Chengcheng Xiao
PhD student @ Imperial College London
ORIGIN - Bloch
- Bloch function: periodic function modulated by plane waves.
\displaystyle
\psi_{n \mathbf{k}}(\mathbf{r})=u_{n \mathbf{k}}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}
- k vectors come from cell periodicity. i.e. for 1D:
\displaystyle
\mathbf{k} = \frac{2\pi}{N\cdot \mathbf{a}} n; n \in (0,1,2,3, \cdots)
\displaystyle
=
\displaystyle
\times
ORIGIN - reverse?
- By doing DFT, we are using basis functions to describe Bloch functions:
- Can reverse this process?
Basis (PW, local set)
DFT calculator
Bloch WFs
Bloch WFs
???
Basis (PW, local set)
- This question is particularly interesting because if we have a DFT WF calculated with PW basis, reverting to a minimum atomic basis will give us much chemical intuition + speed if we want to do other stuff with them.
ORIGIN - FT
- We can use these envelope function to construct localized functions with Fourier transform (AKA superposition of Bloch functions 🤪)
\displaystyle
\begin{aligned}
\mathscr{F}_{\mathbf{k}} \{ \psi_{n \mathbf{k}}(\mathbf{r}) \} (\mathbf{R}) &\sim \int_{-\infty}^{\infty} u_{n \mathbf{k}}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}} e^{i \mathbf{k} \cdot \mathbf{R}} \mathrm{d} \mathbf{k} \\
&= u_{n \mathbf{k}}(\mathbf{r}) \delta(\mathbf{r}-\mathbf{R}) = u_{n \mathbf{k}}(\mathbf{R})
\end{aligned}
- However, we don't have infinite k points, we only have k-points live inside the BZ, this means our delta function is not a delta function, but a modulation with finite width that's localized on the R lattice!
\rightarrow
Definition
- The formal definition of a Wannier function from a single Bloch band:
Wannier transformation
\displaystyle
w_{\mathbf{R}}(\mathbf{r})=\frac{V}{(2 \pi)^{3}} \int_{\mathrm{BZ}} \psi_{\mathbf{k}}(\mathbf{r}) e^{-\mathrm{ik} \cdot \mathbf{R}} \mathrm{d} \mathbf{k}
Gauge freedom
- However, Bloch functions have another degree of freedom: gauge freedom:
\displaystyle
|\tilde{\psi}_{n \mathbf{k}}\rangle=e^{i \varphi_{n}(\mathbf{k})}\left|\psi_{n \mathbf{k}}\right\rangle
- This transformation changes nothing about the physical description of the system.
- However, the "smoothness" of our Wannier function depends on the smoothness of the Bloch function in k space.
Multiband
- For multi-band situation, we can construct another set of bloch functions:
\displaystyle
|\tilde{\psi}_{n \mathbf{k}} \rangle=\sum_{m=1}^{J} U_{m n}^{(\mathbf{k})}\left|\psi_{m \mathbf{k}}\right\rangle
-
U is an unitary matrix so this is a unitary rotation, which means:
- the charge density is not changed
- \(|\tilde{\psi}_{n \mathbf{k}} \rangle \) still form a orthonormal set.
Definition +
- For multi-band situation, the Wannier transformation:
\displaystyle
w_{n \mathbf{R}}(\mathbf{r})=\frac{V}{(2 \pi)^{3}} \int_{\mathrm{BZ}}\left[\sum_{m} U_{m n}^{(\mathbf{k})} \psi_{m \mathbf{k}}(\mathbf{r})\right] e^{-\mathrm{i} \mathbf{k} \cdot \mathbf{R}} \mathrm{d} \mathbf{k}
- The inverse transformation:
\displaystyle
\psi_{m \mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\mathbf{R}} w_{n \mathbf{R}}(\mathbf{r})
Projection
- In order to get an initial unitary matrix, we project the Bloch functions onto a set of localized functions \( |g_n\rangle\) :
\displaystyle
\left|\phi_{n \mathbf{k}}\right\rangle=\sum_{m=1}^{J}\left|\psi_{m \mathbf{k}}\right\rangle\left\langle\psi_{m \mathbf{k}}|g_{n}\right\rangle
- But \(\left|\phi_{n \mathbf{k}}\right\rangle\) does not form a orthornomal set, we need to orthonormalize them by using overlap matrix \( S_\mathbf{k}=\langle\phi_{n\mathbf{k}}|\phi_{n\mathbf{k}}\rangle \):
\displaystyle
|\tilde{\psi}_{n \mathbf{k}}\rangle=\sum_{m=1}^{J}\left|\phi_{m \mathbf{k}}\right\rangle\left(S_{\mathbf{k}}^{-1 / 2}\right)_{m n}
Entanglement
- Normally we don't need this but if we have the following case (graphene, \( p_z\) on C):
No entanglement
Entanglement
Entanglement
We can use another \( U_{mn}^{\mathrm{(pre)}} \) to do the band selection!
