Bachelor Research Project
Drift diffusion simulation of organic semiconducting devices
Dr. R.W.A. Havenith
N.J. van der Kaap
Dr. L.J.A. Koster
This report contains the results of the implementation of a charge injection model and the generalized Einstein relation in a drift-diffusion simulation program for organic semiconductor devices. Charge injection through low energy pathways and image charge barrier lowering showed to have some effect on the current. A non realistic current at zero voltage occurred when the generalized Einstein relation was used. Increasing the number of grid points and decreasing the used tolerance for the numerical integration method were found not to be a solution for this problem. It was discussed that while the used Scharfetter-Gummel discretization scheme is based on Boltzmann statistics, the generalized Einstein relation depends on Fermi-Dirac statistics. It was therefore proposed to generalize the Scharfetter-Gummel discretization scheme.
1 Introduction ... 4
2 Theory ... 5
2.1 Organic semiconductors ... 5
2.2 Charge transport ... 5
2.3 Injection barriers ... 6
2.5 Single carrier device ... 7
2.6 LED ... 8
2.8 Image force lowering ... 9
2.9 Charge injection into organic semiconductors ... 11
2.10 Generalized Einstein relation ... 13
3 Implementation ... 16
3.1 Charge injection and barrier lowering ... 16
3.2 Results and discussion for charge injection and barrier lowering ... 16
3.3 Generalized Einstein relation ... 16
3.4 Results and discussion for the generalized Einstein relation ... 17
5 Conclusion ... 22
6 Acknowledgements ... 23
7 References ... 24
Semiconductors are nowadays used in a wide range of applications like electronic circuits, light emitting diodes and solar cells. Organic semiconductors are an interesting alternative to the traditionally used inorganic semiconductors, because of their relatively easy fabrication, light weight, mechanically flexibility and low cost. A crucial part in the development of organic semiconductor devices is the understanding of charge transport. Understanding of the theory allows to make predictions and give insights in these devices. The theory can be implemented in computer simulations. These simulations can then be used to check the validity of a theory with experimental data.
This report will start with the theory about organic semiconductors. The physical properties of organic semiconductors like the charge transport and the density of states will be discussed. Next, the charge injection into organic semiconductors is discussed. Image charge lowering and a model that describes charge injection through low energy paths are implemented in a drift-diffusion simulation program.
Finally the generalized Einstein relation is discussed. The generalized Einstein relation is a proposed alternative to the currently used classical Einstein relation which has its origin from charge transport in inorganic semiconductors. The main goal of this research will be to implement the generalized Einstein relation in the simulation program, in order to make it possible to check its validity with experimental data.
2.1 Organic semiconductors
Like inorganic semiconductors, organic semiconductors feature a forbidden energy gap between the conduction and valance bands. The origin of the semiconducting properties is the conjugated structure of these materials: the alternation of single and double bonds. In conjugated molecules, the σ-bonds between neighboring carbon atoms form the backbone of the molecule. The Pz atomic orbitals are perpendicular to the backbone of the molecule. A π-bond is formed by two overlapping Pz atomic orbitals (figure 1). The result of the π-bonds is that there will be an alternation of single and double bonds between the carbon atoms, instead of just single bonds. The alternating bonds causes that there will be a small energy gap between the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO), acting as the valance and conduction bands for the material (figure 2a) . Moreover, the conjugation causes electrons to be delocalized over parts of the molecule.
Figure 1: Schematic representation of the atomic orbitals and bonds between two carbon atoms for sp2 hybridization.
2.2 Charge transport
Charge transport in conjugated molecules cannot be described by band transport. The polymer chain exhibits much disorder, caused by kinks and other kinds of defects. The disorder breaks the delocalization of the charge carriers, preventing band transport between different parts of the molecule. Also, inter-chain transport cannot be described using the band model. Instead, charge transport is characterized by hopping: a process in which charge carriers move from an occupied state to an empty state by thermally activated tunneling (figure 2b) . In conjugated polymers, hopping is the slowest transport mechanism and therefore the rate determining step.
The disordered nature of organic semiconductors also causes the conjugated parts to differ slightly in energy. This can be seen in the broadening of the density of states for the HOMO and LUMO. The densities of states for the HOMO and the LUMO are usually approximated by a Gaussian distribution .
