Non Ideal Schottky Barrier Diode’s Parameters Extraction and Materials Identification from Dark I-V-T Characteristics

Several parameters of a commercial Si-based Schottky barrier diode (SBD) with unknown metal material and semiconductor-type have been investigated in this work from dark forward and reverse I-V characteristics in the temperature (T) range of [274.5 K 366.5 K]. Those parameters include the reverse saturation current (Is), the ideality factor (n), the series and the shunt resistances (Rs and Rsh), the effective and the zero bias barrier heights (ΦB and ΦB0), the product of the electrical active area (A) and the effective Richardson constant (A**), the built-in potential (Vbi), together with the semiconductor doping concentration (NA or ND). Some of them have been extracted by using two or three different methods. The main features of each approach have been clearly stated. From one parameter to another, results have been discussed in terms of structure performance, comparison on one another when extracted from different methods, accordance or discordance with data from other works, and parameter’s temperature or voltage dependence. A comparison of results on ΦB, ΦB0, n and NA or ND parameters with some available data in literature for the same parameters, has especially led to clear propositions on the identity of the analyzed SBD’s metal and semiconductor-type.


Introduction
Rectifying metal-semiconductor (MS) contacts, also known as Schottky barrier diodes (SBDs), have received an increasing attention due to their applications in tion energy (E A ), densities (n is ) and energy levels (E is ) of interface states. In addition, photo-response measurements, which involve various spectroscopic techniques, are implemented in order to extract parameters such as band-gap, impurity and doping concentration, layer thickness, surface roughness and texture [29] [30]. Methods from those three groups are combined in some reports [4] [31]- [37].
While several methods from those groups are analytical ones, numerical techniques are also used [1] [2] [9] [38] [39] [40] [41] [42]. Moreover, as shown in some review reports [43] [44] [45] [46], from one method to another, two or several parameters can be simultaneously extracted; dc or ac, static or dynamic, fixed or varying frequency and temperature operation's conditions can be applied; different current transport mechanisms may be taken into account; results can be temperature or voltage dependent; and they may be compared one another when different methods are combined.
The SBD analyzed in this work is a commercial Si-based one from ST Microelectronics, for which neither the metal nor the semiconductor-type (p-or n-) were specified in the relevant catalog. By using dark forward and reverse current-voltage (I-V) characteristics at different temperatures, together with different approaches, our objective was three-fold: firstly to extract different parameters of that structure, secondly to discuss our results, and thirdly to especially come to identify the SBD's metal and semiconductor-type.

Experimental Details
The SBD sample of this analysis is shown magnified in Figure 1 and has the fol-  has been used for that purpose. These include notably a power supply with d.c emf fixed to a maximum value E 0 = 2.5 V; a rheostat mounted with the power supply in such a way to vary the emf, an ammeter and a voltmeter for I-V measurements; an ice bath and an electrical heater to change thermal conditions of the sample, and a thermometer for temperature measurements.

R sh from Reverse I-V Plots
The complete representation of a real diode's I-V characteristic is given by Equ- where I s , n, R s and R sh are the diode's reverse saturation current, ideality factor, series resistance and shunt resistance, respectively, while q is the absolute value of the electronic charge and K B is the Boltzmann's constant. The first term of the sum in Equation (1) For the considered temperature range, the obtained SBD's reverse I-V lines ( Figure 2 is the plot of such a line at T = 286.5 K) were so merged that they have led to an almost constant shunt resistance: R sh = (4.93 ± 0.07) × 10 4 Ω. This has been a proof of the scarcity of crystal irregularities or defects in the bulk and at the edges of our SBD, through which current losses could occur.

