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CODE: Using measured sheet resistance values in batch fitting

Besides optical spectra CODE computes sheet resistance values of layers stacks. A sheet resistance object in the list of integral quantities computes the current value and – with the ‘Optimize’ option turned on – the squared difference to a given target value.

If you type in a measured sheet resistance value as target value CODE optimizes a layer stack to reproduce measured electrical performance (only if the option ‘Optimize’ is switched on, of course). If you check the option ‘Combine fit deviation of integral quantities and spectra’ (File/Options/Fit) you can fit spectra and sheet resistance at the same time, balancing importance by setting proper weights for each quantity.

We have now added the ability to take into account measured sheet resistance values to batch fitting. In order to use this new feature you have to define a sheet resistance object in the list of integral quantities and name it ‘sheet resistance fit’. Set its ‘Optimize’ option ‘On’. Then generate a new batch fit based on the current model – or use an existing one. Here is an example of the results page:

Note the empty line between the last spectrum (‘Reflectance’ in this case) and the line called ‘Fit’. Enter the key phrase ‘sheet resistance fit’ into the first column and type in or copy the measured values for each sample (i.e. for each column). The table should look like this now:

That’s all – you are now ready to start batch fitting. For each sample the measured values will be entered as target values and CODE will optimize both spectra and sheet resistance values.

You might encounter the difficulty that measured and simulated sheet resistance values do not easily agree. In this case you should check if your sample might show a depth gradient of the conductivity. A reason could be a depth dependent density of defects like grain boundaries within the layer. In such a case you should consider dividing the conductive layer into several parts with different damping constants of the charge carriers. You can read a discussion for silver and other conductive layers here.

Optimizing thin film solar cells with CODE

With CODE you can compute the photocurrent in an active layer of a solar cell. The photocurrent is available as integral quantity which can be the target of the layer stack optimization – very likely the target is ‘as high as possible’. Photocurrent objects make use of a special spectrum type in CODE which is called ‘Charge carrier generation’. Since the required steps to setup such a computational scheme are non-trivial (and a little bit hidden in the documentation) we have generated this tutorial which explains all details.

The example is a thin film solar cell with 2 active layers. One is micro-crystalline (mc-Si) and the other one amorphous (a-Si). Light enters through glass from the top. Between glass and the absorber layers a transparent conductive layer (TCO) collects charge charriers. The backside contact is made of aluminum.

The reflectance of the stack is easily calculated – this is standard in CODE:

If you are interested how much light is absorbed in one of the active layers you have to generate a spectrum of type ‘Layer absorption’ in the list of spectra. Please note that this spectrum type works for layer stacks which consist of either ‘thick layers’ or ‘thin films’ – you should not have ‘rough interface’ or ‘thickness averaging’ layers in the stack.

‘Layer absorption’ objects require the selection of the active layer. A control element for this is available in the upper left corner:

In addition, you have to set a parameter called ‘Number of sources’. CODE divides the layer of interest into several sublayers and computes the local electric field inside each sublayer. Then the absorbed power is computed and added up for all sublayers. ‘Number of sources’ is the number of sublayers – it should be selected high enough so that the final results do not change significantly if you change the number of sources somewhat.

In our example the fraction of light absorbed in the a-Si layer is this:

What really counts for the performance of the solar cell is the number of electron-hole pairs generate by the absorbed light. Not every photon may generate an electron-hole pair that contributes to the current of the cell. The spectrum type ‘Charge carrier generation’ is used here: It works like the type ‘Layer absorption’ discussed above, but multiplies the absorbed fraction by an internal efficiency function.

Each spectrum of type ‘Charge carrier generation’ has subobjects called ‘Local absorption’ and ‘Generation efficiency’. The local absorption subobject shows the local absorption of light within the stack:

Starting at z=0 on the left we see the absorption within the Al layer, followed by the a-Si and mc-Si absorption. Finally, on the right side, the TCO absorbs in the UV and the NIR.

