We have measured diffuse scattering in a single crystal of Cd_{0.9}Zn_{0.1}Te using a state-of-the-art laboratory diffractometer. A large-box atomistic simulation of a model crystal is used in conjunction with Monte Carlo modeling and the Kirkwood potential. A combination of structural relaxation in the presence of the dopant and thermal motion results in good qualitative agreement between the computed diffraction patterns of the model crystal and the measured x-ray patterns. This is shown to be rather distinct from the diffuse scattering arising from purely structural relaxations or thermal motion only. The atoms are shown to displace predominantly in ⟨1,1,1⟩ and ⟨1,0,0⟩ type directions. Our approach to Monte Carlo modeling can easily be extended to more complex defect structures to incorporate, e.g., chemical ordering on the Cd/Zn sublattice, more than one species of dopant or vacancies.

## I. INTRODUCTION

Semiconductors with composition A_{1−x}B_{x}C find widespread applications in solid-state devices due to their tunable bandgap. For example, In_{1−x}Ga_{x}As is used in laser and photodetection devices. CdTe is widely used at synchrotron facilities as a detector material displaying high detection-efficiency particularly at high x-ray energies exceeding 100 keV. The bandgap can be tuned and the detection properties can be influenced by substituting Cd for another transition metal, e.g., Zn, Mn, and Hg. This is actively explored to further optimize and tailor materials for detector applications.^{1} Cd_{0.9}Zn_{0.1}Te is a very attractive ternary semiconductor material for x-ray and gamma-ray detectors. It has high density (5.8 g/cm^{3}), a wide bandgap (Eg = 1.6 eV at 300 K), and high-Z constituents [Z(Cd) = 48, Z(Zn) = 30, and Z(Te) = 52] and can effectively operate at room temperature, which makes it perfectly suitable for room-temperature applications. Cd_{0.9}Zn_{0.1}Te is widely used in the homeland security, environmental monitoring, computed tomography, space telescopes, and other nuclear physics products.^{2–6} The presence of a dopant on the Cd site of different sizes leads to a local structural distortion. This has been shown to be well accounted for using the Kirkwood or Keating models and nearest-neighbor bond-stretching and bond-bending forces.^{7} In particular, the lattice parameters of A_{1−x}B_{x}C display a linear relationship between the two endmembers AC and BC. This is known as Végard's law. In contrast, local probes such as EXAFS and pair-distribution function analysis have shown that the AC and BC bonds for various alloys stay close to their natural bond length in the corresponding endmembers.^{8–10} In a series of papers, Thorpe and coworkers developed a rigorous theory based on the Kirkwood model to describe the local structural relaxations and achieved excellent agreement between theory, EXAFS, and pair-distribution function experiments with force constants derived from the elastic constants.^{7,11} Local structural deviations manifest themselves in the form of diffuse scattering in addition to the Bragg peaks in a diffraction experiment.^{12} When using a single crystal, the signal and shapes in the diffuse scattering are three-dimensionally resolved. When single crystals are not available, a powder diffraction experiment is usually carried out and the diffuse scattering is analyzed in the form of the pair-distribution function. Single-crystal diffuse scattering provides very detailed information about the local structure. The appearance has been predicted from the atomistic modeling of the pair-distribution function, but only very few experiments using electron diffraction have been carried out. Here, we close this gap using single-crystal diffuse scattering on Cd_{0.9}Zn_{0.1}Te. To date, synchrotron experiments were usually required to record the rather weak diffuse scattering signal. Here, we demonstrate that modern laboratory x-ray equipment is capable of recording this signal.

In addition, we use the Kirkwood model and parameters from Ref. 11 to carry out Monte Carlo simulations. This is done in stages. First, the atomic coordinates are relaxed using a strict down-hill Monte Carlo simulation. Second, pure thermal motion is modeled using a Monte Carlo approach corresponding to pure CdTe. The diffuse scattering signal coming from either type, i.e., structural relaxation vs thermal motion, is shown to be rather distinct. Finally, a Monte Carlo model incorporating both gives satisfactory qualitative agreement with the experiment. While not as rigorous as the theory presented in Ref. 7, the model can be easily extended to incorporate additional types of disorders such as chemical ordering on the A/B sublattice or other dopants or vacancies on the A and C sites.

