Calculation of Band Structure and COOP of CsPbBr₃ with BAND

On the example of the cubic metal halide perovskite CsPbBr₃, this tutorial illustrates how the analysis of density of states (DOS) and crystal-orbital overlap populations (COOP) provides insights about the nature of chemical bonding within crystalline materials. This also reveals starting points for a systematic tuning of the band gap in such crystals which is of paramount importance for photoelectric and optoelectronic applications. The tutorial is largely based on the following recent publication:

M. G. Goesten and R. Hoffmann Mirrors of Bonding in Metal Halide Perovskites, J. Am. Chem. Soc. 140, 12996-13010 (2018).

The following seminal paper describes in more detail how to extend chemical concepts into the field of solid state physics:

R. Hoffmann How Chemistry and Physics Meet in the Solid State, Angew. Chem. Int. Ed. Engl., 26, 846-878 (1987).

Note

Accurate band gap calculations can be far from trivial problems even within the framework of DFT. Band gaps from semi-local density functional approximations like LDA or GGA are generally underestimated due to spurious artifacts such as the electronic self-interaction error. More accurate results can be obtained from non-local approaches like hybrid DFT or GW methods, which have, however, much higher computational costs. A recent performance assessment study with BAND examines the accuracy of several density functional approximations for the prediction of the band gaps in various crystals.

The original study thereby relies on the meta-GGA functional SCAN, which the authors considered it as a reasonable trade-off between accuracy or computational costs. In contrast, this tutorial will – for the sake of efficiency – employ the GGA functional PBEsol, a smaller basis set, and a frozen core treatment. While more approximative, this approach still yields to the same qualitative results as the original work. An ADFinput file with the same settings as used by Goesten and Hoffmann can be obtained here.

Step 1: Preparations

We first create a separate folder for all calculations of this tutorial

Start ADFjobs

File → New Directory

Name it e.g. BAND__DOS_and_COOP` and press Enter

Click on the folder just created

followed by the preparation of the model structure of the perovskite crystal. In principle such an atomistic model is readily available from structure databases like materialsproject.

Here we will use two preoptimized structures that were relaxed with and without a scalar relativistic treatment, respectively and can be obtained here:

Download CsPbBr3_NR.xyz

Download CsPbBr3_SR.xyz

Note

In custom applications, you might need to optimize the structures first. This also include optimizing the lattice parameters.

Start ADFinput

File → Import Coordinates...

Select the downloaded file CsPbBr3_NR.xyz and click OK

The editor should now show the atoms of the first unit cell as well as the lattice vectors:

Step 2: Calculations

With the file CsPbBr3_NR.xyz loaded as structure in the molecular editor

In the Main panel:

XC-Functional → GGA → PBEsol

Relativity → None

Basis set → TZP

Frozen core → Large

Under Calculate tick both, DOS and Bandstructure

Next we define a specific path through the Brillouin zone along which we want to sample the band structure: From the center Γ we move first to the center X of a facet of the cubic Brillouin zone, then to M in the middle of one of its vertices, then to the corner R to finally return to Γ.

Properties → Band Structure

Interpolation delta-K: 0.02 Bohr^-1

Untick Automatic generate path

In the text box below enter:

0.00 0.00 0.00 G

0.50 0.00 0.00 X

0.50 0.50 0.00 M

0.50 0.50 0.50 R

0.00 0.00 0.00 G

Note

A path consisting of multiple disconnected segments can be defined by leaving an empty line between each block defining a segment

File → Save As...

Save e.g. under CsPbBr3_NR.adf

Next, load the structure in the file CsPbBr3_SR.xyz and enable the scalar relativistic treatment in the Main panel:

Relativity → Scalar

Apart from that, set exactly the same options as for the non-relativistic calculation.

Then save this second input file e.g. under CsPbBr3_SR.adf, switch to ADFjobs and start both calculations.

Step 3: Inspecting the Band Structure

After completion of both calculations”

Right click on the job CsPbBr3_NR in ADFjobs

Select Band Structure

Eventually a pop-up window might appear that asks if the calculation of fat bands can be skipped. We need these quantities in our analysis, therefore:

In the pop-up window click No

After a few seconds you should see a plot of the band structure:

We immediately see that the CsPbBr₃ perovskite exhibits no band gap in our nonrelativistic description as the valence and conduction band touch at the Fermi-level (the thin horizontal grey line) at the point R in the Brillouin zone. The band gap and the Fermi level can be examined in more detail by doing the following:

SCM → Output to switch to ADFoutput

Properties → Band gap info

The picture changes when looking at the corresponding results from the scalar relativistic calculation:

In this case the valence band at R is separated from the conduction band by about 1.2 eV. This value is smaller than the result obtained in the original study due to the different choice of the density functional approximation. In both cases we nonetheless observe a clear effect of the scalar relativistic description: the highest occupied band is lowered in energy and so is another occupied band near the bottom of our plot. Also note that, due to the presence of Pb, a spin-orbit coupling description should be used for relativistic effects when aiming to reproduce experimental results.

