Quantification *

Data can be quantified at a number of levels. The most basic form of quantification is using regions applied to a wide scan spectrum; the results are added to the display using an annotation option. Alternatively, both regions and synthetic components can be prepared over a set of narrow scan spectra and the results specified through a set of expressions involving the names used to label the objects. A report generated in this way can be viewed within CasaXPS or exported as a TAB separated ASCII file for use with a spreadsheet program such as Excel. Figure 1 shows the main toolbar button for the dialog window used to perform these two extremes plus a number of reporting mechanisms in between.

The three property pages used to quantify spectra are labelled "Regions", Components" and "Report Spec".

The first step in quantifying a spectrum is defining one or more energy regions. These specify the data channels that are to be included in the intensity calculation.

Each region requires the following information.

Armed with the values described in the above table, a region is created by either pressing the "Create" button on the "Region" property page or by clicking the column buttons on the scrolled list of regions.

The start and end points for a region are taken from the zoom parameters currently in effect when the region is created. Before pressing a create button, zoom into the spectral feature until the display shows the range which characterises the intensity in question. The new region created at this stage will have start and end energies defined by the display.

After creation, the limits can be adjusted under mouse control. To perform this operation it is necessary that the "Quantification" dialog window is visible and the active-page corresponds to the "Region" property page. Position the cursor near an end point for a region, then hold down the left mouse button. A box will appear showing the extent of the region. The end under the cursor is altered when the mouse is moved whilst continuing to hold down the mouse button. Should the same procedure be followed, but instead of initially positioning the cursor towards an end point, the mouse button is pressed when the cursor is towards the middle; the result is both end points adjusted simultaneously. A box outlining the region is displayed and moving the mouse causes the box to shift both end points for the region.

Backgrounds to spectra can be selected from Tougaard, Shirley, Linear or None. To specify a new background type, it is sufficient to type the first character for the background name. For example, to change from a Tougaard to a Shirley, replace the name "Tougaard" by "S" then press return. The regions will be recomputed with the new background type.

The initial and final values for the background are tied to the end points of the region. Noise in the data often makes the value at the end point unrepresentative of the desired background limits. To reduce this problem, a number of channels may be used to define the background limits. The "Average Width" parameter defines the number of data channels to be averaged when calculating the tie-in points for the background.

The backgrounds are calculated using algorithms based on those presented in a number of published articles. The "Linear" case is simply a straight line between the end points of the region. "None" is just a constant set to the lowest data channel in the region. The more complicated backgrounds are those due to Shirley and Tougaard.

The procedure due to Shirley is essentially a weighted-average of the background limits chosen to tie in with the spectrum at the end points of the region. The weighting is determined from the area between the background and the data. Since the weighting is determined using the quantity being computed, a sequence of iterations are required to arrive at the desired result.

Tougaard has extensively studied the subject of backgrounds to XPS spectra, however the background generated by CasaXPS is simply calculated using the Universal loss function. For this algorithm to work it is necessary to remove all instrumental contributions from the spectral shape before calculating the background. This is seldom possible for practical situations. To allow the procedure to provide a background under less than ideal conditions, an adaptive procedure has been adopted that attempts to fit the background to the given spectrum. The result is a background that "looks" plausible in situations when the data has not received the necessary pre-processing and equal to the background proposed by Tougaard for practical applications when the appropriate adjustments have been made to the data.

To delete a region, first select the region in the scrolled list of regions. Then press the "Delete" button.

Each region defined for a spectrum has a number of statistics gathered from the data and displayed in the scrolled list on the property page in question. The raw intensity (CPS eV) between the background and the data, an estimate for the F.W.H.M. and the position of the maximum count rate recorded are all list below the parameters that define the region. In addition, a percentage concentration is shown at the bottom of each column. The raw intensity is calculated directly from the data, however, the percentage concentration includes R.S.F. adjustments. The results of quantification based upon a wide scan and a set of regions are therefore available through the "Regions" property page.

