MBE: Epitaxy, Many-Body, and Bootstrap Exploration

• MBE is a context-dependent acronym representing molecular beam epitaxy, many-body expansion, and multiplier bootstrap exploration, each characterized by controlled incremental processes.
• Molecular beam epitaxy employs ultra-high-vacuum techniques and in situ diagnostics to enable precise atom-by-atom thin-film growth with tailored electronic properties.
• Many-body expansion and bootstrap-based exploration methods systematically decompose complex interactions and optimize algorithmic exploration, achieving high accuracy in simulations.

MBE is a context-dependent acronym with distinct technical meanings across materials science, numerical analysis, electronic-structure theory, and machine learning. In the supplied literature, it most commonly denotes molecular beam epitaxy, an ultra-high-vacuum thin-film growth method used for semiconductors, oxides, metals, and van der Waals materials; it also denotes many-body expansion in quantum chemistry and learned potential-energy-surface modeling, and Multiplier Bootstrap-based Exploration in bandit algorithms (Slobodskyy et al., 2012, Greiner et al., 2024, Wan et al., 2023). Across these usages, MBE functions as a framework for controlled decomposition or controlled synthesis: atom-by-atom growth in epitaxy, order-by-order interaction decomposition in many-body methods, and uncertainty-driven randomized optimization in exploration algorithms.

1. Molecular beam epitaxy as an overview platform

In the experimental materials literature here, molecular beam epitaxy is presented as a highly controlled growth technique whose standard components include effusion cells with shutters, a main shutter, a cooling shroud, a manipulator, a RHEED setup, and pressure gauges (Slobodskyy et al., 2012). A portable synchrotron-compatible implementation further adds two 90 mm diameter beryllium windows, each 300 μm thick and mounted on CF100 flanges, enabling in situ x-ray diffraction geometries with angular coverage of ±18°, increased to 31° by mechanical inclination (Slobodskyy et al., 2012). The same system uses independently pumped transfer, storage, and growth chambers, base pressures in the low $10^{-11}$ mbar range, a 30 keV RHEED gun, and a tungsten-wire heater supporting up to $900^\circ\text{C}$ (Slobodskyy et al., 2012).

The technique is used across a broad materials set. GaP nanowires were grown on heavily-doped (111) Si substrates, with vertical, uniform growth occurring within a window of V/III flux ratios between 8 and 30 and substrate temperatures in the $610\!-\!630^\circ\text{C}$ range (Sharov et al., 2024). MBE-grown van der Waals films in the transfer study include MoSe$_2$, WSe$_2$, Bi$_2$Se$_3$, Bi$_2$Te$_3$, Fe$_3$GeTe$900^\circ\text{C}$0, and FGT/Bi$900^\circ\text{C}$1Te$900^\circ\text{C}$2 heterostructures on c-plane sapphire (Li et al., 22 Feb 2025). Thin Al layers were grown by MBE on In$900^\circ\text{C}$3Ga$900^\circ\text{C}$4As for superconductor/semiconductor hybrids, with the growth rate itself driving a sharp morphological transition (Elbaroudy et al., 2024). Hybrid MBE was used for adsorption-controlled growth of epitaxial BaTiO$900^\circ\text{C}$5 on an SrO sacrificial layer, using solid-source Ba and volatile TTIP for Ti (Choo et al., 4 Oct 2025). In high-mobility GaAs/AlGaAs heterostructures, MBE supported two-dimensional electron gases with mobility in excess of $900^\circ\text{C}$6 at density $900^\circ\text{C}$7 (Gardner et al., 2016).

This body of work suggests that, in its primary materials-science sense, MBE is not a single recipe but a platform defined by ultra-high-vacuum beam delivery, in situ diagnostics, and unusually strong coupling between source purity, substrate condition, flux ratio, and nonequilibrium kinetics.

