Extremal points and effective mass

This tutorial demonstrates how to find extremal points (maxima, minima, and saddle points) in the Brillouin zone, and calculate effective masses using the band-edge utility. LDA silicon was chosen for simplicity, though it is a trivial example as its extremal points are found on high-symmetry lines. band-edge is particularly useful when searching for multiple extremal points, and/or points distinct from those of high symmetry.

We use Si here, as it is a simple case: the valence band maximum falls at the Γ point, and the conduction band minum falls on the (001) line connecting $\Gamma$ and X, at about 0.85 of the distance to X.

At the end of this tutorial the band-edge manual that documents its usage in more detail.

Preliminaries

This tutorial uses a number of Questaal executables and scripts, e.g. band-edge, blm, lmfa, lmf, pfit, fmain, plbnds, and fplot. They are assumed to be in your path.

Tutorial

The starting point is a self-consistent LDA calculation. You will need the init.si from the basic DFT tutorial for silicon. Starting in a clean directory, paste the init file from that tutorial into init.si and do

blm init.si --simple --wsite --nk,ins --ctrl=ctrl && lmfa ctrl.si && lmf ctrl.si > out.lmfsc


The command above repeats all the steps of that tutorial.

It will also be helpful to refer to the Si band structure, which is generated in this tutorial.

Find the conduction band effective mass

We will now use the band-edge script to accurately locate the position of the conduction band minimum and to calculate the effective mass. This is done in three steps. First do a rough search by ‘floating’ to a point near the minimum. Next, do a more refined search by carrying out a minimization until the gradient is negligibly small. Finally the extremal point is known, calculate the effective mass around this point.

1. Float to low-energy point

band-edge script has a ‘float’ option that is useful for doing a quick search to find a low-energy (or high-energy) region of k-space. You specify a starting point, then the script creates a cluster of points around it and checks what is the lowest-energy point. It then uses the lowest-energy point as the next central point, creates a new set of points around it and again moves to the lowest-energy one. This process is repeated until the central point is the lowest-energy point.

Let’s pick a random point (0.1,0.2,0.3) and let band-edge float downhill. Run the following command:

band-edge -floatmn -maxit=20 -r=0.1 -band=5 -q0=0.1,0.2,0.3 si


‘−floatmn’ tells band-edge to seek a minimum-energy point (see additional exercises for a maximum-energy point example). ‘−maxit’ switch limits the number iterations (number of times a set of points is created) in case convergence is not reach before then, ‘−r=’ sets a range that defines how far from the centre the points are generated, ‘−band’ is for the band considered (here conduction band is 5 since 4 electrons and spin degenerate) and lastly −q0 is the starting k-point. To see what switches band-edge has, invoke it without any arguments, or see documentation below.

You should get an output similar to the following:

         check that "lmf si" reads input file without error ... ok
start iteration 1 of 20
lmf si --band~lst=5~box=0.1,n=12+o,q0=0.1,0.2,0.3 > out.lmf
qnow, E  :     0.100000     0.200000     0.300000       0.3483071   ← q and energy at this q
q*       :     0.160184     0.114033     0.240965                   ← estimate minimum point from gradient
use      :     0.027639     0.147427     0.344721                   ← Lowest-energy point in cluster
...
start iteration 8 of 20
lmf si --band~lst=5~box=0.1,n=12+o,q0=-0.017083,0.009789,0.834163 > out.lmf
qnow, E  :    -0.017083     0.009789     0.834163       0.1956728
gradient :    -0.045677     0.028221    -0.015188       0.055799
q*       :    -0.004553     0.001048     0.844981
cluster center is extremal point ... exiting loop

q @ minimum gradient :    -0.017083     0.009789     0.834163
Final estimate for q :    -0.017083     0.009789     0.834163


Take a look at the first line beginning with lmf. band-edge tells you what command it uses for the energy of each point in the cluster around your starting point. In each iteration qnow gives the current central k-point and its energy in Rydbergs. ‘use’ prints the lowest-energy k-point in the cluster of points around the middle point; this will then be used as the central point in the next iteration. The cluster of 13 k-points and their energies are printed to bnds.si. Take a look and you will see that (-0.017083, 0.009789, 0.834163) is indeed the lowest-energy point in the cluster.

As the iterations proceed, note that the energies at qnow are going down as we float to a low-energy region. After 8 iterations, the following is printed: ‘cluster center is extremal point … exiting loop’. The central q-point is the lowest-energy point and the float routine is finished.

Starting from the low-energy point we floated to, the next step is to do a more refined search using a gradient minimization approach. band-edge creates a new cluster of points, does a quadratic fit and then traces the gradients to a minimum point. Run the following command:

band-edge -edge2 -maxit=20 -r=.04 -band=5 -gtol=.0001 -q0="-0.017083     0.009789     0.834163" si


‘edge2’ specifies what gradient minimization algorithm to use. All the switches are explained in the documentation below.

The output is similar to before but now subsequent iterations (use) tracks the gradient rather than the minimum-energy point in the cluster

qnow, E  :    -0.017083     0.009789     0.834163       0.1956728
q*       :    -0.000123    -0.000442     0.843534
use      :    -0.000123    -0.000442     0.843534
...
qnow, E  :    -0.000045     0.000000     0.845004       0.2122582
gradient :    -0.000004     0.000000    -0.000004       0.000006
q*       :    -0.000044     0.000000     0.845009
use      :    -0.000044     0.000000     0.845009

gradient converged to tolerance gtol = .0001
q @ minimum gradient :    -0.000045     0.000000     0.845004
Final estimate for q :    -0.000044     0.000000     0.845009


Note how the gradient magnitude is decreasing with each step until it falls below the specified tolerance (gtol). The gradient minimization converged in this case, with minimum gradient at (0, 0, 0.846).