The subobject ‘Generation efficiency’ is used to define the spectral function that returns the probability of the generation of an electron-hole pair by the absorption of a photon. Per default setting the internal efficiency is one:

However, you can click on ‘Definition’ to open a list that allows more flexibility:

In this list you can define a spectral dependence of the generation efficiency, using terms that you usually use for dielectric functions. You can use oscillators like this one:

Or a superposition of many oscillators or any user-defined function. The parameters of each term show up as potential fit parameters – this allows to tune the model and get some information about the generation process.

For simplicity in this example the internal efficiency is 1.0.

Having defined ‘Charge carrier generation’ objects for both active layers (mc-Si and a-Si) CODE knows the relation of absorbed photons and generated electron-hole pairs for each wavelength. In order to compute a macroscopic property we have to integrate over the whole spectrum. This is done by objects of type ‘photocurrent’ in the list of integral quantities. For each active layer you have to define one photocurrent object and assign the proper spectrum:

Editing a photocurrent object you can define the spectrum of the incident light and its integrated, total power. Please note that CODE uses the unit mA/cm^2 for the current density:

Next you have to define the spectrum of the incident radiation. Should the graph be missing press ‘Update’ to see it – or press ‘a’ on your keyboard for automatic scaling. Our example uses an AM1.5 spectrum. Do not worry about normalization – CODE will do that for you. You just have to provide the wanted spectral shape:

Finally, the total power of the illumination is entered in W/m^2:

Now CODE knows all parameters and computes the current density generated in the active layer.

In our example we have 2 active layers which may have different layer absorption and also different current densities:

Depending on layer thickness values one or the other active layer produces more current. You can generate graphs like the following doing a parameter variation (which generates all data in the workbook) and a view element showing workbook data:

 

What makes the difference between a good and a very good optical model?

Answering some recent questions about optical modelling, I dug out some old slides. They explain some model improvements that were implemented a long time ago. Combined with today’s fast computer hardware they provide the basis of high quality modelling.

Optical constant models describe the response of matter to the electric fields of a light wave. The main mechanisms are

  • acceleration of free charge carriers in conductive materials (free charge carrier absorption)
  • exciting oscillations of bound charges (vibrational modes)
  • lifting up electrons from an occupied electronic state to an empty state (interband transitions)

Interband transitions

Electronic transitions can be roughly described by simple harmonic oscillators. However, these are not very realistic and should be used far away from the resonance frequency of the oscillator only. For band transitions of crystalline materials the Tauc-Lorentz model is appropriate:

Be prepared to superimpose several of these terms since crystalline materials may have feature many interband transitions, like AlGaAs:

Amorphous semiconductors and almost all oxides and nitrides used in thin film applications can be described by the so-called OJL model. It is based on some very simple assumptions about the interband  transition and states within the band gap which are caused by disorder:

For many materials the OJL model produces the correct shape of n and k with a few parameters only:

This makes fitting optical constants rather easy. In addition, the band gap parameter as well as the tail state exponent (also called ‘disorder parameter’) have physical meaning and characterize the state of the material.

Vibrational modes

Although harmonic oscillators should be the right tool to describe vibrating parts of molecules or crystal lattices they are simplifying too much. Here is an attempt to describe 2 strong and a weak side band of molecules on a metal surface:

The model is ok but the overall shape of the absorption bands is not reproduced very well. The reason is that the harmonic oscillator has a constant damping parameter, i.e. the vibration is slowed down in the same way in the center (where the absorption is strong) as in the regions of small absorption. As the amplitude of whatever vibration we might have is certainly stronger in the resonance region the interaction with the outside world is certainly stronger. So whatever causes the damping will be stronger close to the resonance frequency and weaker far away from it – at least this is very likely. The so-called ‘Kim oscillator’ model uses a Gaussian decay of the damping with distance from the center of the oscillator and it fits many absorption bands very nicely, much better than the harmonic oscillator:

Besides describing vibrational modes the Kim oscillator can also model high energy interband transitions above the OJL model and the broad band absorption of confined charge carriers (which are almost free but are blocked to leave a microcrystal).

