From Atomic Structure to Device-Relevant Parameters in QuantumATK

Silicon band structure, projected density of states, and electron effective mass in Synopsys QuantumATK

Introduction to QuantumATK and Silicon Band Structure

Semiconductor engineers routinely work with quantities such as band gap, density of states, and carrier effective mass. At the device scale, these values often appear as inputs to compact models or continuum simulations. At the atomic scale, however, they emerge from the electronic structure of the material. Understanding that connection is important when established material parameters are unavailable, when a new material is being screened, or when strain, composition, defects, and interfaces begin to change bulk behavior.

This article presents a practical, reproducible QuantumATK workflow for bulk silicon. The purpose is not to rediscover silicon. Instead, silicon serves as a controlled benchmark for learning how calculator settings, convergence choices, and electronic-structure analysis turn an atomic model into device-relevant quantities.

Why Begin with Bulk Silicon?

Silicon is a strong starting point because its crystal structure and electronic properties are well characterized. That makes it possible to distinguish a workflow problem from genuinely new physics. If a simulation does not reproduce the expected qualitative features of silicon, the model setup, numerical convergence, or analysis choices should be checked before the same workflow is applied to a less familiar semiconductor.

The study is designed to answer four questions:

1. Is the calculated band gap direct or indirect, and where are the valence-band maximum and conduction-band minimum located?
2. What electronic states are available near the band edges, and what can orbital projections add to the interpretation?
3. How strongly does the conduction band curve near its minimum?
4. How do longitudinal and transverse electron effective masses differ in silicon?

QuantumATK Silicon Band Structure Simulation Workflow

The workflow follows the current QuantumATK Y-2026.03 introductory tutorial for band structure, projected density of states (PDOS), and effective-mass calculations. The public tutorial recommends an LCAO density-functional-theory calculation with the HSE06 hybrid functional for bulk silicon. It also recommends reducing the electronic broadening from the general default to 300 K for semiconductor calculations. Default numerical settings are useful for starting a workflow, but the final settings used in a published result should be justified through convergence testing.

1. Build and verify the bulk crystalline silicon (diamond structure) in NanoLab.
2. Optimize the geometry or verify that the selected lattice constant is consistent with the chosen electronic-structure method.
3. Converge the k-point sampling and density mesh cutoff for the quantity of interest.
4. Calculate the final electronic structure with the selected LCAO basis and HSE06 exchange-correlation functional.
5. Generate the band structure along the automatically detected high-symmetry route.
6. Restart from the converged bulk configuration to calculate PDOS without repeating the self-consistent calculation unnecessarily.
7. Locate the actual conduction-band minimum in the calculated band structure and evaluate electron effective mass in longitudinal and transverse directions.

Source: Original QuantumATK Y-2026.03 calculations for this study; workflow choices follow the Synopsys introductory tutorial cited in the references.

1. Structure Preparation and Geometry

The bulk silicon structure can be imported from the NanoLab crystal database as Silicon (alpha). Before calculating electronic properties, the geometry must be treated consistently. A band structure obtained from an unverified lattice constant may look reasonable while still shifting the band edges or band curvature.

The HSE06 geometry optimization converged in two steps to a cubic lattice constant of 5.455180 Å and a primitive-cell volume of 40.585 ų. The final maximum force was 2.953 × 10⁻⁴ eV/Å and the reported stress error was 0.00252 GPa. These residuals confirm that the optimized structure was well converged before the electronic-property calculations.

2. Calculator Selection and Convergence

A standard local or semi-local DFT calculation is useful for rapid setup and qualitative checks, but common LDA and GGA functionals underestimate the band gap of silicon. For the final electronic structure, this study uses the HSE06 hybrid functional recommended in the current QuantumATK introductory workflow for Si. The point is not that one functional is universally best; rather, the physical quantity and required accuracy should guide the model choice.

Two numerical parameters deserve explicit attention: k-point sampling and the density mesh cutoff. The total energy may appear stable before the band edge or effective mass is fully stable, so convergence should be evaluated using the output that matters to the engineering question. A practical approach is to converge with a lower-cost functional and then confirm the selected numerical settings in the final HSE06 calculation.

The production calculation used a 9 x 9 x 9 k-point grid and a 125 Ha density mesh cutoff. Increasing the k-point grid from 9 x 9 x 9 to 11 x 11 x 11 changed the indirect gap from 1.156048 to 1.155645 eV – only 0.000403 eV. At 11 x 11 x 11, changing the mesh cutoff from 100 to 150 Ha changed the indirect gap by just 0.000006 eV. The 9 x 9 x 9 / 125 Ha settings therefore provide a well-converged result at lower hybrid-DFT cost.

3. Silicon Band Structure in QuantumATK

The band structure maps allowed electron energies as a function of wave vector along selected high-symmetry directions in reciprocal space. For semiconductor analysis, the most important features are the valence-band maximum (VBM), the conduction-band minimum (CBM), and the energy separation between them.

Silicon should exhibit an indirect fundamental gap: the VBM occurs at the Γ point, while the CBM lies along the path toward X rather than at the same k-point. Reproducing this topology is an important qualitative validation. The numerical band-gap value provides a second, more demanding validation of the selected electronic-structure model.

The final HSE06 band structure gives an indirect gap of 1.155214 eV and a minimum direct gap of 3.335291 eV. With energy referenced to the Fermi level, the VBM is -0.570723 eV at Γ and the CBM is +0.584491 eV at k = (0.425, 0, 0.425), 85% of the Γ-to-X distance. This reproduces the defining topology of silicon: the valence and conduction extrema occur at different wave vectors.