## II. EXPERIMENT

Cd_{0.9}Zn_{0.1}Te single crystals were grown by the vertical Bridgman method in graphitized quartz ampoules from stoichiometric charges of high-purity (6 N) source components. A small amount of In (∼10^{17} at./cm^{3}) was added to the charge. The temperature gradient at solid–melt interface was ∼12 K/сm, the growth rate was 2.2 mm/h, and the cooling procedure was 30 K/h. As a result, the Cd_{0.9}Zn_{0.1}Te single-crystal ingot with a diameter of 20 mm and a length of 60 mm was obtained. 2 mm thick wafers were sliced perpendicular to the crystal axis in the as-grown Cd_{0.9}Zn_{0.1}Te ingot. All the wafers were mechanically polished with 1, 0.3, and 0.05 mm particle size alumina oxide suspension and then chemically polished with 5% bromine–methanol solution.

A small sample suitable for x-ray diffraction was selected and mounted on a MiTeGen pin. A standard single-crystal experiment was performed at room temperature using a Rigaku Oxford diffraction Synergy S instrument and Mo wavelength. A full sphere was recorded with the exposure time set to 28 s per frame, exceeding the exposure time for a standard experiment to just record the Bragg peaks by two orders of magnitude.

Data were processed using CrysAlis PRO, and reciprocal space sections were reconstructed.^{13} The minimum value was used as a background in each frame, and 4 mm Laue symmetry was applied. The crystal structure was refined using JANA2020.^{14} The refinement of the Zn occupancy did not significantly deviate from the expected composition and, thus, was kept at the nominal value expected from the synthesis. The results from the crystal structure refinement are summarized in Table I.

Crystal data . | . |
---|---|

Chemical formula | Cd_{0.9}Zn_{0.1}Te |

M_{r} | 235.3 |

Crystal system, space group | Cubic, $F 4 \xaf3m$ |

Temperature (K) | 293 |

a (Å) | 6.467^{ }64(4) |

V(Å^{3}) | 270.544(4) |

Z | 4 |

Radiation type | Mo Kα |

μ (mm^{−1}) | 18.345 |

Crystal size (mm) | 0.289 × 0.184 × 0.097 |

Data collection | |

Diffractometer | Rigaku Oxford Diffraction XtaLAB Synergy, DualFlex, HyPix detector |

Absorption correction | Numerical Gaussian integration |

T_{min}, T_{max} | 0.055, 0.69 |

No. of measured, independent and observed I > 2σ(I) reflections | 2628, 96, 96 |

R_{int} | 0.0733 |

(sin θ/λ)_{max} (Å^{−1}) | 0.847 |

Refinement | |

R[F^{2 }>^{ }2 σ(F^{2})], wR(F^{2}), S | 0.013, 0.0359, 1.7813 |

No. of reflections | 96 |

No. of parameters | 4 |

No. of restraints | 0 |

Δρ_{max}, Δρ_{min} (e Å^{−3}) | 0.22, −0.19 |

Crystal structure | |

u_{iso} (Cd, Zn) (½, ½, ½) (Å^{2}) | 0.022^{ }75(10) |

u_{iso} (Te) (¼, ¾, ¾) (Å^{2}) | 0.017^{ }67(10) |

Crystal data . | . |
---|---|

Chemical formula | Cd_{0.9}Zn_{0.1}Te |

M_{r} | 235.3 |

Crystal system, space group | Cubic, $F 4 \xaf3m$ |

Temperature (K) | 293 |

a (Å) | 6.467^{ }64(4) |

V(Å^{3}) | 270.544(4) |

Z | 4 |

Radiation type | Mo Kα |

μ (mm^{−1}) | 18.345 |

Crystal size (mm) | 0.289 × 0.184 × 0.097 |

Data collection | |

Diffractometer | Rigaku Oxford Diffraction XtaLAB Synergy, DualFlex, HyPix detector |

