Simulations

  1. Drift-diffusion

Drift-diffusion (DD) is a powerful and widely accepted method for simulating semiconductor devices. The model is based on solving Poisson and drift-diffusion equations simultaneously for electrons, holes, traps, and any other fixed/mobiles charges present in a semiconductor device. The model has been used for simulating solar cells (organic/dye/perovskite), transistors (BJT/FET/MOSFET), diodes, and light-emitting diodes (organic/inorganic). A generalized drift-diffusion model can be defined as [1, 2, 3]:

Where ε is the material dielectric constant, and Φ is the electrostatic potential. q defines the elementary charge, µn the electrons mobility, and µp defines the hole mobility. n, p, Nd+, Na-, Nct and Nan are electron density, hole density, ionized donor density, ionized acceptor density, fixed positive ion density, and fixed negative ion density, respectively. nt+ and nt- describe hole and electron trap densities, respectively. Φn and Φp are the electrochemical potentials of electrons and the holes. The last two equations are the continuity equations for the electrons and holes. R and G describe the net recombination and generation rates, respectively. ∑all represents the collective contributions of a particular type of charges/ions/traps from all possible sources. All these parameters are defined differently in the different parts of a device. For all the mobile charged species, a separate continuity equation needs to be defined.


As an example, for a perovskite solar cell with electrons, holes, traps, and mobile ions (positive and negative), the model can be defined as [4]:


Upon light exposure, electron-hole pairs are generated in the semiconductor material. Optical generation rate (G) can be calculated by using Lambert-Beer model for light absorption :

Where, G(x) is the generation rate at position x, φ(λ) is the solar light intensity. α(λ) is the absorption coefficient of the material at wavelength λ. Recombination (R) can be calculated by summing up contributions from SRH recombination, Bimolecular recombinations, Langevin recombination, Auger recombanations and any other recombination involved. Depending upon the materials and the device architecture, different recombination mechanisms can be present. Direct recombination is a bimolecular process involving electrons and hole, and releases energy in the form of a light photon, governed by:

Rdir = Kdir(np-ni2)

Where, Krad is the bimolecular recombination rate constant, and ni is the equilibrium carrier density. This type of recombination is the desired process in Light-emitting-diodes and laser diodes. Another prominent recombination process in electronic devices is is Shockley-Read-Hall (SRH) recombination. This is a non-radiative, trap-assisted recombination process governed by:

Another prominent recombination process in electronic devices is Shockley-Read-Hall
(SRH) recombination. This is a non-radiative, trap-assisted recombination process governed by:

Where, Et is the trap energy level with respect to the midband energy. τn and τp represent the trapping time constants of electrons and the holes, respectively. It is mainly an undesired recombination process, originated due to trap centers and defects in (bulk and at the interface) a semiconductor material. Langevin recombination ia another type of bimolecular recombination process, occurring in low mobility semiconductors and disordered systems such as organic materials (and hence organic solar cells and LEDs). The Langevin recombination strength is defined as:

where e is the unit charge, ε0 the vacuum permeability, εr is the relative dielectric constant of the semiconductor. γLan is Langevin recombination rate factor. More details about recombination processes can be found here...
The DD model can be used to simulate organic/inorganic/hybrid electronic devices such as solar cells, LEDs, and transistors, etc. DD simulations for a perovskite cell can be found below:

The DD model can be used to calculate electrostatic properties in a solid-state dye-sensitized solar cell. As another example, the calculated orthogonal electric field at the interface of TiO2/Organic HTL is shown in the figure -->:


More information on 3D drift-diffusion simulation for a solid-state dye-sensitized solar cell can be found here. DD simulations for organic field-effect transistors can be found here. DD simulation for organic solar cells can be found here.

3-dimensional drift-diffusion simulation for a solid-state dye-sensitized solar cell can be found here. DD simulations for organic field-effect transistors can be found here. DD simulation for organic solar cells can be found here.

