Calculate Electronic Band Gap with GW Approximation¶
This tutorial page explains how to calculate the electronic band gap of a semiconducting material based on Density Functional Theory. We consider crystalline silicon in its standard equilibrium cubic-diamond crystal structure, and use VASP as our main simulation engine during this tutorial.
What sets the present tutorial apart from the other tutorial on band gap calculations is the employment of the "GW Approximation", which is reviewed in the subsequent paragraph. This method is more compute-intensive, however similarly to the HSE method it yields more accurate results closer to experimental values, thus rectifying the tendency of the GGA to underestimate the size of the band gap. More information on this approximation, together with a demonstration of its application and results on a sample set of materials, can be found in Ref. 1.
The GW Approximation¶
For the sake of this short introduction, it suffices to know that the GWapproximation is obtained using a systematic algebraic approach on the basis of Green function techniques, the most suitable approach for studying excited-state properties of extended systems. It constitutes an approximate expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the crystalline atoms. The implementation of theapproximation relies on a perturbative treatment starting from Density Functional Theory.
We shall now describe the computational implementation of the GW Approximation for computing electronic band gaps on our platform, illustrating the various steps constituting the overall Workflow. For the present explanation, we consider the example case of the VASP modeling engine. Further information on how the GW method is supported by VASP can be retrieved in Refs. 5 and 6.
Workflows performing GW calculations follow a three-step procedure:
1. Preliminary Ground State SCF Calculation¶
The first subworkflow step in the overall GW Workflow is a standard self-consistent field (scf) total ground state energy calculation, providing the ensuing steps of the workflow with the wavefunctions of the material structure under investigation (GW calculations always require a one-electron basis set).
2. Many Bands SCF Calculation¶
A significant number of empty bands is required for GW calculations, such that it is typically better to perform the calculations in two steps, as two separate subworkflows: first the above-mentioned standard ground-state SCF calculation with only a few unoccupied orbitals, and secondly a calculation over a large number of unoccupied orbitals (bands), by setting the
NBANDS VASP tag to a large value.
3. GW Step¶
The actual GW calculation is done in this final subworkflow step. Here different GW flavors are possible and are selected with the
ALGO VASP tag.
The "Single Shot" quasi-particle energies method, often referred to as G0W0, is the simplest GW calculation, and computationally the most efficient one. A single-shot calculation calculates the quasi-particle energies from a single GW iteration by neglecting all off-diagonal matrix elements of the self-energy, and employing a Taylor expansion of the self-energy around the DFT energies.
After a successful G0W0 run, VASP will write the quasi-particle energies into the main "OUTCAR" output file for every k-point in the Brillouin zone of the crystal structure under investigation.
In the present example we calculate quasi-particle energies on the grid of k-points. This might not be the most accurate approach, as points on the grid might not fall exactly onto the band extrema for conduction and valence band, however, it is robust and can provide a very reasonable approximation. An intelligent interpolation technique might be used to further extract band dispersions along symmetry paths.
Creating and Executing Job¶
Apart from this, the same procedural instructions as in the other band gap calculation tutorial should be followed for creating and launching the corresponding GW-based electronic band gap Job through our Web Interface, and for inspecting the associated results.
In the video animation below, we outline the procedure for creating and executing an electronic band gap calculation job via the GW Approximation, considering crystalline silicon as our example material and employing VASP as the main simulation engine. We conclude by inspecting the corresponding results displayed under the Results Tab of Job Viewer.
Computational cost of GW calculations
GW calculations are in general quite computationally demanding. We therefore recommend the employment of at least 8 computing cores. For larger calculations, OF queues will have faster turnaround than the OR queues considered in the video.
Comparison with Experimental Value¶
The calculated value of 1.094 eV for the indirect band gap of silicon is in better agreement with the experimental value for this material (1.17 eV 7) than the alternative case of standard band gap calculations performed with the Generalized Gradient Approximation (GGA), whose shortcomings are assessed in another tutorial page.
This provides an example of how the GW Approximation can result in improved precision in the estimation of important material properties than more traditional approaches within DFT.