In the following we will examine the atomic orbital contributions to the two shifted bands mentioned above. At this point we can make the following assumptions:

• The bands arising from orbitals of the heavy elements Cs and Pb will be most affected by the scalar relativistic correction. Such relativistic effects on heavy atoms are indeed encountered quite frequently.
• The Cs centers in the perovskite crystal will be simple cations and thus their orbitals will not participate in any chemical bonding and only contribute to non-dispersed bands

To visualize the contributions from Pb s levels, do the following steps for both bandstructures, relativistic and non-relativistic:

in the BANDstructure window

Switch from Brillouin Zone to the Atoms perspective in the bottom left corner. This will show the unit cell instead of the Brillouine zone.

Select one of the Pb atoms

Right click on selected atom, select Bands, then tick s

Bands → Options...

Switch to panel Left Y Axis

Change Minimum value to -16 and Maximum value to 0

Switch to panel Curves

Untick Markers: Show

Set details like color or line width of the selected bands

The resulting plots now highlight those band segments with significant contributions from the Pb 6s orbital.

From these plots the following observations can be made:

• Two bands are mainly affected by the relativistic treatment as they are both lowered in energy by 1-2 eV
• These two bands correspond to those with the highest contributions from the Pb6s orbitals
• The Pb6s contribution is about equal for upper and lower band in the non-relativistic case, while Pb6s contributes more to the lower band in the relativistic case
• For both, relativistic and non-relativistic band structures, we find that the two Pb6s bands roughly appear as vertically mirrored images of each other along the path Γ → X → M → R → Γ

We will use crystal-overlap orbital populations (COOP) to examine these two bands further. A COOP is defined between a pair of orbitals (or pair of orbital shells) and directly indicates the bonding character of those crystal orbitals at different energies. In the CsPbBr₃ crystal each Pb atom has six Br centers as nearest neighbors. We therefore focus on COOP plots between Pb and Br orbital shells, which can be obtained as follows:

In the BANDstructure window

DOS → COOP... opens a window to select the COOP pairs and switches from the Brillouin Zone to the Atoms panel on the left side of the BANDstructure window

In the Atoms panel select the Pb atom

In the COOP window under Left click on the + and select Orbitals → s

In the Atoms panel select the three Br atoms

In the COOP window under Left click on the + and select Orbitals → s

Close the COOP window by clicking OK

We repeat the above steps also for the combinations Pbs/Brp, Pbp/Brs, and Pbp/Brp :

as well as for the corresponding pairs in the relativistic case:

We see that the two shifted bands are mainly formed from bonding and antibonding Pb6s/Br4p linear combinations, respectively. The much smaller contributions from antibonding Pbs/Brs orbitals (red curves) can be ignored here.

Interpretation of Results

These results allow us now to rationalize our previous observations:

• Why are only the 6s bands shifted by the relativistic treatment and all others not?

  The Pb6s orbital is lowered in energy by the relativistic treatment and consequently all bands this orbital contributes to. When applying a relativistic treatment, s orbitals are energetically lowered and contracted more than p orbitals, while d and f orbitals are raised and expanded. The relative amount of scalar relativistic effects on p orbitals compared to s levels is thus smaller and can lead to basically unaffected bands as in results obtained here. These relativistic effects are understood well in literature. See for example the following publication:

5. 10. Baerends, W. H. E. Schwarz, P. Schwerdtfeger and J. G. Snijders Relativistic atomic orbital contractions and expansions: magnitudes and explanations, J. Phys. B, 23, 3225-3240 (1990).

• Why do the relative contributions of 6s in the two shifted bands change when going from the non-relativistic to the relativistic case?

  This effect can be explained with basic molecular orbital theory. When two levels interact and form linear combinations, the lower linear combination will have more contributions from the lower level, while the higher one will be formed to a larger extend by the higher level. In the scalar relativistic case the 6s orbitals are essentially lowered with respect to all other levels and will therefore contribute more to the lower band in this case.

• Why do the two bands appear as vertical mirror images of each other?

  The COOP plots above clearly indicate that the two shifted bands are formed by σ-type bonding and antibonding Pb6s/Br4p linear combinations, respectively. We find that each octahedral PbBr₆ unit in the CsPbBr₃ crystal has three Br centers that are located in neighboring unit cells. At the sampled high symmetry points of the Brillouin zone [e.g. X at (0.5 0.0 0.0), see above], the resulting Bloch functions assume values of -1, thus changing the sign of the p orbitals of Br atoms in adjacent unit cells. This leads to either a stabilizing overlap or destabilizing nodal planes between the orbitals, which in turn rationalizes the observed behavior of the bands along the sampled path. The mirroring of the two bands results simply from the fact that they correspond to bonding and antibonding σ crystal orbitals; say e.g. the Pb6s has a positive sign in the σ band and a negative sign in the σ* band. The figure below illustrates these linear combinations at the different points in the Brillouin zone:

• What other interactions are holding the crystal together?

  The σ and σ* bands discussed above are both fully populated so that their bonding and antibonding contributions cancel. However, also the Pb6p/Br4p COOP shows significant bonding characteristics in the energy range between the σ and σ* bands. Indeed, the corresponding antibonding Pb6p/Br4p bands are unpopulated as they are located directly above the Fermi level. Also in this case one observe that the three bonding Pb6p/Br4p bands are mirrored images of their antibonding counterparts, which can be unterstood by the same principles as discussed before. Finally, there are interactions between the Cs ions and the PbBr₆ units, which can be expected to be mostly ionic and will thus not be visible in the COOP plots.