Note: The DD is a semi-classical model where the optical/electrical properties are defined as a point/line/surface/volume representing a physical material. The accuracy depends on the number of points and the mesh size defined at every point. Different recombination models (bimolecular, Shockley–Read–Hall, Auger, Langevin, surface recombination), mobility models (constant, electric field dependent, temperature-dependent, doping dependent), and optical generation models (constant, exponential, Lambert-Beer, transfer-matrix method) can be defined at different points in the material space.

The DD simulation gives information (in different operating conditions) of energy levels (valence band, conduction band, Fermi level), charge density (electron, hole, ions, traps), current densities (electrons, holes, contacts, ions), electric field and electric potential distribution, carrier mobilities (electron, hole, ions), recombination and generation densities at each point defined in the structure at various time scales. The model can be used to simulate organic/inorganic/hybrid solar cells, LEDs, diodes, BJTs, FETs, and MOSFETs. The model can further be integrated with quantum models to simulate quantum dots, quantum wells, nanowires, and nanowire FETs.


Some of the open-source and commercial drift-diffusion tools are listed below:

  1. TiberCAD

  2. Matlab Open-Source DD

  3. AFORS-HET

  4. FLUXiM

  5. SCAPS-1D


For time-dependent 3D simulations at the molecular dynamics level, the DD becomes very complicated and computationally expensive. For this purpose, other tools such as Kinetic-Monte-Carlo (KMC), and/or density-functional-theory (DFT) in conjunction with DD/KMC are employed.


2. Kinetic Monte Carlo Simulation

Kinetic Monte Carlo (KMC) is another simulation model to analyze the kinetic behavior of semiconductor devices. We employ KMC simulations implemented in C++ (can be implemented in any other programming language of choice). In the simulation, the initial state of a system (i.e. distribution of charge carriers) is chosen randomly. Then, using possible transitions (like electron/hole transport, ion hoping), a new state of the system is found. This continues for the defined number of iterations. Hoping between the localized states is described by Miller-Abraham's formula.

Where a0 is an attempt-to-hop frequency given by phonon interaction and rij is localization constant, the distance between two localized states Ɛi and Ɛj. Uncertainty in the energy levels and the energy disorder (σ) are considered as random Gaussian fluctuations. The energy Ei at each point Xn is given by

where Ei0 represents the ideal energy level of the considered molecular orbital (MO), Eiσ is the energetic disorder, EiF is the external electric potential and EiC is the Coulomb energy of state i. Transition time, a lifetime of the state, and the simulation time give information about the system (the solar cell) dynamics. More information...

The Kinetic Monte Carlo technique bridges the gap between macroscopic and microscopic approaches. The time scale problem of the molecular-dynamics (MD) approach is overcome by exploiting the fact that long-time dynamics of the system consist of diffusive jumps from one state to another. Rather than following the trajectory, these transitions are directly treated. This results in acquiring longer time scales because KMC characterizes the system with more underlying macroscopic states where even fast vibrational effects can occur. The dynamic evolution of the system can be seen by transitions between events (long-time events). Such a system is called an infrequent-event system. Hence, a sufficient amount of processing time can be saved by neglecting underlying fast motion effects and considering event transitions. Thus overall simulation time is improved. On the other hand, to overcome the limitation of the DD approach, the KMC algorithm makes use of localized states where the hopping process of particles (excitons, electrons, and holes in case of a solar cell) can take place. To implement the nanoscale morphology, a discretized grid of localized states is implemented on the active layer, where hopping can occur. This gives a better morphology as well as the dynamic process of individual particles at the nanoscale. Overall the KMC can give better insight into the device dynamics but it is computationally expensive.

As an example, dynamics of ionic-movements in perovskite film consisting of grain boundaries can be studied using KMC simulations. Considering grain boundaries with energetics disorder 𐤃EGB, calculated iodine ions distribution for an Illuminated (short circuit) CH3NH3PbI3 solar cell is shown below:

It can be inferred from the figure that, when the grain boundaries have a negative energetic disorder, the iodine ions are confined to travel through the grain boundaries rather than the bulk of the grain. Similarly, the simulations can be done for an open-circuit or an electrically-biased solar cell to calculate ionic distribution, charge carrier distribution, and the JV characteristics.

KMC simulations for organic solar cells can be found here.