Sequential Quenching to Predict Semiconductor Defect Concentrations from Formation & Migration Energies: The Case of CdTe:As Doping

Defect concentrations in semiconductors are strongly influenced by thermal history during growth and cooldown, yet most defect calculations assume either instantaneous quenching from high temperature or that full-equilibrium is maintained - two limiting cases rarely approached in reality. Here, we introduce sequential quenching (SQ) as a 3rd type of defect calculation utilizing defect formation and migration energies to model defect concentrations subject to diffusion-limited kinetics in samples cooled at finite rates. In SQ, the concentration of each defect is frozen at a characteristic temperature determined by its diffusion rate, distance to sources/sinks, and cooling rate. Because different charge-states interact through charge neutrality but freeze at different temperatures, the sequence of freeze-in events is non-commuting. Critically, not all room-temperature SQ solutions can be predicted from full equilibrium (EQ) or full-quenching (FQ) calculations - erroneous predictions are likely without SQ. We illustrate SQ using the example of As-doped CdTe, for which experimental data show differences in doping with cooling rate and between polycrystalline thin-films for photovoltaics and bulk crystals. SQ calculations reveal that fast-diffusing defects such as Cd-interstitials remain mobile to lower temperatures and freeze-in at larger characteristic distances, leading to strong compensation and n-type behavior in rapidly cooled or bulk samples. Slower cooling and reduced characteristic distances suppress donor freeze-in and enhance p-type activation. These results establish SQ as a physically transparent and computationally efficient framework for connecting cooling conditions, sample geometry, and defect kinetics to dopant activation in CdTe and related materials.

💡 Research Summary

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The paper addresses a fundamental limitation in conventional first‑principles defect calculations for semiconductors: they typically assume either full thermodynamic equilibrium (EQ) during infinitely slow cooling or instantaneous quenching (FQ) that freezes all defects at the growth temperature. Real crystal growth and device processing involve finite cooling rates and a wide range of sample geometries, so many defects evolve under diffusion‑limited kinetics that are not captured by these two extremes. To bridge this gap, the authors introduce Sequential Quenching (SQ), a computational framework that combines defect formation energies, migration barriers, cooling rate, and a characteristic diffusion distance (e.g., distance to surfaces, grain boundaries, or dislocations).

In SQ each defect (or charge state) remains in equilibrium down to a freeze‑in temperature (T_f) defined by the condition that its diffusion length λ = √(Dt) equals the characteristic distance L. The diffusion coefficient D follows an Arrhenius law D = D₀ exp(–E_mig/k_BT). Faster cooling (larger γ) shortens the time spent at a given temperature, raising T_f; slower cooling lowers T_f. Once a defect freezes, its concentration is held fixed for all lower temperatures, but because charged defects must satisfy global charge neutrality, the freeze‑in of one species perturbs the Fermi level and forces the remaining mobile defects to re‑equilibrate. Consequently, the order in which defects freeze is non‑commuting and determines the final carrier type and concentration.

The authors illustrate the concept with a toy model containing a fast‑diffusing donor‑like defect D and two acceptors A₁ (high migration barrier) and A₂ (low barrier). Under EQ the system remains intrinsic at room temperature, whereas SQ predicts p‑type behavior when A₁ freezes early while D stays mobile. This simple example demonstrates that SQ can generate defect ensembles that are not contained within either EQ or FQ solutions, even when all calculations start from the same high‑temperature state.

The methodology is then applied to a technologically important system: arsenic‑doped cadmium telluride (CdTe:As), a material used both as a bulk radiation detector and as the absorber layer in thin‑film photovoltaics. Experimentally, bulk crystals can be p‑type doped with As, while polycrystalline thin films often show low acceptor activation and strong n‑type compensation. Prior DFT studies suggested that As on Te sites (As_Te) might adopt an AX‑donor configuration, but more recent work—including spin‑orbit coupling—indicates that As_Te remains an acceptor and that Cd interstitials (Cd_i), together with As_Te‑Cd_i complexes, are the primary compensating donors.

Using SQ, the authors calculate the freeze‑in temperatures for the relevant defects. Cd_i has a low migration barrier (~0.5 eV) and therefore remains mobile to lower temperatures than the acceptor defects. For fast cooling rates (typical of rapid crystal growth or quenching of bulk samples) Cd_i does not freeze until near room temperature, providing abundant electrons that compensate the As acceptors and drive the material toward n‑type behavior. In contrast, slow cooling or a small characteristic distance (as in thin films with micron‑scale grains) forces Cd_i to freeze at a higher temperature, limiting its compensating effect. Consequently, As_Te acceptors dominate, leading to higher hole concentrations and stronger p‑type activation. Grain boundaries in polycrystalline films act as sinks that effectively reduce the diffusion length, further promoting early freeze‑in of Cd_i and explaining the experimentally observed low p‑type activation in such films.

The SQ approach requires only a modest set of inputs—formation energies, migration barriers (obtainable from standard DFT or hybrid functional calculations), the cooling rate, and an estimate of L (film thickness, grain size, or dislocation spacing). These parameters are readily available or can be measured, making SQ a computationally inexpensive alternative to full drift‑diffusion‑Poisson‑reaction simulations, which would otherwise demand multidimensional PDE solvers and extensive kinetic data. The authors have implemented SQ within the KROGER framework, an open‑source tool for defect thermochemistry, enabling rapid “what‑if” studies of processing conditions.

In summary, the paper establishes Sequential Quenching as a physically transparent, analytically tractable, and computationally efficient method for predicting non‑equilibrium defect populations in semiconductors. By explicitly accounting for diffusion‑limited freeze‑in, it reconciles long‑standing discrepancies between theory and experiment in CdTe:As doping, and it offers a generalizable strategy for other material systems where cooling history, sample size, and microstructure critically influence electrical properties. The work underscores that defect formation energies alone are insufficient for realistic device modeling; kinetic considerations must be integrated to achieve predictive power in semiconductor materials design.