Effective mass of electrons in silicon

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It is well known that DFT methods, or to be more specific the LDA and GGA exchange-correlation functionals, are not very adept at predicing the band gap of semiconductors. They do, however, in many cases give rather accurate curvatures of the bands. We can use this to compute the effective mass of holes and electrons by fitting a parabola to the minimum/maximum of the conduction/valence bands.

In this tutorial we will, primarily, show how we can compute the electron effective mass for silicon. The effective mass is a parameter that depends on several assumptions and definitions, so it will be important to understand properly what goes on, instead of just using it as a black box. For other materials, these assumptions may be different!

Background

The lowest conduction band of Si has 6 equivalent minima, located along (101) (called Δ) and its permuted equivalent directions in reciprocal space. The minimum is located at (x,0,x) where x is about 0.425, or 85% of the distance to the first Brillouin zone boundary at X=(1/2,0,1/2). At this point, the energy isosurfaces are ellipsoids, thus you get different values for the effective mass depending on if you look in the longitudinal (along Δ) or transverse (perpendicular to Δ) directions. See Refs. [1-2] for more details.

By (1/2,0,1/2) we mean a k-vector K=(G_A+G_C)/2, where G_A, G_B, G_C are the three unit vectors that span the reciprocal unit cell. If we express this point in Cartesian coordinates (still in reciprocal space) we find that the Δ direction is parallel to K_Y. Obviously then, K_X and K_Z, which are parallel to G_B+G_C and G_A+G_B, respectively, are perpendicular to K_Y. Thus the transverse directions are (011) and (110), and the longitudinal one is (101).

What we now need is a script that can

generate a sequence of points k around a specified point k₀, in a given direction,
compute the energy eigenvalues E(k) for the lowest conduction band for these k-points
fit a parabola E = d|k-k₀|² + e|k–k₀| + f to the dispersion relation
compute the effective mass as m* = ħ²/2dm_e where m_e is the free electron mass

If we do this for the longtitudinal (L) and transverse (T) directions, we will obtain three values, m*_L, m*_T1, and m*_T2 (actually, for Si m*_T1 = m*_T2 = m*_T), which we can insert in the well-known expressions for the conductivity and density of states effective masses

m*_DOS = 3 / (1/m*_L + 2/m*_T)
m*_C = 6^2/3 (m*_L m*_T²)^1/3

where the factor 6 which enters the expression for the conductivity mass is the number of equivalent Δ valleys.

DFT calculation

The first step is to do a self-consistent DFT calculation for Si.

The details involved in such a task are described in a separate tutorial. There, the extended Hückel method was used, so you should modify the calculator to use ATK-DFT instead. Save the results in a new NetCDF file, not the same one as the Huckel calculation, to avoid mixing up the methods when analyzing the results.

Inspecting the so-obtained band structure (figure to the right; click the picture for bigger version) it's clear that the band gap is severly underestimated, but the shape is essentially correct; the gap is indirect and we do have a conduction band minimum at about 85% of the distance to X.

Compute the effective mass

We will now compute the effective mass of the lowest conduction band in Si.

Download the Effective Mass Analyzer (right-click and "Save as...") to a convenient location.
Drop the downloaded script on the Analyzer icon· on the main VNL toolbar.
Before proceeding, untick the "Auto update" checkbox, to avoid calculating results we do not need anyway.
Drop the NetCDF file with the Si DFT calculation on the drop-zone labeled "Drop NetCDF file here".
Change the k-point to [0.425, 0, 0.425] and the direction to [1,0,1] for the longitudinal mass.
Click "Update", and read off the computed effective mass 0.90 in the status panel at the bottom.
Change the direction to [1,1,0] for the transverse mass, and click "Update" again. This time we get the result 0.19.

The plots on the right-hand side of the Analyzer are used to verify that we indeed have a reasonable point around which to fit. The top left plot actually contains two curves, the band structure and the fitted parabola. Usually they are however too similar to distinguish, and therefore the lower left-hand plot shows their difference instead. The right-hand plot illustrates the nearest (in energy) bands above and below the one we use for the fitting, for the same k-points as those used to compute the effective mass. It is possible to zoom in/out in these plots.

With these values we can now compute the density of states mass m*_DOS=1.05 and m*_C=0.26, very close indeed to the reference values 1.08 and 0.26 (see e.g. http://ecee.colorado.edu/~bart/book/effmass.htm). Thus DFT actually does reproduce the effective mass of the electrons in the Δ valley very well.

You can now also try the same calculation with the extended Hückel method; if you followed the Si band structure tutorial you already have the necessary NetCDF file. The band minimum is not at x=0.452 this time, but closer to x=0.445. The results are very similar, however (m*_L=1.00 and m*_T=0.17).

Going further

The tool contains a couple of additional parameters which we have not mentioned.

It is possible to select the number of k-points that will be used to generate the band structure against which we fit the parabola, while "delta" controls how far away from the chosen k-point we go. The default is typically fine; if the band structure is strongly non-parabolic, the value can be decreased a bit perhaps.
If the NetCDF file contains more than one BulkConfiguration (perhaps the same calculation but several calculations, with different parameters), the "object id" can be used to select the relevant one.
Finally, we can use this analyzer to also compute the effective masses of different bands. In a semiconductor, the bands with index up to N/2, where N=the number of electrons in the unit cell, are always filled, and the script uses this to determine the band index of the lowest conduction band, which corresponds to a relative band index of 0, and this is the default. Thus, if we wanted to obtain the effective masses of the top three degenerate valence bands, we would set this parameter to -1, -2, and -3, respectively.

Hole masses in Si

For Si, we compute a light hole mass (at the Γ point [0,0,0]) m*_LH=0.16 (relative band index=-3), and for the two degenerate heavy hole bands the result is m*_HH=0.25. These values are obtained in the directions of the Cartesian axes (i.e. if we use the same directions as for the electron, along Δ from G to X). If, however, we look in the Λ direction from Γ to L (by setting the direction in the Analyzer to [1,1,1]), we find m*_HH=0.64 and m*_LH=0.09, a result of fact that the constant-energy surfaces for holes are strongly warped in Si, especially for the heavy hole bands [1-2].

To get proper hole masses that can be compared to experiments one should however also include spin-orbit interaction.