Absorption correction | Numerical Gaussian integration |

T_{min}, T_{max} | 0.055, 0.69 |

No. of measured, independent and observed I > 2σ(I) reflections | 2628, 96, 96 |

R_{int} | 0.0733 |

(sin θ/λ)_{max} (Å^{−1}) | 0.847 |

Refinement | |

R[F^{2 }>^{ }2 σ(F^{2})], wR(F^{2}), S | 0.013, 0.0359, 1.7813 |

No. of reflections | 96 |

No. of parameters | 4 |

No. of restraints | 0 |

Δρ_{max}, Δρ_{min} (e Å^{−3}) | 0.22, −0.19 |

Crystal structure | |

u_{iso} (Cd, Zn) (½, ½, ½) (Å^{2}) | 0.022^{ }75(10) |

u_{iso} (Te) (¼, ¾, ¾) (Å^{2}) | 0.017^{ }67(10) |

## III. MODELING

_{0.9}Zn

_{0.1}Te is made up of two interpenetrating FCC cubic lattices with Cd and Zn occupying one sublattice and Te occupying the other. There are four sites for each sublattice in the unit cell. In the ideal lattice, each species is located at the center of a regular tetrahedron. The average distance between the nearest neighbors corresponds to

*a*√3/4 with

*a*being the lattice parameter of the cubic cell. The natural-nearest-neighbor distances for Cd–Te and Zn–Te deviate from this average bond length due to the different sizes of Cd and Zn. The Kirkwood potential can be written as

^{7}

*α*is the bond-stretching force constant for atom pairs

_{ij}*i*and

*j*,

*L*is the corresponding bond length, $ L i j 0$ is the natural (unstrained) bond length,

_{ij}*β*is the bond-bending force constant for the triangle of atoms

_{ijk}*i,j,k,*and

*θ*is the bond-angle. The value −1/3 is the cosine of the ideal tetrahedral angle.

_{ijk}*L*is the average bond length. The parameters for the average unit cell and natural bond lengths have been computed previously

_{e}^{11}and parameterized using the following relationship:

Parameters for the various bond lengths and force constants are taken from Tables I and II of Ref. 11. We note that different force constants are reported dependent on the experimental data used to derive them.^{15} A model crystal of size 32 × 32 × 32 unit cells was used in all simulations. This corresponds to 262 144 atoms. Zn was randomly distributed in this cell to give a close match to the nominal composition. The diffuse scattering patterns were calculated for various reciprocal space sections using 200 lots of size 10 × 10 × 10 with the Bragg peaks subtracted^{16,17} in order to obtain smooth diffuse scattering patterns free of Fourier termination ripples. We note that a crystal size of 3 × 3 × 3 containing 22 Zn atoms would, in principle, suffice. However, the corresponding computed diffraction patterns and atomic distributions would appear rather noisy. Various simulations were carried out to explore different aspects of the model.

## IV. RESULTS AND DISCUSSION

Initially, a Monte Carlo simulation was carried out where the coordinates of the atoms were relaxed. The aim was to minimize the total energy of the crystal. This corresponds to static distortions without the inclusion of thermal motion. After initial trial-and-error, the production run was using 2 × 10^{6} Monte Carlo cycles. In each cycle, each atom is visited once on average and displaced in a random direction by a random amount. If the local energy is lower, the move is accepted. Forty runs with different random seeds were run. The minimization took 4 days per configuration. The resulting diffuse scattering patterns calculated from one of the relaxed configurations are shown in Fig. 1 along with the patterns obtained from the experiment.

Diffuse features in the (h,k,0)-plane appear comparable to the relaxation from a single defect (Fig. 4 in Ref. 15). The individual features display an asymmetric bow-tie shape around the Bragg peaks. This corresponds to Huang scattering arising from the relaxation of the atoms away from a dopant. The intensity of Huang scattering has been shown to be proportional to cos^{2}(φ) giving it a bow-tie or double drop shape.^{15} Here, φ is the angle between the $ Q \u2192$-vector of the image pixel and $ q \u2192= Q \u2192\u2212 B \u2192$ is the vector pointing from $ Q \u2192$ to the nearest Bragg point $ B \u2192$. Notably, such scattering signatures are absent in the measured data. Inspection of the resulting distribution of atoms around a given site of the relaxed structure shows near-spherical, slightly distorted distributions for Cd and Zn (Fig. 2). The Te atom displacements are more pronounced and are mostly along the ⟨1,1,1⟩ and ⟨1,0,0⟩ directions and a blob in the center. The latter corresponds to largely undisturbed regions of mostly Cd–Te composition. This would be expected from a size-effect distortion. A similar distribution was obtained in In_{1−x}Ga_{x}As.^{18} Figure 3 shows a histogram of pair distances. The nearest-neighbor Cd–Te and Zn–Te peaks just below 3 Å are clearly split and close to their natural bond lengths.