4. Density of States and Orbital Character

The density of states answers a different question from the band structure. Instead of showing how energy changes along a reciprocal-space path, it shows how many electronic states are available in each energy interval. The absence of states across the band gap provides a useful cross-check on the band-structure result.

Projected density of states separates the total DOS into selected atomic, shell, or orbital contributions. In elemental bulk silicon the interpretation is comparatively simple, but the same analysis becomes much more valuable for compounds, interfaces, dopants, and defects. At an interface, for example, site-projected DOS can help reveal whether new states appear inside an otherwise clean material band gap.

The tetrahedron-method PDOS calculation reports an indirect gap of 1.155486 eV, within 0.000272 eV of the band-structure value. The shell projection shows predominantly p-like character on the valence side. Above the conduction edge, the supplied basis decomposition contains mixed s-, p-, and substantial d-like polarization contributions. Because these curves are projections onto localized basis functions, the shell weights should be interpreted as analysis components rather than perfectly pure atomic orbitals.

5. Electron Effective Mass

Effective mass measures the local curvature of an electronic band. In one dimension, 1/m* = (1/ℏ²)(d²E/dk²). A strongly curved band corresponds to a smaller effective mass; a flatter band corresponds to a larger effective mass.

The silicon conduction-band valleys are anisotropic, so a single scalar electron mass is not sufficient. The mass along the valley axis is different from the mass perpendicular to it. QuantumATK can evaluate these directions from the actual CBM selected in the band-structure analyzer.

At the calculated CBM, k = (0.425, 0, 0.425), QuantumATK reports a longitudinal electron mass of 0.850 mₑ and a transverse mass of 0.186 mₑ. Their ratio, 4.57, quantifies the strong valley anisotropy. Using the conductivity-mass relation m_c = 3/(1/mₗ + 2/mₜ) gives 0.251 mₑ. Including silicon’s six equivalent valleys gives a density-of-states mass of 1.02 mₑ.

The longitudinal direction follows the valley axis; the transverse direction is perpendicular to it. Conductivity and DOS masses are derived from the two fitted curvatures.

Results Summary

The verified result set is summarized below. Band structure and PDOS independently reproduce the same approximately 1.155 eV indirect gap, while the directional masses capture the anisotropy of silicon’s conduction-band valleys.

What These Outputs Mean for Semiconductor Engineering

Band Gap
The band gap establishes the energy separation between occupied valence states and available conduction states. It influences carrier generation, leakage mechanisms, tunneling windows, and optical absorption. A band-gap prediction must therefore be evaluated with an electronic-structure method appropriate for the intended use. The minimum direct electronic gap provides an energy scale for vertical transitions, but predicting absorption also requires optical matrix elements and, when relevant, excitonic effects.

Density of States
DOS describes the availability of electronic states as a function of energy. PDOS adds spatial or orbital information. The same workflow used here for bulk silicon can later be applied to determine which material, layer, atom, or orbital contributes a state near an interface or inside a nominal band gap.

Effective Mass
Effective mass connects the curvature of the electronic bands to the way carriers respond to an applied force. It is relevant to density-of-states models, injection, confinement, and transport calculations. Effective mass is not, by itself, mobility. A mobility prediction must also account for scattering by phonons, impurities, defects, interfaces, and other mechanisms.

A Validated Starting Point for Larger Workflows
The value of this benchmark is procedural. Once the structure, calculator, convergence, and analysis sequence are trusted, the same logic can be extended to strained silicon, SiGe alloys, compound semiconductors, two-dimensional materials, interfaces, and nanoscale devices. The numerical cost and physical complexity increase, but the validation discipline remains the same.

Practical Lessons from the QuantumATK Silicon Band Structure Workflow

1. Converge the property you plan to report. Total-energy convergence alone does not guarantee a stable band edge or effective mass.
2. Separate geometry and electronic-structure decisions. Record which method produced the lattice constant and which method produced the final bands.
3. Use hybrid or other band-gap-aware methods when a quantitative semiconductor gap is required; do not present a known GGA underestimate as a material prediction.
4. Select the effective-mass point from the calculated CBM. Verify the k-point coordinates and longitudinal/transverse directions for the actual cell convention.
5. Use restart capability to add PDOS or other analyses to an already converged bulk configuration when possible.
6. Treat electron and hole effective masses differently. Hole masses near degenerate valence bands can require spin-orbit coupling and careful control of the finite-difference step.

Engineering and Business Value

For established silicon, the primary value of this study is workflow validation. For a new semiconductor or engineered interface, the same calculations can answer material questions before a complete device is fabricated. Candidate materials can be compared on a consistent basis, uncertainties in model choice can be identified early, and atomistic outputs can support higher-level transport or device modeling.

QuantumATK brings structure building, electronic-structure calculation, analysis, visualization, and Python-based automation into one environment. That integration can reduce the manual handoffs that often separate atomic models from the properties engineering teams need to interpret.

From Silicon Benchmark to Semiconductor Design Workflow

A silicon band-structure calculation is simple enough to be approachable, but rich enough to expose the decisions that determine whether an atomistic result is credible. The essential steps are to verify the geometry, converge the numerical settings, select an electronic-structure method appropriate for the band gap, and interpret band structure, PDOS, and effective mass as complementary outputs.

The next step is to carry this validated bulk-material workflow into a more application-focused study. Two natural extensions are the optical response of silicon and an atomistic NEGF analysis of a silicon p-n junction under bias.

Need help with semiconductor materials modeling in QuantumATK?

SimuTech Group can help your team use Synopsys QuantumATK to evaluate electronic structure, band gaps, density of states, effective mass, and other atomistic material properties that support semiconductor and device-level modeling.

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