Determination of the mean inner potential of cadmium telluride via electron holography Open Access

CdTe is a II-VI compound semiconductor material, with attractive properties for a number of energy¹ and imaging applications.² These properties include a direct bandgap in the infrared region (1.5 eV), and a relatively high density of 5.85 g/cm³ (which confers enhanced stopping power for high energy radiation).² At present, CdTe is being intensively researched as a direct detector material for tomographic medical imaging,³ but to date it has not seen mainstream application.² Growth of CdTe single crystals is a difficult and expensive process, and present devices are affected by relatively high defect densities and performance instability.² Metal-semiconductor junction formation also plays a critical role in achieving usable devices, particularly the formation of a high-quality Schottky contact to suppress leakage.⁴ To underpin the technological development of CdTe, deeper understanding of both its intrinsic properties, as well as the impact of defects and device processing, is highly desirable. Determination of the mean inner potential is important in both respects.

The mean inner potential, V₀, is a volumetric average of the electrostatic potential in a solid-state material. It arises from opposing electric field contributions of positive atomic nuclei in the material, and partial screening by dispersed electron clouds, and is thus always a net positive potential. The mean inner potential may be formulated generally as⁵ 

where Ω is the volume of the unit cell and V(r) is the variation of electrostatic potential, as a function of position inside the crystal. V(r) is, in general, not known, being strongly influenced by redistribution of the outer electrons caused by atomic bonding in the solid state.⁵ Determination of the mean inner potential can thus provide insight into bonding processes that dominate material properties, such as valence electron densities, directionality of covalent bonding, and ionic charge transfer.⁶ 

Experimentally, the mean inner potential plays a direct role in propagation of electron waves through matter and is thus of central importance for electron diffraction and electron holography. In the absence of magnetic and dynamical diffraction contributions, the phase shift $Δϕ$ experienced by an electron wave on passing through a solid specimen with the mean inner potential V₀ can be expressed as⁶ 

where $CE$ is a known constant and t(x,y) refers to the specimen thickness. $Vext(x,y)$ describes the contribution of any additional electric fields (arising, for example, from residual charge, engineered semiconductor junctions, or applied external bias). The phase shift $Δϕ(x,y)$ can be quantitatively mapped with high resolution by electron holography. Assuming that $t(x,y)$ can be determined by independent methods, Eq. (2) allows direct mapping of the internal potential distributions within a material. In the first instance, in the absence of external potentials, electron holography can be utilized to determine V₀ (the subject of the current work). Furthermore, once the mean inner potential is known, more advanced studies can proceed to analyze the fields associated with extrinsic nanoscale features, such as patterned electrical junctions⁷ or crystallographic defects.⁸ 

For execution of experiments, unprocessed single crystal CdTe samples were obtained from Acrorad, Ltd. (Okinawa, Japan).⁹ Electron transparent samples were prepared using an FEI Helios 650 Nanolab Focused Ion Beam (FIB) system. Electron Energy Loss Spectroscopy (EELS) and holography experiments were performed using an FEI Titan transmission electron microscope, operated at 300 kV. Electron Energy Loss Spectroscopy (EELS) data were acquired in STEM mode, utilizing a post-column Gatan Quantum 966. Holography data acquisition employed a Lorentz mini-lens and an electrostatic biprism mounted in the selected area aperture plane (off-axis configuration⁶). Full experimental details are included in the supplementary material.

As mentioned briefly, specimen thickness must be determined independently. EELS is particularly suited to the task of mapping specimen thickness with high precision,¹⁰ but the absolute thickness measurement requires knowledge of the inelastic scattering probabilities, embodied by the mean free path, λ_i. Therefore, an initial sequence of experiments was performed to determine the inelastic mean free path of CdTe at an accelerating voltage of 300 kV, thereby enabling calibrated EELS thickness mapping for subsequent mean inner potential calculation.