Figure 2: (a) Schematic representation of the energy distribution, showing a Gaussian density of states for the HOMO and LUMO and an energy gap that separates them (b) Inside the HOMO/LUMO the charges move by hopping transport.
The charge carrier mobility characterizes how quickly a charge can move through a material when pulled by an electric field. The mobility of a charge carrier is defined as: µ =𝑉𝐸𝑑, where 𝑉𝑑 is the drift velocity of the charge carrier and 𝐸 is the electric field. An analytically derived solution for the carrier mobility is not available for hopping based transport. However, Pasveer et al. made a model to describe the mobility in energetically disordered systems . This model is based on a numerical solution of the master equation for hopping transport. For conjugated systems it was found that the mobility depends on the electric field strength, the charge carrier densities and the temperature.
2.3 Injection barriers
When a metal and semiconductor are brought into contact, their Fermi levels align and thermal equilibrium is established. The Schottky-Mott rule predicts that the barrier height for electrons to move from the metal into the conduction band of the semiconductor is given by:
(1) 𝑞𝜑𝑏 = 𝑞 𝜑𝑚− 𝜒
Were 𝑞𝜑𝑏is the barrier height, 𝑞𝜒 is the electron affinity of the semiconductor and 𝑞𝜑𝑚is the work function of the metal (figure 3) . The work function of a metal is the minimal required energy for an electron to escape from an initial energy at the Fermi energy into vacuum. If the barrier is much higher than the thermal energy 𝑘𝑇, the contact is called non-ohmic: the current is injection limited.
For small barriers the contact is ohmic, as it follows ohms law: the current is limited by buildup of space charge in the effective layer. This results in a current that depends quadratically on the applied voltage .
Figure 3: Energy-band diagram of a metal-semiconductor interface.
Source: Sze, S.M. Physics of semiconductor devices. Wiley-Interscience, 1969.
2.4 Device Structure
A typical device for a conjugated polymer systems that can be used for a LED or a solar cell consist of an active layer of conjugated molecules that is sandwiched in between two electrodes. The top electrode is usually a metal with a low work-function, like Aluminum, Calcium or Barium. Below this is the active layer which consist of conjugated molecules. These layers are placed on a transparent electrode that is usually made of Indiumtin oxide (ITO). Finally on the bottom there is a transparent substrate .
2.5 Single carrier device
If a device has ohmic contacts for holes and non-ohmic contacts for electrons, it is called a hole only device. If this is the other way around it is called an electron only device (figure 4). These single carrier devices rely on the charge transport of only one type of charge carrier. The use of these single carrier devices makes it possible to study the transport in a device without having to worry about recombination effects. Using single carrier devices it is possible to experimentally determine the electron and hole mobility. Graph 1 shows the voltage plotted against the current density for a single carrier device. Parasitical currents between the electrodes causes a leakage current at low voltages.
If the voltage is increased, the current gets dominated by diffusion. In this diffusion regime the current has an exponential dependence on the voltage. At voltages larger than the build-in voltage the current follows a quadratically dependence on the voltage. This is the space charge limited regime where the drift currents are dominating.
Figure 4: Schematic representation of an electron only device.
Graph1: Current density (J) against the voltage (V) for a single carrier device showing the different current regimes: (1) leakage, (2) diffusion and (3) drift current.
Source: G. A. H. Wetzelaer, L. J. A. Koster, and P. W. M. Blom. Validity of the Einstein relation in disordered organic semiconductors. Phys. Rev. Lett., 107:0666605, Aug 2011.
A light-emitting diode(LED), has a low injection barrier for electrons on one side and a low injection barrier for holes on the other side (figure 5). So that electrons are injected into the LUMO on one side and holes into the HOMO on the other side. Applying an voltage on the device causes the charge carriers to move in opposite directions. When an electron and a hole meet each other in the bulk of the device, they form an exciton. Recombination can then take place and a photon will be emitted.
Figure 5: Schematic representation of a LED.
