X-ray Beam Induced Current Measurements for Multi-Modal X-ray Microscopy of Solar Cells

The key advantage of using lock-in amplification for XBIC measurements is the dramatic increase of the signal-to-noise ratio as compared to measurements with standard amplification. The measurement settings that are particularly critical for successful lock-in-amplified XBIC measurements will be discussed in the first five sections. They are: (a) signal modulation; (b) pre-amplification; (c) signal mixing in the LIA; (d) low-pass filter frequency of the LIA; (e) low-pass filter roll-off of the LIA.

Illustrations of the impacts of these settings are demonstrated in Figure 3, Figure 4, Figure 6. For the measurements, a laboratory setup used a red laser () in place of an X-ray beam, modulated at  2177.7 Hz by an optical chopper. Fluorescent tubes served as a source for bias light. The DUT was a thin-film solar cell with a Cu(In,Ga)Se₂ (CIGS) absorber. Although different measurement settings would be chosen for other DUT, the general guidelines described here to find suitable settings are valid for a variety of DUT such as solar cells with different absorber layers or nanowires. The PA was used with an amplification factor of . The effects discussed here apply equally to other pre-amplifiers. If nothing else is specified, the low-pass filter roll-off of the LIA was 48 dB/oct.

The following sections (f)-(i) show exemplary results to display the possibilities and challenges of XBIC measurements in conjunction with other measurement modes. In (f), specific challenges of XBIC measurements in fly-scanning mode are discussed. In (g), XBIC and XRF measurements of a CIGS solar cell are combined, and the effect of lock-in amplification is discussed with bias voltage applied. In (h), XBIV is added as a measurement mode for a CIGS solar cell. In (i), XBIC and compositional data from XRF of a CdS nanowire are shown. For all XBIC measurements in sections (f) to (i), we used a PA and a LIA as specified in the Table of Materials and Reagents.

(a) Modulation of the Incoming Signal

Figure 3 shows the pre-amplified DUT response measured by a scope without (top row) and with (bottom row) bias light turned on. As the PA converts currents to voltages, the displayed signal is in volts. It is negative due to the contacting of the solar cell, with the p- and n-type contacts connected to the shield and core of the input of the PA, respectively. In XBIC measurements, the solar cell contacting is governed by the necessary grounding of the front contact as discussed in section 1.3.6. of the protocol. 

Comparing Figure 3A and Figure 3D, we note an offset signal on the order of 8 mV that is shifted to -65 mV by turning on the bias light from fluorescence tubes. Furthermore, the signal variation on short timescales is significantly enhanced by the bias light. Such a bias offset of roughly 70 mV can prove problematic, due to limits in the acceptance range of the PA and LIA. As we would like to use the full range of the PA, a small offset as in Figure 3A-C is preferable. Therefore, all sources of unintentional bias, such as ambient lighting, should be eliminated.

Adding a chopped photon source, as displayed in Figure 3B,C,E,F, increases the induced signal by the same amount – roughly 66 mV – for both with and without bias light, when the beam passes through the chopper blade; when the beam is blocked by the blade, the signal remains at the level of the respective offset, as is expected. The frequency of the chopper is distinct in the signal of Figure 3B and 3E with a period of  ms.

In Figure 3D-F, we note an additional modulation at a frequency of 90 kHz. The source of this high-frequency modulation is the electronic ballast of the fluorescent tube, which is driven at 45 kHz. Although lock-in amplification is capable of differentiating the contributions from different modulation frequencies, as will be shown in Figure 6, the reduction of noise signal is paramount for a good measurement. Ambient light is just one possible source, but other electronics can also induce noise, which would then be superimposed onto the signal. Note that bias light is not always unwanted noise, but often bias light is applied on purpose to set the DUT into operating conditions.

In Figure 3B,C,E,F, we note further that the response of the DUT upon change of the irradiation intensity is delayed. These rise-time effects will be discussed in greater detail in the next section and originate here from two distinct effects: first, the steep increase and decrease of the DUT response upon the 2177.7-Hz modulation is delayed by the low-pass filter in the PA. Second, the signal continues to increase/decrease at slower time scales (e.g., visible between 0.68 and 0.80 ms in Figure 3C), which we attribute to the occupation kinetics of defect states in the solar cell.

(b) Pre-Amplification

The PA not only amplifies the modulated signal of the DUT but can significantly change its wave form. As detailed above, the contacts of the solar cell are such that a negative voltage is measured upon illumination. No bias light was added for the measurements shown in Figure 4.

The measurements were taken with increasing filter rise times to demonstrate their effects when amplification strength is held constant. In many cases, filter rise times are hardware-coupled to the amplification. The stronger the amplification is, the longer the response time is, and the smaller is the cut-off frequency of the low-pass filter in the PA^36,37.

With a filter rise time of 10 µs as in the top panel of Figure 4, the signal is barely delayed, spans the nominal peak-to-peak range from roughly 10 mV to -65 mV, and reaches plateaus at the peak values. With 100 µs filter rise time, delay effects are visible in the modulated signal but the modulation is still distinct and the amplitude is in a similar range as for 10 µs. A filter rise time of 1 ms is longer than the period of the modulation (0.46 ms). Therefore, the modulation is suppressed to amplitudes below 10 mV and the shape reflects only the beginning of the rising and falling edge, which is obviously not suited for quantitative XBIC measurements. This connection between gain and filter rise time has to be kept in mind particularly for the combination of fast modulation frequencies, , with strong amplification.

(c) Signal Mixing

The key difference between standard signal amplification and lock-in amplification is the mixing of the DUT signal with a reference signal and the subsequent suppression of high frequencies by a low-pass filter.

The signal path for the mixing is depicted in Figure 5. For the discussion of the signal mixing, a few simplifications are made. The reference signal can be described as a sinusoidal signal

(6)      ,

where   is the amplitude and  is the modulation frequency of the reference signal. The modulated signal of the DUT fed into the LIA can be represented in a similar fashion as

(7)     ,

where  is the amplitude and  is the modulation frequency of the DUT signal, and  is a phase offset of the DUT signal to the reference signal.

Following from (1) and (2), the mixed signal is:

(8)     . 

The modulation frequency of the DUT is the reference frequency, . Therefore, the trigonometric principle

(9)      

can be used to rewrite  as the sum of two terms with different frequencies:

(10)      . 

The low-pass filter mitigates the fast signal  such that the lock-in amplified signal can be approximated^38,39 as 

(11)      . 

The DUT signal mixed with the reference signal is called the in-phase component , and the DUT signal mixed with the 90 ° phase-shifted reference is called the quadrature component :

(12)      

(13)     . 

From Eq. (12) and (13), the RMS amplitude

(14)     

as well as the phase

(15)      

of the mixed signal can be obtained with the two-argument arcus tangent function. Many LIA have an internal phase adjust to set  to zero during measurements.

(d) Low-Pass Filter Frequency

Figure 6 shows the effect of bias light and different low-pass filter settings on the lock-in amplified RMS amplitude, . We used a LIA that allowed us to record the signal resulting from different filter parameters simultaneously.

The cut-off frequency  of a low-pass filter defines the frequency, at which the signal is attenuated to 50%. While lower frequencies are transmitted, higher frequencies are suppressed. Figure 6A,E show the direct signal with  = 466.7 kHz, which effectively does not eliminate noise or lower-frequency modulations but lets them pass with the raw signal. The conversion of the raw pre-amplified signal to the RMS amplitude  leads to an additional factor of  for frequencies sufficiently below . For example, a constant input voltage of  is output as .

Whereas the average offset in Figure 6E is negligible without bias light (in average 2 mV), it increases to an average of around 75 mV with bias light (Figure 6A). The difference is of comparable strength as between Figure 3A and Figure 3D, but beware that these were separate measurements. In both cases, turning on the chopping source leads to a significant increase in , and the peak-to-peak variation of  corresponds to the peak-to-peak variation of the raw signal shown in Figure 3B and Figure 3E.