Free charge carrier excitation

The absorption of light by the acceleration of free charges in a metal or doped semiconductor can be described by the classical Drude model. It has 2 parameters only – the plasma frequency is directly related to the concentration of charge carriers whereas the damping constant is connected to the mobility (or the inverse mean free path).

The only class of materials for which we need more than the Drude model is that of highly doped semiconductors, i.e. those materials which are used for transparent but conductive layers (TCOs). The following example shows the difference – the transmission spectrum of a ITO layer is not adequately reproduced by the (in this case too simple) Drude model:

Again, like in the harmonic oscillator case, we need to introduce frequency dependent damping. As it turns out we need to have a smooth transition between 2 levels of damping – high in the mid infrared, and low in the near infrared:

The 3 extra parameters of this model describe the difference of the damping constants, the width as well as the spectral position of the transition zone.

For many applications TCOs need to be rather thick (a few hundred nm). Be prepared to face inhomogeneity in depth of such layers. In this case you need to use 2 or more sublayers which differ in the concentration of charge carriers.

Mixed materials

Surface roughness, porous materials or mixtures of 2 phases require the computation of effective optical constants.

Unfortunately, there is not a unique way of mixing optical constants. Our software packages have the following implementations for effective material properties:

  • Bruggeman model (also known as EMA = Effective Medium Approximation)
  • Maxwell Garnett model
  • Looyenga model
  • Bergman representation (very general, very powerful, but very difficult to apply). In fact, the Bergman representation includes all simple concepts, i.e. Bruggeman, Maxwell Garnett and Looyenga are special cases within the Bergman formalism.

For some composites it does not really matter which approach you take. Here is the reflectance of a SiO2-air mixture, computed using the simple expressions given above:

That should not lead to the wrong conclusion that this is always the case. Especially metals show a large variety of mixed properties. For a silver-air mixture the simple solutions give very different results (in fact, all of them will be quite wrong and you have to use the difficult Bergman approach):

Which model should I use?

Finally, here are some recommendations which model to choose:

 

New start behaviour

Using SCOUT and CODE as OLE servers in your own applications you had – up to now – no chance to verify a correct start of the software. Starting with object generation 5.28 the start-up routine gives information about the attempt to start the software remotely.

To enable response of the server to a calling OLE client the server has to survive a failing start, at least in such a way that it still can react to client requests. Whereas in the past the software would be closed when no proper license was identified it will now continue to run in a restricted mode (and show up in Task Manager).

We have implemented 4 OLE routines to retrieve status information:

  • license_status: returns an integer number charcterizing the state of the program
  • license_status_string: returns a line of text describing the status of the program
  • license_status_details: returns more detailed information about the program status
  • available_program_instances: returns the number of program instances that can be started in addition to the already running instances (most licenses have a limitation concerning the number of instances that can be operated in parallel).

 

WOSP-DESKTOP-RT system improved

Our desktop system for absolute R and T measurements has been modified: You can now use a combination of the WOSP-LEDO-P light source and an Avantes spectrometer to record spectra in the range 270 … 1050 nm. A polarizer is an optional add-on.

The system needs a table of dimensions 1000 mm * 700 mm or larger (table not included in the system). The new design allows reflectance measurements at 8° angle of incidence which was not possible with the previous version.

WOSP-LEDO-P

Following customer demands we have added a small halogen light source to the LEDO system. This enhances the intensity at the NIR end of the spectrum. Although in this case the light source is not based on LEDs alone we still call it LEDO (actually: LEDO-P): The built-in halogen bulb has an average lifetime of 20,000 hours which is in the range of the expected LED lifetime.

There is a strong increase in counts in the NIR range due to the halogen lamp (blue vs. green in the graph below):

Consequently, the NIR noise is reduced and we can extend the useful spectral range up to 1050 nm (100% line, measurements have been done with an Avantes EVO spectrometer):

Spectra taken with the new light source (200 ms integration time, 25 averages, 5 seconds total measuring time):

Stability in time: The graph below shows 25 sample spectra taken during a period of 3 minutes: