Electron and hole mobilities in organic electronics: charge transfer integrals
******************************************************************************

ADF OLEDs Photovoltaics OFETs Charge transfer integrals Electronic
coupling Marcus theory Reorganization energy Hopping rate Effective
mass tensor


Calculation of charge mobilities
================================

Charge mobility is of crucial importance for efficiency in organic
electronic devices such as field-effect transistors (OFETs), light-
emitting diodes (OLEDs) and photovolatic cells (OPVs). In the so-
called hopping regime, charges move incoherently from site to site,
with a rate determined by the charge transfer integrals, which are
straightforwardly calculated with the unique fragment-based approach
in ADF. More approximate charge transfer integrals can be calculated
very fast with DFTB.

This guided example explains how to calculate charge mobility with
ADF.

See also:

  * Python example for computing charge transfer integrals with ADF

  * Python example for computing reorganization energies

  * ADF manual on charge transfer integrals

  * DFTB manual on charge transfer integrals


Define the fragments for charge transfer
========================================

As a simple example we will study electron and hole mobility between
two naphthalene molecules. Let us start with downloading the ".xyz
file" of the naphthalene dimer, which was cut out from the
experimental crystal structure (for your own organic semiconductor
material, you can just import the cif file, generate a supercell, and
cut out dimers to calculate the charge mobility in different
directions). Then:

   **1.** Start up **AMSinput**
   **2.** Select **File** → **Import Coordinates** in the menu bar and import the .xyz file you just downloaded
   **3.** In the **panel bar**, select **Model** → **Regions**
   **4.** Select all atoms of one naphthalene molecule and add a new region by clicking on [image: AddButton][image]
   **5.** Select all atoms of the other naphthalene molecule and add a new region by clicking on [image: AddButton][image]
 [image]

   **1.** In the panel bar, select **Multilevel** → **Fragments**
   **2.** Tick the **Use fragments** check box
 [image]

You can read more about ADF's fragments approach in the ADF manual or
in the ADF fragments tutorial.


Transport properties and settings
---------------------------------

   **1.** In the panel bar, select **Properties** → **Other: Etot, Charge Transport, ...**
   **2.** Tick the check box **Charge transfer integrals (for transport properties)**
   **3.** Go back to the **Main** panel
   **4.** Select the XC functional **GGA:PW91**
   **5.** Select Relativity **None**
   **6.** Select the **TZP** basis set
   **7.** Set the Frozen core at **None**
   **8.** Run the calculation (Ctrl/cmd + r) (Save file and acknowledge use of NOSYM)

When using molecular fragments in ADF, multiple calculations will be
(automatically) performed: first, each fragment is computed
individually (in this case, the individual naphthalene molecules).
Then the calculation of the full system is performed in which the
Hamiltonian matrix elements are expressed in the basis of the
molecular orbitals of the fragments. It is within this framework that
the site energies, charge transfer integrals and overlap integrals
between fragment orbitals are computed. More information on this can
be found in the charge transfer integrals section  of the ADF manual
and in the following publications by Senthilkumar et al. [1] [2].

When the job is finished, we search for the charge transfer integrals
in the output:

   Open the output file by clicking at **SCM** → **Output**
   In the **menu bar**, select **Properties** → **Charge transfer integrals**


Generalized Charge Transfer Integrals
-------------------------------------

The electronic coupling V (also known as effective or generalized
charge transfer integral J_eff) that goes into Marcus theory is
calculated as (see e.g. Wen et al. [3] )

   V = \frac{J - \frac{S(\varepsilon 1 + \varepsilon 2)}{2}}{1 - S^2}

J is the transfer integral, S the overlap, and ε the site energy.

The generalized charge transfer integrals  and the components are
printed in the output:

[image]

The electronic coupling between these two naphthalene molecules is
thus calculated as 0.00766 eV for hole transport and 0.03780 eV for
electron transport.

Look at the t_ab results in Coropceanu et al. [4] for more
information. The signs of J and S depend on the phase of orbitals
(which is arbitrary). Since V^2 is used in the Marcus equation, the
sign does not affect the hopping rates.