Specimens with well-defined geometries were prepared using FIB and analyzed using EELS. In Figure 1(a), one can see a plan view SEM image, illustrating the geometry of the sample after FIB milling (in this case, a 15° wedge sample). In Figure 1(b), one can see a side-view TEM image of the same specimen, in which the protective capping layers and the contrast variation associated with the wedge thickness can be observed. In Figure 1(c), a STEM image is shown, with Label “SI” indicating a spectrum image region from which an EELS spectrum was acquired at each pixel. A typical low-loss EELS spectrum from CdTe is shown in Figure 1(d). It is evident that CdTe exhibits a rather broad plasmon feature centered at approximately 14 eV energy loss, as well as a relatively large background, arising from the excitation of single outer shell electrons.¹⁰ The details of the energy loss spectrum are not the focus of the current work, but rather that the relative quantities of loss (inelastically scattered) and zero-loss (elastically scattered) electrons allow the relative specimen thickness, t/λ_i, to be determined. In this context, I_T (blue) refers to the total integrated counts from electrons that passed through the specimen and were detected by the spectrometer. I_ZL refers to the integrated counts in the zero-loss peak (which is determined by fitting the I_T data using a pre-acquired vacuum reference). The relative thickness, t/λ_i, can be computed according to the equation shown in Fig. 1(d).¹⁰ It is worth pointing out that the spectrometer collection (or acceptance) angle can directly affect the EELS thickness measurement in the small collection angle regime, before reaching a stable value at larger angles.¹² With this in mind, we performed supporting experiments to calibrate the collection angle and to explore the effect of the collection angle on the EELS measurement. As expected, in the large angle regime, the EELS data showed no significant dependence on the collection angle (see Fig. S3, supplementary material, for example), and thus, the maximum available collection angle of 40.5 mrad was utilized for all subsequent measurements.

A map of relative thickness can be created by processing spectra in each pixel of region “SI,” as shown in Fig. 1(e). The t/λ_i values vary in a linear fashion with specimen thickness, as shown in Fig. 1(f), which shows a line profile extracted along QQ′. No problems with linearity were evident in the thin region (where surface plasmon effects might contribute strongly¹⁰) or in the thick region (where multiple scattering may cause non-linear behavior¹¹). These data confirm the validity of EELS measurements of CdTe thickness in this range. As this method relies on prior knowledge of the specimen geometry, and real milled samples may deviate slightly from the intended mathematical design, seven samples with different thicknesses and geometries were made to improve the measurement quality. Figure 2 shows the aggregation of the EELS data from these seven samples. The linearity and mutual agreement, from samples with markedly different thicknesses and ion-milling geometries, are quite good. This EELS dataset, in conjunction with known specimen thickness, leads to a mean value for λ_{i(CdTe, 300 kV)} of 192 nm, with a standard deviation of 10 nm. This value is required in the current work to enable calibrated thickness mapping but is also of great benefit for other electron-beam techniques, such as quantitative EELS for compositional or bonding studies. The relatively large CdTe value is also advantageous for those techniques that require a single-scattering distribution (t/λ < 0.3), in that reasonably thick samples can be investigated.

For comparison purposes, the mean free path for silicon (with much lower density and atomic number) has been previously reported to be 180–200 nm, via EELS.^11,12 Additionally, Gan et al. recently measured the mean free path of ZnTe (a rather similar material to CdTe) to be 46 nm (Ref. 13) via holography amplitude measurements. Therefore, to place the CdTe value in a wider context, it is important to clarify the definition of mean free path, and the electron scattering processes that underpin it. In the current work, the EELS spectrometer did not resolve any energy loss events below 5 eV, so small energy losses, such as those associated with phonon excitations, were not detected. The majority of detected energy loss events were then primarily plasmon excitations and single-electron transitions.¹⁰ The scattering probabilities for these events depend on, in a very complex way, the electronic band structure of the material,¹⁴ and it is not unexpected that CdTe, despite being a high-Z material, will exhibit a relatively long mean free path. Holography measurements of mean free path, on the other hand, are sensitive to even very low energy loss events (such as phonon scattering), given the stringent coherency requirements for interference to occur.⁶ Furthermore, blocking of elastically scattered electrons by system apertures (scattering absorption) is an important consideration for holographic measurements of mean free paths. The theoretical background and pertinent experimental considerations have been explained in detail.^20,21 The ZnTe measurements by Gan et al.¹³ utilized a very small aperture of 0.75 mrad, presumably resulting in significant scattering absorption. Considering both of these effects, it is therefore logical that the holographic measurements of mean free path, as reported by Gan et al. for ZnTe, are much shorter. A key point is that in considering mean free path for electron scattering, caution must be exercised regarding the types of scattering event, or energy range, that contribute to the measurement.

