MCIA Modeling for Epitaxial Interfaces

• MCIA modeling is a quantitative, geometry-driven approach that minimizes the 2D coincident-site lattice area to predict energetically favorable epitaxial orientations.
• The framework computes rigid in-plane rotations and exhaustively searches for integer matching matrices, providing a clear metric (A_CSL) to rank film/substrate pairs.
• While effective for screening heteroepitaxial interfaces under kinetic growth conditions, MCIA must be integrated with thermodynamic models to account for chemical and high-temperature effects.

The minimal coincident interface area (MCIA) modeling framework is a quantitative, geometry-driven approach to predicting preferred epitaxial orientations during the heteroepitaxial growth of crystalline thin films on lattice-mismatched substrates. Central to the scheme is the hypothesis that, when presented with competing film-substrate orientation alignments, the configuration that minimizes the two-dimensional coincident-site lattice (CSL) supercell area yields the most energetically favorable interface, especially under kinetic growth regimes where adatom mobility is limited. MCIA thereby provides a tractable, predictive metric for orientation selection, as demonstrated in the controlled epitaxial growth of rare-earth-doped TiO$_2$ thin films on III-V semiconductor substrates such as GaAs(001) and GaSb(001) (Hammer et al., 5 Nov 2025).

1. Theoretical Foundations

MCIA modeling is grounded in the coincident-site lattice (CSL) concept, which analyzes the two-dimensional lattice superstructure resulting from juxtaposing two crystalline solids of dissimilar lattice constants and/or orientations. When projected onto the putative interface plane, overlapping lattice points between film and substrate define a CSL with primitive cell area $A_{CSL}$. A smaller $A_{CSL}$ indicates a denser registry of coincident sites, which reduces local elastic and chemical mismatch due to improved atomic correspondence across the interface. Within the MCIA framework, each candidate film/substrate plane pair is evaluated for the smallest possible CSL supercell (subject to a misfit tolerance $\delta$), and orientations are ranked by their minimal $A_{CSL}$ values. The orientation with the minimal MCIA is postulated to be the easiest to nucleate epitaxially.

2. Mathematical Formalism

Let $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ denote the Cartesian lattice vectors of the film and substrate, respectively. For given film and substrate planes, $(hkl)_n$ and $(HKL)_s$, lattice vectors $\{u_1, u_2\}$ and $\{v_1, v_2\}$ spanning the interface are identified. A rigid in-plane rotation $R$ brings the film plane normal into coincidence with the substrate plane normal. The search is performed over integer $2 \times 2$ “matching” matrices $M$ satisfying

$M \cdot [u_1\ u_2] \approx R \cdot [v_1\ v_2]$

subject to $\|M\cdot u - R\cdot v\|/\|v\| < \delta$, with typical mismatch tolerances $\delta \approx 3$–$5\%$ and $|\det M|$ limited (e.g., to $20$) to restrict supercell size. For the optimal $M$, the coincident supercell area is given by

$$A_{CSL} = |\det([u_1\ u_2] \cdot M)| = |\det([u_1\ u_2])| \cdot |\det M|\,.$$

This area is proportional to the product of the primitive film surface unit cell area, $|u_1 \times u_2|$, and the number of coincident sites per film cell, $N_{CSL} = |\det M|$.

3. Modeling Workflow

The implementation of MCIA modeling proceeds as follows:

1. Plane Selection: Specify candidate film (hkl) and substrate (HKL) planes; extract in-plane lattice vectors.
2. Rotation Alignment: Compute the rigid rotation $R$ aligning the film and substrate normals; rotate the in-plane film vectors accordingly.
3. CSL Matrix Search: Exhaustively search over integer matrices $M$ up to prescribed $|\det M|$ for those which, under $R$, yield residual lattice mismatches below $\delta$.
4. Area Calculation: For each admissible $M$, compute $A_{CSL} = |\det([u_1\ u_2] \cdot M)|$.
5. Orientation Ranking: Record the minimal $A_{CSL}$ for each orientation pair; this value is the MCIA used to rank epitaxial orientation preference.