SPread function
- Even tough now we have a unary matrix, Wannier functions are still non-unique and depend on the original projectors used.
- Introducing - spread function:
\displaystyle
\Omega=\sum_{n}\left[\left\langle\mathbf{0} n\left|r^{2}\right| \mathbf{0} n\right\rangle-\langle\mathbf{0} n|\mathbf{r}| \mathbf{0} n\rangle^{2}\right]=\sum_{n}\left[\left\langle r^{2}\right\rangle_{n}-\overline{\mathbf{r}}_{n}^{2}\right]
- Essentially, this is a squared deviation from mean in real space. We can keep on changing the unitary matrix to minimize this function.
- To calculate this we need (details not shown):
\displaystyle
M_{m n}^{(\mathbf{k}, \mathbf{b})}=\left\langle u_{m \mathbf{k}} \mid u_{n, \mathbf{k}+\mathbf{b}}\right\rangle
Maximally localized Wannier function!
What DO we need?
- So, what do we need from the DFT calculation to get the Wannier functions:
- projection coefficients for initial guess
- overlap matrix for calculating spread function
Procedure -VASP
- Run a single point calculation, write charge density and wave functions files.
- "Fat band" calculation to determine the energy window and bands to Wannierize.
- Write a wannier90.win file.
- Continue previous DFT calculation with:
- ALGO = None
- LWANNIER90 = .T.
- Run wannier90.x to get unitary matrices.
- Do whatever post processing you want 🪄.
Things needeD
- From DFT calculations:
- .amn file: projection coefficients
- .mmn file: overlap coefficients
- UNK.* files (optional): periodic part of the Bloch functions for plotting purpose.
- From input (wannier90.win file):
- number of target Wannier functions
- DFT bands: how many, what to exclude
- initial projection (guiding) functions
A typical .WIN file
# band indeices starts from 0
num_bands = 7
num_wann = 4
exclude_bands = 8-16
# total window
dis_win_max = 20.0
dis_win_min = -7.5
# frozen window
dis_froz_max = 6.2
dis_froz_min = -7.5
# disentanglement control
#dis_num_iter = 120
#dis_mix_ratio = 1.d0
num_iter = 0
num_print_cycles = 10
# projection section
Begin Projections
c=0.0,0.0,0.0:sp3
End ProjectionsLets calculate!
Energy Windows
- There are 4 tags to control the energy windows:
- dis_win_max
- dis_win_min
- dis_froz_max
- dis_froz_min
- blue region: disentanglement
- Orange region: no disentanglement
Energy Windows
- Use fat band analysis determine the windows.
- For this example, by projecting on to s and p orbitals of silicon, we need a total of 8 bands (two atoms, each with 4 orbitals).
Energy Windows
- So, we need dis_win_max and dis_win_min to contain more than 8 bands.
- dis_froz_max and dis_froz_min should contain less than 8 bands and the bands inside this range should have more apparent contribution from s and p orbitals from Si atoms.
Energy Windows
- dis_win_max: 17 eV
- dis_win_min: -7.5 eV
- dis_froz_max: 10 eV
- dis_froz_min: -7.5 eV
Projection only
- Often times running spread minimization procedure causes the lost of symmetry in Wannier functions or other undesired behaviors.
- To avoid this, we can run "projection-only" Wannierzation by setting the following in wannier90.win:
- num_iter = 0
Plotting WF
- To plot Wannier function, put the following to your INCAR along side LWANNIER90=.T. when generating Wannier inputs:
- LWRITE_UNK=.T.
- And in wannier90.win file, add something like :
- wannier_plot = true
- wannier_plot_list = 1 2 3
- wannier_plot_supercell = 5 5 1
- the Im/Re ratio for each of the Wannier functions are written in the wannier90.wout file.
Quanlity control
- What attributes can we determine if we have a good Wannier functions or not?
- Real function: check Im/Re ratio of the WFs.
- Minimum spread: check spread < cell.
- Can reproduce DFT band structure.
- Generally, matches your chemical intuition. 👻
Tips & Tricks
- If a projection only calculation is viable, do that!
- Play with projection (guiding) functions, try changing their orientation and shape (see manual, chapter 3).
- Place s orbital on bond center and run minimization procedure.
- use guiding_centres=.T. in wannier90.win
Resouces
- Wannier90's Manual & Tutorials 😹
- Nicola Marzari, et al, Rev. Mod. Phys. 84, 1419, 2012
- Some names to remember:
- G. H. Wannier: inventor of Wannier function
- N. Marzari and D. Vanderbilt: localization criteria
- I. Souza: disentanglement
Other stuff
☄️Some of my work that has something to do with Wannier90:
thank you 🤟
Questions?
Find these slides:
Download examples:
2021-11-06-W90 short tutorial
By Chengcheng Xiao
2021-11-06-W90 short tutorial
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