The property page labelled "Components" manages the creation and optimisation of synthetic components.

Synthetic components are specified by name, line shape, R.S.F., position, F.W.H.M. and area (CPS eV).

The name is the means of identifying a component. A quantification report specified using expressions makes use of the name to define how the intensities are to be combined. Further, components with the same name are summed together before an expression is evaluated.

Line shapes and R.S.F. values are stored in the element library and these values are used when a new component is created. A new component is created using the buttons above the list on the "Components" property page. If the element library dialog is active and a transition is selected from the "Element Table" property page, then when a "create" button is pressed the values for the synthetic component are read from the transition so specified. If the element library page is not active then the VAMAS species/transition label is used to pick out the parameters for the component from the element library.

Line-shapes that are available at present are either Gaussian-Lorentzian product or sum functions augmented by an asymmetric tail shape. These are specified as follows, GL(n) where 0<=n<=100 for a product function or SGL(n) for a sum function. A value of "n = 0" provides a pure Gaussian while "n = 100" results in a pure Lorentzian. A tail is introduced by appending T(x) where x > 0 and is typically between 0.5 and 10.0. For example, an asymmetric line-shape for Al 2p might be "GL(30)T(2.0)", see Figure 2 for an example of such a component.

Clicking on the corresponding item in the scrolled list and entering new values can specify peak positions, widths and areas. Alternatively, when the "Components" property page is on top of the dialog window, the mouse can be used to pick up a component and move it to a new position, area or width. If the cursor is located near the top of a peak then the position and area are adjusted. If the cursor is located near the side of the peak then the width will change. Note that the height also changes when the width is adjusted using the mouse. These two parameters are modified simultaneously so as to preserve relationships between peak intensities.

Constraints must be specified for position, width and area. These may be either an interval (for example "0.5,1.3") or the constraint is with respect to another component. In the latter case, the parameter for one peak can be fixed with respect to another by entering the column label into the constraint box. For example, to fix the position of a peak in column "B" with respect to a peak in column "A", type "A" for the position constraint in column "B". The system responds by setting the constraint to be "A + x", where x is equal to the initial offset between the two peaks. The value for x can be modified by hand. To provide an offset of 2 eV, column "B" constraint item should read "A + 2". Area may be constrained similarly, but instead of plus use times, e.g. "A * 0.666". Read, column B component is 0.666 times the area of column A.

There are several types of quantification report. The property page labelled "Report Spec" allows peak intensities to be compared to one another either via information derived from regions, components or both regions and components. In addition, a set of named formulae allows intensities to be mixed and matched in any way that an arithmetic expression can define.

A report generated from regions and/or components is presented in the form of a scrolled list. For each experimental variable from the VAMAS file, the set of quantification items is listed. Information including peak position, F.W.H.M., R.S.F. and raw-area appear in separate columns. The final column is the percentage concentration for each set of quantified items with the same experimental variable.

When the report is based upon regions and/or components, each item is included in the report on a separate line even if a region has the same name as another region or component. This allows the statistics associated with each quantification item to be included in the list. Items with the same name are treated in a different way when formulae are used to generate the report.

If a report is based on regions and/or components, and not all the regions are to be included in the calculation for the percentage concentration, then an R.S.F. of zero should be entered for those items that are to be excluded form the results.

The second type of report is generated using the table of named formulae. Each formula takes the form of an arithmetic expression. The variables in these expressions are the names given to the regions and components, such that the data blocks are currently selected in the Browser view for the active VAMAS file. The list entitled "Quantification item names" shows the set of names currently available for generating a report.

Clicking on the column header buttons labelled "Name" or "Formula" creates a new name/formula pair. The entry is adjusted via a dialog window. Right click over the name item for which changes are required. A dialog window allows the name and/or the formula to be modified or deleted.

