MHCD-tTLG : Marcus-Hush-Chidsey DOS rates in Twisted Trilayer Graphene system

Implement MHC-DOS kinetics from the Julia-based ElectrochemicalKinetics.jl package. Needs density of states of the solid as input. We employ the low-energy momentum space model to get DOS for the twisted trilayer graphene system. See code descriptions for instructions for data reproducibility.

Contact

Mohammad Babar : mdbabar@umich.edu

Code descriptions

1. angles.py takes in DOS data files and outputs q_dict.mat with twist angle data. Will be input for script.jl

2. script.jl main Julia calculation script that outputs .mat file with oxidation rates (kox_list), reduction rates (kred_list), $\theta_{12}$ (q12_list) and $\theta_{23}$ (q23_list) variables for specified parameters $A$ , $\lambda$ and $\eta$.

where $\lambda$ = reorganization energy (eV), $\eta$ = applied overpotential (V) and $A$ = proportionality constant for MHC-DOS theory. Other input parameters:

i. C_dl : EDL capacitance (F)

ii. V_dl : EDL voltage (V)

iii. C_q : Quantum capacitance (F)

iv. V_q : Quantum capacitance voltage (V)

v. Vq_min / Vq_max : Min/Max range of Quantum capacitance voltage for interpolation (Eq. 3 in paper)

vi. kT : Thermal energy to temperature setting (0.26 eV at 300 K)

vii. ef : Fermi energy of the electrode

The output file has a format: k_data_{A}_λ_{}_η_{}.mat. Run Julia files with this command:

> julia script.jl

3. eta_run_script.jl runs script.jl at a range of eta values. Parallelized for HPC in \Eo_var\

The output mat file has a format: k_data_{A}_λ_{}_η_{}.mat.

4. /sweep/ folder contains the .mat DOS files of the tTLG system at a range of $\theta_{12}$ and $\theta_{23}$.

5. /ttlg_dos/ repository contains data and instructions to generate twisted trilayer graphene DOS at a range of $\theta_{12}$ and $\theta_{23}$ (1-5 degrees).

Formal potential of Ruthenium Hexamine, E = -0.25 V vs. Ag/AgCl electrode and reorg. energy λ=0.82 eV Ref.

Formal potential of twisted graphene, E = -0.18 V vs. Ag/AgCl electrode Ref.

Hence Eo = -0.25 - (-0.18) = -0.07 V is used for Ruthenium Hexamine. The kinetic rate files for Figure 2b are stored in /Eo_var/_0.07/ folder.

Data for Figure 4 in paper is in /Eo_var/0.3/ at equilibrium k_data_1.0_λ_0.82_η_0.0.mat.

6. /trilayer_stacked/ repository contains Bernal stacked (ABA) trilayer graphene DOS data and MHC-DOS kinetic rates. The ABA rates are used as reference for color maps in paper (Fig. 2a, Fig.4, SI Fig. 3). The rate value is specifed in sweep_dos.m. See /trilayer_stacked/README.md for instructions.

7. sweep_dos.m is an analysis script that uses the rate file .mat to generate colormap of k ${red/ox}$ or DOS with twist angles $\theta{12}$ and $\theta_{23}$.

8. /Eo_var/ folder contains kinetic rates for a range of $\eta$ and $E_{o}$ (formal potential of redox couple wrt electrode). Also contains parallelized input eta_run_script.jl for calculation on 128 cores.

9. uncertainty.jl and uncertainty_par.jl are input files for calculating uncertainty propagation from DOS. See section in supplementary information. The latter file is parallelized, whose output is stored in /DOS_unc_files/ in format: k_err_{A}_λ_{}_η_{}.mat. Output files can be analyzed using analyze_mat.jl

Specify surface vector v in lines 38-42 to either kox_list for oxidation rates, kred_list for reduction rates or dos_max for maximum DOS values (Figure 2a) as shown below.

x = q12_list;
y = q23_list;
v = kox_list;
v = kred_list;
v = dos_max;

k_aba = 2.632e-5 on line 44 of sweep_dos.m is the equilibrium rate constant ko for ABA (Bernal) stacked trilayer graphene. It is used as reference for other twist angle rates. For calculation of k_aba see repository trilayer_stacked.

8. /SI_fig3_data/ folder contains kinetic rate files at η = 0.0 and Eo = -0.07 V for two reorganization energies λ = 0.2 eV and λ = 1.2 eV. These files can be loaded into sweep_dos.m to plot SI figure 3.

Steps to reproduce data

1. Generate DOS data by following instructions in /ttlg_dos/ repository. Data is stored in /sweep/ folder.

2. Calculate reference ABA trilayer graphene DOS and corresponding kinetic rates in /trilayer_stacked/ repository. Use k_aba value in sweep_dos.m

3. Use script.jl or eta_run_script.jl files to generate kinetic rates in bulk for all twist angles (parallelized).

4. Input rate file and run sweep_dos.m to visualize rates in a color map with $\theta_{12}$ and $\theta_{23}$ as x and y axes respectively.