et al. [21] is also shown in (c)

Figure 4.10 Raman spectrum for CZTS measured at 300 K.

Figure 4.11 Room-temperature Raman spectra for coevaporation-grown CZTSe thin films with Cu/(Zn + Sn) = 0.71–1.26. For comparison, the Raman spectrum for Cu2Se film taken from Minceva-Sukarova et al. [28] is shown.

Figure 4.12 Room-temperature Raman spectra for o-Cu2ZnSiS4 and t-CZTS measured by Levcenco et al. [44] and Altosaar et al. [21], respectively

Figure 4.13 Values of frequency ωq (A or A1 mode) versus inverse molecular weight 1/M for some Cu2–II–IV–S4 (II = Zn, Cd, and Hg; IV = Si and Sn) and Cu2–II–IV–Se4 (II = Zn and Cd; IV = Si, Ge, and Sn) quaternaries at 300 K. The solid lines represent the least-squares fit results given by Equation 4.33

Figure 4.14 Relations between (a) frequency ωq and 1/MII (MII = mass of group II atom) and (b) ωq and 1/MIV (MIV = mass of group IV atom) for some Cu2–II–IV–S4 and Cu2–II–IV–Se4 quaternaries at 300 K. The solid lines represent the linear least-squares fit results with respect to 1/MII and 1/MIV

Figure 4.15 Four different types of long-wavelength phonon mode behavior in ternary alloy: (a) one mode, (b) two mode, (c) one–two mode (1), (d) one–two mode (2). Note that this scheme can also be used in Cu2–(IIxII1–x)–IV–VI4, Cu2–II–(IVxIV1−x)–VI4, or Cu2–II–IV–(VIxVI1−x)4 pentanary alloy system

Figure 4.16 Plots of the strongest Raman peak frequency A (A1) at ~330 cm−1 observed in Cu2ZnxCd1−xSnS4 at 300 K versus alloy composition x. The endpoint quaternary data taken from Tables 4.10 and 4.14 are also plotted by the open circles. The solid line shows the linear least-squares fit result given by Equation 4.36.

Figure 4.17 Variation with composition x of long-wavelength (A- or A1-mode) Raman frequencies in Cu2ZnSn(SxSe1−x)4 pentanary alloy measured at 300 K. The experimental data are taken from Grossberg et al. [25] (open circles), Mitzi et al. [69] (open triangles), and He et al. [70] (solid circles). The solid lines represent the linear least-squares fit results given by Equation 4.37

Figure 4.18 (a) Plots of long-wavelength (A- or A1-mode) Raman frequencies in Cu2ZnSn(SxSe1−x)4 pentanary versus alloy composition x at 300 K. The solid lines correspond to those represented by Equation 4.37. (b) Raman spectrum for an ingot of the Bridgman-grown Cu2ZnSn(SxSe1−x)4 pentanary alloy measured at 300 K by Das and Mandel [72]. From (a) and (b), the alloy composition x for Cu2ZnSn(SxSe1−x)4 pentanary alloy can be determined to be ~0.5, in agreement with EDX result of x = 0.505 [72]

Figure 4.19 (a) Perturbation picture for the one-phonon Raman scattering. Here, ωi and ωs represent the angular frequencies of the incident and scattered lights, respectively. HeR and HeL also represent the electron–radiation and electron–lattice (phonon) perturbation Hamiltonian, respectively. (b) Schematic energy-band diagram for the first-order Raman scattering process occurring in direct band-gap semiconductor

Figure 4.20 (a) Room-temperature Raman spectra for CZTS measured by excitation at λex = 325, 514.5, and 785 nm [80]. Note that the maximum scattering intensity of each spectrum is normalized to unity. (b) Schematic energy-band structure at the Γ, X, and P points of CZTS. E0 represents the lowest direct band-gap energy

Figure 4.21 Raman scattering and far-IR transmittance spectra for (a) t-CZTS, (b) t-Cu2CdSnS4, and (c) t-Cu2HgSnS4 measured at 300 K.

Figure 4.22 Lorentzian line shape fit for the Raman spectra of CZTS measured at three different temperatures T = 98, 233, and 338 K (solid lines). The experimental Raman spectra are taken from Sarswat et al. [24] (open circles). The Lorentzian line shape function is defined by Equation 4.64 (Eq. 4.78). The fit-determined parameters are given in the text

Figure 4.23 Variation with temperature T of long-wavelength (A-mode) Raman frequency in CZTS. The experimental data are taken from Sarswat et al. [24] (solid circles) and Singh et al. [91] (open circles). The dashed lines show the results calculated using Equation 4.53. The solid lines also represent the linear least-squares fit results given by Equation 4.66

Figure 4.24 Lorentzian linewidth (Γ) versus lattice temperature T for the first-order Raman scattering in CZTS. The experimental data are taken from Sarswat et al. [24]. The solid line represents the fitted result using Equation 4.67 (Eq. 4.68), whereas the dashed line shows the result calculated using Equation 4.69. The inset shows the same results, but those measured by Singh et al. [91]

Figure 4.25 Strain-induced phonon frequency shift Δωq versus elastic strain e for the A-symmetry vibration modes in CZTS at 300 K. The elastic strain (e) was estimated from the XRD measurements. The solid lines show the results calculated using a quadratic expression defined by Equation 4.70.

Figure 4.26 (a) Schematic representation for the phonon dispersion curves and phonon DOS (P-DOS) along the principal symmetry directions of bulk c-Si. (b) Modeled Raman scattering spectra for bulk Si and nanocrystalline Si calculated using Equations 4.78 and 4.79, respectively

Figure 4.27 (a) XRD traces and (b) Raman scattering spectra for nanocrystalline CZTS samples with diameters of 2 and 7 nm measured at 300 K. The Raman FWHM values are indicated in (b).

Figure 4.28 (a) XRD traces and (b) Raman scattering spectra for nanocrystalline CZTS samples with diameters of 3 and 10.5 nm measured at 300 K. The Raman FWHM values are plotted against nanocrystalline dot size in (c). The solid line shows the result calculated using Equation 4.72.

Figure 4.29 Room-temperature Raman spectra for (a) bulk Si and (b) Si nanowire array formed by catalytic etching. The dashed and solid lines show the results calculated using Equations 4.78 and 4.79, respectively

Figure 4.30 Variation with composition x of long-wavelength (A- or A1-mode) Raman frequencies in t-Cu2ZnSn(SxSe1−x)4 pentanary alloy measured at 300 K. The experimental data are taken from Ou et al. [120] (open circles) and Li et al. [123] (solid circles). The solid lines represent the linear least-squares fit results given by Equation 4.80. The dashed lines also show the results calculated using Equation 4.37 (large-crystalline Cu2ZnSn(SxSe1−x)4 data)

Figure 4.31 Raman spectra for (a) WZ-CZTS and (b) WZ-CZTSe nanocrystals, together with those for monograin-powdered t-CZTS and t-CZTSe samples measured at 300 K. The experimental Raman spectra for the WZ nanocrystals are taken from Singh et al. [122]. Those for the tetragonal CZTS and CZTSe samples are taken from Altosaar et al. [21] (see also Grossberg et al. [25]). See the dashed line in (b) in the text

Figure 4.32 (a) Long-wavelength optical phonon frequencies ωLO and ωTO versus x for w-CdSxSe1−x ternary alloy measured at 300 K. The experimental data are taken from Chang and Mitra [50]. The solid lines show the results calculated using Equations 4.81 and 4.82. (b) Long-wavelength optical phonon frequencies ωq versus x for WZ-Cu2ZnSn(SxSe1−x)4 pentanary nanocrystals measured at 300 K. The experimental data are taken from Fan et al. [121] (open triangles) and Singh et al. [122] (open and solid circles). The heavy solid and dashed lines show the results calculated using Equations 4.83 and 4.84, respectively. The light solid lines represent the LO phonon frequencies of bulk ZnS (ωLO ~ 350 cm−1) and ZnSe (ωLO ~ 252 cm−1)

Chapter 05

Figure 5.1 Electronic energy-band structures of CZTS and CZTSe in the kesterite (upper parts) and stannite crystal structures (lower parts) calculated using a state-of-the-art self-consistent GW approach. Numerical values correspond to the lowest direct band-gap energy occurring at the Γ point in each material.

Figure 5.2 (a) Relativistic energy-band structure of kesterite-type CZTS calculated in the DFT with the Heyd–Scuseria–Ernzerhof hybrid functional. The corresponding total DOS curve is shown on the right-hand side, (b). The locations of several interband CPs are indicated by the vertical arrows

Figure 5.3 (a) One-electron (Kohn–Sham) band structure and (b) total DOS curves for stannite-type CZTSe calculated using a DFT code, along with the local density approximation.