Each entry in the "Custom Report" corresponds to the experimental variable in the active VAMAS file. The columns are entered two per named formula, one for the raw intensity calculated from the expression and one for the percentage concentration. Thus a table as follows would result in a report with five columns. One for the experimental variable and two pairs headed "Oxygen CPS eV", "Oxygen %" and "Aluminium CPS eV", "Aluminium %".

The variable names derive from an Oxygen region "O 1s region", and three synthetic components, two "Al 2p" (from a doublet of aluminium metal peaks both named "Al 2p") and one "Al 2p Oxide". The "Al 2p" name was repeated so that the intensities calculated for both peaks for the metal doublet were implicitly summed before the explicit sum defined via a formula computes the total contribution from the aluminium.

Fitting synthetic components to a data envelope is probably the most useful tool in XPS data reduction. It is also one of the more difficult procedures to perform owing to the complex nature of the underlying peak structure and the question of how to account for the background to the data. The line-shapes used to characterise a peak are only approximations and there can be no doubt about the uncertainty associated with the different background algorithms. It is therefore very possible to fit a set of synthetic components with good statistics, yet without any physical meaning.

To provide a guide to peak fitting, an option on the processing dialog window labelled "Test" can be used to replace the true data using one of a set of known peak structures. These structures derive from work presented by Seah and Brown. The relative intensities and separations are those presented in table 4 within that article; a GL(50) line shape has been used to generate the components. It is useful to exercise the fitting procedure using these structures with different options for the synthetic components, especially when a background has been added.

The following table provides the characteristics of the artificial peak structure. Note that the data replaced determines step size, energy position and count rate.

Table 1: Artificial Peak Structures. Carbon C 1s peaks are used to form data envelopes with the offsets and relative sizes list below.

It is very useful to try to fit the artificial peaks without any constraints with respect to the other peaks and then gradually include the relationships shown in the above table. The chi-square value for the curve fit should go to zero. This is the case since the artificial data does not contain noise. Ordinarily, in the presents of noise, the chi-square value should be about equal to the number of degrees of freedom (also shown on the "Components" property page).

A good exercise is to attempt to reproduce the peak envelope for PIB. What should be observed is how difficult it is to produce a good fit for PIB. The three peaks are very hard to identify unless additional information is given to the fitting procedures. If the offsets are first provided then the fit improves, but the exact match does not materialise until both the correct intensity ratios are supplied as well as the relative positions.

Another useful feature of fitting this artificial data is that the strengths of the two fitting algorithms on offer can be seen. PIB is a stern test for the Marquardt method. The uncertainty as to which of the many combinations of similar peak parameters presents a plateau in the parameter space that fails to give any good direction towards the optimum values. On the other hand, the Simplex algorithm, once close and sufficiently constrained, will march straight to the exact fit. The Marquardt method works best when it has a clear view of the target such as with PVC (Figure 4).

The option for switching between the two-optimisation methods is on the "Components" property page.

The optimisation procedure labelled "Marquardt" is actually not a pure Levenberg-Marquardt method. Linear and non-linear parameters are separated. Each step of the optimisation procedure includes the solution of sub-problems introduced by the presence of constraints. The Marquardt method is used to establish the next set of non-linear parameters for the current set of optimal linear ones. Although no optimisation procedure should be viewed as universally applicable, these procedures provide a robust method for determining the peak parameters.

The Marquardt method uses information about both the function and its derivative. There are situations where the informed steps produced by this method are slower that the "lets guess here" approach of the Simplex method. In fact, once the Marquardt method stops making significant improvements in the chi-square value there is no harm in switching to the Simplex method for one last try. This can sometimes push the parameters away from a local minimum that has trapped the Marquardt method or on occasion serve to confirm the optimum has been found.

Marquardt method with constraints can help with the progress towards physically significant peak parameters. However, if the constraints prevent the synthetic peaks reaching an obvious optimum without an alternative available then the algorithm tends to be slow. That is, if no natural optimum lies within the range of the constrained parameters and yet an optimum is visible out side the parameter range, then the algorithm will labour. The problem lies in the numerous probes outside the range that will cause backtracking to the boundary values. Understanding this fact is useful. A well-posed problem will tend to converge quickly, while an inadequate model may result is sluggish behaviour.