As the next step involves phase measurements, the topic of dynamical diffraction must be introduced. Dynamical diffraction refers to repeated electron wave scattering upon propagation through a crystal. It directly affects wave amplitude and phase and is particularly pronounced in single crystalline materials oriented close to a Bragg condition.¹⁵ It is a complex phenomenon, which can cause very significant electron wave phase shifts that vary dramatically with only slight lattice orientation or thickness changes.¹³ It has been studied systematically by Lubk et al.,²² who explored the effect of dynamical diffraction on the mean inner potential measurements, as a function of simulation methodology, microscope parameters and specimen thickness, orientation, and composition. They found that dynamical scattering results in a systematic underestimation of the mean inner potential, and that the dynamical contribution was reduced as a function of tilt away from low-index crystallographic zone axes. Therefore, for CdTe experimental measurements, a brief study was conducted to ensure that such dynamical contributions were minimized (Fig. 3). The tilt angles of three primary poles ([100], [110], and [111]), and numerous weakly diffracting orientations, were identified using converged beam electron diffraction (CBED), on a CdTe needle sample. Electron holograms were also acquired at each tilt angle according to the usual off-axis scheme (further details on acquisition, reconstruction, and sign conventions are included in the supplementary material). In Fig. 3(a), one can see the CBED patterns acquired at each tilt angle, while in Figure 3(b), the locations of each acquisition are indicated on a stereographic projection (exact tilt angles are tabulated in Table S1, supplementary material). In Figure 3(c), the impact of dynamical diffraction on the phase value is illustrated. Line profiles have been extracted from the phase images, along the long axis of the needle, at each of the tilt angles. When oriented to a low-index zone axis (red), one can see that the measured phase shows very large changes with the position, which is consistent with strong dynamical diffraction effects. When the crystal is tilted away from low-index zone axes (blue), however, one can see that the phase value changes in a stable and monotonic fashion with thickness, which is a confirmation that dynamical diffraction effects are not contributing significantly. Thus, it can be concluded that the phase data collected at these tilt angles (A-E) may be safely used in Equation (2) for computation of the mean inner potential.

This computation of the mean inner potential requires both thickness and phase data, and a practical issue concerns the different pixel sizes, dimensions, and orientations of the images produced by the rastered STEM-EELS (Fig. 4(a)) and optical Lorentz-holography (Fig. 4(b)). With this in mind, a short Matlab code was written to resample and align the thickness and phase 2D data arrays and to perform the computation (details in the supplementary material). The computed mean value for the mean inner potential was 14.0 V, with a standard deviation of 0.9 V (Fig. 4(c)).

Several theoretical estimates of the mean inner potential of CdTe have been reported previously. Schowalter et al.¹⁶ performed ab initio calculations of V₀ in CdTe, utilizing density functional theory (DFT), and obtained values of 13.61 V and 13.67 V (depending on the exact DFT formalism used). This methodology had already been experimentally verified for some other materials (GaAs, ZnTe¹³) and also seems to work well for CdTe. Several analytical calculations have also been reported, based upon the atomic Cd and Te reference data from the literature and by applying approximations to deal with electron redistribution when atoms are bonded in a crystalline lattice. Dvoryankina et al.¹⁷ performed calculations which utilized published values for Cd and Te mean square atomic radii as the starting point, and assumed that electron redistribution effects can be ignored for mean inner potential calculations for covalently bonded materials. They obtained a value of 14.765 V for CdTe. This is somewhat higher than the DFT value¹⁶ but is still well within the experimental distribution. Another established methodology⁵ involves (i) utilizing Doyle-Turner atomic scattering factors; (ii) estimating an upper limit by assuming that the lattice is composed of distinct, non-bonded, neutral atoms (the “isolated atom” or “non-binding” approximation); and (iii) estimating a lower limit by applying an empirically derived correction factor (the “Ross-Stobbs” approximation). Applying this methodology to CdTe, in the same fashion as for ZnTe,¹³ yields an upper value of 14.2 V and a lower value of 10.3 V. The experimental value certainly falls within this range, but it is clear that the experimental distribution is centered very close to the non-binding value. This lends weight to the assumption that the non-binding approximation yields accurate values for covalently bonded materials.¹⁷ 

The standard deviation of 0.9 V in the experimental data, while considered acceptable for this Letter announcing first measurements of this parameter, should be further reduced by conducting a more detailed study and by addressing a number of specific issues. In particular, the contribution from the specimen surface (lattice orientation, oxidation state, and adsorbed species) warrants a dedicated study.^18,19 A more systematic study of the contributions of dynamical diffraction would also be beneficial.^13,22 The roles of synthesis method, crystal quality, and trace impurities should also be investigated. Measurements could also be acquired at multiple accelerating voltages, temperatures, electron dose rates, and specimens with different specimen geometries, to obtain a fuller picture of the mean inner potential for this semiconductor material.

In conclusion, we have applied electron energy loss spectroscopy, converged beam electron diffraction, and off-axis electron holography to obtain values for the EELS mean free path λ_i (at 300 kV) and the mean inner potential, V₀, of CdTe. The obtained values provide basic insights into the electronic structure of the material and should be useful for those engaged in simulation or microscopy studies of this material. CdTe shows significant promise for energy and imaging applications, and the reported values will facilitate efforts to further understand and develop this material.

See supplementary material for additional experimental details, holography data, and tabulated specimen tilt angles.

The provision of CdTe specimens by Acrorad, Ltd., and by Dr. Shin'ichiro Takeda (OIST) is gratefully acknowledged. The authors would also like to thank Dr. Steven Aird for useful discussions and editing the manuscript.