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Crystals and the Electronic Band Structure

This week we are going to start doing some calculations on solids, i.e., periodic crystals. Many of the principles will be the same, but as you will see there are a few things that need to be done differently.

As before, all the inputs and scripts you need can be found in /opt/MSE404-MM/docs/labs/lab04 and you should copy the folder to your home directory.

Basic input for Diamond :material-diamond-outline:

As our first example of a crystalline solid we're going to look at diamond. You can find the input file at 🔗C_diamond.in, here I'll give a brief overview of it:

&CONTROL
   pseudo_dir = '.' 
   disk_io = 'none' 
/

&SYSTEM
   ibrav = 2 #(1)!
   A = 3.567
   nat = 2
   ntyp = 1
   ecutwfc = 20.0
/

&ELECTRONS
   conv_thr = 1.0E-6
/

ATOMIC_SPECIES
 C  12.011  C.pz-vbc.UPF

ATOMIC_POSITIONS crystal #(2)!
 C 0.00 0.00 0.00
 C 0.25 0.25 0.25

K_POINTS automatic #(3)!
  4 4 4 1 1 1

  1. ibrav=2 specifies a FCC unit cell (for a complete list of ibrav, see 🔗input descriptions).

  2. crystal specifies that the atomic positions are given in fractional coordinates of the unit cell vectors (defined by ibrav and A).

  3. We are using automically generated k-point grid with a 4$\times$4$\times$4 grid size, 1 1 1 means to shift the grid by one half of a grid spacing in each direction. By default, the k-point grid is generated such that the grid is centered around the origin of the reciprocal lattice vectors ($\Gamma$ point).

    ??? note "K-point Shift" The k-point shift is a trick that can speed up the calculation by including less k-points in the calculation. This is because only k-points in the irreducible Brillouin zone are calculated, and shifting the grid can help to sample the entire Brillouin zone with fewer points.

    <figure markdown="span">
  ![kpt_shift](assets/kpt_shift.png){width="500"}
</figure>

k-points

One important difference between periodic crystals and molecules is that the electronic states are not localised and their wavefunction is given by Bloch's theorem:

$$ \psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}), $$

where the electronic states are labelled by both the band index $n$ and the k-point $\mathbf{k}$. As discussed in the lecture, $\mathbf{k}$ lies in the first Brillouin zone.

The additional card K_POINTS in the input file specifies the k-point grid. The first three numbers 4 4 4 represent how many k-points are generated along each direction of the reciprocal lattice vectors.

The fineness of the k-point grid is a convergence parameters and we must make sure that it is sufficiently fine such that physically meaningful results are obtained.

Structure Parameters for Crystals

Now let's take a look at how the atomic positions in the unit cell are specified in the input file.

Quantum Espresso allows us to express the atomic positions either in absolute Cartesian coordinates $(x,y,z)$ or alternatively in crystal coordinates $(x_c,y_c,z_c)$. The position vector of the atom can then be obtained from the lattice vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ as follows:

$$ \begin{align} \mathbf{r} = x_c \mathbf{a} + y_c \mathbf{b} + z_c \mathbf{c} \end{align} $$

For diamond, which has the same atomic structure as 🔗Zinc Blende, the atomic structure looks like this:

![Diamond primitive cell](assets/C_diamond_light.png#only-light){width="200"} ![Diamond primitive cell](assets/C_diamond_dark.png#only-dark){width="200"}

To specify the shape of the primitive unit cell, we first set ibrav=2, i.e. face-centred cubic (fcc) Bravais lattice. Internally, with ibrav=2, Quantum ESPRESSO sets the the fcc lattice vectors as:

$$ \begin{align*} \mathbf{a} &= \frac{A}{2}(-1,0,1)\\ \mathbf{b} &= \frac{A}{2}(0,1,1)\\ \mathbf{c} &= \frac{A}{2}(-1,1,0) \end{align*} $$

!!! warning Note that here we are using the experimentally measured lattice constant A of 3.567 Å which might not be the same as the DFT optimized value. In later labs we'll see how to find the lattice constant predicted by DFT.

In terms of these lattice vectors, the crystal coordinates of the two carbon atoms are (as we see in the input file, indicated by ATOMIC_POSITIONS crystal):

$$ \begin{align*} \mathbf{r}_c^{C1} &= (0,0,0) \\ \mathbf{r}_c^{C2} &= (\frac{1}{4},\frac{1}{4},\frac{1}{4}) \end{align*} $$

Hence, the absolute Cartesian coordinates for the two carbon atoms are given by:

$$ \begin{align*} \mathbf{r}^{C1} &= \frac{A}{2}(-1,0,1) \times 0 + \frac{A}{2}(0,1,1) \times 0 + \frac{A}{2}(-1,1,0) \times 0\\ &= (0,0,0)\\ \mathbf{r}^{C2} &= \frac{A}{2}(-1,0,1) \times \frac{1}{4} + \frac{A}{2}(0,1,1) \times \frac{1}{4}+ \frac{A}{2}(-1,1,0) \times \frac{1}{4} \\ &= (\frac{A}{4},\frac{A}{4},\frac{A}{4}) \end{align*} $$

!!! example "Task 1 - Examining input & output files"

Run `pw.x` for the carbon diamond inside the `01_carbon_diamond` directory.
There are a couple of extra things to notice in the output file:

- The output lists the automatically generated k-points. How many k-points 
  are there and why?

