ABX3 Halide Perovskite Alloys

• ABX₃ halide perovskite alloys are mixed-cation/anion materials derived from a corner-sharing BX₆ framework, enabling systematic tuning of structural, electronic, and optical properties.
• Advanced computational methods, including DFT, SQS, and machine learning, reveal precise composition–property relationships and guide band-gap and stability optimization.
• Targeted alloying at A, B, or X sites permits engineering of lattice, dielectric, and thermodynamic characteristics for improved photovoltaic, optoelectronic, and photodetector performance.

ABX₃ halide perovskite alloys are mixed-cation or mixed-anion crystalline materials derived from the archetypal perovskite structure (formula ABX₃, where A is a monovalent cation, B a divalent cation, and X a halide anion). Alloying at A, B, or X sites enables systematic tuning of structural, electronic, optical, and thermodynamic properties, providing a highly versatile platform for photovoltaic, optoelectronic, and photodetector applications. Advanced density functional theory (DFT), special quasi-random structures (SQS), multi-fidelity machine learning, and high-throughput experimental studies have yielded a unified framework for understanding and predicting the composition–property–stability relations in these alloys, with rigorous attention to phase stability, band-gap bowing, defect tolerance, and design principles.

1. Crystal Chemistry and Alloying Strategies

ABX₃ halide perovskites adopt a three-dimensional corner-sharing BX₆ octahedral framework, stabilized by appropriately sized A-site cations. Alloying is most commonly implemented at one lattice site at a time (A, B, or X), though multi-site mixing is possible:

• A-site: Inorganic (e.g., Cs⁺, Rb⁺, K⁺) and organic (methylammonium MA⁺, formamidinium FA⁺, azetidinium Az⁺) cations are employed; tolerance factor

$t = \frac{r_A + r_X}{\sqrt{2}(r_B + r_X)}$

governs formability. Mixed A-site alloys tune t to promote phase stability and framework rigidity (Yang et al., 2023, Lehmann et al., 2018).

• B-site: Group IV (Ge²⁺, Sn²⁺, Pb²⁺), IIB (Cd²⁺), and IIA (Ca²⁺, Sr²⁺, Ba²⁺) elements are relevant for alloying. B-site substitution enables strong band-gap tuning via direct s- and p-orbital mixing (Dalpian et al., 2018, Zhang et al., 31 Jan 2026).
• X-site: Halide anions I⁻, Br⁻, and Cl⁻ are routinely alloyed. Substitution tunes optical gaps, dielectric properties, and stability, subject to solubility limits (e.g., MAPb(I₁₋ₓBrₓ)₃ exhibits complete miscibility, while MAPb(I₁₋ₓClₓ)₃ has a wide miscibility gap) (Lehmann et al., 2018, Magdaleno et al., 2022).

Special Quasirandom Structures (SQS) are used in DFT calculations to approximate statistical alloy disorder, enabling reliable computation of mixing energetics and property trends (Yang et al., 2023, Dalpian et al., 2018).

2. Structural and Lattice Property Evolution

Upon alloying, ABX₃ lattice metrics generally obey Vegard’s law, with lattice constants varying (often linearly) with composition:

• For CsPb(I₁₋ₓBrₓ)₃, the lattice parameter decreases linearly:

$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$

with $a_\mathrm{I}=6.297$ Å and $a_\mathrm{Br}=5.874$ Å, closely matching experiment (<1% error), and indicating faithful reproduction of lattice mixing by virtual atom pseudopotentials (Yu et al., 2019).

• Similar Vegard-type shifts occur for X-site and A-site alloys, with bowing or negative deviation possible near solubility limits or at miscibility gaps (Lehmann et al., 2018). For example, MAPb(I₁₋ₓBrₓ)₃ exhibits nearly linear evolution, while MAPb(I₁₋ₓClₓ)₃ shows a negative deviation at miscible compositions.
• Unit-cell volume contracts as ionic radius decreases (I⁻→Br⁻→Cl⁻ or MA⁺→Cs⁺); bulk modulus increases, reflecting stronger B–X bonds upon substitution of smaller or less polarizable ions (Yu et al., 2019, Lehmann et al., 2018).
• Octahedral tilting and polymorph selection are sensitive to alloying; SQS and DFT studies reveal persistent, composition-tunable octahedral-tilt phonon instabilities that drive α→β→γ phase polymorphism, with the stability regimes mapped in (P, T, x) space (Yang et al., 2023, Yu et al., 2019).

