Full case study: /case-studies/alloy-sampling

Observable

Band gaps and electronic structure computed for ~50 Cu₂Ge(S₁₋ₓSeₓ)₃ configurations sampled across compositions x = 0 to 1 in periodic supercells.

Claim

Computed band-gap vs. composition trends are consistent with Boltzmann-weighted ensemble behavior. Single-configuration properties do not represent the material—thermal averaging required.

Load-Bearing Constraints

Axiomatic

  • Composition discards atomic arrangement information (many configs per stoichiometry)
  • At synthesis temperatures, k_B T ~ ΔE_config (multiple arrangements thermally accessible)

Measurement

  • Experimental spectra average over grain boundaries, defects, configuration distributions
  • No single-structure probe available

Fabrication

  • Growth conditions can lock in non-equilibrium configurations (not sampled here)
  • Impurity content varies across synthesis batches

Statistical

  • Sampled ~50 from >10⁶ combinatorially possible arrangements
  • Band-gap spread at fixed composition: up to 0.3 eV within sampled set
  • Rare motifs with high optical weight may be absent

Computational

  • PBE: underestimates absolute gaps by ~1 eV (affects calibration, not relative trends)
  • Supercell size: 48-96 atoms (limits configuration diversity)
  • Convergence varies with local atomic order

Primary Limiting Factor

Finite sampling of configuration space.

~50 structures cannot represent full ensemble. Conclusions hold only for sampled model class under stated assumptions.

What This Ruled Out

  • Treating any single structure as representative of bulk behavior
  • Attributing band-gap shifts to composition alone (configuration entropy matters)

What Remains Non-Identifiable

  • Actual configuration distribution in synthesized samples (growth-condition-dependent)
  • Relative contributions: composition vs. local order vs. impurities in experimental spectra
  • Quantitative gap values (PBE systematic error limits experimental comparison)

What Would Help

  • Expand sampling (cluster expansion, active learning, Monte Carlo at synthesis T)
  • Local-order probes (EXAFS, pair distribution function analysis)
  • Higher-level theory (HSE06, GW) for subset of representative configurations
  • Synthesis-protocol-specific sampling with known temperature/pressure history

Methods Referenced

Related constraints: g-C₃N₄ optical (similar structural sampling limitations)

Analysis date: Spring 2024
My experience: First computational materials project, learning DFT and thermodynamic sampling from scratch

This was my first project where I realized composition ≠ structure. I initially thought “pick one structure per composition” was fine. It’s not—when configurational entropy competes with electronic energy, you need ensemble thinking. The 50-structure sample size was pragmatic (what we could compute) not rigorous (what we needed for convergence).

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