9 2.7 Numerical device model
Numerical device models can be used to simulate the behavior of organic semiconductors. The results in this report are obtained by using a drift-diffusion simulation model . In this model, organic semiconductor devices are represented by the Metal-Insulator-Metal (MIM) model. This model solves the basic equations for semiconductor devices. These equations are the Poisson equation, the current continuity equations and the drift diffusion equations. The Poisson equation is given by:
𝜕𝑥2 = 𝑞
𝜖0𝜖𝑟 𝑛(𝑥) − 𝑝(𝑥)
Were 𝑉(𝑥) is the potential, 𝑥 is the distance, 𝑞 is the elementary charge, 𝜖0 is the vacuum permittivity, 𝜖𝑟 is the relative dielectric constant of the bulk material, 𝑛 𝑥 is the electron charge density and 𝑝(𝑥) is the hole charge density. The Poisson equation is used to relate the potential to the corresponding charge carrier densities.
The current continuity equations are used to obtain the current densities:
𝜕𝑥 = 𝑞𝑈(𝑥)
𝜕𝑥 = −𝑞𝑈(𝑥)
Here, 𝐽𝑛(𝑥) is the electron current density, 𝐽𝑝(𝑥) is the hole current density and 𝑈(𝑥) is the net generation rate.
In order to solve these equations, it is necessary to relate 𝐽𝑛, 𝐽𝑝 to 𝑛(𝑥), 𝑝(𝑥) and 𝑉(𝑥). This is done by using the drift diffusion equations:
(4a) 𝐽𝑛 = −𝑞𝑛µ𝑛𝜕𝑉
𝜕𝑥 + 𝑞𝐷𝑛𝜕𝑛
(4b) 𝐽𝑝 = −𝑞𝑝µ𝑝𝜕𝑉
𝜕𝑥 − 𝑞𝐷𝑝𝜕𝑝
Were µ𝑛 is the electron mobility, µ𝑝 is the hole mobility, 𝐷𝑛 is the electron diffusion coefficient and 𝐷𝑛 is the hole diffusion coefficient. This equation can be used for hopping based systems because we know a relation for the mobility on its dependencies (e.g. the Pasveer model).
In order to numerically solve the system of equations, the discretization scheme of Scharfetter and Gummel is used . The discretization is performed on a one-dimensional grid, were each gridpoint contains the voltage and charge carrier densities. Between these grid points, the current density, carrier mobility and diffusion coefficients are calculated.
2.8 Image force lowering
Boundary conditions near the metal-semiconductor interface require the electric field lines to be perpendicular to the metal contact. When using a one-dimensional model the electric field lines only have one direction component. Therefore it is not possible to let the program find the correct
electric field. A method to solve this problem is by using an imaginary charge with an opposite sign that is in the metal at an distance that is equal to the distance that the original charge is from the interface (figure 6) .
Figure 6: The use of an image charge makes it possible to get the correct electric field lines at the interface.
This image charge results in an attractive force:
(5) 𝐹 = −𝑞2
That lowers the barrier by:
(6) ∆𝜙 = 4𝜋𝜀𝑞𝐸
the maximum of the barrier is not at the metal semiconductor interface but at some distance 𝑥𝑚
inside the semiconductor (figure 7).
Figure 7: The effect of an image charge on the effective barrier height. The solid line shows the new potential. The maximum barrier is now at a distance 𝑥𝑚 inside the semiconductor.
Source: Sze, S.M. Physics of semiconductor devices. Wiley-Interscience, 1969.
2.9 Charge injection into organic semiconductors
In conventional inorganic semiconductors, two relevant injection mechanisms that are usually considered are: Richardson-Schottky emission and Fowler-Nordheim tunneling (figure 8a, 8b). In Richardson-Schottky emission an electron is thermally excited from the Fermi level of the metal and travels over the maximum of the electrostatic potential. Richardson-Schottky emission requires the scattering length of the charge carriers to be much larger than the interatomic separation. However, this is not the case for organic semiconductors. In Fowler-Nordheim tunneling, an electron at the Fermi level of the metal tunnels through the potential barrier. Charge injection into disordered organic semiconductors is different from that of inorganic semiconductors. In disordered organic semiconductors a thermally activated jump raises an electron from the Fermi level of the electrode to the tail state of the density of states distribution. A necessary condition is that the site to which the electron has been excited, has at least one hopping neighbor at an energy equal or lower than that of the site. This condition ensures that charge carriers move away from the interface, rather than recombine . An injection model should therefore take the hopping transport into account.
Also the Gaussian density of states and the image charge lowering should be taken into account (figure 8c).
Figure 8: Charge carrier injection from a metal electrode into a semiconductor: (a) via Richardson- Schottky emission, (b) via Fowler-Nordheim tunneling and (c) via hopping transport in disordered organic semiconductors.