Reorganization energies
=======================

The reorganization energy λ is defined as the energy difference
between the charged and neutral systems at the two different
geometries (adiabatic PES). E.g. for electron transfer:

   \lambda = (E^\text{anion}_\text{neutral geometry} -
   E^\text{neutral}_\text{neutral geometry}) +
   (E^\text{neutral}_\text{anion geometry} -
   E^\text{anion}_\text{anion geometry})

We therefore need to perform four calculations: two geometry
optimizations and two single point calculations.

First, set up the **geometry optimization for the neutral
naphthalene** (from which we will obtain
E^\text{neutral}_\text{neutral geometry}):

   Open a new **AMSinput**
   **1.** Select **File** → **Import Coordinates** in the menu bar and import "naphthalene_dimer.xyz"
   **2.** Delete one of the two naphthalene molecules
   **3.** Symmetrize the molecule by clicking on [image: SymmTool][image]
   **4.** We will use the same calculation settings as in the charge-transfer integrals job: in the main panel, set the XC functional to **GGA:PW91**, the basis to **TZP**, the frozen core to **None** and relativity to **None**
   **5.** Change the task to **Geometry Optimization**
   **6.** Click on **File → Save as** and save it as **neutral_geometry_optimization**
 [image]

Then, set up the **geometry optimization for the anion** (from which
we will obtain E^\text{anion}_\text{anion geometry}):

   In **AMSinput** (main panel)
   **1.** Set the **Total charge** to "-1"
   **2.** Tick the **Unrestricted** check box
   **3.** Set the **Spin Polarization** to "1"
   **4.** Click on **File → Save as** and save it as **anion_geometry_optimization**

Run the two geometry optimizations:

   Open **AMSJobs**
   **1.** Select the jobs **neutral_geometry_optimization** and **anion_geometry_optimization** and click on **Job** → **Run**

We now need to run the two single point calculations:

   **1.** Run a **single point** calculation for a neutral naphthalene (i.e. **Total charge = 0** and **Spin Polarization = 0**) at the optimized geometry of the anion (i.e. the final geometry of the **anion_geometry_optimization** calculation). From this we will obtain E^\text{neutral}_\text{anion geometry}

   **2.** Run a **single point** calculation for the naphthalene anion (i.e. **Total charge = -1** and **Spin Polarization = 1**) at the optimized geometry of the neutral naphthalene (i.e. the final geometry of the **neutral_geometry_optimization** calculation). From this we will obtain E^\text{anion}_\text{neutral geometry}

Collect the four bonding energies from either the output files or the
log files (see tutorial on 10 Ways to Get the Energy and Other
Properties).

E^\text{anion}_\text{anion geometry} = -119.3514 \text{[eV]}

E^\text{anion}_\text{neutral geometry} =  -119.2451 \text{[eV]}

E^\text{neutral}_\text{neutral geometry} = -119.4217 \text{[eV]}

E^\text{neutral}_\text{anion geometry} = -119.3142 \text{[eV]}

and calculate λ which is around 0.21 eV at this level of theory.


Hopping rates from Marcus theory
================================

The Marcus rate for charge hopping between two sites is:

   k = \frac{V^2}{\hslash}  \sqrt{\frac{\pi}{\lambda k_{b}T}}   e^{
   \left( \frac{-\lambda}{4k_{b}T} \right) }

In our example (V = 6.06 10^-21 J, λ = 3.36 10^-20 J) the Marcus
hopping rate for electrons is 6.9 10^12 s^-1 at 300K.

Macroscopic charge mobilities can be obtained from these hopping rates
via Monte Carlo (e.g. Kwiatkowski et al. [5] ) or analytical
approaches to calculate the diffusion coefficient that goes into the
Einstein relation (e.g. Wen et al. [3] ).


Further considerations for charge mobilities
============================================

As shown by Sutton et al. [6] , the transfer integrals depend strongly
on the amount of exact exchange. The GGA functional PW91 has often
been used with good results. Likewise, the site energies and
reorganization energies will also depend on the functional used. For λ
often hybrids are used in the literature. Other technical settings may
affect the accuracy of the calculation as well: basis sets, numerical
quality...

