Band Structure and COOP
***********************

BAND COOP Crystal Perovskite

Using the perovskite CsPbBr_3 as an example, this BAND 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 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, but these
  methods have much higher computational costs. A 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 relies on the meta-GGA functional SCAN, which the
authors considered to be 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 approximate, this
approach still yields to the same qualitative results as the original
work. An **AMSinput** 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 AMSjobs
   **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.

To speed up the calculations required in this tutorial, we provide two
structures whose geometries were optimized with (SR) and without (NR)
a scalar relativistic treatment.  These structures 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 AMSinput
   **File → Import Coordinates (System) →  From File(s)...**
   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:

   [image]


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 **PDOS** and **Bandstructure**

   [image]

To enable the COOP calculation:

   Click the [image: MoreBtn][image] button next to **PDOS**
   Tick **COOP**

   [image]

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**.

   [image]

   **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*

   [image]

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.ams"

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

   Delete the previous molecule by selecting all atoms (e.g., "CTRL+A" and pressing "delete")
   **File → Import Coordinates...** and choose "CsPbBr3_SR.xyz"
   **Relativity → Scalar**

Apart from that, set exactly the same options as for the non-
relativistic calculation.  This is most easily done as described above
-- by first filling in the right settings for one molecule, deleting
that molecule, loading a new molecule, and changing the single setting
(**Relativity**) that is different between the two calculations.

Save this second job, then run both calculations:

   **File → Save As...**
   Save e.g. under "CsPbBr3_SR.ams"
   **SCM → AMSjobs**
   Select the two jobs and then **Job → Run**


Step 3: Inspecting the Band Structure
=====================================

After completion of both calculations

   Right click on the job "CsPbBr3_NR" in AMSjobs
   Select **Band Structure**

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

   [image]

We immediately see that the CsPbBr_3 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 AMSoutput
   **Properties → Band gap info**

   [image]

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

   [image]

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 AMSbandstructure 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 Vertical 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.

   [image]

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 Pb*6s* orbitals

* The Pb*6s* contribution is about equal for upper and lower band in
  the non-relativistic case, while Pb*6s* contributes more to the
  lower band in the relativistic case

* For both, relativistic and non-relativistic band structures, we find
  that the two Pb*6s* 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
CsPbBr_3 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 AMSbandstructure 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 AMSbandstructure 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 **Right** click on the **+** and select **Orbitals → s**
   Close the COOP window by clicking **OK**
 [image][image]

We repeat the above steps also for the combinations Pb*s*/Br*p*,
Pb*p*/Br*s*, and Pb*p*/Br*p*. When reopening the COOP window, the
previously selected atoms and orbitals are remembered, so you
typically only need to adjust the orbital selection:

   [image]

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

   [image]

Remark: graph editing options can be accessed by clicking **DOS →
Options...** or by clicking outside of the plot, so in the area around
the left or bottom axes. This allows one to adjust the line width of a
curve, for example. The relative sizes of the individual graphs in the
window can be changed if one drags the line that is in between these
graphs with a mouse.

We see that the two shifted bands are mainly formed from bonding and
antibonding Pb*6s*/Br*4p* linear combinations, respectively. The much
smaller contributions from antibonding Pb*s*/Br*s* 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 Pb*6s* 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.

   [image]

* **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 Pb*6s*/Br*4p* linear
  combinations, respectively. We find that each octahedral PbBr_6 unit
  in the CsPbBr_3 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 Pb*6s* 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:

   [image]

* **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 Pb*6p*/Br*4p* COOP shows significant bonding characteristics in
  the energy range between the σ and σ* bands. Indeed, the
  corresponding antibonding Pb*6p*/Br*4p* bands are unpopulated as
  they are located directly above the Fermi level. Also in this case
  one observe that the three bonding Pb*6p*/Br*4p* bands are mirrored
  images of their antibonding counterparts, which can be understood by
  the same principles as discussed before. Finally, there are
  interactions between the Cs ions and the PbBr_6 units, which can be
  expected to be mostly ionic and will thus not be visible in the COOP
  plots.