    ??? success "Answer"

        We requested a 4$\times$4$\times$4 grid but instead in the ouput
        file indicates 10 k-points are being used. This is because
        Quantum Espresso uses crystal symmetries to relate certain k-points
        and to reduce the computational load.

- What are the eigenvalues?

    ??? success "Answer"
        For periodic systems, we have a set of band energies for each 
        k-point. And these are given in the output file:
        ```
              k =-0.1250 0.1250 0.1250 (   116 PWs)   bands (ev):
    
        -7.3461  11.5621  13.5410  13.5410
    
              k =-0.3750 0.3750-0.1250 (   116 PWs)   bands (ev):
    
        -5.1246   6.0725   9.6342  12.3836
    
              k = 0.3750-0.3750 0.6250 (   117 PWs)   bands (ev):
    
        -2.0454   1.1023   9.9094  10.6497
    
              k = 0.1250-0.1250 0.3750 (   120 PWs)   bands (ev):
    
        -6.2574   8.8031  11.2205  12.0763
    
              k =-0.1250 0.6250 0.1250 (   118 PWs)   bands (ev):
    
        -4.0419   6.4510   8.7237   9.1414
    
              k = 0.6250-0.1250 0.8750 (   111 PWs)   bands (ev):
    
         0.0174   2.6697   5.4037   7.5509
    
              k = 0.3750 0.1250 0.6250 (   115 PWs)   bands (ev):
    
        -2.9709   4.0228   7.6281   9.9651
    
              k =-0.1250-0.8750 0.1250 (   114 PWs)   bands (ev):
    
        -0.7739   3.2191   6.5088   8.0627
    
              k =-0.3750 0.3750 0.3750 (   114 PWs)   bands (ev):
    
        -4.0297   3.1416  11.7036  11.7036
    
              k = 0.3750-0.3750 1.1250 (   114 PWs)   bands (ev):
    
        -1.0562   2.2032   6.0516   9.9570
        ```

Convergence Test for K-Point Grid Above, we used a uniform

4$\times$4$\times$4 k-point grid to sample the first Brillouin zone. However, to really converge a periodic system, an additional convergence test with respect to the k-point sampling is necessary.

To test the convergence of our results with respect to the size of the k-point grid, we need to calculate the total energy for different grid sizes.

!!! example "Task 2 - Convergence with respect to k-point sampling and cut-off energy"

- The directory `02_convergence` contains input files to calculate the total
  energy. Change the k-point grid size in the input file, run the DFT
  calculation and see how the total energy changes. For example, perform a
  series of calculations with k-point grids set to `2 2 2`, `4 4 4`, `6 6
  6`, ..., all the way to `30 30 30` and see how the total energy changes.
  If you have any trouble doing so, you can always go back to
  [:link:lab03](../lab03/readme.md) for help.

    ??? success "Result"

        To obtain a **total energy per atom** which is converged to within
        10 meV/Atom, we need at least a 10$\times$10$\times$10 k-point grid. 

- For every periodic system you simulate, you should converge **both** the
  plane-wave cut-off energy and k-point grid size. To do this, one usually
  starts with one parameter set to a very high value and then varies the
  other one. Then one repeats this with switched roles. Try do this yourself
  and find the best set of parameters for diamond.

    ??? success "Tips"
        Try starting with `ecutwfc` of ~60.0 Ry and converge the k-points. 
        Or start with k-points of 30$\times$30$\times$30 and converge the
        plane-wave cutoff.

The Electronic Band Structure

What is the Electronic Band Structure?

We know that, while the electronic density obtained from DFT is meaningful, the Kohn-Sham states are not strictly the electronic states of the system. Nonetheless, they are in practice often a good first approximation of the electronic states of a system, so can be useful in understanding the properties of a system.

We have seen how to converge our calculations with respect to the k-point grid size (task 2), and have found in task 1 that the calculated energy eigenvalues are a bit different at each calculated k-point. Now we want to study how exactly these eigenvalues change as we move from one k-point to the next.

Examining how the Kohn-Sham energies change from one k-point to the next can tell us useful things such as if a material is likely to have a direct or indirect optical gap. For this it is useful to visualize how the energies change along a k-point path in the first Brillouin zone. The usual way this is done is to plot the band energies along lines between the various high-symmetry k-points in the Brillouin zone. For example, a high symmetry path for a face-centred cubic (FCC) lattice (see figure below) could be Γ—X—U|K—Γ—L—W—X:

![FCC BZ](assets/FCC_BZ.png){ width="250" }

??? "Finding High Symmetry Points and Paths" The details of how such path can be found is beyond the scope of this course, but an outline is given 🔗 here.