2. Growth regimes, diagnostics, and process control

A recurrent theme is that MBE growth windows are narrow and often strongly nonlinear. In GaP nanowires, Be doping can proceed through side-facet VS growth or through the Ga droplet in self-catalyzed VLS growth, but high Be flux causes droplet inflation, suppression of axial growth, or droplet loss and nanowire coalescence; stable high-doping VLS growth was achieved by raising $900^\circ\text{C}$8 to $900^\circ\text{C}$9 and setting V/III $610\!-\!630^\circ\text{C}$0 (Sharov et al., 2024). For Si-doped n-type GaP, direct VLS growth did not yield high carrier concentrations because Si is amphoteric in GaP and has low probability of incorporation on gallium sites under Ga-rich conditions; a two-step core/shell protocol with a VLS core and a VS shell was therefore introduced (Sharov et al., 2024).

In Al on InGaAs, morphology changes abruptly with Al flux. At 0.1 Å/s and 0.5 Å/s, growth remained in a 3D island mode for nominal 100 Å deposition, while at 1.5 Å/s and above the growth quickly transitioned into island coalescence and a uniform 2D Al layer (Elbaroudy et al., 2024). The switch occurred over an Al flux increase of less than 1%, and higher flux also reduced the effective thermal load because deposition finished faster (Elbaroudy et al., 2024). The interpretation given is kinetic: higher Al arrival rates increase nucleation density and accelerate coalescence.

For $610\!-\!630^\circ\text{C}$1-Ga$610\!-\!630^\circ\text{C}$2O$610\!-\!630^\circ\text{C}$3 homoepitaxy, the MEXCAT-MBE study identifies the substrate offcut as a central kinetic control parameter. Offcuts of 0°, 2°, 4°, and 6° along [001] introduce (-201) step edges on (100) terraces, with terrace widths decreasing as offcut increases; growth rate rises monotonically with offcut, reaching around 1.5 nm/min and approximately 45% Ga incorporation at 6° (Mazzolini et al., 2020). The interpretation is that step edges act as preferential nucleation sites, favoring step-flow when terrace width approaches the adatom diffusion length (Mazzolini et al., 2020).

RHEED remains the principal in situ diagnostic tool in several of these studies. A deep-learning pipeline for GaAs deoxidation monitoring compresses each RHEED frame into a latent vector using an unsupervised convolutional autoencoder and classifies oxidation state from sequences of latent vectors using a supervised convolutional classifier (Khaireh-Walieh et al., 2022). The method used 7644 images recorded at 24 fps while the substrate rotated at 12 rpm, reached essentially 100% validation and test accuracy for $610\!-\!630^\circ\text{C}$4 and $610\!-\!630^\circ\text{C}$5, and detected deoxidation within a few seconds of expert human judgment without requiring auxiliary information such as rotation angle or temperature (Khaireh-Walieh et al., 2022). This suggests that MBE process control is increasingly being formalized as a sequence-classification problem rather than a purely operator-dependent visual skill.

3. Electrical, structural, and functional outcomes in MBE-grown materials

A central reason for using MBE is that growth conditions map directly onto electronic and functional properties. In GaP nanowires, conductive AFM combined with Silvaco Atlas modeling showed that doping concentration dominated current magnitude and device behavior; the reported strategies reached $610\!-\!630^\circ\text{C}$6 for GaP:Si core/shell nanowires and $610\!-\!630^\circ\text{C}$7 for GaP:Be nanowires (Sharov et al., 2024). For n-type GaP, photo-assisted Kelvin Probe Force Microscopy established n-type conductivity by positive photovoltage under UV, and lithographically defined Ni/Pd/Si/Al ohmic contacts were validated up to $610\!-\!630^\circ\text{C}$8 without shorting (Sharov et al., 2024).

In ultrathin Al films, MBE growth reduced static and dynamic defect signatures relative to electron-beam evaporation. Ten-nanometer MBE-grown Al on sapphire with an in situ 3 nm Al$610\!-\!630^\circ\text{C}$9O$_2$0 capping layer showed lateral grain size of about 42 nm and $_2$1-rocking-curve FWHM of 0.015°, compared with about 9 nm grains in electron-beam films (Sahu et al., 2 Jul 2025). At 300 K, the noise magnitude in MBE-grown films was about three times lower, corresponding to $_2$2 values about ten times lower than in electron-beam films; the analysis linked $_2$3 noise to thermally activated TLS at grain boundaries (Sahu et al., 2 Jul 2025). For quantum and low-noise electronics, the result is not merely better crystallinity but a lower density of dynamic defects in the relevant activation window.

