3.1. Numerical Simulations
In this section, we analyze several examples characterizing different surface conditions of samples described by the efficiency of the surface source of free carriers and/or by band bending. We perform numerical simulations of the charge dynamics solving one-dimensional drift-diffusion and Poisson’s equations. Both electrons and holes are involved. The free carrier interaction with impurity levels is described by the Shockley-Read-Hall model [
14]. Details of the approach may be found in [
15]. Since the relaxation time
τ is sufficiently fast here, the full time-integration involving the electron and hole transients is used. The nonsymmetrical boundary conditions simulating the contactless setup are distinguished by an ideal metal contact on one side representing the back electrode and a free surface as the air gap on the other side. The metal contact is characterized by the work function of the metal defining the band bending. Free carriers are permanently in the thermodynamic equilibrium at the semiconductor-metal (MS) interface in the drift-diffusion model. An eventual appearance of an insulating layer forming the metal-insulator-semiconductor (MIS) structure is omitted. The air gap is defined by a prescribed zero current across the interface.
Two types of defect levels are considered. Level A defines defects homogeneously spread through the whole volume of the sample. Level B is confined to a narrow surface layer below the air gap. The generation of free carriers at the free surface may be mediated in this way. Each defect level is characterized by its position in the band gap, defect density and electron and hole capture cross-sections.
With the aim to illustrate the peculiarities of transient effects at the contactless setup, we performed all simulations on a p-type CdTe at temperature 295 K characterized with fixed transport parameters and the defect structure of the bulk. The chosen parameters represent a typical high-resistivity single-crystalline detector-grade sample, with the Fermi level close to the mid-gap, experimentally observed mobilities of electrons and holes [
16] and a deep level fixing the Fermi level while guaranteeing sufficiently a long lifetime of the free carriers. The parameters used are as follows: energy gap
, Fermi energy
, equilibrium electron and hole density
and
and electron and hole mobility
. We use a single deep level defined by the density
, energy
and equal capture cross-section of electrons and holes
. The resistivity results
. The degeneracy factors of all considered defect levels are equal to one for simplicity. The thickness of the simulated sample
L = 2 mm and the thickness of the air gap
are chosen, which define the ratios of respective capacities and charges as
The relaxation time corresponding to chosen ρ and is . Surfaces of semiconductor samples typically contain higher concentrations of defects and connected energy levels in the bandgap. Therefore, even in a detector-grade sample with a low concentration of deep mid-gap levels ( in the bulk, the surface concentration of these levels can be much higher (. We included such a thin layer in some examples and discuss its effects to the charge relaxation and evaluated mobility.
We start the simulations with an ideal case, when the leading interface with flat bands forms an ohmic contact with the back electrode, i.e., it provides a sufficient supply of free holes to the sample, keeping their equilibrium concentration during the whole time of the sample response to the applied voltage step. The results of the simulation and its single exponential fit are presented in
Figure 2. It is evident that the response can be well-described by a single exponential fit, and the evaluated resistivity is practically the same as the input one. The slight increase of the resistivity is caused by a depletion of minority electrons, which are not formed in sufficient amounts at the ideal defect-free surface (cathode) adjacent to the air gap.
The following two simulations describe the impact of the surface source of free carriers on the response of the sample to a voltage step of 5 V and on the evaluation of relaxation time
τ and resistivity
ρ. Oppositely to other examples shown in this section, the anode is set to the free surface, which thus becomes the leading interface here.
Figure 3a shows the basic scheme of a sample with a surface source of free carriers and with flat energy bands.
Figure 3b,c presents the results of the numerical simulation of the relaxation response for two surface sources with concentrations of level B—
(source I) and
(source II) positioned at the same energy as the bulk defects distributed in a 10-µm thin layer near the surface. The capture cross-sections for electrons
and holes
equal to those in the bulk level are used. The level is half-by-half filled by electrons and defined initially neutral so that no space charge affects the free carrier supply. Nearly the same results were obtained with a thinner layer when the defect density was increased proportionally.