3. Calculate effective mass

Now that we have accurately determined the conduction band minimum, we can calculate the effective mass. This is done by fitting a quadratic form to a set of points around the conduction band minimum. Run the following command:

band-edge -mass -alat=10.26 --h5 -r=.0005,.001,.002,.003 -band=5 -q0=0,0,0.845 si

• -mass tells band-edge to estimate the effective mass tensor.
• -alat specifies the lattice constant (found in various places such as the lmf stdout, init or site file). The lattice constant is needed to convert to atomic units since the code reports k-points in units of $2\pi/\mathrm{alat}$.
• -r specifies the radii of the four clusters of points around the central point; each radius has 32 points (points and faces of an icosohedron). The extra points improve the accuracy of the quadratic fitting.

The last line of the output prints the three effective mass components in atomic units. So for silicon, the effective mass is anisotropic with lighter masses in two directions and a heavier effective mass in the third direction.

     0.941601     0.187815     0.187815


The masses are fairly close to experimental values for Si.

To see the principal axes (eigenvectors of the mass tensor) do the following:

echo princ | pfit 2 bndsa.si


The three columns below eval are the three principal axes (eigenvectors of the quadratic form). The first points along z, while the latter two (which are degenerate) lie in the x-y plane.

The band-edge manual

band-edge is shell script that calls lmf (or any Questaal executable that can generate energy bands through the --band switch.

Its purpose is to find band extrema and it operates in four distinct modes. All modes start from a reference q point, and compute energy bands in a cluster of points around the reference. One band in particular is selected out. The cluster of points are arranged in some regular polyhedron with 12, 20, or 32 points in a shell. You specify the radius of the polyhedron (or radii of multiple shells if desired). Energy bands are generated by invoking lmf or similar code with the --band switch, which is invoked in a special-purpose box mode that makes bands for clusters of q points centered around a reference.

The four modes are:

1. (-floatmx or -floatmn) Update the current q point by indentifying whichever point in the cluster has the maximum value (-floatmx) or minimum value (-floatmn). If the central point is the extremal point, band-edge exits. Otherwise, the extremal point becomes the new central point and the cycle is repeated.
2. (-edge) The selected energy band at a given cluster of points is subject to a least-squares fit and the gradient in q extracted. The q point and gradient are fed to a Broyden minimization algorithm built into the fmin utility. fmin returns a new estimate for the extremal point. The process is iterated until the gradient falls below a specified tolerance.
3. (-edge2) The selected energy band at a given cluster of points is fit to a quadratic polynomial in qx, qy, qz and the extremal point is estimated from the extremal point of the parabola. This new point becomes the reference point; the process is iterated until the gradient falls below a specified tolerance.
4. (-mass) The selected energy band at a given cluster of points is fit to a quadratic polynomial. The normal matrix is diagonalized; its eigenvalues are the tensor effective masses and eigenvectors are the principal axes of the mass tensor.

Usage : band-edge switches ext

ext is the extension in ctrl.ext, the input file the band code uses to generate $E_n(\mathbf{q})$. By default band-edge uses lmf (but see -cmd below).

Choose between one of the following modes :

• -edge
Seek extremal q by calculating $\nabla_q E_n$ and minimizing it using a Broyden algorithm

• -edge2 | -edge2=fac
Seek extremal q using 1st and 2nd derivatives of $\nabla_q E_n$, from a polynomial fit to $E_n$ on the cluster.
Calculates qnew = (1-fac)*qold + fac*qmin where qmin is extremal point estimated from quadratic form.
fac is used if it is specified; otherwise fac=1. Independently of fac, the change in q might be limited by dqmx (below)

• -floatmx
Update reference q with largest $E_n$ in cluster. Iterate until the reference point (cluster center) is the maximum point.

• -floatmn
Update reference q with smallest $E_n$ in cluster. Iterate until the reference point (cluster center) is the minimum point.

• -mass
Fit bands around q0 to estimate effective mass tensor.

Required switches:

• -q0=#,#,#
Starting value of reference q

• -band=n
Specify band index n to optimize

• -r=#[,#2,#3,…]

• -alat=a
Lattice constant a (a.u.). It is not needed except in the calculation of effective mass (required there because q has units 2πa).

Optional switches:

• -h | --h | --help
show this message

• -cmd=strn
use strn in place of default command to generate bands, e.g.
-cmd="mpirun -n 16 lmf"
Default cmd is lmf.

• -n=12 | -n=20 | -n=32
Specify number of points in shell of radius r

• -dqmx=#
Maximum allowed change n any component of q for one iteration

• -gtol=#
Convergence tolerance in the gradient (applies to -edge and -edge2)

• –maxit=#
Maximum number of iterations to attempt

• –spin1
replace --band with -band~spin1

• –spin2
replace --band with -band~spin2

• –noexec
Show what cmd will be executed, without executing it

• –h5’ Tell the bands maker to write bands in hdf5 format

Although we know the valence band maximum for silicon is at the $\Gamma$ point, we will use it as an example for finding a maximum point. Try starting from the point (0.1,0.1,0.1). In the float step, change the floatmn (for minimum) to floatmx (for maximum) and change the band number to 4 (for valence band). Notice the energies are going up. Now do a gradient minimization from the point you floated to, remember to change the band number to 4. The valence band maximum is in a flatter region so try using a smaller excursion radius, say 0.001. Then calculate the effective mass. Your values should be around (-0.34, -0.56, -0.88). The negative signs indicate a maximum point (−mass = hole mass); a saddle point would have at least one positive sign.