# Silke Steingrube

### Research interests:

See also my PhD thesis:*Recombination models for defects in silicon solar cells*.

## 1. Solar Cell Modelling

### FEM simulations of solar cells

Many parameters influence the performance of a solar cell. In particular, high-efficiency cells require a relatively large number of process steps, and any not perfectly optimized step might reduce the final conversion efficiency. Often, it is difficult to find the main losses from measurements alone. Also, the potentials for improvement result from the complex interplay of many influences and are hard to predict without detailed calculations.

Therefore, a current topic in the photovoltaics research community are numerical simulations of solar cells and their properties. Thereby, the development of suitable models plays a crucial role, since the models must describe the measured properties as accurate as possible, while keeping the numerical demands low. As well the computer precision as also the models have been improved strongly during recent years. Therefore, it is possible today to predict the behaviour of complete solar cell structures very precisely within suitable computation time. This development is of major interest for the industry, since reliable simulations provide a tool to optimize the solar cells efficiency with low effort and low financial expenses.

At our institute we have several finite element method (FEM) simulators (Sentaurus, COMSOL multiphysics) available which we apply to improve the efficiency of solar cells in collaboration with our experimental colleagues.

### Surface recombination

#### Surface recombination basics

If high-quality material or extremely thin wafers are used for silicon solar cells, the recombination losses of charge carriers at electronically active surface states becomes important. Therefore, one focus in current solar cell research is concerned with the electronic passivation of the cells surfaces.Recombination loss means that light generated charge carriers recombine before they reach the electrical contacts. Since for recombination, both an electron and a hole, are required, the recombination probability increases with the product n*p. Thereby, recombination is enhanced by defects at energies close to the middle of the forbidden band-gap which capture electrons or holes, respectively. In conclusion, three major contributions to surface recombination have to be considered:

- Defect density
- Capture cross sections of the defects
- The carrier product n×p

2.) While at low illumination levels, the capture cross section for minorities dominates the recombination rate, under high injection conditions, the majority capture cross section dominates the recombination rate.

3.) If the product n×p can be reduced in regions where many defects are present, recombination can be reduced strongly. At the interface, the defect density is typically much larger than inside the volume of the cell. Thus, depositing charged dieelectrica at the interface helps to repel one carrier type from the interface while attracting the other one. Thereby, the product of n×p is reduced at the interface. In addition, these dielectrica often serve as a source of hydrogen and thereby reduce the interface defect density.

**FIG 1:**Schematic band-diagram to explain the band bending caused by fixed interface charges Q

_{f}.

#### Surface recombination advanced

Under illumination, the recombination rate U_{s}at the interface is rather proportional to the excess carrier density Δn. Therefore, it is suitable to consider the surface recombination velocity:

_{s}/Δn.

_{eff}= U

_{s}/Δn(x=z

_{scr}),

_{scr}denotes the position of the end of the surface charge region where the energy bands are flat.

**FIG 2:**At Δn < 10

^{15}cm

_{-3}, recombination in the surface damage region dominates the total S

_{eff}. Lines correspond to calculations, symbols correspond to measurements [3].

_{eff}at low injection conditions. However, in particular for SiNx passivated surfaces, a strong injection dependence is observed (FIG 2). We explain this behaviour by introducing a damaged region in a thin layer underneath the interface [1,2]. We assume that the volume density of defects is maximal at the interface and decreases exponentially into the bulk. Assuming the capture cross-sections to be independent of depth, this is equivalent to an exponential depth-profile of the SRH lifetime parameters &tau

_{n}(z) and &tau

_{p}(z). The reduced bulk-lifetime enhances Seff by the additive term

_{deg}= &int

_{z=0}

^{zdeg}U

_{SRH}(z)/Δn(x=z

_{scr}) dz n,

_{SRH}is the recombination rate in the bulk. The existence of the damage region is experimentally verified by HR-TEM images. It is very likely that this damage is connected with a strong hydrogen density measured close to the interface. Hydrogen is well known to improve surface passivation by the saturation of silicon dangling bonds. On the other hand, H forms recombination active defects and voids if present in an excessive amount in the cristalline Silicon network.

[1] S. Steingrube et al., Phys. Stat. Sol. RRL 4, 91 (2010).