In Figure 6B,F, the RMS amplitude  is displayed after using a low-pass filter with 1000 Hz. Again an offset can be observed in Figure 6B due to the bias light, but the offset is smaller with around 18 mV on average. This offset is caused by the 100 Hz modulation of the fluorescent light, whereas the 90 kHz modulation is blocked by the low-pass filter. Furthermore, the noise level of the 'beam on' state is still significant with a peak-to-peak variation around 46 mV, while the average signal value amounts to 32 mV. Without bias light (Figure 6F) the peak-to-peak variation amounts to about 17 mV during 'beam on' with an average value of 23.5 mV. The average offset during 'beam off' is smaller than 0.5 mV. These measurements show that the combination of a low-pass filter with 1000 Hz and a chopping frequency of 2177.7 Hz is not ideal: the signal carrying the modulation frequency is only partially removed but not entirely suppressed by the low-pass filter. The remaining part leads to significant peak-to-peak variations of  during the 'beam on' state. When bias light is present, the 100 Hz modulation due to net frequency of the fluorescence lamps further increases the peak-to-peak values.

In Figure 6C,G, the influence of the bias light can be seen as minimal: the 10.27 Hz low-pass filter cuts off most noise and modulation of the fluorescent light, and a clear beam-induced signal can be extracted. Albeit hardly visible here, the offset and spread of noise are still slightly greater with bias light. This can be caused by stray light passing through the chopper wheel onto the DUT. Therefore, it is advisable to implement the chopper far upstream to avoid the modulation of stray light.

Figure 6D,H are a zoom into the change from 'beam on' to 'beam off' after 6 s in Figure 6B,C,F,G, respectively. The superimposed modulation at 100 Hz (fluorescence lamps frequency) is visible in Figure 6D for the low-pass filter with 1000 Hz. Note also the delay in the signal after the filter with 10.27 Hz compared to the signal after the filter with 1000 Hz, when the beam is turned off. Similar to the case for slow rise times of the PA, low  of the low-pass filter in the LIA cause slower adaptation of to signal changes. 

Altogether, we have found that a low-pass filter with 10.27 Hz and a roll-off of 48 dB/oct (see next section) offers in this case the best compromise between fast scanning speed (in favor of high  values) and suppression of bias light or noise (in favor of low  values, most importantly below the grid frequency 50 Hz).

(e) Low-Pass Filter Roll-off

As many digital lock-in amplifiers, the model that was used here employs so-called discrete-time RC filters or exponential running average filters whose characteristics are very close to those of an analog resistor-capacitor RC filter⁴⁰. Apart from the filter cut-off frequency that has been discussed in the previous section, there is only one free parameter, the filter order , that defines the slope of the cut-off as  dB/oct.

Figure 7A shows the effect of the filter order on the frequency-dependent attenuation for different cut-off frequencies that correspond to time constants  ms and  ms. Time constants between these two extremes are suitable for most XBIC measurements. The filter attenuation has been calculated⁴⁰ in the frequency domain as the absolute value squared  of the complex transfer function

(16)      

as a function of the frequency  and a filter of order  with a time constant . Transfer functions of higher order filters are obtained by multiplication of the transfer functions of the serially connected individual filters. Similar to , we define  and  as the frequencies, at which the attenuation is 5% and 95%, respectively. The product of these frequencies and is constant and given in Table 1 for the conversion between the cut-off frequencies and the filter time constant.

In the time domain, the filter response  for  is recursively calculated from an input signal  that is defined at discrete times , , , etc., spaced by the sampling time :

(17)      

The response of filters with  is calculated by multiple iteration of Eq. 17 with  calculated from  and . The filter response to an increasing (at time 0) and decreasing step function (at time ) is shown in Figure 7B for filter orders 1 to 8, as a function of the time in units of . Note that the response is delayed with respect to the input signal and that this delay increases with . The delay is quantified in Table 1 as the times , , and , within which the transmitted signal reaches 5%, 50%, or 95%, respectively.

