GPU-Accelerated Tight-Binding Models

• GPU-accelerated tight-binding models are computational methods combining tight-binding theory with GPU parallelism to simulate electronic structures accurately and efficiently.
• They utilize compact atomic orbital frameworks, sparse matrix techniques, and iterative solvers like Chebyshev expansions to tackle complex phenomena in large-scale systems.
• These models achieve significant performance gains in quantum transport and heterostructure simulations, enabling linear scaling for systems with millions of atoms.

GPU-accelerated tight-binding (TB) models fuse the quantum-mechanical foundation of the TB method with the computational capabilities of modern GPUs to achieve high-throughput, scalable, and accurate electronic structure calculations across a range of material systems. Recent research demonstrates that GPU architectures substantially increase the performance and tractability of TB calculations, enabling the simulation of complex phenomena in large-scale devices, heterostructures, and correlated systems.

1. Tight-Binding Models and Their Formulation

TB models employ a compact basis, typically using atomic orbitals as a linear combination (LCAO framework). The Hamiltonian elements $H_{ij}$ consist of on-site energies (diagonal) and hopping terms (off-diagonal), with the form

$H_{ij} = \epsilon_i\, \delta_{ij} + t_{ij}$

where $\epsilon_i$ is the onsite energy and $t_{ij}$ is the hopping amplitude between orbitals on atoms $i$ and $j$. Modern TB methods leverage maximally localized Wannier functions for ab-initio parameter extraction (Javvaji et al., 19 Apr 2024), incorporate symmetry constraints (1305.60892103.15655), and extend to multi-band systems with spin-orbit coupling (SOC) (Liu et al., 2013).

GPU acceleration is particularly well-suited for TB models due to:

• Small Hamiltonian matrix sizes per $k$-point and high density of independent calculations.
• Exploitation of symmetry-imposed structure for efficient matrix assembly.
• Sparse matrix storage in extended systems for optimal memory usage (1311.60821705.01387Wang et al., 8 Sep 2025).

2. Parallelism and Scaling in GPU-Accelerated TB Workflows

The embarrassingly parallel nature of TB calculations involves separate diagonalizations or time propagations at each $k$-point in the Brillouin zone or for each atom/bond in large systems. This facilitates the development of hybrid codes (CUDA, PyTorch, Cython/Fortran for critical components (Li et al., 2022Wang et al., 8 Sep 2025)) with linear scaling properties in system size:

• Sparse matrix-vector multiplications for iterative solvers and time evolution (1705.01387Wang et al., 8 Sep 2025).
• Chebyshev polynomial expansion for quantum propagators and kernel polynomial method (KPM) for spectral functions (1705.01387Wang et al., 8 Sep 2025).
• Message-passing neural networks for environment-dependent TB parameter prediction (Wang et al., 8 Sep 2025).

Performance benchmarks indicate that frameworks such as GPUTB can compute the density of states (DOS) for pristine graphene up to 100 million atoms, achieving $O(N)$ scaling and successfully addressing device-scale configurations (Wang et al., 8 Sep 20251705.01387Li et al., 2022).

3. Model Specifics: Orbitals, Symmetry, and Multi-band Effects

TB models derive accuracy and physical fidelity from basis selection (e.g., $d_{z^2}$, $d_{xy}$, $d_{x^2-y^2}$ for MX$_2$ (Liu et al., 2013)) and symmetry constraints imposed by the crystal (trigonal prismatic coordination, $D_{3h}$ point group symmetry (Liu et al., 2013)). Nearest-neighbor (NN) and third-nearest-neighbor (TNN) hopping parameters capture low-energy electronic structure near valley points (±$K$), Berry curvature, and band dispersion.

For SOC, a block-diagonal formulation is used, with onsite coupling: $$H_{SOC}(k) = I_2 \otimes H_0(k) + \frac{\lambda}{2} \begin{pmatrix} L_z & 0\ 0 & -L_z \end{pmatrix}$$ where $L_z$ is the orbital angular momentum matrix in the relevant basis (Liu et al., 2013). GPU implementations efficiently handle small Hamiltonian blocks, parallel diagonalization, and postprocessing for quantities such as Berry curvature.