The Simplex method on the other hand works based upon maintaining a set of function values at the vertices of an N-dimensional simplex. One of a set of prescribed transformations for the simplex is employed depending on what function value is founded at a probe point.

As it is prescribed, there is very little that can go wrong with the Simplex algorithm. However, convergence to the optimum set of N-parameters is not guaranteed especially when a non-smooth function is used (i.e. when constraints are introduced). It does seem to work well for the situations typically found by CasaXPS, although it has been pointed out that for some optimisation problems, the Simplex method fails to converge when the number of parameters exceeds a problem-dependent value (even for smooth functions). The conclusion is, don’t try to fit too many peaks using the Simplex method unless the parameters are already close to the optimal values.

Once a representative spectrum has been modelled using regions and components, the next question is how to transfer this model to similar spectra within the same VAMAS file and other files. Casa XPS provides a propagation mechanism to accomplish this task.

The spectrum for which the component model has been constructed must be displayed in the active tile view. Any VAMAS blocks that require the equivalent model must be selected via the Browser View for each of the VAMAS files involved. Once these conditions are met, the mouse cursor is placed over the active tile and the right mouse button pressed. A dialog window appears for propagating the quantification objects.

Check boxes are offered on the "Propagate" dialog window. These allow a choice of what actions are propagated through the selected VAMAS blocks. To transfer the quantification model the "Regions" and the "Components" check box should the ticked. Then press the "Ok" button. A progress dialog window appears that allows the propagation process to be terminated (see Processing).

Curve fitting assumes a statistical model for the noise recorded with the data. This model can sometimes be invalidated by the presents of unrepresentative spikes that are due to the detector system and have nothing to do with the true electron yield. It is best to remove such artefacts before attempting to optimise the parameters for the synthetic components.

A data editor is provided on the quantification dialog window. The abscissa and ordinates for the data displayed in the active tile are listed on a property page labelled "Data Editor". The value for an ordinate can be changed. By right-clicking the mouse whilst the cursor is over the corresponding abscissa a dialog window is brought up that allows the ordinate to be edited.

A check box allows the user to specify that an ordinate value should be permanently altered. That is to say, if the data is written back to disk then the adjusted value will be used in the file. It should be noted that some actions in the processing window can cause edits made to the data to be reverse. This occurs if the check box is not ticked and the processing history is used to change the state of the processing (either by resetting or applying a selection of processing to the data). The history mechanism always refers to the unprocessed data before taking the requested action.

A number of authors have proposed methods for identifying the underlying peaks responsible for a measure spectrum. Some of these ideas can be tried out on the known peak structures for PMMA, PVA, PVC and PIB.

PMMA has four peaks, two well resolved and two that merge together to from a broad structure towards the lower binding energies. Spotting the shoulder associated with the third and fourth peaks and determining the locations for these poorly resolved peaks is critical for constructing a physically meaningful model. The presence of a background also interferes with identifying the true structure, although the data in Figure 5 has been prepared without this complication.

Differentiating the artificial data once and then twice can help to see the underlying peak structure (see Figure 5). The shoulder of the third peak in the PMMA envelope can be seen in the second derivative where a kink in both derivatives height-lights the shoulder in the data.

The procedure for quantifying a set of spectra will be explained using a VAMAS file. This file includes measurements for four transitions taken after etch cycles have been used to change the state of the sample. The problem is to create suitable models for each of the transitions using a combination of regions and components, then to define the way these quantities should be combined in the form of a report.

The multiple-document-interface (MDI) used by CasaXPS requires a new document before a file can be selected. When CasaXPS is first run, the initial state is an empty document and the "Open" menu option on the "File" menu is in an active state. If all the documents have been used or none are on offer then the "Open" option as well as the corresponding toolbar button will be inactive. To activate the "Open" option, press the "New" menu button; an empty VAMAS document frame will be created.