Source: H. Bässler and A. Kohl. Charge transport in organic semiconductors. Top. Curr. Chem. 2012
Burin and Ratner made a hopping transport based injection model . This model is based on the idea that transport occurs through one dimensional straight paths. An assumption made in deriving this model is that the current is in the injection limited regime.
13 2.10 Generalized Einstein relation
The Einstein relation is the relation between two important transport parameters for charge carriers, the diffusion coefficient 𝐷 and the mobility µ . The classical Einstein relation can be written as follows:
Were 𝐷 is the diffusion coefficient, µ the mobility, 𝑇 the temperature and 𝑘 the Boltzmann constant (for all the equations shown here the Boltzmann constant will be given in eV/K and the energies will all be given in eV). A consequence of the disordered nature of disordered organic semiconductors is that they don’t have the same physical properties as classical semiconductors. Therefore, the validity of the classical Einstein relation is under debate [13-14]. The generalized Einstein relation was proposed for disordered organic semiconductors :
𝑛is the carrier density and 𝜕𝐸𝑓𝜕𝑛 is the carrier density differentiated with respect to the Quasi Fermi energy.
The energy distribution of the HOMO and the LUMO can be approximated by a Gaussian density of states . The Gaussian density of states (DOS) is given by:
(9) 𝐷𝑂𝑆 𝐸 =𝑁𝑐
𝜎 2 2
In this equation Nc is the effective density of states, E is the energy, E0 is the center of the Gaussian energy distribution and sigma (𝜎) is the variance. The Fermi-Dirac distribution gives the probability that a fermion will have an energy E:
(10) 𝑓 𝐸 =1
exp (𝐸−𝐸𝑓)/𝑘𝑇 +1
Applying an external voltage causes the concentration of holes in valence band and electrons in conduction band to be out of equilibrium: the carrier concentrations can then nob be described by just one Fermi level. If the disturbance caused by the voltage is not large and not changing too fast, the conduction and valence bands each relax to a state of quasi equilibrium. In this state, a separate Quasi-Fermi level can be used for each band . This changes the Fermi-Dirac distributions for electrons in the conduction band and holes in the valence band to:
(11a) 𝑓 𝐸 =1
(11b) 𝑓 𝐸 =1
Were 𝐸𝑓𝑛 and 𝐸𝑓𝑝 are the Quasi-Fermi levels for the electrons in the conduction band and holes in the valence band respectively.
The charge concentration can then be calculated by integrating over the density of states and the Fermi-Dirac distribution (figure 9):
(12) 𝑝 =−∞∞
𝐷𝑂𝑆 𝐸 𝑓 𝐸 𝑑𝐸
Figure 9: (a)Gaussian density of states (b) Fermi-Dirac distribution and (c) carrier concentration
For a Gaussian density of states the electron and hole densities become:
(13a) 𝑛 =𝑁𝑐
exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 +1
(13b) 𝑝 =𝑁𝑣
exp 𝐸𝑓𝑝−𝐸 /𝑘𝑇 +1
𝐸𝑣 are the centers of the Gaussian distribution for the conduction and valence band respectively. Calculating the charge carrier densities in this way, makes the law of mass action invalid .
The derivative of the charge carrier densities with respect to the Quasi-Fermi levels are:
exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 +1 2
exp 𝐸𝑓𝑝−𝐸 /𝑘𝑇 exp 𝐸𝑓𝑝−𝐸 /𝑘𝑇 +1 2
Using equations 13 and 14, the general Einstein relation (equation 8) can now be written as:
𝑒𝑥𝑝 − 𝐸−𝐸𝑐 𝜎 2
exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 +1𝑑𝐸
𝑒𝑥𝑝 − 𝐸−𝐸𝑐 𝜎 2
exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 exp 𝐸−𝐸𝑓𝑛 /𝑘𝑇 +1 2𝑑𝐸
𝑒𝑥𝑝 − 𝐸−𝐸𝑣 𝜎 2
exp 𝐸𝑓𝑝 −𝐸 /𝑘𝑇 +1𝑑𝐸
𝑒𝑥𝑝 − 𝐸−𝐸𝑣 𝜎 2
exp 𝐸𝑓𝑝 −𝐸 /𝑘𝑇 exp 𝐸𝑓𝑝 −𝐸 /𝑘𝑇 +1 2𝑑𝐸
When the center of the Gaussian distribution is located far from the Fermi energy and the variance is low, the Fermi-Dirac distribution can be approximated by the Boltzmann distribution. The generalized Einstein relation then approaches the classical Einstein equation.
