Contributions of point defects, chemical disorder, and thermal vibrations to electronic properties of Cd${}_{1-x}$Zn${}_x$Te alloys

We present a first-principles study based on density functional theory of thermodynamic and electronic properties of the most important intrinsic defects in the semiconductor alloy ${\mathrm{Cd}}_{1-x}{\mathrm{Zn}}_x\mathrm{Te}$ with $x<0.13$. The alloy is represented by a set of supercells with disorder on the Cd/Zn sublattice. Defect formation energies as well as electronic and optical transition levels are analyzed as a function of composition. We show that defect formation energies increase with Zn content with the exception of the neutral Te vacancy. This behavior is qualitatively similar to but quantitatively rather different from the effect of volumetric strain on defect properties in pure CdTe. Finally, the relative carrier scattering strengths of point defects, alloy disorder, and phonons are obtained. It is demonstrated that for realistic defect concentrations, carrier mobilities are limited by phonon scattering for temperatures above approximately 150 K.

Figure 1

Free energy of mixing of CZT as a function of Zn concentration and temperature. Symbols denote the enthalpies of mixing calculated using the total energies from the supercell approach. The red solid line is the result of a quadratic polynomial fit to the lowest energy configurations within the gray area ($\le$13$\%$). The remaining lines represent the free energy of mixing up to 400 K, as generated by adding the entropy contribution for a random solution.

Figure 2

Deviation from linear dependence of band gap as a function of Zn concentration. The solid yellow line and the red open circles denote the VCA and supercell results, respectively. The calculated values are shifted with a linear function to coincide with the room-temperature experimental band gap of CdTe and ZnTe at $x=0$ and $x=1$, respectively. Experimental values recorded at room temperature are shown with blue symbols, see Refs. 43, 44, 45, 46, 47. Data recorded below 10 K are shown with green symbols, see Refs. 48 and 49. The latter have been shifted by a constant to coincide with the room-temperature data.

Figure 3

Dependence of the formation energies on the electron chemical potential under Te-rich conditions for the most important intrinsic defects in ${\mathrm{Cd}}_{1-x}{\mathrm{Zn}}_x\mathrm{Te}$ ($x=0.08$). Each line corresponds to a separate random Zn configuration. The electronic (charge state) transition energies are denoted by the symbols.

Figure 4

(a) For each defect and charge state, the formation energy at the VBM (${\mu }_e=0$) for Te-rich conditions is plotted for the set of configurations for five different concentrations ranging from 5.5 to 13.0$\%$ (left to right, indicated by the arrow). (b) For each defect, the electronic transition levels are plotted for the set of configurations for five different concentrations ranging from 5.5 to 13.0$\%$ (left to right, indicated by the arrow).

Figure 5

(a) $V_{\mathrm{Te}}$ formation energy at the VBM (${\mu }_e=0$) as a function of average $V_{\mathrm{Te}}$-nearest-neighbor distance. The Zn concentration is indicated by the different symbols. Red and green symbols correspond to charge states $q=0$ and $q=2$, respectively. Dashed line indicates separation of configurations with small ($d_{<}$) and large ($d_{>}$) $V_{\mathrm{Te}}$-NN distances. (b) $V_{\mathrm{Te}}$ formation energy for $q=0$ at the VBM (${\mu }_e=0$) as a function of displacements of all atoms with respect to the defect-free system. Again, symbols denote Zn concentration and colors indicate small ($d_{<}$) and large ($d_{>}$) $V_{\mathrm{Te}}$-NN distances. The line is drawn to guide the eye.

Figure 6

Formation energy at the VBM (${\mu }_e=0$) as a function of average Cd${}_{i,\mathrm{Te}}$-nearest-neighbor distance. The symbols and colors indicate the Zn concentration and the number of Zn atoms in the next-nearest-neighbor shell.

Figure 7

Formation energies at the VBM (${\mu }_e=0$) under Te-rich conditions for the six most important intrinsic defects in CdTe as a function of compressive strain (lines) and in CZT as a function of volume (symbols).

Figure 8

Kohn-Sham defect levels for the most important intrinsic defects in CZT. For each defect, levels of all the randomized cells are shown for five different concentrations ranging from 5.5 to 13.0$\%$ (left to right, indicated by the arrow). Red, blue, and green line colors denote an empty, half-filled, and fully occupied level, respectively. The solid black lines show the level positions in the pure CdTe case.

Figure 9

Relative scattering strengths [see Eq. (8)] for the most important intrinsic defects in CZT. For each defect, scattering strengths are shown for five different concentrations ranging from 5.5 to 13.0$\%$ (left to right, indicated by the arrow). The solid black lines show the relative scattering strengths for the pure CdTe case. The last column (Alloy) shows the effect of alloy scattering at the same level of approximation. The data are arbitrarily normalized to the value of the strongest intrinsic defect scattering channel. We emphasize that the defect concentrations for a 216-atom cell are at least a factor of 1000 too large in comparison to realistic concentration. Thus, alloy scattering clearly dominates over defect scattering.

Figure 10

Mean-square displacements of the Cd and Te sublattices from the harmonic model, MD simulations, and experiment.[55] The lines are drawn to guide the eye.

Figure 11

Relative thermal scattering strengths (unscaled, see the Appendix) for CdTe as a function of temperature, calculated using both a harmonic approximation (solid line) and molecular dynamics simulations (dashed line).

Figure 12

Transition densities, as defined by the integrands of $L_1$ ($n=m+1$, red filled curve) and $L_2$ (blue solid line) of Eqs. (A1) and (A2), respectively.

Figure 13

Transition amplitudes $L_1$ (red solid line) and $L_2$ (blue dashed) as function of the vibrational quantum number $n$. Their ratio approaches $4/\pi$ (green dashed-dotted line, right y-axis).