I s , R s and n from Forward lnI d -V Plots
With the assumption of thermoionic emission (TE) as the prevailing charge transport mechanism in the SBD, the forward diode's I-V characteristic is given by [19] In Equation (3), the reverse saturation current (I s ) is expressed as where T is the diode's absolute temperature, A is the junction's electrically active  [43]), one extracts R s from the gap ∆V (on the V-axis) between the actual lnI d vs V curve and the diffusion line. Figure 3 is the plot of that curve for our SBD at one fixed temperature. On that plot, as different R s are obtained at different I d in the related region, a mean R s -value is extracted. Values of n, I s and R s determined according to the previous methodology are summarized in Figure 4.
n(T) data from Figure 4 [51]. Nevertheless, values of n for our SBD are higher than those commonly observed for c-Si solar cells [44]. Moreover, the ideality factor of our sample has values in good agreement with those of other investigated Si-based SBDs, e.g.: 1.2 2.7 n < < for Pt/p-Si [16].   The average series resistance (R s ) of our SBD is much higher than R s -values commonly observed for c-Si solar cells [43]. This is likely due to high resistivity (ρ) of Si-based SBDs (e.g.: ρ is equal to 15 Ω cm for Tb/p-Si; 8 Ω cm for Ru/n-Si and 1 Ω cm for Pt/n-Si) [52], compared to resistivity of p-Si and n-Si materials (and thus of p-n Si junctions). As examples, ρ lies in the ranges [4 × 10 −4 ; 3 × 10 −2 ] Ω cm and [10 −4 ; 6 × 10 −2 ] Ω cm for n-Si and p-Si, respectively, when the doping concentration decreases from 10 +21 cm −3 to 10 +18 cm −3 [6].
At its side, the reverse saturation current (I s ) may increase with increasing temperature according to Equation (4). That theoretical trend is not well evidenced by results of Figure 4.

Φ B and AA ** from the Activation Energy Method
From Equations (4) and (5), assuming n ≈ 1, the SBD's forward I-V characteristic is expressed as and thus The activation energy method is based on the plot of experimental ln(I/T 2 )-(1/T) data at a given voltage bias . From such an Arrhenius (or Richardson) plot, the effective Schottky barrier height (Φ B ) and the AA ** product (of the contact's electrically active area and the effective Richardson constant) can be derived from the negative slope and the intercept of the expected resulting straight line, respectively. Based on experimental forward I-V-T data on the SBD of this analysis, results of such a determination are shown in Figure 5. The SBHs in Figure 5 are estimates of actual Φ B since too many approximations are used in the present method. Moreover, experimental data points have been found scattered in each ln(I/T 2 )-(1/T) plot. Nevertheless, a clear increase of the SBH with increasing bias voltage is noticed in agreement with theory (e.g. in Section 3.4).
The AA ** mean product is equal to 2.35 × 10 −6 A/K 2 . Using that result and the effective Richardson constants of 12 and 32 × 10 4 A·m −2 ·K −2 for n-Si and p-Si, respectively [6] [52], one finds the contact's electrically active area A of the SBD equal to 2.23 μm 2 and 7.81 μm 2 for n-Si and p-Si, respectively. Those values correspond to diameters of 1.69 μm and 3.15 μm, respectively, which are clearly lower than the measured diameter (=0.7 mm) of each terminal's wire.

Φ B0 from the SBH's Bias Dependence Behavior
The basic equation used to estimate the SBH within the TE theory is obtained by combining Equations (3) and (4): As the SBH is strongly dependent on the electrical field in the depletion region and thus on the applied bias, Φ B is commonly expressed as [16] [19] where Φ B0 is the barrier height at zero bias (or the asymptotic barrier height) and β is assumed to be a positive constant over the region of measurement. That means an increase of the SBH with increasing bias voltage. This trend is experimentally observed from results in Figure 5. If the ideality factor in defined as in Equation (11) then the forward I-V characteristic of Equation (9) becomes: where For bias voltage , Equation (12) reduces in the following simple form From Equation (13), the SBH at zero bias is expressed as: where I 0 is the reverse saturation current extrapolated at zero bias. Equation (15) offers a way to determine Φ B0 using AA ** data from Figure 5 and I s -values from since extrapolated at zero bias). The results of such a determination are shown in Figure 6.
In accordance with Equation (10), values of Φ B0 are lower than those of Φ B from Figure 5. Nevertheless, Φ B0 -T data exhibit a wavy trend whereas results from other works state that the SBH and its value at zero bias slightly increase with increasing temperature [19] [54].
Using the AA ** mean value of Figure 5 and experimental forward I-V data at a given temperature, allows one to get H(I) data from Equation (22). The plot of those H(I) data (Equation (21)) leads to a straight line from which R s and nΦ B can be extracted as the slope and the intercept, respectively. Figure 7 shows a curve of [dV/d(lnI)]-I data at a fixed temperature. Its points are quite scattered, whereas H(I) data present a good linear behaviour at bias voltages as illustrated in Figure 8.
The values of n, R s and Φ B derived by using the previous procedure are presented in Figure 9.
It     A comparison of R s -results of Figure 4 and Figure 5 shows that [dV/d(lnI)]-I plots yield higher values (mean R s = 0.56 Ω), followed by data from H-I plots (mean R S = 0.45 Ω), lnI d -V plots leading to lower results (mean R s = 0.30 Ω).