The time evolution of the collected charge after the application of the bias with source I can be well-fitted by a single exponential. The evaluated relaxation time is nearly the same as the true sample input time . This situation corresponds to the case when the surface generation source suffices to supply carriers (in our model case, majority holes) that are extracted from the surface by the applied bias. The evaluated resistivity is correct.
Figure 3c shows the evolution of the collected electric charge for the weaker source II. The dependence is a complex function. The reason for this behavior can be described as follows. The source of the holes is insufficient to resupply all holes drained from the sample volume after application of the bias. The reduced hole density slows down the relaxation. Simultaneously, a negative space charge is formed due to the hole extraction. An excellent fit was obtained using Equation (5) with parameters
,
,
and
, producing a curve practically undistinguishable from the fitted one. The considerable deviation of both
(
i = 1 and 2) from the correct
is eminent.
Let us now discuss the situation when the efficiency of the surface source is modified by the band bending. Next, the examples show the case of a p-type semiconductor with (i) bands bent up at the anode adjacent to the base electrode, injecting holes into the volume of the sample (
Figure 4a), while (ii) bands bent down block the transport of holes to the volume (
Figure 5a).
The surface source defined in example (I) above is used again to store holes in the thin surface layer after relaxation. This option is important when the simulations are performed at a low bias and the Boltzmann-type free carrier distribution markedly spreads into the sample’s volume. The presence of the surface layer allows us to reach the final charge defined by the setup geometry even at the low .
If, in this situation, the bands are bent up, the back electrode is injecting holes in the volume. The simulated dependence of the collected charge on the time after application of the bias (
Figure 4b) is characterized by a single exponential increase with a time constant smaller than the input
τ. Therefore, the evaluated apparent resistivity is smaller than the real value.
If the bands are bent down, the surface is blocking the holes. Depending on the value of the band bending, a situation can occur when the surface source is no longer able to resupply the holes extracted after application of the bias inducing hole depletion. The transport and its characteristics have similar characters, as described in
Figure 3c (flat bands and weak surface source of free carriers).
It is apparent that single exponential fits do not provide sufficient agreement with the simulation data in many cases (
Figure 3c,
Figure 4b and
Figure 5b), and the resistivity evaluated through Equation (1) markedly deviates from the correct value. Therefore, we applied trial functions (4) and (5), which involve two exponentials and compared the results of the fit with the true input relaxation time
. The goal was to assess whether this approach can lead to a better evaluation of the true relaxation time and this way to an estimate of the sample resistivity when its response to a bias step is nonexponential. We performed simulations in dependence of the applied bias in the range of 0.3–10 V for defect models with the hole injection and depletion schematized in
Figure 4a and
Figure 5a, respectively, and evaluated the results of the simulation using respective trial functions (4) and (5). We focused on the pertinent
representing the relaxation at the later time where the disturbance caused by the band banding should fade away. That is, the longer (shorter) time denoted as
in the case of injecting (blocking) an anode. The evaluated
is presented in
Figure 6. Simultaneously, we show the time
τ received by the single-exponential fit.
We may see that
τ deviates significantly from the right relaxation time
represented by the horizontal black line in
Figure 6 when the high bias is used. Decreasing of the bias leads to a substantially improved agreement between the evaluated and true relaxation times, which is also consistent with the approaching of the relaxation curve to the single-exponential shape. The evaluation of the relaxation by the double-exponential fits allowed us to approach to the
even at the large bias. The deviation is mostly less than 30%. Nevertheless, as it is apparent at the injecting contact with
, anomalies caused by an interference between exponentials resulting in unwelcome instabilities may appear.
