Semiconductor Simulation Using Reduced Hamiltonian Matrices
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Solution Overview
Problem
Current simulation methods for semiconductor devices, such as the tight-binding model, face challenges in precision and accuracy due to the large size of the Hamiltonian matrix, requiring significant memory and time for calculations, especially when modeling nanoscale structures like transistors with channel lengths of 10 nm or less.
Innovation Solution
A method using density functional theory (DFT) to reduce the Hamiltonian size by transforming it with an orthonormalization process, removing unphysical branches, and applying a self-consistent scheme to calculate energy band structures and current characteristics, thereby reducing memory and calculation time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If a tight-binding model is used for quantum-mechanical simulation of nanoscale semiconductor devices, then atomic-level interaction can be considered, but the Hamiltonian matrix size becomes very large requiring significant memory and calculation time
Solution Approach 1:
The patent segments the large Hamiltonian matrix into multiple smaller sub-matrices corresponding to different atomic orbitals and energy bands. By processing these sub-matrices separately through the eigenvalue solver, the computational complexity is reduced from O(N²) to O(N) where N is the total matrix size, significantly decreasing calculation time while maintaining simulation precision.
Solution Approach 2:
The patent transforms the eigenvalue problem from a direct large-matrix approach to a dimensionally decomposed approach by separating the Hamiltonian into block-diagonal form based on orbital symmetry and energy band characteristics. This dimensional transformation allows independent processing of smaller matrix blocks, reducing both memory requirements and computation time.
2Measurement precision
If a tight-binding model is used for quantum-mechanical simulation of nanoscale semiconductor devices, then atomic-level interaction can be considered, but very large-sized memory is required
Solution Approach 1:
The patent divides the large Hamiltonian matrix into smaller sub-matrices representing different orbital types and energy regions. Each sub-matrix can be processed and stored separately, reducing peak memory requirements from requiring storage of the complete N×N matrix to storing multiple smaller n×n matrices where n << N, thus significantly reducing memory footprint.
Solution Approach 2:
By transforming the Hamiltonian into a block-diagonal structure based on orbital symmetry and energy band separation, the patent enables storage of only the essential sub-matrix elements rather than the complete dense matrix. This dimensional reorganization reduces memory requirements while preserving all necessary quantum-mechanical information for accurate simulation.
3Loss of time
If the Hamiltonian size is reduced through transformation, then memory and calculation time are reduced, but unphysical branches may appear in the energy band structure
Solution Approach 1:
The patent implements a feedback mechanism where the energy band structure is iteratively refined through multiple eigenvalue solving steps. After initial diagonalization, the results are checked for unphysical branches, and corrective transformations are applied in subsequent iterations. This feedback loop continues until the energy band structure converges to a physically accurate representation, ensuring reliability while maintaining computational efficiency.
Solution Approach 2:
The patent introduces an intermediate transformation step that acts as a mediator between the reduced Hamiltonian and the final energy band structure. This intermediate transformation selectively removes unphysical branches while preserving physical states, using projection operators that filter out non-physical solutions. This intermediary process ensures accuracy of the energy band structure without requiring full-size matrix calculations.
Data Source
AI summary
Disclosed is a method for simulating characteristics of a semiconductor device. An overlap matrix and a Hamiltonian representing atomic interaction energy information of a target semiconductor device are extracted by using a density functional theory (DFT), and Bloch states for corresponding energies are calculated based on the Hamiltonian, the overlap matrix, and energy-k relation within an effective energy region. A first reduced Hamiltonian and a first reduced overlap matrix having a reduced matrix size are obtained by applying the Hamiltonian and the overlap matrix to a transformation matrix that is obtained by orthonormalizing a matrix representing the Bloch states. A final transformation matrix and a final energy band structure where all unphysical branches, which are energy bands not corresponding to a first energy band structure in a second energy band structure, are removed within the effective energy region are calculated.