Calculating the Electronic Band Structure

The directory 03_bandstructure contains input files to calculate and plot the band structure of diamond. This a four-step process:

Step 1 - SCF Calculation

The first step is to calculate a converged electron density with a standard self-consistent field (SCF) calculation. In this step, the electron density is optimized in order to minimize the total energy of the system. The input file can be found at 🔗01_C_diamond_scf.in.

!!! example "Task 3.1 - SCF Calculation" Run the input file 🔗01_C_diamond_scf.in to calculate the ground state electron density. pw.x < 01_C_diamond_scf.in > 01_C_diamond_scf.out

Step 2 - NSCF(bands) Calculation

The second step is to use the obtained electron density to construct the Kohn-Sham Hamiltonian at a set of k-points along the path we want to study and to calculate the Kohn-Sham eigenvalues at those k-points. This is called a non-self-consistent field (NSCF) calculation as the charge density is kept fixed.

A brief overview of the 🔗input file is given below:

&CONTROL
 pseudo_dir = '.'
 calculation = 'bands' #(1)!
/

&SYSTEM
   ibrav =  2
   A = 3.567
   nat =  2
   ntyp = 1
   ecutwfc = 30.0
   # Also add 4 additional bands (unoccupied states)
   nbnd = 8 #(2)!
/

&ELECTRONS
/

ATOMIC_SPECIES
 C  12.011  C.pz-vbc.UPF

ATOMIC_POSITIONS crystal
 C 0.00 0.00 0.00
 C 0.25 0.25 0.25

# Path here goes: Γ X U|K Γ L W X
K_POINTS crystal_b #(3)!
  8
  0.000 0.000 0.000 30 Γ
  0.500 0.000 0.500 30 X
  0.625 0.250 0.625 00 U
  0.375 0.375 0.750 30 K
  0.000 0.000 0.000 30 Γ
  0.500 0.500 0.500 30 L
  0.250 0.500 0.750 30 W
  0.500 0.500 1.000 00 X
  1. calculation = 'bands' specifies that we are calculating the band structure.
  2. nbnd = 8 specifies that we want to calculate 8 bands. 4 more bands than the default value of 4. We add these bands so that we can calculate the band gap later.
  3. K_POINTS crystal_b specifies that we express the high-symmetry k-points in crystal coordinates. The number of high-symmetry k-points is 8, followed by the crystal coordinates of each k-point and the number of points to generate along the section of the path that connects this k-point to the next one in the list.

Since diamond has a face-centred cubic (FCC) lattice, we have chosen the path Γ-X-U|K-Γ-L-W-X where U|K means no k-point is sampled between U and K.

!!! example "Task 3.2 - NSCF Calculation" Run the input file 🔗02_C_diamond_nscf.in to get the eigenvalues of each band at each k-point. pw.x < 02_C_diamond_nscf.in > 02_C_diamond_nscf.out Take a look at the output, can you find where it says the charge density is read? ??? success "Answer" The potential is recalculated from file : ./pwscf.save/charge-density

Step 3 - Extracting Band Energies

Now we need to extract the Kohn-Sham energies from the output file and convert them into a dataset we can plot.

To do this, we use the bands.x tool from the Quantum Espresso package. The 🔗input file for bands.x contains only a BANDS section. For more fine-grained control please refer to 🔗bands.x input description.

!!! example "Task 3.3 - Extracting band energies" Run the input file 🔗03_C_diamond_bands.in with bands.x to extract and organize the eigenvalues calculated in the last step. bands.x < 03_C_diamond_bands.in > 03_C_diamond_bands.out

Step 4 - Plotting the Band Structure

Finally, we are ready to plot the band structure. The band structure is typically plotted with the energy on the y-axis and the high-symmetry k-points on the x-axis. The energy is usually shifted so that the valence band maximum is at 0 eV. The directory 03_bandstructure contains a python script (plotband_shifted.py) that can be used to plot the band structure.

!!! example "Task 3.4 - Plotting the band structure" Run the python script to plot the band structure of carbon diamond. python plotband_shifted.py Download the output file C_diamond_bands.png and take a look at it. Is carbon diamond a metal or an insulator? At which k-points is the valence band maximum and the conduction band minimum? How large is the band gap (and is it direct or indirect)?

??? success "Final result"
    <figure markdown="span">
      ![Diamond primitive cell](assets/band_structure.svg){ width="500" }
    </figure>

    From the graph we can see that the valence band maximum is at Γ (the
    first point on our path), the conduction band minimum is located between
    Γ and X and the size of the indirect band gap is around 4 eV (note that
    this is significantly smaller than the experimentally measured band gap
    of diamond). Note that here we have shifted the entire spectrum so that
    this point is at 0 eV.

Summary

  • In this lab we learned how to:
    • achieve k-point convergence in solids.
    • calculate the electronic band structure of a solid.
  • We have seen how several calculations may be chained together where the output of one is used as an input for the next one.
  • We should always keep in mind that the Kohn-Sham eigenvalues obtained from a DFT calculation do not correspond to the energy levels of the real interacting electrons, but are often useful as a first approximation.