The choice of the correct filter roll-off is as critical as of the cut-off frequency when designing the experiment. In application 1 presented in section (g), high-quality XBIC measurements have been obtained with a chopper frequency of 1177 Hz, dwell time of 100 ms, and cut-off frequency of 40 Hz at filter order 8. With the numbers from Table 1, this translates into , and . This time is considerably shorter than the dwell time such that no delay-artifacts are introduced.

(f) Dwell Time Correction

In classical step-mode measurements, the scanning stage moves to the nominal position, and the start of the measurement at that pixel position is triggered after the precise position is reached. For short dwell times, the settling time becomes limiting for the overall scan time, which motivates so-called fly-scan or continuous measurement modes: there, the scan stage moves continuously, and the measurement data is attributed to pixels with the encoded stage position in post-processing. However, this can lead to additional issues as shown in Figure 8. In this case, the motors of the sample stage were not moving uniformly in the  direction, resulting in varying dwell times per pixel (see Figure 8A). The dwell-time variations directly translate into variations in XBIC measurements, as seen in Figure 8C. Therefore, the XBIC signal needs to be normalized to the dwell time, the results of which are shown in Figure 8D. Similarly, variations in beam intensity (displayed in Figure 8B) often need to be accounted for by normalization to the photon flux. XBIC signal normalized to the photon flux can be seen in Figure 8E; for minimal error on the absolute XBIC quantification, the photon flux itself has been normalized to its median value. Figure 8F shows the XBIC map normalized to the dwell time as well as to the photon flux, which reduced the impact of most measurement artifacts. Finally, Figure 8G shows the XBIC data after conversion from a count rate to the current using Eq. (1).

(g) Application 1: XBIC of a Solar Cell with Bias Voltage and XRF

Figure 9A-B shows the impact of lock-in amplification on the signal-to-noise ratio in X-ray beam induced current measurements. The noisiness of the direct signal is apparent in Figure 9A: strong intensity contrasts from line to line are indicative of measurement artifacts, and fine XBIC variations from the DUT get buried in the arbitrarily changing signal. On the other hand, these fine features are clearly visible in Figure 9B. Note that the noise level in Figure 9A is unusually high for unknown reasons despite the optimization of the setup prior to the measurements. In such cases, the signal-to-noise ratio improvement by lock-in amplification is dramatically higher than in cases of already high signal-to-noise ratio with standard amplification (e.g., application 3 in section (i)), where lock-in amplification would only lead to marginal improvements.

With the PA, forward (Figure 9C) and reverse (Figure 9D) bias voltages of -50 mV and +50 mV, respectively, were applied to the sample and the area of Figure 9A–B rescanned. The dominant features visible in Figure 9B are still visible in Figure 9C and Figure 9D, but they are less distinct as the maps are noisier. This is because the application of bias voltage or bias light induces a direct current that is often orders of magnitude larger than the modulated XBIC signal. Ultimately, the ratio of direct to modulated signal limits the applicability of lock-in amplification. Despite the poor signal-to-noise ratio, it is worth pointing out that lock-in amplification enables mapping of the solar cell performance at the nanoscale with bias voltage and bias light applied, which would hardly be possible otherwise³⁰.

As the performance of the CIGS solar cell is correlated to the absorber layer composition^7,41, we measured the XRF signal simultaneously with the XBIC. In Figure 9E–F, the concentrations of Ga and In are presented. Both elements are part of the absorber layer and their ratio is deemed to be of great influence to the performance of the solar cell⁷. The statistics of Ga are much greater than for In, which is due to the higher absorption coefficient and less self-absorption at the excitation energy of 10.4 keV. Due to the low statistics, features in the In map are almost invisible, whereas the Ga concentration is clear enough to be correlated with the electrical performance in Figure 9B. For a higher In signal, one could either choose longer dwell times or choose an absorption energy with larger In absorption cross section. This illustrates the importance of a sufficiently long dwell time as well as the tailoring of the beam energy to the elements of interest.

With long dwell times and large maps, another point has to be kept in mind: during measurements spanning multiple hours, sample drift can become a critical issue. Thermal fluctuations (particularly after sample change or large motor movements with poor heat dissipation) and the instability of mechanical stage components often lead to sample drift as can be seen by comparing the vertical positions of Figure 9D and Figure 9B.

