3.1. Main Effects of Optical Irradiation
We report in
Figure 3 four Pockels images all under 600 V but at specific times of the sequence (marked with red dots in
Figure 2), corresponding to these situations: just after the 600 V bias is applied (panel a); just before the optical perturbation (b); at the end of the irradiation interval (c), which lasts 5 min; and 15 min after the end of the irradiation (d). The complete sequence of this experiment can be seen as a movie in the
Supplementary Materials where all images have been normalized by
.
It can be immediately noted that the two first panels (
Figure 3a,b) look similar to each other, with a bright region extending vertically from the anode side and uniform along the horizontal direction. The brighter region close to the bottom anode represents the presence of the electric field in that region, imaged through the electro-optic effect. By comparison, after the application of the focused optical beam, the map shows a substantially different situation (
Figure 3c). The Pockels map becomes highly perturbed, showing a central dark area where before it was light, and most interestingly, a number of fringes close to the bottom anode and at central x. The fringes, according to Equations (1) and (2), now indicate the presence of local electric fields larger than in the first two maps, and increasing up to different multiples of
E0. Substantially, the main consequence of the optical irradiation, which occurs on the cathode side (and whose section is represented as a red segment in
Figure 3b) is a huge increase in the electric field close to the anode, maximum at the transverse position of irradiation x
irr and, as expected, laterally symmetric with respect to this axis. Furthermore, a region of negligible electric field has formed almost circularly on the cathode side.
For the same instants, the vertical crosscuts of the maps were used to retrieve the central electric field profiles that we report in
Figure 3e–h. The two profiles corresponding to the pre-irradiation times (
Figure 3e,f) are basically linear and their slopes tend to slightly increase with time, consistently with the bias-induced polarization due to the hole emission from the deep level (see Equation (A5)).
Looking at the field profiles at the end (
Figure 3g) and after the irradiation (
Figure 3h), we note again that the effect of irradiation is to shrink the field towards the anode, where it becomes as large as 75 kV/cm, whereas the field becomes negligible across most of the detector thickness, except for a weak build-up close to the cathode. However, the most striking result is that the strong perturbation of the electric field persists almost unaltered after the irradiation, in every point of the detector, as can be seen by comparing
Figure 3c,g with
Figure 3d,h. The process is reversible: a voltage reset, by erasing the space charge, quickly restores the initial conditions and a successive irradiation experiment under bias produces once again the same results.
The shrinking effect of the electric field towards the anode has been already reported in the case of uniform optical irradiation [
16,
17] and it is coherent with the same charge state modification of the deep level occurring under dark, i.e., with the increase in its negative space charge. Such an increase is due to the great number of photo-generated electrons initially flowing from the irradiated cathode region. Whereas the hole emission is the dominant process under dark conditions, it is under irradiation that electron capture plays a major role, and its rate is dependent on the electron concentration. In other words, the space charge evolution is still described by Equation (A4) for the same deep level, but the rate is given by
instead of
. Hence, the deep level is able to communicate with both the valence and conduction band like a pure recombination center, but still retains the two charge states typical of traps.
Just after the light is switched off, the electric field appears to be only slightly affected. Then, it remains almost unaltered for 15 min after the end of irradiation (see maps in
Figure 3c,d and profiles in
Figure 3g,h), for almost any spatial point, which indicates the stability of the space charge profile set at the end of the irradiation. Actually, after switching off high optical fluences (integrated irradiance during the time of exposure), residual electric field variations at the anode smaller than a 0.5% were measured in experiments lasting one hour or more. This is consistent with the acceleration in the space charge increase provided by the electron capture: when irradiation stops at
tstop, the space charge evolves with the slow rate
following Equation (A5), starting from the initial condition
. This keeps further changes in the space charge limited to
, which can be possibly very small after exposure to large fluences.
When the diode is biased under dark, hole emission is the dominant, temperature-activated, process [
11]. This is further confirmed by measurements carried out at different temperatures, from 20 to 50 °C (see
Figure S2), where, after each voltage step, the electric field transients are faster at higher temperatures. By comparison, when light is shined on the device, the time constant of the electric field is very fast and does not undergo appreciably change at higher temperatures. This is ascribed to the temperature-independent electron capture process, which prevails over the thermal hole emission. After irradiation, the maps of the electric field remain practically frozen to the last instant of the optical irradiation for all temperatures.