MBE-grown van der Waals films have also been integrated post-growth through polymer-assisted dry transfer. Using PCL, PDMS support, TRT, and roller-assisted peeling, full-film transfer achieved areas up to $_2$4 with yield $_2$5 (Li et al., 22 Feb 2025). AFM showed RMS roughness increases of only about 1 nm, Raman signatures of MoSe$_2$6 and WSe$_2$7 were unchanged after transfer, and for Fe$_2$8GeTe$_2$9/Bi$_2$0Te$_2$1 the Curie temperature remained at $_2$2 K while hysteresis loop shape and coercivity were largely unchanged (Li et al., 22 Feb 2025). This addresses a long-standing limitation of epitaxial MBE films, namely strong substrate adhesion that impedes heterogeneous integration.

Other studies emphasize that MBE does not automatically optimize every property. In MBE-grown MoTe$_2$3 and MoSe$_2$4, abundant grain boundary and edge defects accelerated carrier recombination by about 50 times relative to exfoliated samples; measured lifetimes changed from 1087 ps to 16 ps in MoTe$_2$5 and from 2206 ps to 47 ps in MoSe$_2$6 (Chen et al., 2018). In few-layer MoTe$_2$7 grown on CaF$_2$8 and GaAs, GI-XRD indicated grain sizes of about 90 Å with significant twinning and XPS showed Te:Mo ratios greater than 2, while 2H films were stable only up to about $_2$9 and oxidized rapidly in air (Vishwanath et al., 2017). In FeSe, MBE-grown films on YAlO$_2$0(110) were superconducting with $_2$1 K but exhibited microcracks down to the substrate and roughness of about 100 nm, in contrast to smoother sputtered films (Venzmer et al., 2015). These results clarify a common misconception: epitaxial control does not by itself guarantee optimal morphology, defect landscape, or surface stability.

4. Molecular beam epitaxy as a continuum model and numerical-analysis subject

In numerical analysis, MBE denotes a class of fourth-order gradient-flow PDEs used to model thin-film evolution. One formulation considered is the no-slope-selection model

$_2$2

with energy

$_2$3

For this model, a third-order BDF3/EP3 discretization was shown to satisfy a modified discrete energy dissipation law under the mild constraint $_2$4, $_2$5, and to possess unconditional uniform energy boundedness for any $_2$6 without stabilization terms or fictitious variables (Li et al., 2021). The same work reports a third-order error bound $_2$7 under sufficient regularity (Li et al., 2021).

A distinct sinc-type MBE model replaces the classical slope-selection nonlinearity with

$_2$8

the $_2$9-gradient flow of

$_3$0

Its advantage is that all derivatives of the nonlinearity are uniformly bounded (Cheng et al., 2021). For this model, a first-order IMEX scheme satisfies discrete energy dissipation for $_3$1, and a BDF2 scheme satisfies a modified energy dissipation property for $_3$2, while both admit unconditional time-step-independent uniform energy bounds (Cheng et al., 2021).

A further extension introduces memory through a Caputo derivative in the time-fractional MBE model with slope selection: $_3$3 For a variable-step L1 discretization, the paper establishes an asymptotically compatible $_3$4-norm error estimate under the CSS-consistent step-size condition

$_3$5

and shows preservation of volume conservation, variational energy dissipation, and $_3$6-norm boundedness (Yang et al., 2022). In this mathematical lineage, “MBE” names a PDE family whose analysis is organized around discrete energy laws, high-order semi-implicit schemes, and compatibility with physical invariants.

