# 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.

At the end of this tutorial the band-edge manual 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

#### Self-consistent density and energy bands

The starting point is a self-consistent LDA density, you may want to review the DFT tutorial for silicon. Cut and paste the following lines into file init.si:

LATTICE
ALAT=10.26
PLAT=    0.00000000    0.50000000    0.50000000
0.50000000    0.00000000    0.50000000
0.50000000    0.50000000    0.00000000
# pos means cartesian coordinates, units of alat
SITE
ATOM=Si   POS=    0.00000000    0.00000000    0.00000000
ATOM=Si   POS=    0.25000000    0.25000000    0.25000000


Run the following command to obtain a self-consistent density:

$blm --express si --nk=4 --gmax=5$ cp actrl.si ctrl.si
$lmfa si$ cp basp0.si basp.si
$lmf si  It will be helpful to have a band structure to refer to when finding the extremal points, you may want to review the silicon band plotting tutorial. Create a symmetry file syml.si joining L, Γ, X, W, and Γ again. lmchk si --syml~n=41~lbl=LGXWG~q=.5,.5,.5,0,0,0,1,0,0,1,.5,0,0,0,0  Generate the energy bands (file bnds.si) $ lmf si --rs=1,0 -vnit=1 --band~fn=syml


The following two commands will plot and display the bands:

$echo -6,6,10,15 | plbnds -fplot -ef=0 -scl=13.6 -lbl=L,G,X,W,G bnds.si$ fplot -f plot.plbnds && gs fplot.ps


Take a look at the band structure plot. The valence band maximum falls at the $\Gamma$ point while the conduction band minimum lies between $\Gamma$ and X, at about 0.85 of the distance to X.

#### 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. Lastly, you 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 0.3728323 gradient : -0.154879 0.182604 -0.061755 0.247276 q* : 0.160109 0.113958 0.240887 use : 0.027639 0.147427 0.344721 ... 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 0.2202239 gradient : -0.045667 0.028215 -0.015231 0.055799 q* : -0.004562 0.001050 0.845041 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. ##### 2. Gradient minimization 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’ part specifies what gradient minimization algorithm to use. All the switches are explained in the documentation below

The output is similar to before but now we will pay attention to the gradient line which prints the x, y and z gradient components and the last column gives the magnitude. Note how the gradient magnitude is decreasing with each step until it falls below the specified tolerance (gtol). At this point, the gradients minimization is converged and the following is printed ‘gradient converged to tolerance gtol = .0001’. It found the minimum gradient to be (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 --bin -r=.0005,.001,.002,.003 -band=5 -q0=0,0,0.846 si


mass tells band-edge to estimate the effective mass tensor. alat is the lattice constant (found in various places such as the lmf output, 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 gives 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.

    0.918580    0.185604    0.185532


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). In this case they are just the Cartesian axes.

### 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

• –bin’ Tell the bands maker to write bands in binary 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, a saddle point would have at least one positive sign.