Figure 5.4 Electronic energy-band structures of (a) kesterite-type CZTSe, (b) stannite-type Cu2CdSnSe4, and (c) stannite-type Cu2HgSnSe4 quaternary semiconductors calculated using the screened exchange local density approximation functional.

Figure 5.5 (a) Electronic energy-band structure and (b) total DOS for WZ-stannite-type Cu2ZnSiS4 quaternary semiconductor calculated in the DFT method.

Figure 5.6 CB and VB structures of (a) ZB-type (Δso = Δcr = 0), (b) ZB-type (Δso ≠ 0, Δcr = 0), (c) WZ-type (Δso = 0, Δcr ≠ 0), and (d) WZ-type (Δso ≠ 0, Δcr ≠ 0) crystals at k = 0 (Γ), together with those of (e) Cu2–II–IV–VI4 crystals. The group-symmetry representations are taken from Koster et al. [22]. K, kesterite; S, stannite; WZ-S, WZ-stannite

Figure 5.7 Optical transition selection rules for (a) stannite-type and (b) kesterite-type crystals at k = 0 (Γ) with (Δso > 0, Δcr > 0) and (Δso < 0, Δcr > 0), together with those for zincblende crystal

Figure 5.8 Optical transition selection rules for WZ-stannite crystal at k = 0 (Γ), together with those for zincblende and wurtzite crystals

Figure 5.9 CB and VB structures at the Γ point of (a) WZ-stannite crystal, (b) WZ-modeled Cu2ZnSiS4 (Δso ~ 0, Δcr ≠ 0), and (c) Cu2ZnSiSe4 in the quasicubic approximation model (Δso ≠ 0, Δcr ≠ 0). The vertical arrows represent the optical transition selection rules at the Γ point

Figure 5.10 Lowest direct band-gap energy E0 versus effective cubic lattice constant aeff for some group IV, III–V, and II–VI semiconductors at 300 K. The solid line represents the least-squares fit result given by Equation 5.30. This equation yields Eg ~ 3.1 and ~2.4 eV for CZTS and CZTSe (vertical arrows), respectively

Figure 5.11 Schematic energy bands for the s and p states of tetrahedrally bonded semiconductors. The lowest direct band-gap energy (E0) versus lattice constant (a) data are also shown by the vertical arrows for Ge, Si, 3C-SiC, and diamond and by the solid circle for α-Sn (E0 ~ 0 eV)

Figure 5.12 Lowest direct band-gap energy E0 versus molecular weight M for some group IV, III–V, and II–VI semiconductors at 300 K. The solid line represents the inverse proportional relation given by Equation 5.31

Figure 5.13 Lowest direct or indirect band-gap energy Eg versus molecular weight M for some Cu2–II–IV–VI4 quaternaries at 300 K. The solid lines show the results calculated using Equations 5.32 and 5.33

Figure 5.14 Variation of Eg in some Cu2–II–IV–VI4 quaternaries with respect to a kind of the group IV cation atom (a and b) or group VI anion atom (c) at 300 K

Figure 5.15 Variation of Eg versus (a) Cu/(Zn + Sn) or (b) Zn/Sn atomic ratio for CZTS at 300 K. The experimental data are gathered from various sources

Figure 5.16 Variation of Eg versus (a) Cu/(Zn + Sn) or (b) Zn/Sn atomic ratio for CZTSe at 300 K. The experimental data are gathered from various sources

Figure 5.17 Variation of Eg versus alloy composition x for Cu2ZnxCd1−xSnS4 pentanary alloy measured at 300 K by Xiao et al. [74] (open circles). The solid line represents the linear least-squares fit result given by Equation 5.35. The solid circles show the endpoint Cu2CdSnSe4 (x = 0) and CZTS (x = 1.0) values taken from Table 5.4

Figure 5.18 Variation of Eg versus alloy composition x for Cu2ZnGexSn1−xSe4 pentanary alloy measured at 300 K by Morihama et al. [75] (open circles). The solid line represents the linear least-squares fit result given by Equation 5.36. The solid circles show the endpoint CZTSe (x = 0) and Cu2ZnGeSe4 (x = 1.0) values taken from Table 5.4

Figure 5.19 Variation of Eg versus alloy composition x for t-Cu2ZnGe(SxSe1−x)4 pentanary alloy measured at 300 K by Heinrich et al. [76] (open circles). The solid line represents the linear least-squares fit result given by Equation 5.37. The solid circles show the endpoint Cu2ZnGeSe4 (x = 0) and t-Cu2ZnGeS4 (x = 1.0) values taken from Table 5.4

