Calculation of single-crystal silicon via Density Functional Theory#

We introduce Density Functional Theory (DFT) calculation for electronic condition of single-crystal silicon (Si) with Advance/NanoLabo, an integrated GUI for nanomaterials.

We carried out structural optimization of unit cell and estimated physical quantities like lattice constant and cohesive energy theoretically. Also, we analyzed electronic structures such as band dispersion and density of states (DOS).

The entire process from modeling to running calculation and analysis of results was executed on Advance/NanoLabo.

Note

We used Quantum Espresso¹ as a solver of calculation for electronic states, but you can use Advance/PHASE as a solver too if you obtain the license separately.

Calculation model#

Advance/NanoLabo has a function to search external materials databases (Materials Project², PubChem³) for crystal and molecular structures. You can search structure information files on the databases from substance name or chemical formula and configure calculation models.

We intended to calculate single-crystal silicon, so we typed “Si” in the search window on Advance/NanoLabo. In the figure below, various search results are shown, and we selected diamond primitive cell which is the most energetically stable.


The search screen for structures	SCF calculation conditions

Structural optimization#

Next, we executed structural optimization. In structural optimization, the given lattice structure is relaxed to equilibrium state by reducing forces working on each atom sufficiently.

The structural optimization of single-crystal Si converged in 4 steps, and the lattice constant of the relaxed structure⁴ is larger than that of the given structure by about 0.1 Å.

By analyzing the relaxed structure, the optimized lattice constant of single-crystal Si was obtained as 5.49 Å. Relative error between the reported experimental lattice constant of 5.43 Å⁶ and the calculated lattice constant is 1.0 %, so we can see that the calculated value is comparable with the experimental value.

Also, we obtained cohesive energy of 5.71 eV which is calculated by taking difference between energy of the optimized structure per an atom and energy of an isolated atom⁵. Cohesive energy of single-crystal Si reported experimentally is 4.63 eV⁶, and the calculated value is overestimated by about 20 %.


Structural optimization conditions	Configuration of the calculation job	Running calculation

List of results	Change of energy	Change of lattice constant

Analysis of electronic structure#

Finally, we calculated electronic structures such as band dispersion and density of states (DOS) from the optimized structure of single-crystal Si.

DOS means the number of states electrons allowed to occupy at a given energy and its unit is number of states/energy. The calculated DOS shows band gap of about 0.7 eV around Fermi level. Band gap of single-crystal Si reported experimentally is 1.17 eV⁶, so we can see that the calculated value is underestimated compared to the experimental value.

In band dispersion diagram, the horizontal axis corresponds to wave vector, and the vertical axis corresponds to energy eigenvalue. The wave number space is three-dimensional, so high symmetric points of Brillouin zone such as Γ-point and X-point are selected and band dispersion diagram is plotted along them. The calculated band dispersion diagram shows wide bands in valence band and conduction band which comes from sp orbit.


DOS calculation conditions	DOS	Band calculation conditions

Band dispersion diagram

Calculation of single-crystal silicon via Density Functional Theory#

Calculation model#

Structural optimization#

Analysis of electronic structure#

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