Measuring nanometre-scale electric fields in scanning transmission electron microscopy using segmented detectors
Introduction
The development of modern materials and devices requires precise control over the electric and magnetic characteristics of these materials and an understanding of how these properties correlate with structural features of the material. Electron microscopy is well suited to characterising these materials. Precise quantification of specimen electric fields has been demonstrated in electron holography [1], [2]. However, it would be advantageous if such electric field quantification could also be performed in scanning transmission electron microscopy (STEM), as this would permit simultaneous acquisition of complementary STEM modes for imaging, such as high-angle annular dark-field, and spectroscopic techniques for elemental mapping, such as energy dispersive X-ray analysis.
Differential phase contrast (DPC) STEM has been demonstrated for studies probing magnetic fields at micrometre and nanometre resolutions [3], [4], [5], [6], [7], [8], and electric fields at both nanometre [9], [10], [11], [12] and atomic [13], [14], [15] resolutions. Conceptually, the field within the specimen deflects the trajectory of the electron probe, resulting in a shift of the bright field disk across the diffraction plane. DPC STEM and associated methods seek to use this deflection information to map out the variation of electromagnetic field strength within the sample. This can be done most effectively if the full scattering distribution is available, and the new generation of fast pixel detectors [16], [17], [18] provides a means for acquiring the full scattering distribution at each position in a STEM scan raster. However, DPC STEM has to date mostly been undertaken using segmented detectors, which offer sensitivity to beam deflection through the increase in signal in some segments and the decrease in others. This more established technology allows much faster scan speeds and live imaging [11], [19], meaning it is still the most practical method for industrial application. However, because segmented detectors only give a coarse sampling of the scattering distribution, more care is required for quantitative analysis. This paper explores several issues that relate to achieving quantitative measurement of nanoscale-electric fields in STEM using segmented detectors.
As our case study, we use DPC STEM data for the GaAs p-n junction presented in Ref. [11]. This specimen was 290 nm thick with a symmetrical p-n junction between 1019 cm p-doped (Zn) and 1019 cm n-doped (Si) GaAs. Fig. 1(a) shows unprocessed segmented detector STEM images of this p-n junction acquired with a 16 segment JEOL detector [19]. The camera length was chosen such that the bright field disk was situated mid-way through the third annular ring of the segmented detector, as shown schematically in Fig. 1(b). Image contrast at the p-n junction is clearly visible in the STEM images from the detector segments in the horizontal direction, consistent with the expected increase or decrease in electrons incident upon those segments due to the deflection (towards the right in the figure) of the bright field disk caused by the junction electric field. Faint contrast is also visible for the detector segments in the vertical direction, due to a combination of “absorption contrast” (intensity scattering outside the bright-field disk; hence consistently dark contrast for detectors within the bright field disk), detector inhomogeneity, and residual defocus contrast. Shibata et al. [11] used a model-based analysis to quantify the thickness-integrated electric field strength along the beam direction, which was possible because of the simple geometry of the p-n junction. However, general reconstruction of field distributions of unknown structure and symmetry requires more direct reconstruction methods. This paper compares three reconstruction approaches, exploring consequences of thick specimens and limitations of implicit assumptions in reconstruction algorithms using a Fourier basis.
The paper is organised as follows. In Section 2 we investigate three methods of reconstructing the p-n junction projected electric potential and electric field distributions without recourse to a model-based approach: DPC via the first moment or “centre of mass” approach of Waddell and Chapman [20] and Müller et al. [21] as approximated by the segmented detector, and two segmented detector ptychography (SDP) approaches, one due to Brown et al. [22] and the other to Landauer et al. [23]. These methods make potentially limiting assumptions: approximating the centre of mass using segmented detectors in DPC and the weak object approximation in SDP. This means that discrepancies in the quantitative reconstruction of the projected electric field are found when compared with the earlier model-based analysis. In Section 3 we show that by extending the DPC approach using an experimental calibration of the centre of mass response of the detector system we are able to quantitatively retrieve the p-n junction projected electric field within error bars. We further demonstrate that plasmon scattering contributes appreciably to the required calibration correction, and in Section 4 we show how some of the SDP methods can be modified to account for plasmon scattering. In Section 5 we show how to overcome the distortion that enforcing periodic boundary conditions introduces to the reconstructed potential away from the junction.
Section snippets
Reconstruction of the electric field and potential
We will compare three different approaches for reconstructing the projected electrostatic potential of the p-n junction using the experimental data in Fig. 1. One approach can be used to calculate the electric field information directly, but the other two are phase-retrieval-type approaches, geared towards calculating the projected electrostatic potential, from which the projected electric field can then be determined. The starting point for all three methods is the phase object approximation:
Calibration of the detector centre of mass
To motivate an empirical correction to the centre of mass approach to DPC, consider the simplified model shown in Fig. 3, in which the probe diffraction pattern and the detectors Dj(k) are all rectangular with width w and with lengths d and D, respectively. When, as shown in the figure, is displaced by any amount δ (provided only the diffraction pattern does not extend beyond the bounds of the detector), Eq. (5) gives the centre of mass of the intensity measured on
Accounting for plasmon scattering in segmented detector STEM phase retrieval algorithms
In the previous section we saw that with the specimen in the path of the beam the gradient of that experimentally calibrated detector centre of mass response was diminished relative to that in the no sample case (i.e. that due to detector geometry alone). This suggests that scattering mechanisms beyond that captured in the transmission function in Eq. (1), with the scattering potential due to the long-range electric field of the p-n junction, need to be considered.
Consider Bragg scattering, by
Refinements for the construction of the projected electrostatic potential
As shown in Fig. 2, all methods gave similar values for the field strength in the vicinity of the p-n junction. In Sections 3 and 4 it was shown how, by appropriately calibrating the response of the detector system, quantitative reconstruction of projected electric fields was possible in DPC STEM. In this section we return to consider the reasons for the discrepancies between the methods in the reconstructed projected electrostatic potential away from the p-n junction, with the goal of refining
Comparison between methods
Ref. [11] presented a comparison between the model-based DPC reconstruction and an off-axis holographic analysis of this same sample (drawing results from Ref. [46]), showing the two methods to be in good agreement. It is worth briefly commenting on the relative strengths of the different approaches. Holography is capable of recording large fields of view simultaneously, and consequently has great sensitivity to small local potential gradients which accumulate to large potential differences
Conclusion
Using STEM images of a p-n junction recorded with a segmented detector as a test-case, we have explored quantitative electric field mapping based on three reconstruction methods: DPC based on centre of mass analysis and two SDP approaches.
In the form usually written down, the three methods significantly underestimated the magnitude of the built-in electric field relative to a previous model-based analysis. For this thick specimen, plasmon scattering was found to contribute significantly to the
Acknowledgements
We thank Daniel Taplin and Dr Laura Clark for helpful discussions. This research was supported by the Australian Research Council Discovery Projects funding scheme (Project DP160102338). N.S. acknowledges support from SENTAN, JST and JSPS KAKENHI Grant number JP26289234 and JP17H01316.
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