I(mA)
The auxiliary Cheung's functions method leads also to lower SBHs (mean Φ B = 0.198 eV) than the activation energy method (in Figure 5, mean Φ B = 0.325 eV). Moreover, a mean trend of SBH data from the Cheung's method is a decrease with increasing temperature and with increasing ideality factor. This is in accordance with statements from other works [14] [16] [49] [51].

V bi from the Maximum Forward Current Method
At a given temperature, the SBD's maximum forward current (I d = I max ) is recorded at bias voltage V equal to the junction's built-in potential (V bi ), for which Equation (14) becomes [19]  parameters are extracted from the slope and intercept, respectively. Estimates obtained by using that procedure and n-values of Figure 9, for the SBD and temperature range of this analysis, have been: V bi = 0.496 V and I s = 0.34 × 10 −4 A, respectively.

N A or N D from Reverse I-V-T Data
In forward bias conditions, the SBH increases with increasing bias voltage as shown in Figure 5 (Section 3.3) and Section 3.4. At the opposite, in reverse bias case, the main effect is the lowering of the SBH with the applied bias voltage V . In that case, the reverse current is expressed as [19] where I 0 is the reverse current at zero bias and the E quantity is given by with ε s the semiconductor's dielectric constant. If where the α parameter is expressed as According to Equation (29), by using experimental reverse I-V data at a given temperature, together with the bi V -value stated in section 3.6, 0 11.9 s ε ε = for Si [19], and results obtained by following that procedure are given in Figure 11.
A comparison of our SBD reverse saturation current's results shows that the reverse I-V-T data method leads to slightly lower values ( Figure 11, mean I s = 1.31 × 10 −4 A) than those from the forward.
lnI d -V plots ( Figure 4, mean I s = 1.7 × 10 −4 A). Moreover, the reverse I-V-T data method appears to be better than the lnI d -V one for I s -parameter extraction, since its results clearly exhibit an increase of reverse saturation current with increasing temperature, in accordance with theory.
Furthermore, for the SBD and the temperature range of this analysis, the semiconductor (Si)'s average doping concentration is found equal to 6.06 × 10 18   Reverse I-V-T data Decreases with increasing T cm −3 . This indicates that, either n-or p-type, the actual Si material has a resistivity ρ of about 10 −2 Ω cm [6].

Results Summary
For the SBD and the temperature range of this analysis, Table 1 shows in synthesis the obtained parameters' mean values and the extraction methods implemented so far.

Device Materials Identification
On one hand, the following data on some SBDs are reported amongst others in literature: for Pt/p-Si [16]; 4) with a doping concentration of 10 +18 cm −3 , the resistivity, ρ = 3 × 10 −2 Ω cm and 6 × 10 −2 Ω cm for n-Si and p-Si materials, respectively [6]. On the other hand, the parameters' results obtained for the SBD and the temperature range of this study are presented in Table 1. A comparison of results on the same parameters in those two sets of data, allows one to certify that the SBD of this analysis is either Pt Si/p-Si or Au/p-Si.

Conclusion
As shown in the synthesis of Table 1, from I-V-T measurements and the use of different methods, up to nine parameters have been extracted on a Si-based MS contact with unknown metal and semiconductor-type materials. Two of those parameters, i.e. the shunt resistance (R sh ) and the semiconductor doping concentration (N A or N D ), have been derived from reverse I-V-T data. All the remaining seven parameters have been determined from forward I-V-T data.
Those are the ideality factor (n), the series resistance (R s ), the reverse saturation current (I s ), the effective Schottky barrier height (SBH, Φ B ), the SBH at zero bias (Φ B0 ), the product of the contact's electrical active area (A) and the effective Richardson constant (A ** ), and the built-in potential (V bi ). Some of those seven parameters have been extracted by using two or three different approaches. The main features of each approach, including prevailing current transport mechanism, operation conditions and other assumptions, have been clearly stated. From one parameter to another, results have been discussed in terms of structure performance, parameter's temperature or voltage bias dependence, accordance or discordance with data from other works, and comparison on one another of results obtained from different methods. Furthermore, a comparison of results on the n, Φ B , Φ B0 , and N A or N D parameters of Table 1 with some available data on the same parameters in literature, has led to state that the analyzed SBD is either Pt Si/p-Si or Au/p-Si.