4. Environment-Dependent and Machine Learning TB Models

Transferability across materials, strain, and bonding environments is achieved through environment-dependent models (1311.60822509.06525). Onsite and hopping parameters are functions of atomic positions, bond lengths, and environment descriptors: $$E_{il} = \epsilon_{il} + \sum_{j \in NN} k_l \exp[-p_l (R_{ij}/R^{(0)} - 1)]$$ for onsite energies, and similar forms for couplings. Chebyshev polynomial expansions and message-passing neural networks are used to encode local atomic environments, training directly on ab-initio or experimentally relevant datasets (Wang et al., 8 Sep 2025). The Slater–Koster formalism provides the necessary conversion coefficients for directionality and orbital overlap (Wang et al., 8 Sep 2025).

This adaptability enables accurate simulations of alloys, strained nanostructures, heterojunctions (e.g., h-BN/graphene), and finite-temperature structures. The approach also enhances mapping to different basis sets and exchange-correlation functionals.

5. Quantum Transport and Large-Scale Simulations

Quantum transport calculations in GPU-accelerated TB frameworks are built on linear-scaling techniques (LSQT), which incorporate:

• Velocity autocorrelation functions for conductivity:

$$\sigma(E, t) = \frac{2e^2}{\Omega} \, \text{Tr}[ \delta(E-\hat{H})\, \text{Re}( \hat{V} \hat{V}(t) ) ]$$

• Mean-square displacement and running conductivity for localization and diffusive regime analysis (1705.01387).
• Random phase approximations for trace evaluation, sparse matrix-vector multiplication, and time evolution via Chebyshev expansion.

Disorder is modeled via random onsite potentials (Anderson disorder), variations in hopping integrals, and more complex environmental modifications (1705.01387). GPUQT and GPUTB demonstrate linear scaling with system size and enable studies of diffusive and localized transport in lattices containing millions of sites (1705.01387Wang et al., 8 Sep 2025).

6. Representative Applications and Benchmarks

Recent GPU-accelerated TB works demonstrate applicability to:

• Quantum wells and oxide heterostructures, where TB eigenvalue spectra accurately capture confinement and subband quantization (Zhong et al., 2013).
• Large-scale III–V nanowire MOSFETs and TFETs, via mode-space (MS) and NEGF techniques, achieving up to $10,000\times$ speedup over real-space methods (Afzalian et al., 2017).
• Metal alloys and nanowires (Cu, AuAg), enabling predictive simulations of ballistic conductance, effective resistivity, and electron scattering (Hegde et al., 2013).
• Warm dense matter simulations with transferable TB molecular dynamics, including orthogonal and nonorthogonal Hamiltonians and LCAO basis sets (Medvedev, 2019).

In graphene, GPUTB reproduces carrier concentration and room-temperature mobility relationships observed in experiments while maintaining accuracy for band structures in h-BN/graphene junctions (Wang et al., 8 Sep 2025).

7. Implementation Considerations and Limitations

The modularity and parallelism offered by GPU-accelerated TB frameworks provide substantial speedups; however, certain factors require attention:

• Memory and data layout must be optimized to avoid uncoalesced accesses and maintain high bandwidth efficiency (Hegde et al., 2013).
• Sparse matrix formats (CSR, ELLPACK) need careful management to handle varying coordination in extended and disordered systems (1705.01387).
• Strain, environment, and temperature dependence should be parameterized via either physical scaling laws or learned descriptors, with validation against ab-initio or experimental references (Wang et al., 8 Sep 2025).
• For time-resolved correlated simulations, iterative and analytical solvers must gracefully handle dynamic Hamiltonian updates and expanding basis sets (Medvedev, 2019).

While GPU acceleration enables large-scale computation and rapid postprocessing, the accuracy of TB models fundamentally depends on quality of parameterization, orbital selection, and inclusion of relevant physical effects (SOC, strain, many-body corrections).

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As evidenced by recent research (1305.60891311.6082Afzalian et al., 20171705.01387Medvedev, 2019Hu et al., 2021Li et al., 2022Candiotto, 8 Jan 2024Huang et al., 13 Mar 2024Javvaji et al., 19 Apr 2024Lee et al., 23 May 2025Wang et al., 8 Sep 2025), GPU–accelerated tight–binding models provide a powerful computational platform with linear scaling and adaptability, allowing precise modeling of electronic, transport, and many–body physics in complex and technologically relevant materials across length scales ranging from the atomistic to the device level.