In the case where the leading interface is injecting holes (bands bent up 50 meV,
Figure 4a), the value of
is larger than
at the higher applied bias, while, in the case of contact blocking, the majority holes (bands bent down 50 meV,
Figure 5a)
are shorter than
τ at the higher applied bias. Using a low bias,
approaches to the correct value and, at a very low bias, it even surpasses it. The deviation of
τ from
at the lowest bias is caused by the variation of the total equilibrium resistivity due to the band banding. In the case of blocking the anode, the depleted region contributes by a slightly damped relaxation. Conversely, the injecting contact yields an opposite effect, as is seen in
Figure 6.
While the total relaxation velocity depicted by τ in the single-exponential fit accords with the prediction of the accelerated (decelerated) relaxation at the carriers’ injection (depletion), the evaluated with the two-exponential fit results oppositely, showing an extended (shortened) at these settings. This seeming disaccord comes from the interference of the two exponentials at the two-exponential fit. The exponential depicting the initial part of the relaxation through the relaxation time accommodates the principal period of the relaxation affected by the band bending and adopts partly also the latter stage of the relaxation at the quasi-ohmic regime at a low bias. Consequently, the second exponential is affected by this interplay, and the effect manifests in the observed behavior of . This feature might be conveniently used at the rough simple estimation of the resistivity of the samples, considering that depicting the correct relaxation velocity should lie in the interval limited by and τ. While fitted may be used at the estimation of a correct , the value of τ puts an additional clause on the error bar of , as follows. If , then . If , then .
The results shown in
Figure 6 can be qualitatively understood from the simulated I-V characteristics for the studied example using flat bands, the injecting and the blocking regimes (
Figure 7). Each point in the showed dependence was calculated by integrating the drift-diffusion equation 1 ms after the switch-on of the bias. The plotted current thus represents the initial current appearing at the contactless setup shortly after the biasing. We may see that the deviation from ideal ohmic-type I–V characteristics induced by band bending appears as early as at the biasing by 0.3 V. The deviation of the curves corresponding to the injecting and blocking regimes is substantial for the charge relaxation at the contactless setup. The non-ohmic character of the I–V characteristics results in the complex character of the charge relaxation.
We have used a rather small band bending of 50 meV at the leading interface in the simulations. If the band bending was larger, the observed effects of the carrier injection or depletion would be stronger. We also note that the reduction of the bias to reach the right τ may be partly avoided by measuring a thick wafer, at which the electric field related to the chosen bias is lowered. The maximum bias characterizing the ohmic part of the I–V characteristics is then increased.
The principal contribution to the discussion to determine reliably the resistivity of high-resistive materials by the contactless method ensues from
Figure 8. It shows the I–V characteristics of the material characterized by parameters equal to those used in previous simulations in
Figure 7, with an exception of the deep level density, which was increased to
, i.e., 1000× compared to the previous case. We may see that, in this case, the nonlinearity induced by the band bending moved to a larger bias that is well-above the bias applied usually during measurements (typically, 5–10 V). Consequently, a single exponential relaxation producing the correct resistivity is obtained.
The reason for this effect is a much narrower width of the space charge region (depletion width) in the material, with the enhanced deep-level density and related suppressed contribution of the contact’s resistance [
17] to the total resistance in the circuit predicted by the diffusion theory of metal-semiconductor contacts [
18]. The high density of the level simultaneously with the chosen capture cross-section also accords with the model of the strong deep trap mentioned as an alternative charging channel in the theory section. The carriers’ injection or depletion is then accommodated by the trap, and the current attains the ohmic-like character for the sufficient time to reach the relaxation.
In detector-grade CdTe-based materials, the density of the mid-gap level is very low (
[
19]. Simultaneously, the captured cross-sections of the mid-gap levels are typically less than
[
20]. The requirement for long-carrier lifetimes is, according to the Shockley-Read-Hall model [
14], in direct contradiction to the efficient thermal generation of electron-hole pairs via a deep level in the volume of the sample. Therefore, the alternative charging by Shockley-Read generation via deep levels does not take place.