3.1 Charge injection and barrier lowering
An injection model that lets the user choose different barrier lowering mechanisms was implemented in the simulation program. This model first calculates the induced barrier lowering at the boundaries and then recalculates the carrier concentrations at the boundaries using the new barriers. After this is done, the potentials at the boundaries are also adjusted to the new barriers.
At the moment a choice can be made between no barrier lowering, image charge lowering and the Burin and Ratner barrier lowering. Also the possibility is given to calculate the contact densities using Boltzmann or Fermi-Dirac statistics. Other barrier lowering mechanisms can now easy be implemented using this injection model if necessary.
3.2 Results and discussion for charge injection and barrier lowering
In graph 2 the effects of the different injection models on the currents are shown. An electron only device with an energy band gap of 3.0 eV and injection barriers for electrons and holes of 0.5 eV and 2.0 eV respectively was simulated. It can be seen that both image charge lowering and the Burin and Ratner model have some effect on the current.
Graph 2: A single carrier device simulated with different injection models.
3.3 Generalized Einstein relation
At the beginning of the program, the charge carrier concentrations are calculated at the contacts using Fermi-Dirac statistics and a Gaussian density of states (equation 13). This is done by assuming thermal equilibrium at the contacts and by setting the Fermi energies equal to that of the relevant injection barriers. The integral in equation 13 is numerically solved by using Romberg’s integration method . Using exponential interpolation, a guess is made for the charge carrier concentrations
1,0E-12 1,0E-10 1,0E-08 1,0E-06 1,0E-04 1,0E-02 1,0E+00
0 0,5 1 1,5 2 2,5 3
no barrier lowering
Image charge lowering
Burin and Ratner
at the grid points between the contacts. After this is done, the program starts calculating the real carrier concentrations at the grid points between the contacts by making use of the Scharfetter- Gummel discretization scheme.
The diffusion coefficients are calculated in between grid points. This requires that the charge carrier concentrations are also calculated in between grid points. The fluctuations in carrier concentrations between grid points can be very large. For this reason one cannot simply take the average carrier concentration of two grid points. Therefore the growth function was used :
(16a) 𝑔 𝑥 =1−exp
𝜓 𝑖+1−𝜓 𝑖 𝑘𝑇
𝑥 −𝑥 𝑖
∆𝑥 1−exp 𝜓 𝑖+1−𝜓 𝑖
(16b) 𝑛 𝑥 = 𝑛𝑖
1 − 𝑔(𝑥) + 𝑛𝑖+1
Using the in between grid points concentrations and the derivative of the carrier concentration with respect to the Quasi-Fermi level (equation 14), the diffusion coefficients can be calculated with the generalized Einstein relation (equation 8).
Above a cutoff concentration of 𝑛 =12𝑁𝑐, the generalized Einstein relation is not explicitly calculated anymore. The enhancement factor is set equal to the value at 𝑛 =12𝑁𝑐. This is done because it is not expected that the diffusion coefficient increases above the cutoff concentration .
3.4 Results and discussion for the generalized Einstein relation
Simulation of an device that has an energy band gap of 3.0 eV and injection barriers for electrons and holes of 0.2 eV and 2.0 eV respectively gave the current-density voltage plot that can be seen in graph 3. At zero voltage the device is in thermal equilibrium. This implies that there shouldn’t be a current flowing through the device. However, the implemented generalized Einstein relation gives a non zero current at zero voltage. This non realistic current at zero voltage requires some further investigation.
Graph 3: The Current density plotted against the voltage for the classical Einstein relation and the generalized Einstein relation. The generalized Einstein relation gives an unrealistic current at zero voltage.
The first step in trying to solve this problem was to put the generation and recombination of the charge carriers off in order to simplify the problem.
The integral for calculating the charge carrier concentrations and the derivative of the carrier densities with respect to the Fermi energy is calculated using a numerical integration method. This numerical integration method is an approximation of the real value of the integral. Therefore it was checked if the tolerance of this integral was set low enough. However, decreasing the tolerance of the numerical integration did not give a significant different result for the current density (graph 4).
