Electron Phonon Coupling for Molecular Crystals

Installation:

Requirement: Gaussian 09 or 16

We recommend using conda to create virtual environment.

    git clone https://github.com/moule-group/ElPh.git

    conda create -n elph
    cd Elph
    pip install .

Environment variables in .bashrc

export PATH="your_path/catnip/build/":$PATH
export GAUSS_EXEDIR="your_path/g16"
export GAUSS_SCRDIR="$your_path/GaussianScratch"
export PATH="$GAUSS_EXEDIR:$PATH"
export PATH="$GAUSS_SCRDIR:$PATH"

For installing Catnip (ChArge TraNsfer Integral Package), please refer to https://joshuasbrown.github.io/docs/CATNIP/catnip_downloads.html

Usage:

Transfer Integral

First step: Input the number of molecules (defaults to 3) to be extracted. The code will generate monomer and dimer structure files.

Second step: Prepare input files in the folder: POSCAR (VASP structure format) ; FORCE_SETS from Phonopy simulation; phonopy_disp.yaml from Phonopy simulation.

Note: We consider 2D plane (high mobility plane of organic semiconductors) and only pick 3 nearest neighbors in this 2D plane. The 3 numbering monomers will be pair A (monomer 1 and 2); pair B (monomer 1 and 3); pair C (monomer 2 and 3), pair A and pair B will be transversed pairs and pair C will be parallel pairs (the shorter lattice parameter in 2D plane).

elph -w 1

Prerequisite: Finish transfer integral simulation first.

Electron Phonon Coupling

elph -w 2 

Transient Localization Theory Charge Carrier Mobility

Prepare mobility.json as the input, then run:

elph -w 3

Arguments

-w --workflow: Workflow selection (1: non local electron phonon coupling 2: SVD phonon mode projection 3: TLT mobility)

-q --mesh: Defining a mesh grid. (Defaults to [8,8,8])

-n --nmol: The number of molecules will be extracted (Defaults to 3)

-b --basis: Gaussian basis sets (Defaults to ['6-311G*','6-311G**']) for local and non-local simulation

-f --functional: Gaussian functional (Defaults to['b3lyp','b3lyp']) for local and non-local simulation

-s --supercell: The supercell matrix (Defaults to [2,2,2])

-homo --homo: P-type semiconductors: HOMO; N-type semiconductors: LUMO. (Defaults to True)

-o --output: Mobility calculation output name (Defaults to tlt_mobility.json)

-svd --svdqpts: Number of qpoints that SVD projection will apply (Defaults to 1)

mobility.json

In order to run mobility calculation, there are variables need to be specified. Please see the example here. View Example File

Theory:

This will divide into 3 parts. First part is transfer integral J, the second part is electron phonon coupling parameter g and the last part is transient localization theory.

Transfer integral J

The method we use is called dimer projection method (DIPRO) (Note: Some people call it Fragment orbital Method (FO)), it is proposed by D. Andrienko group in 2010 research paper. Transfer integral between 2 molecules i and j: $J_{ij} = <i|H|j>$ The reason why we cannot simply the equation above is because overlap matrix S is not 0 in molecular crystals. Therefore, we have to apply effective transfer integral

Effective Transfer Integral: $J_{eff}= \frac{J_{ij}-\frac{S_{ij}(E_i+E_j)}{2}}{1-S_{ij}^2}$

In order to use DIPRO to calculate transfer integral, we have to run 3 quantum-chemical simulations (2 monomers and 1 dimer), there are 9 quantum-chemical simulations in total.

Electron Phonon Coupling Parameter g

It can be further written as $J_{ij} = J_{ij}^0 + \sum_{I} g_{ij}^IQ_{I}$, where $J_{ij}$ is transfer integral at equilibrium geometry. $g_{ij}^I$ is the electron-phonon coupling parameter, $Q_{I}$ is normal coordinate at vibrational mode I.

In order to efficiently calculate the electron-phonon coupling parameter $g_{ij}^I$, we can do the conversion as below.

$g_{ij} = \frac{\partial J_{ij}}{\partial Q_{I}}$

Using chain rule, we get

$g_{ij} = \nabla J_{ij}\frac{\partial x_{k}}{\partial Q_{I}}$

where $x_{k}$ is Cartesian coordinate of the molecule. And $\frac{\partial x_{k}}{\partial Q_{I}}$ represent how each Cartesian coordinate changes with the normal mode coordinate, which is calculated by Phonopy modulation. The equation of modulation (creation of crystal structures with displacements) is

, where A is the amplitude (Defaults to 0.01 Angstrom), $N_{a}$ is the number of atoms in the supercell, $m_{j}$ is the mass of j-th atom. $r_{jl}$ is the position of the j-th atom in the l-th unit cell, $e_{j}$ is the j-th atom part of eigenvector, and $\phi$ is the phase.

To evaluate $\nabla J_{ij}$, a displacements -0.01 Å and 0.01 Å for each direction (x,y,z) of the gradient have been employed.

Transient Localization Theory (TLT)

The mobility equation is shown below: $\mu = \frac{e}{kT} \frac{L^2_{x(y)}}{2\tau}$

where $\tau$ is the relaxation time, $L^2_{x(y)}$ is squared localization length, e is the charge, T is the temperature in K, k is the Boltzmann constant. The mobility unit is in $\frac{cm^2}{Vs}$