The effect of pure thermal motion without any structural relaxation was simulated again using the Kirkwood potential and only one set of force constants for bond-stretching and bond-bending forces corresponding to a pure composition of CdTe. The Monte Carlo simulation was aiming not to minimize the energy of the crystal but rather to achieve a target thermal displacement, here of the Cd site corresponding to the square root of u_{iso} from Table I.^{12} This is essentially a balls-and-springs model. After each MC cycle, the atoms are projected back into one unit cell and the force constants are adjusted with a fixed ratio between them until the desired target square root of u_{iso} is achieved. Five hundred MC cycles were carried out taking about 2 min. The force constants oscillate until the crystal is equilibrated. The resulting diffraction patterns are shown in Fig. 4. Comparison of the simulated patterns with the data reveals better agreement compared to the energy minimized structure. Weak lines along ⟨1,1,0⟩ directions and maxima underneath the Bragg peaks appear. However, these are essentially straight lines displaying maxima where they intersect and directly underneath the Bragg peaks. However, the data in Fig. 1 display more curvature along these lines. Hence, we combined the essential features of relaxation around dopants and thermal motion.

The final simulation was built on the thermal motion simulation but now incorporating the size effect allowing for the Cd–Te and Zn–Te bond lengths to be different according to Eq. (2). The resulting diffraction patterns are shown in Fig. 5. These compare very favorably to the experimental patterns.

The resulting atomic distributions for Cd, Zn, and Te are shown in Fig. 6.

The thermal motion has clearly broadened the distributions. Cd and Zn appear rather more isotropic and Te is slightly distorted from being spherical. However, the Cd–Te and Zn–Te distances are still split due to the size effect, leading to an asymmetric bond-length distribution for the nearest-neighbor as shown in the histogram of the pair distances (Fig. 7).

## V. CONCLUSIONS

In this paper, we show that state-of-the-art laboratory equipment allows to record single-crystal diffuse scattering in addition to the Bragg peaks, making this technique accessible in a laboratory setting. Synchrotron experiments will still be superior. To qualitatively model the data, we use a large-box Monte Carlo modeling approach exploring various aspects. These are

Energy minimization of the Kirkwood potential.

Pure thermal motion of the atoms.

Combining pure thermal motion and size-effect distortion.

The pure size effect corresponding to model (i) is found to mostly affect the Te atoms with preferential displacements along ⟨1,1,1⟩ and ⟨1,0,0⟩ directions. This is leading to Huang scattering in the form of bow-tie shaped diffuse scattering. Pure thermal motion, model (ii), leads to weak diffuse lines along ⟨1,1,0⟩ connecting the Bragg peaks. Finally, model (iii) gives the best qualitative agreement. The present models are purely mechanistic in nature. Electronic terms are not included explicitly.^{19} Apart from the energy minimization, the Monte Carlo simulations can be run in a matter of minutes. These run times would be very similar when simulating other compositions. Furthermore, they can easily be extended to incorporate more than one kind of dopant on both sites and chemical ordering on each sublattice. It is hoped that the box of atoms obtained can be used to compute electronic properties such as the bandgap to further compare and optimize device performance.^{19}

## ACKNOWLEDGMENTS

We kindly acknowledge access to the SCARF supercomputing resources at the Rutherford Appleton Laboratory. Access to the x-ray diffraction facility at the Materials Characterisation Laboratory at the ISIS spallation neutron source is acknowledged. G.L.P. acknowledges the funding from the Ministry of Research, Innovation, and Digitalization within Program 1—Development of National Research and Development System, Subprogram 1.2—Institutional Performance-RDI Excellence Funding Projects, under Contract No. 10PFE/2021. V.M. was supported by Romania National Council for Higher Education Funding, CNFIS, under Project No. CNFIS-FDI-2024-F-0155.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose

### Author Contributions

**M. J. Gutmann:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **O. Kopach:** Conceptualization (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **V. Kopach:** Conceptualization (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **V. Mykhailovych:** Conceptualization (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **G. L. Pascut:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **P. Fochuk:** Conceptualization (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{1−x}ZnxTe compounds designed for radiation detectors

*Handbook of II-VI Semiconductor-Based Sensors and Radiation Detectors*

*Diffuse X-ray Scattering and Models of Disorder*

*CrysAlis PRO*