(h) Application 2: XBIC of a Solar Cell with XBIV and XRF

Figure 10 shows a multi-modal scan of a CIGS solar cell, where the cell is operated under short-circuit condition measuring XBIC in Figure 10A, and under open-circuit condition measuring XBIV in Figure 10B. The XRF measurement shown in Figure 10C was taken simultaneously with the XBIV measurement. To collect enough XRF counts, the dwell time per pixel was 0.5 s for Figure 10B-C as compared to 0.01 s in Figure 10A. Accordingly, a lower cut-off frequency in the low-pass filter for the XBIV measurement could be used compared to the XBIC measurement (10.27 Hz vs. 501.1 Hz, both with roll-off 48 dB/oct). For XBIV measurements alone, we could have used the same dwell time and low-pass filter settings as for the XBIC measurement with similar signal-to-noise ratio. However, it was overall more time-efficient to combine XBIV with XRF measurements with the XRF measurement governing the dwell time, than performing separate XBIV and XRF measurements. 

Comparing Figure 10A, and Figure 10B, we note that the short-circuit current , measured as XBIC, and the open circuit voltage , measured as XBIV, are correlated: large high- and low-performing areas are visible in both measurement modes. This indicates that local thickness variations and/or recombination dominate the performance here, rather than bandgap variations, which would lead to opposite trends in XBIC and XBIV²⁸.

Further, taking Figure 10C into account, one can see that certain areas with low performance such as at  correlate with low Cu count rate, whereas performance is not correlated with the Cu count rate in other areas.

(i) Application 3: XBIC and XRF of a Nanowire

Beyond solar cells, contacted nanowires²⁴ or nano-sheets, as well as quantum dots, are other examples of DUT that can profit from lock-in amplified XBIC measurements. For demonstration, Figure 11A shows the elemental distribution from XRF measurements, and Figure 11B the corresponding XBIC map of a CdS nanowire. The two contacts made of Pt and the CdS wire are clearly distinguishable, and the XBIC signal shows a matching electrical response. Particularly noteworthy is the fact that XBIC can unveil the electrical performance of the nanowire underneath the Pt contact, which is unique to X-ray nanoprobes and attributable to the high penetration depth of hard X-rays. The complementation of material composition and electrical properties of the nanowire exemplarily demonstrates the advantages of multi-modal X-ray measurements.

Figure 1: Setup for lock-in amplified X-ray beam induced current (XBIC) measurements on a device under test (DUT). The beam path is depicted in red. The green forms indicate optional X-ray fluorescence (XRF) and area detectors for multi-modal measurements, yellow indicates optional bias light. Hardware components for XBIC measurements are colored black, while XBIC signal paths are blue with signal outputs and inputs shown as filled and empty circles, respectively. Before the data acquisition (DAQ), the DC (direct current) and AC (alternating current) signal is converted from a voltage to a frequency (V2F). For alternative signal paths we refer to part (a) of the discussion section. Please click here to view a larger version of this figure.

Figure 2: Example of a kinematic sample holder optimized for multimodal X-ray microscopy measurements including X-ray beam induced current. Thin copper wires are mounted onto the front and back contacts of a Cu(In,Ga)Se₂ (CIGS) solar cell with silver paint, and connected to the PCB contacts. Polyimide tape is used to separate the wires, avoiding short-circuiting of the sample. Please click here to view a larger version of this figure.

Figure 3: Pre-amplified solar cell response upon irradiation with bias light and modulated beam. Top row without bias light, bottom row with bias light: A & D – beam off; B & E – beam on; C & F – zoom into the red rectangle of B & E. Please click here to view a larger version of this figure.

Figure 4: Solar cell response after pre-amplification with three different filter rise times (10 µs – blue, 100 µs – red, 1 ms – green) in the pre-amplifier. Please click here to view a larger version of this figure.