3. Electronic Structure and Band-Gap Engineering

ABX₃ perovskite alloys exhibit systematic band-gap tunability via composition:

• For CsPb(I₁₋ₓBrₓ)₃, band gap increases with Br content based on a bowing fit:

$$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$

yielding $E_g=1.75$ eV (x=0) to $2.38$ eV (x=1), consistent with experiment (Yu et al., 2019).

• In MAPb(I₁₋ₓBrₓ)₃, the band gap varies nearly linearly from $\sim1.46$ eV (x=0) to $\sim2.32$ eV (x=1) with negligible bowing ($b\approx0$) (Magdaleno et al., 2022). For ternary systems, bowing is explicitly modeled:

$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$0

with $a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$1–$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$2 eV for MAPb(I₁₋ₓBrₓ)₃ (Lehmann et al., 2018).

• B-site alloying (Sn↔Pb) introduces larger bowing ($a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$3–$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$4 eV), enabling gaps as low as $a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$5–$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$6 eV for optimal single-junction PV absorbers (Dalpian et al., 2018). A-site mixing produces smaller band-gap modulation (e.g., $a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$7–$a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$8 eV for FA/Cs) (Dalpian et al., 2018).
• Rare upward band gap bowing (alloy gap exceeding all endmembers) can be realized in multi-component alloys with group IVB/IIB B-site mixing (e.g., Cs₄[GeSnPbCd]I₁₂): upward gap bowing ($a(x) = 6.242\,\text{Å} - 0.378\,\text{Å}\cdot x$9) and negative mixing enthalpy (thermodynamic stability) originate from cross-band-gap s–s repulsion between IVB and IIB cations (Zhang et al., 31 Jan 2026).

4. Optical and Dielectric Properties

Optical response functions (absorption, reflectivity, dielectric constants) are alloy- and composition-dependent:

• The absorption edge in CsPb(I₁₋ₓBrₓ)₃ blue-shifts with increased Br content; reflectivity peak decreases and static dielectric constant $a_\mathrm{I}=6.297$0 falls nearly linearly ($a_\mathrm{I}=6.297$1 for BSE-EXC) (Yu et al., 2019).
• Optical constants are extracted from the frequency-dependent dielectric function:

$a_\mathrm{I}=6.297$2

$a_\mathrm{I}=6.297$3

Under alloying, these properties can be linearly interpolated except in cases where local disorder or phase separation alters the electronic structure (Yu et al., 2019, Magdaleno et al., 2022).

• Experimentally, mixed-halide MAPb(I₁₋ₓBrₓ)₃ films on paper show continuous blue-shifting of the absorption onset from $a_\mathrm{I}=6.297$4 nm (x=0) to $a_\mathrm{I}=6.297$5 nm (x=1), with no detected phase segregation and fast photodetector response times ($a_\mathrm{I}=6.297$6–$a_\mathrm{I}=6.297$7 s across $a_\mathrm{I}=6.297$8) (Magdaleno et al., 2022).

5. Thermodynamic Stability and Mixing Energetics

Thermodynamic (decomposition) stability of ABX₃ alloys is quantified via enthalpy and free energy of mixing and decomposition:

• The decomposition energy relative to the competing AX + BX₂ phases is

$a_\mathrm{I}=6.297$9

Only compounds with $a_\mathrm{Br}=5.874$0 are thermodynamically resistant to breakdown (Yang et al., 2023, Yang et al., 2023).