Our results indicate that, under both dark conditions and optical activation, the space charge tends towards the same steady state (Q
ss), which is set by the applied voltage, eventually corresponding to the full ionization of the deep acceptor
, and so does the electric field distribution. Under dark conditions, the rate of the process is very slow, increasing with temperature. Under an optical beam, the rate raises substantially, depending on the irradiance level. Indeed, the transients of the electric field close to the anode in
Figure 4 show both a larger step and speed with increasing levels of irradiance.
Figure 4 also shows that, as a consequence of the increased negative space charge, the electric field levels reached at the anode increase tending to a saturation. Saturation behavior is also observed for the associated space charge under high irradiance levels, as shown in the inset of
Figure 4. This is a signature that, under reaching such a circumstance, the deep level becomes completely electron filled in the space charge region. Again, cathode irradiation looks like a kinetic factor that accelerates the process of achieving the stationary condition. After switching the light off, the space charge remains there under dark conditions, while its residual build-up process is so slow that measures prolonged over hours only showed very small variations.
A small and fast bump is noticed in
Figure 4 at the end of the optical irradiations, which grows with irradiance. This is related to the free carriers present on the cathode side being quickly swept out, thus leaving the electric field being only determined by the fixed charges. This point will be further addressed in the Numerical Simulation Subsection.
In summary, what is usually called bias- or radiation-induced polarization, represents in both cases the evolution towards the full ionization of the deep level responsible for the electrical compensation, resulting in an electric field strongly confined under the anode. The detectors work the best when the electric field extends uniformly throughout its whole thickness. However, because voltage is applied, either under dark conditions and under irradiation, they work in a non-stable situation, which degrades during operation only by gradually shifting it towards the stationary point. In terms of radiation-induced polarization, it should be remarked that some difference might be expected when dealing with X-rays, which are characterized by penetration depths that are much longer than the optical photons considered here.
From a different perspective, these results highlight the potentialities offered by controlling the space charge upon application of an optical excitation. The biased detector works as a reservoir of space charge, which can be activated and drawn locally close to the anode side in correspondence to the irradiated position on the opposite cathode. The persistence of the induced charge when irradiation is switched off enables an optical memory functionality. Additionally, with voltage kept applied, successive optical irradiations will add further local space charge, until the maximum level Qss is achieved. Such a property can be exploited, for example, in dose-meter applications because the total space charge depends on the integrated optical flux. Conveniently, as a read-out tool, the Pockels effect allows us to directly monitor the local electric field, and hence the local space charge, in any instance, without affecting it.
When switching off the voltage bias, independently of temperature or voltage, the net space charge and the electric field is nulled everywhere in the device (except built-in values very close to the electrodes), resulting in completely erasing the memory of previous optical irradiations. Notably, multiple optical irradiations at different x coordinates can be exploited to control the spatial distribution along the
x-direction (see
Figure S3 as an example of two successive irradiations at different points). If irradiation was carried out using focused spots instead of lines, the space charge could be arbitrarily written with resolution in the x-z plane instead of along the
x-direction only, while still keeping the charge integration functionality.
3.2. Electric Field and Space Charge Maps
The Pockels effect has been extensively used to evaluate the electric field profiles between planar electrodes (i.e., field only along the
y-direction, as in our case before the applied optical perturbation) in CdTe-based radiation detectors [
16,
18]. In a recent paper by Dědič [
19], a non-homogeneous x-y electric field was analyzed and imaged in 111 CdTe crystals, as in our case. A strip electrode was present along the
z-direction, rather than our line-shaped optical perturbation, in order to introduce the non-uniformity. As noticed by Dědič [
19], the vertical component
Ey(
x,
y) only is obtained from Equations (1) and (2) when using the diagonal configuration, i.e., with the probe beam linearly polarized at 45° with respect to the vertical direction y. Dědič also calculated, for the 111 CdTe crystal, the numerical relation between the angle of the first polarizer (the analyzer still being perpendicular to it) and the weights of the
Ex and
Ey electric field components that are contributing to build up the electro-optic image intensity.