Figure 5: Signal processing by the lock-in amplifier³¹.  is the signal input from the DUT and  is the reference signal from the chopper. Please click here to view a larger version of this figure. 

Figure 6: Lock-in amplified RMS amplitude  with low-pass filter cut-off frequencies 466.7 kHz (blue), 1 kHz (purple), 10.27 Hz (red), and constant filter roll-off 48 dB/oct. The DUT was a Cu(In,Ga)Se₂ solar cell with (A, B, C, D) and without (E, F, G, H) bias light applied. The times when the chopped photon beam was turned on and off are indicated in the figures as vertical dashed lines. Please click here to view a larger version of this figure.  

Figure 7: Effect of low-pass filter settings in the lock-in amplifier. A – Attenuation by the low-pass filter in the frequency domain for two time constants ( ms and  ms) and for filter orders 1 to 8. B – Transmitted signal response of the low-pass filter in the time domain, in units of the time constant , for filter orders 1 to 8 upon step-like change of the input signal from 0 to 1 at time 0 and from 1 to 0 at time . Please click here to view a larger version of this figure.  

Figure 8: Fly-scan measurement of a Cu(In,Ga)Se₂ solar cell at beamline P06 at PETRA III, taken at 15.25 keV photon energy with a focused flux of about  ph/s. The PA was used with  = 10⁶ V/A, and the LIA with  Hz (48 dB/oct). A – dwell time, B – photon flux, C – X-ray beam induced current (XBIC); XBIC map normalized to: D – dwell time, E – photon flux normalized to its median value, F – dwell time and normalized photon flux. G – normalized XBIC signal after conversion from the count rate to the current using Eq. (1). Please click here to view a larger version of this figure.

Figure 9: X-ray beam induced current (XBIC) and X-ray fluorescence (XRF) measurements of a Cu(In,Ga)Se₂ solar cell, taken at the beamline ID16B at the European Synchrotron Radiation Facility with a focused flux on the order of  ph/s. The PA was used with  V/A, the LIA with Hz (48 dB/oct). The beam energy was 10.4 keV, the chopper frequency was 1177 Hz, and the low-pass filter cut off at 40 Hz. The dwell time was 100 ms and the pixel size was 40 nm x 40 nm. The maps A, B, E and F were all taken at the same time; C and D are retakes after 50 min and 113 min, with 50 mV forward and reverse bias voltage applied, respectively. Please click here to view a larger version of this figure.     

Figure 10: Multi-modal measurement of a Cu(In,Ga)Se₂ solar cell, taken at beamline P06 at PETRA III with a focused flux of about  ph/s. The beam energy was 15.25 keV, the chopper frequency was 8015 Hz, and the pixel size 50 nm x 50 nm. A – X-ray beam induced current (XBIC) measured with a dwell time of 0.01 s, a PA with  = 10⁶ V/A, and a LIA with Hz (48 dB/oct); B – X-ray beam induced voltage (XBIV) covering the same area as panel A, measured with a dwell time of 0.5 s and a LIA with Hz (48 dB/oct); C – Cu count rate from an X-ray fluorescence (XRF) measurement, taken simultaneously with the XBIV measurement. Please click here to view a larger version of this figure.         

Figure 11: Multi-modal measurement of a CdS nanowire with Pt contacts, taken at beamline 26-ID-C of the Advanced Photon Source with a beam energy of 10.6 keV. A – Pt and Cd distribution from an X-ray fluorescence measurement. B – X-ray beam induced current (XBIC) measurement taken simultaneously with the XRF measurement, without lock-in amplification. Please click here to view a larger version of this figure. 

Table 1: For discrete-time RC filters of orders 1 to 8, the product of the time constant and the frequency, at which the signal is attenuated by 5% (), 50% (), and 95% (), is constant and given in the top part. In the lower part, the time delay is given, within which the signal reaches 5% (), 50% (), and 95% (), in units of the time constant  and of the inverse cut-off frequency . Please click here to download this excel file.

Table 2: Qualitative assessment of X-ray beam induced current (XBIC), electron beam induced current (EBIC) and laser beam induced current (LBIC).