A file dialog window allows the file system to be browsed and a VAMAS file to be chosen. Take care within the file dialog. It is the standard MFC dialog with drag and drop functionality. It is therefore possible to move files around simply by dragging the icon for the file over the icon for a folder.

Once read, the VAMAS file structure is displayed in the right-hand side of the Document view. Each transition appears as a column and each row is populated by spectra acquired with the same etch time.

Table 2: Structure of the Browser view for a depth profile.

The first step towards creating a model for each of these transitions is to identify a representative spectrum for each column of the browser. Click the column header for "C 1s". All the carbon spectra then become the browser selection. Pressing the toolbar button for displaying the blocks one per tile; the spectra are displayed in the left-hand scrolled view.

Choose a carbon spectrum that shows a well formed C 1s envelope. The limits for the energy range and hence the background to the spectrum needs to be well defined. A good idea of where the carbon peaks are located is essential for establishing a realistic background shape.

Before proceeding further ensure that your element library is loaded. See the chapter on the element library.

Another important consideration is intensity calibration. Before quantifying the raw data it may be necessary to apply a transmission adjustment. See the processing chapter for a discussion of the issues involved with intensity calibration.

The next decision determines what values are used in the region. In the event that only the total counts for carbon is required, no synthetic peaks are necessary and the name plus the R.S.F. values will be the key information used by the quantification report. On the other hand, if the chemical states within the carbon envelope are to be profiled, then the R.S.F. within the region can be set to zero. The name for the region should be chosen to be different from any that will be used to identify the components.

Let us say the only chemical state information required is that related to the "Al 2p" transitions. For all other transitions, only the total intensity is needed to construct the profile. So for "O 1s" and "Mg 2p" repeat the following steps described for "C 1s".

1. Select all the spectra for "C 1s" by clicking the column header.
2. Enter the spectra into the scrolled list by pressing the button that displays one spectrum for each of the tiles (Figure 7).
3. Scroll to the spectrum that best illustrates the character of the data and click in the tile with the left-hand button of the mouse. This makes that tile the active tile.
4. Bring up the dialog window for quantifying the data (Figure 1).
5. Add a region via the "Region" property page and adjust the parameters until the background and the integration limits look right (what ever that may mean).
6. Ensure that only the set of carbon transitions is still selected in the Browser view of the VAMAS file. Then right-click with the mouse cursor over the active display tile. The "Propagate Actions" dialog window will appear.
7. Select the check box for "Regions" and press "Ok". The result is that each spectrum in the set of carbon spectra now has a region defined to be the same as the spectrum displayed in the active tile.
8. Scroll through the carbon spectra and check that sensible regions have been defined and that the backgrounds are indeed representative of the data. Repeat steps 6 through 8 until satisfactory results are obtained.

Once the above sequence of steps have been performed for "C 1s", "O 1s" and "Mg 2p" it is then time to generate synthetic components for the "Al 2p" data. Figure 2 shows a representative spectrum taken from a sample for which aluminium metal and aluminium oxides were present.

To create a model for an aluminium envelope, the first step is to create a region. In this case the region is best named "Re: Al 2p" and a value of zero should be entered for the R.S.F. value. By entering zero for the R.S.F. value, any quantification report will not include the intensity calculated from the region in the percentage concentration values.

Two pairs of components named "Al 2p Metal" and "Al 2p Oxide" describes the data seen in Figure 2. These correspond to two sets of doublets for aluminium. This knowledge can be introduced into the model via the constraints for the component parameters. The relative spacing and size for such doublets is often known. When included the result is a more meaningful physical model and also one that the optimisation routines work more efficiently.

Armed with the component model for one spectrum from the "Al 2p" set, the other members of the set can be given similar models automatically. Ensure that the "Al 2p" set is selected in the Browser view then, in the tile view, right-click on the active "Al 2p" spectrum. Choose both the "Region" and "Component" check boxes, then press the "Ok" button. Each spectrum within the set is fitting with the same region and components as the model that was prepared earlier.