[2] S. Steingrube et al., J. Appl. Phys. 108, 014506 (2010).

[3] M. J. Kerr and A. Cuevas, Semicond. Sci. Technol. 17, 166 (2002).

### The a-Si/c-Si interface

Hydrogenated amorphous silicon or is a promising material which can be used for surface passivation. It causes a band bending at the interface caused by the different work functions at the interface and effectively saturates the dangling bonds by H atoms.
However, the interface defects are amphoteric, since H may act as well as a donor as also as an acceptor. The defect distribution at the interface is for example described by the defect pool model [1,2] and is well approximated by a Gaussian distribution. The amphoteric nature of the defects requires a complicated recombination statistics which has no closed form solution 3. The investigation of this recombination statistics and a suitable approximation is a current topic of our work [4,5].

A detailed study regarding this topic is currently under review for publication at a journal. More information will follow as soon as the review process is completed.

[1] S. C. Deane and M. J. Powell, Phys. Rev. Lett. 70(11), 1654 (1993).

[2] M. J. Powell and S. C. Deane, Phys. Rev. B 53(13), 10121 (1996)

[3] F. Vaillant and D. Jousse, Phys. Rev. B 34(6), 4088 (1986).

[4] S. Steingrube et al., phys. stat. sol (c), 7 (2), 276 (2010)

[5] S. Steingrube et al., phys. stat. sol (c), doi: 10.1002/pssa.201127277 (2011)

### Free Energy Loss Analysis (FELA)

Modeling and understanding the transport and recombination of electrons and holes in solar cells is crucial for investigating and optimizing experiments. Here, we derive equations for the free energy balance for thermalized electrons and holes in a solar cell. Equations for the loss rates of free energy due to recombination and transport of carriers are derived. The well known expression for Joule heat dissipation also holds for the free energy loss by diffusive transport. Usually, recombination losses are quantified as an areal current density which makes it difficult to compare them directly to transport losses. In the Free Energy Loss Analysis (FELA), all loss rates are quantified in units of an areal power density (typically mW/cm^2), and can therefore be easily compared in magnitude. The impact of various loss mechanisms on the areal power density extracted from the cell becomes directly apparent [1].

**FIG 1:**The cell volume V of the cell is bounded by contacted surface elements δV

_{e}, δV

_{h}, and non-contacted surface elements δV

_{nc}. Light generated electrons and holes contribute to the electron and hole currents

^{&rarr}j

_{Q,e}and

^{&rarr}j

_{Q,h}, respectively. A part of the generated charge carriers is extracted through the respective contacts.

_{e}to an

*n*-type semiconductor, the hole contactδVh to a

*p*-type semiconductor, and the noncontacted surface δV

_{nc}. The free energy extracted per time is the sum of the free energy flux carried by electrons and holes through their respective contacts.

The FELA is implemented in the open source MATLAB based software CoBoGUI (Conductive-Boundary-model Graphical User Interface). The CoBoGUI uses MATLAB functions for the two-dimensional simulation of solar cells using COMSOL Multiphysics [2]. The CoBoGUI can be downloaded here.

[1] R. Brendel, APL 93, 173503 (2008)

[2] R. Brendel, Prog. Photovoltaics, doi:10.1002/pip.954 (2010)

### Fitting: Genetic algorithms

*The following text appears partly taken from the preprint for a contribution for the 25th PVSEC/ 5th WCPEC, which can be downloaded here.*

The reproduction of measurements, either of complete solar cells or of sample structures, may contribute significantly to the improvement of solar cells. To obtain reliable predictions, the model must rely on physically meaningful parameters, ideally obtained from independent experimental data.
The exact values of several parameters is not known and may, therefore, be determined in a fitting procedure. Several physical meaningful constraints can be imposed to carefully restrict the parameter range.
Nevertheless, the number of local minima in the NxN parameter space increases rapidly with the number of free parameters N, which makes deterministic optimization procedures unsuitable.
Genetic algorithms meet the challenge of being insensitive to local minima and yet to converge relatively quickly. Such algorithms mirror biological evolution in which the fitness of a population is increased by the processes of selection, crossover, and mutation. The fitting algorithm is initialized by setting a seed population. One or more parents are generated with an initial set of reliable parameter values. During each simulation step, a part of the population is replaced by newly generated children. In each step, the current population is evaluated using a fitness function that is based on the least-square-method. Only the "fittest" fraction of the population survives, and is able to propagate during the next simulation step. During propagation, children are generated by mixing the parameter values of two arbitrarily chosen parents, and adding a Gaussian noise to these parameter values. If the fitness of the fittest member of the population does not improve over several simulation steps, the Gaussian noise is reduced gradually to refine the fitting procedure close to the currently optimal parameter sets.

