# Extremal points and effective mass

This tutorial demonstrates how to find extremal points (maxima/minima) in k-space and calculate effective masses using the band-edge program. This is done for silicon starting from a self-consistent LDA density. Silicon is a trivial example as its extremal points are found on high-symmetry lines. The band-edge program is particularly useful when looking for multiple extremal points that do not fall on high-symmetry lines.

```
$ blm si --nk=4 --gmax=5 && cp actrl.si ctrl.si && lmfa si && cp basp0.si basp.si && lmf si > out.lmfsc
$ lmf si --rs=1,0 -vnit=1 --band~fn=syml #calculate bands
$ echo -6,6 / | plbnds -fplot -ef=0 -scl=13.6 -lbl=L,G,X,W,G bnds.si #set up plotting
$ fplot -f plot.plbnds && gs fplot.ps #plot and view
$ band-edge -floatmn -maxit=20 -r=.1 -band=5 -q0=0.5,0,0 si #float step
$ band-edge -edge2=1 -maxit=20 -r=.04 -band=5 -gtol=.0001 -q0=0.82,0,0 si #gradients step
$ band-edge -mass -alat=10.26 -r=.0005,.001,.002,.003 -band=5 -q0=0.847,0,0 si #calculate mass
```

### Tutorial

The starting point is a self-consistent LDA density, you may want to review the DFT tutorial for silicon. Copy the following lines to a file called *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 si --nk=4 --gmax=5 && cp actrl.si ctrl.si && lmfa si && cp basp0.si basp.si && lmf si > out.lmfsc
```

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* containing the following lines:

```
41 .5 .5 .5 0 0 0 L to G
41 0 0 0 1 0 0 G to X
21 1 0 0 1 .5 0 X to W
41 1 .5 0 0 0 0 W to G
0 0 0 0 0 0 0
```

Run the following command to generate the bands:

```
$ lmf si --rs=1,0 -vnit=1 --band~fn=syml
```

Now plot the bands by running the following two commands:

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

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.

We will now use the band-edge tool to accurately locate the position of the conduction band minimum and to calculate the effective mass. This is done in three steps, you first do a rough search by **‘floating’** to a point near the minimum. From here, you 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

The 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 sets the number of iterations (number of times a set of points is created), **‘−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.

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=.1,n=12+o,q0=.1,.2,.3 > out.lmf
qnow : 0.100000 0.200000 0.300000 0.3724182
gradient : -0.153734 0.181962 -0.063093 0.246425
q* : 0.160163 0.114053 0.240440
use : 0.027639 0.147427 0.344721
...
start iteration 8 of 20
lmf si --band~lst=5~box=.1,n=12+o,q0=-0.017083,0.009789,0.834163 > out.lmf
qnow : -0.017083 0.009789 0.834163 0.2195189
gradient : -0.045263 0.028025 -0.016453 0.055721
q* : -0.004925 0.001103 0.846697
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 **(0.027639, 0.147427 0.344721)** is indeed the lowest-energy point in the cluster around (0.1, 0.2, 0.3).

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 k-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=1 -maxit=20 -r=.04 -band=5 -gtol=.0001 -q0=-0.02,0.01,0.83 si
```

The **‘edge2’** part specifies what gradient minimization algorithm to use (run **‘band-edge –h’** for more information), the r is the excursion range (step size in minimization), the **‘gtol’** is the tolerance in the gradient and the **q0** is the starting point that we obtained from the float step.

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 **(0,0,0.847)**.

#### 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=0,0,0.847 si
```

The **mass** tells **band-edge** to do effective mass calculation. **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 $2\pi/alat$. Here, the 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.

```
0.926148 0.185508 0.185469
```

### Additional Exercises

**1) Valence band maximum and effective mass**

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. 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 **(-0.33, -0.56, -0.88)**. The negative signs indicate a maximum point, a saddle point would have at least one positive sign.

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