• In CsPb(I₁₋ₓBrₓ)₃, the critical pressure $a_\mathrm{Br}=5.874$1 and decomposition temperature $a_\mathrm{Br}=5.874$2 rise with Br, expanding the (P,T) stability envelope:

$a_\mathrm{Br}=5.874$3

$a_\mathrm{Br}=5.874$4

Br-rich compositions resist decomposition at much higher temperature/pressure (Yu et al., 2019).

• Miscibility gaps, particularly on the Cl–I or Cs–MA mixing line, are mapped precisely via high-resolution diffraction and optical characterization: miscibility is complete in MAPb(I₁₋ₓBrₓ)₃, but partial or marginal (<3–4%) for MAPb(I₁₋ₓClₓ)₃ and Rb–MA alloys (Lehmann et al., 2018).

6. Data-Driven Discovery and Computational Design Principles

High-throughput DFT and machine learning have enabled the rapid prediction and screening of thousands of ABX₃ alloys for targeted optoelectronic performance:

• Datasets comprising hundreds of DFT-calculated compositions are used to train regression and neural network models to predict band gap, stability, defect energetics, and optical absorption across the ABX₃ space (Yang et al., 2023, Mannodi-Kanakkithodi et al., 2021).
• Multi-fidelity modeling incorporating DFT (GGA-PBE, HSE06+SOC) and experimental data streamlines the screening of >150,000 hypothetical ABX₃ compounds; surrogate models combined with genetic algorithms for inverse design identify thousands of compositions with $a_\mathrm{Br}=5.874$5 eV/f.u., $a_\mathrm{Br}=5.874$6 eV $a_\mathrm{Br}=5.874$7 eV, and photovoltaic efficiency (SLME) $a_\mathrm{Br}=5.874$8 (Yang et al., 2023).
• Screening and design rules:
  • A-site: favor FA/MA for stability; avoid excessive K or Rb.
  • B-site: partial Sn substitution enables optimal $a_\mathrm{Br}=5.874$9; Ge lowers gap but destabilizes; Ba/Sr/Ca stabilize but widen $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$0.
  • X-site: I–Br mixture for $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$1–$$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$2 eV and stability; avoid Cl fraction $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$3 for $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$4 eV (Yang et al., 2023, Yang et al., 2023).
  • Tolerance factor $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$5–$$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$6, octahedral factor $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$7–$$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$8 are essential for cubic stability (Yang et al., 2023).

7. Distinct Physical Mechanisms and Alloy Bowing Behavior

The physical origins of compositional bowing in electronic and energetic properties can be decomposed as follows (Dalpian et al., 2018):

• Band gap bowing: Strongest for B-site alloying due to direct mixing of Sn/Pb/Ge s- and p-orbitals; minimal for A-site alloying (mainly geometric).
• Mixing enthalpy: Positive bowing signals a propensity towards phase separation (miscibility gap); A-site alloying typically incurs higher positive $$E_g(x) = 1.750\,\text{eV} + 0.454\,\text{eV}\cdot x + 0.180\,\text{eV}\cdot x^2$$9 than B-site mixing.
• Upward gap bowing and negative mixing enthalpy: Realized in multi-B-site (IVB/IIB) cubic alloys via cross-gap s–s repulsion; enables alloys whose band gap exceeds all constituent binaries while being stable (ΔH<0), an exceptional scenario relevant for barrier/tunnel layers (Zhang et al., 31 Jan 2026).
• Born–Haber cycle decomposition: Octahedral distortion removal, volume deformation, charge exchange, and structural relaxation are quantitatively separable energetic contributions in SQS DFT frameworks (Dalpian et al., 2018).

---

These findings frame ABX₃ halide perovskite alloys as a model system for rational band-gap engineering, stability optimization, and optoelectronic property control, with robust computational-experimental convergence enabling targeted material design for photovoltaics and beyond (Yu et al., 2019, Yang et al., 2023, Dalpian et al., 2018, Lehmann et al., 2018, Magdaleno et al., 2022, Yang et al., 2023, Zhang et al., 31 Jan 2026).