Computational parameters used for TiO$_2$ on GaAs/GaSb include lattice constants from rutile ($a=4.593\ \textrm{Å}$, $c=2.959\ \textrm{Å}$) and anatase ($a=3.785\ \textrm{Å}$, $c=9.514\ \textrm{Å}$) TiO$_2$ polymorphs, and III-V substrates ($a=5.653\ \textrm{Å}$ for GaAs, $a=6.096\ \textrm{Å}$ for GaSb), mismatch tolerance $\delta\approx5\%$, and maximum $|\det M|=20$ corresponding to supercells up to a few nm across (Hammer et al., 5 Nov 2025).

4. MCIA Predictions: Specific Results for TiO₂ on GaAs(001) and GaSb(001)

MCIA ranks candidate orientations by their coincident supercell area:

Substrate	Orientation	$A_{CSL}$ (Å$^2$)
GaAs(001)	anatase TiO$_2$(001)	64 (global minimum)
GaAs(001)	rutile TiO$_2$(110)	≈494
GaAs(001)	rutile TiO$_2$(210)	≈530
GaAs(001)	anatase TiO$_2$(101)	≈680
GaSb(001)	anatase TiO$_2$(001)	≈175
GaSb(001)	rutile TiO$_2$(001)	≈175
GaSb(001)	rutile TiO$_2$(110)	≈320
GaSb(001)	anatase TiO$_2$(100)	≈330

On GaAs(001), anatase TiO$_2$(001) shows the smallest interface area (64 Å$^2$), predicting strong orientation selection even at modest adatom mobility. On GaSb(001), anatase(001) and rutile(001) orientations are nearly degenerate ($A_{CSL}\approx175$ Å$^2$), suggesting additional temperature or chemical parameters influence the resultant phase.

5. Comparison with Experimental Observations

Experimental characterization using RHEED, XRD, and Raman spectroscopy demonstrates strong correspondence between MCIA predictions and observed film orientation:

• On arsenic-capped GaAs(001) at low temperatures ($\approx390$ °C), smooth, epitaxial anatase TiO$_2$ films form, with structural signatures matching anatase(001).
• At elevated temperatures ($>450$ °C) or on oxide-desorbed GaAs, growth yields polycrystalline rutile, with dominant rutile(110) XRD reflections, corresponding to the next-lowest MCIA configuration.
• On GaSb(001), both anatase(001) and rutile(001) can be realized under different growth conditions, consistent with their near-degenerate MCIA values. Low-temperature growth favors anatase(001), while other parameters may stabilize rutile.

Deviations from purely geometric MCIA predictions arise due to factors not captured by the model, such as buffer-layer thickness, oxygen-vacancy accumulation, and details of the substrate surface chemistry. Sub-optimal interface preparation (e.g., oxide desorption) can suppress orientation selectivity.

6. Utility and Limitations in Heteroepitaxial Integration

MCIA offers a physically motivated, computationally tractable metric for orientation selection in lattice-mismatched epitaxy. Its successful application to Er$^{3+}$:TiO$_2$ films on III-V substrates provides a rational approach to engineering smooth, orientation-controlled interfaces necessary for hybrid quantum photonic devices (Hammer et al., 5 Nov 2025). MCIA modeling is most predictive under kinetic growth conditions dominated by geometric registry and limited adatom migration.

The framework, however, is constrained by its geometric nature; it does not capture interface energetics arising from dissimilar bonding, surface reconstructions, or interfacial diffusion. As evidenced by discrepancies under high-temperature or chemically dynamic conditions, thermodynamic and non-stoichiometric factors may override the MCIA preference.

A plausible implication is that MCIA is a reliable first-pass screening tool for orientation preference, but comprehensive predictions for heteroepitaxy interfaces—particularly in technologically relevant systems combining disparate materials functionalities—require integration of MCIA with atomistic or electronic structure computations.