When components are propagated the parameters are automatically fitted to the target data. If there is a trend in the data, for example, the oxide components dominate at the surface but smoothly diminish with depth, then the component parameters for the previous spectrum in the set of "Al 2p" narrow scans are a better starting point for the optimisation process. The "Propagate Action" dialog window includes a check box labelled "From previous block". If the check box is ticked, the parameters for successive blocks are taken from the preceding block. Otherwise the parameters derive from the block displayed in the active tile.

Again, it is advisable to view the results of these automatic fits by scrolling through the spectra in the tile view. Once satisfied, the VAMAS file is now ready to generate a quantification report.

The first step is to ensure that all the spectra to be included in the report should be selected within the Browser view.

A full report detailing parameters such as position, F.W.H.M., R.S.F., raw-area and percentage concentrations for all the quantification units is obtained by pressing the "Combined" button on the "Report Spec." page. Regions and components can be reported in a similar format by pressing the buttons on the same property page but baring the appropriate names.

A custom report for the same data is usually in a format more suited to showing trends. To generate such a report it is necessary to complete a table of names and formulae.

The first column for a custom report is a list of the experimental variable values. In this example the column will contain etch time in seconds; the units are defined by the VAMAS file format. Subsequent columns are defined by the formulae, and are labelled by the corresponding names.

Pressing the "Reset" button initialises the list of names/formulae (Table 3). The entries are taken from the regions defined for the spectra currently selected within the Browser view. Typically this provides the right number of quantification names, but additional ones may be created or existing ones remove via the edit dialog window. Right-click the mouse button over a name field to display the edit dialog window.

Table 3: Name/Formula List.

Name	Formula
C 1s	C 1s
O 1s	O 1s
Re: Al 2p	Re: Al 2p
Mg 2p	Mg 2p

In the current example, three of the entries are appropriate for the custom report; only the "Re: Al 2p" region needs to be edited. Move the cursor over the name field and then press the right-hand mouse button. The dialog window for editing the name/formula field appears and the current values for these two items are entered into the text-edit fields. Change the name from "Re: Al 2p" to "Al 2p", then edit the formula to read "Al 2p Metal + Al 2p Oxide". Note that the list above the name/formula table contains all the names defined for the report. On the edit dialog window, press the button labelled "Ok"; observe that the name/formula for aluminium has changed to the desired values (Table 4).

Table 4: Name/Formula List after editing.

Name	Formula
C 1s	C 1s
O 1s	O 1s
Al 2p	Al 2p Metal + Al 2p Oxide
Mg 2p	Mg 2p

The custom report is generated by pressing the "Apply" button on the "Report Spec" page. Nine columns of data are displayed in a scrolled list view. The first column is the experimental variable and is followed by eight columns corresponding to the names/formulae previously prepared. The first four of these eight columns list the raw areas that were specified via the formulae, and the second set of four columns represents the percentage concentrations for the same items.

The values within this report may be written to file in an ASCII format. Each column of data is TAB separated. This allows the text file to be read into a spreadsheet program such as Excel, where the data can be formatted for printing either as a table or in graphical form.

When a report is generated the data is displayed in a scrolled list. This represents an alternative view for a VAMAS file document. If a report view is the active MDI frame (indicated by the frame title bar colour), then the main menu only offers "File" and "Window" menus. The options on the "File" menu under these circumstances are restricted to "Save As …"; this allows a File Dialog to be used to specify a text file to receive the report.

Once a report has been saved to disk, Excel can open the text file simply by selected the file name via the "File/Open" option on the Excel main menu. A Wizard for loading text-files into Excel guides the way through the available options. On completion the columns of data appear in a spreadsheet format.

Excel provides many tools for presenting spreadsheets in both tabulated forms as well as graphically. These far exceed the sophistication that could be implemented in CasaXPS and so the use of a spreadsheet for presenting the results seems most appropriate.