Here, we follow an alternative single measurement approach to obtain the missing Ex(x,y) component, starting from the Ey(x,y) one, which is that imaged in the configuration with the first polarizer set at 45°. We rely on the property of the electric field being a conservative field, meaning that it can be described as the derivative of a scalar potential. Upon integrating the Ey component along the y-direction for each vertical profile (i.e., at any x), we can hence build back the full potential map V(x,y). We have used for the initial condition the approximation that the field is null at the top cathode electrode. It is then possible to apply the derivative of the potential along the x-direction to obtain the missing component, Ex(x,y) = −dV(x,y)/dx.
To firstly achieve the
Ey(
x,
y) component, we initially normalized the sequence of cross-polarized images
by using the parallel configuration image. At this point, it is needed to reverse apply the modulation formula, but also to unwrap the E values across multiple fringes. This is not always straightforward. Furthermore, due to experimental limitations, the visibility of the fringes can be notably reduced in the region of their maximum density (due, for example, to resolution factors). This would strongly affect the field retrieval. Thus, we decided instead to fit the normalized experimental image along the vertical profiles. Upon using a two-factor function directly for the electric field, on top of which we applied the EO modulation formula, we aimed for the procedure to match the experimental target image. One example of a normalized image (corresponding to the situation at the end of the optical excitation, i.e., to data in
Figure 3c,g) and its fit are presented in
Figure 5a,b, respectively. We note the importance of the experimental minima and maxima in the Pockels image, where they are unambiguously associated with multiple values of
E0.
Figure 5c reports the distribution of the electric field intensity as a false color map, showing its higher concentration close to the bottom anode. The direction of the electric field is represented by the superimposed streamlines, which were retrieved using the condition on the conservative nature of the field as described above (see also
Figure S5 for a map of the two components). Finally, we applied the divergence to the vector electric field in order to achieve a representation map for the excess spatial charge, as shown in
Figure 5d. Interestingly here, the maximum localization of the space charge happens to be at some finite distance from the bottom anode electrode. Its transverse profile (horizontal cut across the maximum) is well reproduced by a Gaussian curve.
The calibration of the electric field units was performed by matching the field integral condition , where V = 600 V is the applied voltage, whose uniformity is well observed at the lateral boundaries. The retrieved E0 is 10–13.7 kV/cm, in very good agreement with the value expected from Equation (2). To compute the spatial charge map N(x,y) we used the formula of the field divergence N(x,y) = ε/q × (dEy/dy + dEx/dx), with ε = (10.3 × 8.85) 10−14 F cm−1.
3.3. Numerical Simulations
Two-dimensional numerical simulations was performed using the semiconductor device simulator “Sentaurus”, part of the Technology CAD software package provided by Synopsys, Inc. [
20]. By starting from a reliable model for the CdTe semi-insulating diode in the dark, a uniform optical irradiation through a 150 μm wide window on the cathode was then implemented.
Without attempting overly onerous best-fitting procedures, the simulations allowed the inference of meaningful ranges for some critical parameters within a two-level model (shallow donor and deep acceptor) that was able to reproduce not only the main experimental features observed in the time sequences, but also in the electric field profiles and, reasonably, in the maps. In particular, this was obtained with concentration differences , with Nsd being a few 1013 cm−3. It was found to be crucial to use comparable values of deep level capture cross-sections around . As capture cross sections directly affect the rates of space charge variations (see Equation (A2)), they are key parameters in our experiments dominated by transient effects.
We remark that, in addition to the present experimental results, different analyses [
21,
22] of similar material showed that the deep level was able to communicate with both valence and conduction bands, which implies comparable capture cross-sections.
According to previous experiments on similar materials, the energy of the deep acceptor was fixed to
Eda = 0.725 eV from the valence band [
22] and the electron Schottky barriers for indium and the platinum contacts at 0.5 and 0.8 eV [
21], respectively. We note that the semi-insulating property within our two-level model is mainly ensured by the proper combination of the
Nsd,
Nda, and
Eda parameters [
21]. With the Fermi level being around mid-gap, the considered Schottky barrier values account for the hole-blocking nature of the Indium contact and the slightly hole-injecting character of the Pt contact [
23]. As previously shown [
21], such a combination of parameters accounts for the completely different electric field profiles experimentally observed among In/CdTe/Pt and Pt/CdTe/Pt detectors.
In order to proper simulate the optical irradiation, the spectral distribution of the incident radiation and the CdTe absorption coefficient [
24] are accounted for by the simulation. A scaling factor was introduced in the simulated optical irradiance to account for the contact transparency.