5. Many-body expansion in electronic-structure theory and learned potentials

In quantum chemistry and molecular modeling, MBE denotes many-body expansion, a decomposition of a total energy into one-body, two-body, three-body, and higher-order contributions. In the fragment-based graph-neural-network framework FB-GNN-MBE, the total energy of an $_3$7-fragment system is approximated as

$_3$8

with recursive definitions of $_3$9 and $_2$0 (Chen et al., 10 Apr 2026). The approach uses ab initio QM for 1B energies and FB-GNNs for 2B and 3B terms, reporting chemical accuracy across water, phenol, and mixture benchmarks; for example, PAMNet-MBE achieved MAE 0.2766 kcal/mol and $_2$1 for water 2B energies, and MAE 0.0109 kcal/mol and $_2$2 for water 3B energies (Chen et al., 10 Apr 2026). A teacher–student transfer-learning protocol was then used to adapt a heavy-weight teacher trained on mixed-density water clusters to a light-weight student fine-tuned on $_2$3, enabling prediction of 2B and 3B energies for variously sized water clusters without retraining (Chen et al., 10 Apr 2026).

MBE also appears in multireference wavefunction theory. The MBE-CASSCF method replaces the exponentially scaling CASCI step of CASSCF with an MBE-FCI expansion over orbital subsets, allowing approximate construction of RDMs or the generalized Fock matrix from small CASCI increments (Greiner et al., 2024). Orbital optimization uses a hybrid first-order strategy combining Super-CI and quasi-Newton L-BFGS steps (Greiner et al., 2024). The method was benchmarked up to active spaces of 50 electrons in 50 orbitals in iron(II) porphyrin and found to converge to within chemical accuracy in smaller calibration cases (Greiner et al., 2024).

A further large-scale ab initio realization is multi-GPU MBE(3)-OSV-MP2, where the third-order many-body expansion decomposes local MP2 amplitudes into 1-body, 2-body, and 3-body localized orbital clusters (Liang et al., 17 Mar 2026). The GPU implementation incorporates Jacobi-Pipek-Mezey localization, randomized OSV generation, direct integral regeneration, and custom CUDA kernels for local operations (Liang et al., 17 Mar 2026). It reports empirical $_2$4 scaling, 84% parallel efficiency up to 24 GPUs, a 40-fold wall-time speedup over canonical RI-MP2 for $_2$5, and full MBE(3)-OSV-MP2 energies for a 784-atom insulin peptide in 24 minutes with cc-pVDZ and 6.4 hours with cc-pVTZ on eight NVIDIA A800 GPUs (Liang et al., 17 Mar 2026). Across these works, many-body expansion functions as a systematic locality-exploiting reduction strategy for otherwise intractable electronic-structure calculations.

6. Multiplier Bootstrap-based Exploration and the acronym’s broader disciplinary scope

In sequential decision theory, MBE denotes Multiplier Bootstrap-based Exploration, a bandit algorithm applicable to any reward model amenable to weighted loss minimization (Wan et al., 2023). Its core step is a weighted empirical-risk objective augmented by pseudo-rewards 0 and 1: $_2$6 For $_2$7-armed bandits with sub-Gaussian rewards, the method is proved to achieve both instance-dependent and instance-independent rate-optimal regret bounds (Wan et al., 2023). The paper emphasizes that naïve multiplier bootstrap without pseudo-rewards can incur linear regret, whereas pseudo-reward injection guarantees persistent exploration (Wan et al., 2023). Empirically, MBE matched or nearly matched Thompson Sampling in simulations and performed competitively on real-data problems including cascading bandits, combinatorial semi-bandits, and multinomial logit bandits (Wan et al., 2023).

The coexistence of molecular beam epitaxy, many-body expansion, and Multiplier Bootstrap-based Exploration under the same acronym is not merely a terminological accident. A plausible implication is that “MBE” now functions as a strongly field-local abbreviation whose interpretation must be inferred from disciplinary context, accompanying notation, and surrounding methodology. In materials and device papers it typically refers to beam-based epitaxial synthesis; in quantum chemistry it refers to order-truncated interaction expansions; in learning theory it refers to bootstrap-driven exploration. The technical literature therefore treats MBE less as a single concept than as a family of domain-specific formalisms sharing an emphasis on controlled incremental structure.