This means that the used tolerance is good enough for the approximation of the integral.
1,0E-11 1,0E-09 1,0E-07 1,0E-05 1,0E-03 1,0E-01 1,0E+01 1,0E+03 1,0E+05 1,0E+07
0 0,5 1 1,5 2 2,5 3
J ( A/m² )
V ( V )
Graph 4: The current density as a function of the numerical integration tolerance at zero applied voltage.
The number of grid points used was gradually increased, in order to check if this could get the current approach to zero. Graph 5 shows that this was not the case. Instead, a rather random fluctuating pattern occurred. Therefore increasing the number of grid points does not improve the outcome.
Graph 5: The current density as a function of the number of grid points at zero applied voltage.
The sigma (σ) of a Gaussian density of states characterizes its width. A smaller sigma means that the Gaussian density of states becomes smaller. The influence of the sigma on the current density at zero voltage was tested (graph 6). For small sigma’s the current density approaches zero. If the sigma is increased, the current density increases and thus deviates more from the expected value.
2,2 2,4 2,6
1,0E-06 1,0E-05 1,0E-04 1,0E-03
J ( A/m² )
Tolerance of numerical integration
J at V=0
0,1 1 10 100
100 200 300 400 500 600 700
J ( A/m²
Number of grid points
J at V=0
Graph 6: The current density as a function of the sigma at zero applied voltage.
A possible explanation for the trend in graph 6, is the assumption of Boltzmann statistics in the currently used discretization scheme for the continuity equation. This dicretization scheme is based on the work of Scharfetter and Gummel . At the moment only the contact charge carrier concentrations are calculated using Fermi-Dirac statistics and a Gaussian density of states. It was assumed that the Scharfetter Gummel discretization method would then automatically calculate the right carrier concentrations in between the contacts.
At this moment it cannot yet be concluded that the generalized Einstein relation doesn’t apply to organic semiconductors. This is because the problem of a current at zero voltage also occurred when the generalized Einstein was not used, but instead only the contact concentrations where calculated using Fermi-Dirac statistics and a Gaussian density of states. This calculation should be done in any case, even when the generalized Einstein relation wouldn’t be valid. The currently used calculation of these contact concentrations is done with Boltzmann statistics, which is just an approximation of Fermi-Dirac statistics (graph 7).
The proposed solution of the above described problem is to make use of a generalization of the Scharfetter-Gummel scheme as was done by Koprucki and Gartner .
1,0E-09 1,0E-07 1,0E-05 1,0E-03 1,0E-01 1,0E+01 1,0E+03
0 0,1 0,2 0,3
J ( A/m² )
σ ( eV )
J at V=0
Graph 7: The concentration plotted against the Fermi energy. The graph shows the exponential dependence for the carrier concentration on the Fermi energy in the case of Boltzmann statistics.
Calculation of the concentration using Fermi-Dirac statistics and a Gaussian density of states shows that only for low sigma’s the Boltzmann approximation is valid.
1,0E-10 1,0E-04 1,0E+02 1,0E+08 1,0E+14 1,0E+20
-0,5 0 0,5 1 1,5
concentration (n) against Fermi Energy (Ef)
Sigma = 0.2 eV
Sigma = 0.01 eV
An injection model that lets the user choose between different barrier lowering mechanisms was implemented in a drift-diffusion simulation program. Image charge lowering and a model made by Burin and Ratner that describes charge injection through low energy pathways showed to have some effect on the current.
Also the generalized Einstein relation was implemented in the simulation program. The simulation that used the generalized Einstein relation gave a non realistic current at zero voltage. Attempts to solve this by increasing the number of grid points and decreasing the tolerance of the used numerical integration method had no success. It was discussed that while the used Scharfetter-Gummel discretization scheme for the continuity equation is based on Boltzmann statistics, the generalized Einstein relation depends on Fermi-Dirac statistics. Therefore it was proposed that a generalized Scharfetter-Gummel discretization scheme that takes into account Fermi-Dirac statistics should be implemented.
I would like to thank my supervisors Jan Anton Koster and Remco Havenith. Furthermore I would like to thank my daily supervisor Niels van der Kaap for supervising my research and answering my questions. Also, I would like to thank Gert-Jan Wetzelaer and the rest of the Photophysics and OptoElectronics group.
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