The whole time sequence, as reported in
Figure 2, was simulated for a proper comparison with the experimental results.
The results of numerical simulations reported in
Figure 6, which show the time evolution of the electric field at the anode in correspondence with the center of irradiation (x
irr) under similar conditions to those corresponding to
Figure 4, show a good agreement with the experimental results. In particular, this concerns the voltage steps under dark conditions and the slow transience observed at 600 V, then the increase in the electric field under different levels of optical irradiance, both in terms of time constant and growth level. As in
Figure 4, the inset of
Figure 6 reports the space charge computed at the anode at the end of the irradiations, which confirms the agreement with the experiments. Importantly, the stability of the electric field after the irradiations is confirmed by the simulations. Electric field profiles at different instants are also comparable with the experiments (see
Figure S4). In particular, a secondary feature is also confirmed, which consists in the weak field build-up close the cathode, which is especially noticed under irradiation. The effect, more pronounced in the experiments than in the simulations, is associated with and sensitive to the slight upward band bending expected for the platinum contact [
23]. Finally, we report in
Figure 7 the maps of the electric field and space charge simulated under the same experimental conditions of
Figure 5. In particular,
Figure 7a refers to the module of the electric field at the end of optical irradiation, which thus can be directly compared with
Figure 5c. Analogously, in
Figure 7b, the map of the simulated space charge can be compared with the experimental one in
Figure 5d.
When comparing the two maps, we should take into account that the map of the space charge in
Figure 5d is subject to some limitations due to both the heuristic fitting functions and because these are applied to experimental images, whose fringe visibility is resolution limited in the region of the most intense field.
However, it can be seen that the agreement is also favorable in 2D space. Remarkably, the simulations confirm the localization of the field close to the anode and its peak density value, in addition to the localization of the space charge with a maximum at a few tens of µm internal position. In
Figure 7c, the horizontal crosscut of the space charge maps is plotted; for the sake of comparison, both simulated and experimental profiles are reported. One main difference can be noted in that the experimental profile is that of a Gaussian shape, whereas the numerical one is slightly flattened in the central zone. We ascribe this to a non-perfect tuning of the parameters, which in this case are seemingly describing a central saturation of the space charge. Simulations at a lower irradiance show a Gaussian transverse space charge profile. On another note, it should be mentioned that the non-uniform space charge vertical profile (i.e., an electric field not linearly varying in space) in the space charge region is not predicted in the approximate full-analytical model expressed by Equation (A7). This is arguably related to the boundary conditions at the anode affecting the charge terms in the Poisson equation.
Many other interesting features emerge from the numerical simulations but their analysis is behind the scope of this paper. Here, we just point out the role of carrier diffusion.
In previous papers, spatially uniform pulsed [
17] and constant [
16] irradiations were performed on the cathode of CdTe diode-like detectors, and it was inferred that electron diffusion was the main transport mechanism close to the cathode. In the present work, the maps in
Figure 5c and
Figure 7a show a circular region around the optical irradiation window where the electric field becomes negligible and diffusion prevails. However, during the optical irradiation, a great number of electron hole pairs are created and simulations show that the hole concentrations (see
Figure S6a) and their associated diffusion currents (
Figure S6b) compete with the electron ones in such a region. Moreover, the net concentration of free carriers is well balanced by the fixed charges provided by the deep level, except when very close to the cathode, where we have already noticed a weak field build-up, indicative of a small positive charge (see the inset of
Figure S6a). This is consistent with the large quasi-neutral region extending around the irradiation window. When the optical irradiation is switched off, the excess of free carriers quickly disappears, either by free carrier recombination or by the deep level trapping, still maintaining the charge neutrality in the same region as during the optical irradiation. As the electric field is mainly determined by the negative space charge in the depletion region under the anode, no appreciable changes occur at the anode when switching off the irradiations. However, the simulations show that complex carrier dynamics is occurring, especially just after the light switching. Hence, it is not surprising that, in contrast to the simulations, appreciable variations are detected in the anode electric field at the irradiation switch off. Moreover, non-idealities, as thin interfacial layers, which are known to be present at the contacts [
11,
25], could, for example, play a role by increasing the surface recombination velocity and thus distorting the electric field distribution.