Pavanello et al. [7] have implemented electronic couplings with the
frozen-density embedding framework in ADF. This approach also allows
the inclusion of multiple molecules which will affect the electronic
couplings. It is formally linear-scaling with the number of
subsystems, hence enabling the study of environment effects on
electronic transport in amorphous crystals. You can watch Pavanello’s
web presentation on electronic couplings with FDE.

When many charge transfer integrals are needed (e.g. an amorphous
crystal with many different dimer orientations), the calculation may
be speeded up. Energy decomposition may be switched off, the SCF
cycles can be set to 0 (with typically a small effect on the
integrals), and the preparation and analysis can be streamlined by
using PLAMS or the AMSprep and AMSreport tools.

Alternatively, one can do very fast, but more approximate, charge
transfer integrals calculations with DFTB.


Charge transfer integrals with DFTB
===================================

More approximate charge transfer integrals can be calculated in a very
fast way with DFTB, see the DFTB manual on charge transfer integrals.
Note that the DFTB parametrizations typically give too low charge
transfer integrals if they are compared with ADF results. Thus one may
need to scale them, or use specifically optimized DFTB parameters (not
available in AMS).

   Start up **AMSinput**
   Switch to **DFTB**: [image: ADFPanel][image] **→** [image: DFTBPanel][image]
   Import or create a system with 2 molecules, and make 2 separate regions
   Select the **Model** and possibly the **Parameter directory** of choice
   In the panel bar, select **Multilevel** → **Fragments**
   Tick the **Use fragments** and **Charge transfer integrals** check boxes
   Run the calculation
   Open the output file by clicking at **SCM** → **Output**
   In the **menu bar**, select **Properties** → **Charge transfer integrals**


References
==========

[1] K. Senthilkumar, F.C. Grozema, F.M. Bickelhaupt, and L.D.A.
    Siebbeles, *Charge transport in columnar stacked triphenylenes:
    Effects of conformational fluctuations on charge transfer
    integrals and site energies*, Journal of Chemical Physics 119,
    9809 (2003).

[2] K. Senthilkumar, F.C. Grozema, C. Fonseca Guerra, F.M.
    Bickelhaupt, F.D. Lewis, Y.A. Berlin, M.A. Ratner, and L.D.A.
    Siebbeles, *Absolute Rates of Hole Transfer in DNA*, Journal of
    the American Chemical Society 127, 14894 (2005)

[3] S. Wen, A. Li, J. Song, W. Deng, K. Han, W.A. Goddard III, *First-
    Principles Investigation of Anistropic Hole Mobilities in Organic
    Semiconductors*, The Journal of Physical Chemistry B 113, 8813
    (2009)

[4] V. Coropceanu, R.S. Sánchez-Carrera, P. Paramonov, G.M. Day, J.
    Brédas, *Interaction of Charge Carriers with Lattice Vibrations in
    Organic Molecular Semiconductors: Naphthalene as a Case Study*,
    The Journal of Physcial Chemistry C 113, 4679 (2009)

[5] J.J. Kwiatkowski, J. Nelson, H. Li, J.L. Bredas, W. Wenzel, C.
    Lennartz, *Simulating charge transport in tris(8-hydroxyquinoline)
    aluminium (Alq3)*, Physical Chemistry Chemical Physics 10, 1852
    (2008)

[6] C. Sutton, J.S. Sears, V. Coropceanu, J. Brédas, *Understanding
    the Density Functional Dependence of DFT-Calculated Electronic
    Couplings in Organic Semiconductors*, The Journal of Physical
    Chemistry Letters 4, 919 (2013)

[7] M. Pavanello, T. Van Voorhis, L. Visscher, J.Neugebauer, *An
    accurate and linear-scaling method for calculating charge-transfer
    excitation energies and diabatic couplings*, The Journal of
    Chemical Physics 138, 054101 (2013)