Figure 5.20 Variation of Eg versus alloy composition x for Cu2ZnSn(SxSe1−x)4 pentanary alloy at 300 K. The experimental data are taken from He et al. [77], Gao et al. [78], and Levcenco et al. [79]. The solid and dashed lines show the results calculated using Equations 5.38–5.40

Figure 5.21 Temperature variation of the lowest direct band-gap energies Eg for Cu2ZnSiS4 quaternary crystallizing in the WZ-stannite structure with polarizations E⊥c and E||c measured by Levcenco et al. [35]. The solid lines show the fitted results using the Varshni formula of Equation 5.6 (see fitting parameters in the text). The heavy solid line also shows the temperature dependence of the A-exciton transition energies in ZnO calculated using Equation 5.6 (see fitting parameters in the text; see also Adachi [9])

Figure 5.22 Temperature variation of the lowest indirect band-gap energies Eg for Cu2ZnSiSe4 quaternary crystallizing in the WZ-stannite structure with polarizations E⊥c and E||c measured by Levcenco et al. [36]. The solid lines show the fitted results using the Varshni formula of Equation 5.6 (see fitting parameters in the text). The heavy solid line also shows the experimental lowest indirect band-gap energy versus T plots for GaP given by Equation 5.6 (see fitting parameters in the text; see also Adachi [9])

Figure 5.23 Temperature variation of the lowest indirect band-gap energies Eg for Cu2ZnGeS4 quaternary crystallizing in the WZ-stannite structure with polarizations E||a and E||b measured by Levcenco et al. [38]. The solid lines show the fitted results using the Varshni formula of Equation 5.6 (see fitting parameters in the text). The heavy solid lines also show the lowest indirect and direct band-gap energies for AlAs and ZnTe, respectively. These curves are calculated using Equation 5.6 with the Varshni parameters given in the text (see also Adachi [9])

Figure 5.24 Temperature variation of Eg for CZTS and CdTe. The experimental data are taken for CZTS from Sarswat and Free [87] and for CdTe from Adachi [88]. The theoretical curves are calculated using Equation 5.6 for CZTS (heavy solid line; Eg(0) = 1.64 eV, α = 1.0 × 10−3 eV/K, and β = 340 K) and CdTe (heavy solid line; Eg(0) = 1.60 eV, α = 5.0 × 10−4 eV/K, and β = 150 K), using Equation 5.9 for CZTS (thin solid line; EB = 1.73 eV, αB = 0.09 eV, and ΘB = 250 K) and, using Equation 5.11 for CZTS (dashed line; Eg(0) = 1.63 eV, αp = 7.7 × 10−4 eV/K, Θp = 260 K, and p = 2.7)

Figure 5.25 Temperature variation of Eg for CZTSe [89], together with those for InP and CuInSe2. The experimental data are taken for InP from Chung et al. [90] and for CuInSe2 from Hong et al. [91]. The theoretical curves are calculated using Equation 5.6 with Eg(0) = 1.422 eV, α = 5.5 × 10−4 eV/K, and β = 334 K for InP and Eg(0) = 1.187 eV, α = 8.57 × 10−4 eV/K, and β = 129 K for CuInSe2. See detailed CZTSe data in Ref. [89]

Figure 5.26 Electron effective mass versus E0 (Eg) for some III–V and II–VI semiconductors crystallizing in the hexagonal structure. The experimental data are taken from Adachi [9]. The solid line represents the least-squares fit result with the relation given by Equation 5.49. The CZTS and CZTSe values estimated from this equation are indicated by the solid circles

Figure 5.27 Electron effective mass versus lattice constant a for some group IV, III–V, and II–VI semiconductors. The solid line represents the least-squares fit result given by Equation 5.50. This equation yields and 0.09m0 for CZTS and CZTSe, respectively (see vertical arrows)

Figure 5.28 (a) Lowest direct band-gap energy E0 at 300 K and (b) Γ-valley electron effective mass for some III–V alloy semiconductors. The relations between E0 and for CZTS and CZTSe are also indicated by the solid lines. These E0 versus relations predict and 0.06m0 for CZTS and CZTSe, respectively

Figure 5.29 Electron effective mass versus alloy composition x for Cu2ZnSn(SxSe1−x)4 pentanary alloy. The dashed and solid lines show the results