**FIG 4:**Selection, propagation and mutation during the genetic algorithm. The selection is based on a fitness function which selects the optimal parameters. In this example, the fit parameters are the colors of heads and legs.

## 2. Neuroscience: Neural Chaos Control and Robotics

Chaotic attractors which consist of many unstable periodic orbits (UPOs) of different periods in phase space, can be transformed into various stable dynamics by applying chaos control. Using such a controllable chaotic system as a central pattern generator (CPG) in robotics offers the opportunity to select many different actions from a broad behavioral spectrum.### Neural chaos control

Our research focuses on adaptive chaos control of artificial neural networks. We developed a control system which is implemented in a modular structure and is, thus, easily applicable to various neural circuits [1,2]. We demonstrate the flexibility of this chaos control module by using it to control an autonomous robotic system, which requires simultaneous targeting *and* stabilizing in real time.
The so called *adaptive neural chaos control* (ANC) method is designed for the purpose of stabilizing an orbit of an arbitrary target period *p* if no knowledge about the UPOs is provided. Control is achieved by simultaneous detection and stabilization of UPOs.
The controller is modelled as a small neuromodule which allows for flexible and simple integration into larger systems with modular architecture such as sensory-motor networks of complex robots. Furthermore, the coupling strengths between the single units of the neuromodule are plastic, i.e. they are temporally adapting. Integrating an adaptation rule that adjusts these coupling strengths in dependence of the desired period allows to stabilize a large number of different orbits of almost arbitrary periods.

**FIG 5:**Schematics of the application of the neural chaos controller to an artificial two neuron network.

*p*periodic points

*cyclically*such that each orbit point on the target orbit is subject to control. In general, the control strength has to be adjusted for each period and each controlled system, separately. The ANC now overcomes this problem by considering the control strength as a dynamic variable that adapts via synaptic weight changes between the control unit and the controlled network. These synaptic weights are adapted according to an adaptation rule which takes into account the empirical dependencies of the control strength on the period

*p*.

[1] S. Steingrube et al. Nature Physics 6, 224 (2010)

[2] S. Dreissigacker, Diploma Thesis, University of Goettingen (2007).

[3] P. Schmelcher and F. K. Diakonos, Phys. Rev. Lett. 78, 4733 (1997).

[4] P. Schmelcher and F. K. Diakonos, Phys Rev E 57, 2739-2746 (1998).

### Autonomous Robots

*The following text is identically to the abstract of [1]*

Controlling sensori-motor systems in higher animals or complex robots is a challenging combinatorial problem, because many sensory signals need to be simultaneously coordinated into a broad behavioural spectrum. To rapidly interact with the environment, this control needs to be fast and adaptive. Present robotic solutions operate with limited autonomy and are mostly restricted to few behavioural patterns. Here we introduce chaos control as a new strategy to generate complex behaviour of an autonomous robot. In the presented system, 18 sensors drive 18 motors by means of a simple neural control circuit, thereby generating 11 basic behavioural patterns (for example, orienting, taxis, self-protection and various gaits) and their combinations. The control signal quickly and reversibly adapts to new situations and also enables learning and synaptic long-term storage of behaviourally useful motor responses. Thus, such neural control provides a powerful yet simple way to self-organize versatile behaviours in autonomous agents with many degrees of freedom.

**FIG 6:**Chaotic dynamics helps the hexapod robot AMOS to free itself from a hole trap. Single leg dynamics (visualized recording an attached light diode) switches from periodic wave gait to chaotic search behavior when the leg falls into the trap.

[1] S. Steingrube et al. Nature Physics 6, 224 (2010)

[2] S. Dreissigacker, Diploma Thesis, University of Goettingen (2007).