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

M. Bashahu^{}, D. Ngendabanyikwa^{}, P. Nyandwi^{}

Department of Physics and Technology, Institute of Applied Pedagogy, Bujumbura, Burundi.

**DOI: **10.4236/jmp.2022.133020
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Department of Physics and Technology, Institute of Applied Pedagogy, Bujumbura, Burundi.

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 (*I _{s}*), the ideality factor (

Keywords

SBD, Dark Forward and Reverse *I-V-T* Characteristics, Different Parameters Extraction Methods, Identification of the Structure’s Metal and Semiconductor-Type

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Bashahu, M. , Ngendabanyikwa, D. and Nyandwi, P. (2022) Non Ideal Schottky Barrier Diode’s Parameters Extraction and Materials Identification from Dark *I-V-T* Characteristics. *Journal of Modern Physics*, **13**, 285-300. doi: 10.4236/jmp.2022.133020.

1. Introduction

Rectifying metal-semiconductor (MS) contacts, also known as Schottky barrier diodes (SBDs), have received an increasing attention due to their applications in photovoltaic solar cells [1] [2] [3] [4], field effect transistors (FETs) [5], infrared high-speed detectors, electronic switching and other high-frequency devices [6] [7] [8] [9]. Many reports on SBDs physical properties have been proposed in order to better understand the performance of those structures and related devices. The four SBD’s key parameters are the reverse saturation current (*I _{s}*), the ideality factor (

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.

2. Experimental Details

The SBD sample of this analysis is shown magnified in Figure 1 and has the following actual specifications: blue color, BAT 48 as trade mark, diameter and length of the central part equal to 0.15 cm and 0.3 cm, respectively, diameter and length of each of the two terminal’s wires equal to 0.7 mm and 3.0 cm, respectively.

Reverse and forward *I-V-T* characteristics of the diode were measured in dark conditions over bias voltage and temperature ranges of [−2.5; +0.5] V and [274.5; 336.5] K, respectively. A common experimental arrangement of simple equipments

Figure 1. Magnified view of the SBD of this study.

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.

3. Extracted Parameters, Used Methods, Results and Discussion

3.1. *R _{sh}* from Reverse

The complete representation of a real diode’s *I-V* characteristic is given by Equation (1) [47]:

$I={I}_{s}\left\{\mathrm{exp}\left[\frac{q\left(V-{R}_{s}I\right)}{n{K}_{B}T}\right]-1\right\}+\frac{V-{R}_{s}I}{{R}_{sh}}$ (1)

where *I _{s}*,

${G}_{p}=\frac{1}{{R}_{sh}}=\frac{\Delta {I}_{p}}{\Delta V}$ (2)

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

3.2. *I _{s}*,

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] [47] [49]

Figure 2. Plot of reverse *I-V* data for the SBD at 286.5 K.

${I}_{d}={I}_{s}\left[\mathrm{exp}\left(\frac{qV}{n{K}_{B}T}\right)-1\right]$ (3)

In Equation (3), the reverse saturation current (*I _{s}*) is expressed as

${I}_{s}=A{A}^{**}\left(\frac{-{\Phi}_{B}}{{K}_{B}T}\right)$ (4)

where *T* is the diode’s absolute temperature, *A* is the junction’s electrically active area,
${A}^{**}$ is the effective Richardson constant and
${\Phi}_{B}$ is the effective Schottky

barrier height (SBH). When the forward bias $V>\frac{3{K}_{B}T}{q}$, Equation (3) is given by [50]

${I}_{d}={I}_{s}\mathrm{exp}\left(\frac{qV}{n{K}_{B}T}\right)$ (5)

According to Equation (5), *n* and *I _{s}* parameters can be, respectively, extracted from the slope and the intercept of the linear region (diffusion line) in the plot of the experimental ln

${I}_{d}={I}_{s}\mathrm{exp}\left[\frac{q\left(V-{R}_{s}I\right)}{n{K}_{B}T}\right]$ (6)

Using Equation (6) and following the Cowley and Sze method (ref. no. 2 in [43] ), one extracts *R _{s}* from the gap ∆

Values of *n*, *I _{s}* and

*n*(*T*) data from Figure 4 exhibit a somehow wavy trend, while results from other works state a slight decrease of the ideality factor with increasing temperature [19] [22] [36] [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.:
$1.2<n<2.7$ for Pt/p-Si [16].

Figure 3. Plot of ln*I _{d}*-

Figure 4. Plots of *n*, *I _{s}* and

The average series resistance (*R _{s}*) of our SBD is much higher than

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. Nevertheless, those results do not depart too much from

3.3. Φ* _{B}* and

From Equations (4) and (5), assuming *n* ≈ 1, the SBD’s forward *I-V *characteristic is expressed as

$I=A{A}^{**}{T}^{2}\mathrm{exp}\left[-\left(\frac{{\Phi}_{B}-qV}{{K}_{B}T}\right)\right]$ (7)

and thus

$\mathrm{ln}\left(I/{T}^{2}\right)=\mathrm{ln}\left(A{A}^{**}\right)-\left(\frac{{\Phi}_{B}-qV}{{K}_{B}}\right)\frac{1}{T}$ (8)

The activation energy method is based on the plot of experimental ln(*I*/*T*^{2})-(1/*T*)

data at a given voltage bias $V>\frac{3{K}_{B}T}{q}$. From such an Arrhenius (or Richardson)

plot, the effective Schottky barrier height (Φ* _{B}*) and the

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(

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.

3.4. Φ_{B}_{0} 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):

${I}_{d}=A{A}^{**}{T}^{2}\mathrm{exp}\left(\frac{-{\Phi}_{B}}{{K}_{B}T}\right)\left[\mathrm{exp}\left(\frac{qV}{{K}_{B}T}\right)-1\right]$ (9)

Figure 5. Results of Φ* _{B}* and

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]

${\Phi}_{B}={\Phi}_{B0}+\beta V$ (10)

where Φ_{B}_{0} 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)

$\frac{1}{n}=1-\beta $ (11)

then the forward *I*-*V* characteristic of Equation (9) becomes:

${I}_{d}={I}_{0}\mathrm{exp}\left(\frac{qV}{n{K}_{B}T}\right)\left[1-\mathrm{exp}\left(\frac{-qV}{{K}_{B}T}\right)\right]$ (12)

where

${I}_{d}=A{A}^{**}{T}^{2}\mathrm{exp}\left(\frac{-{\Phi}_{B0}}{{K}_{B}T}\right)$ (13)

For bias voltage $V>\frac{3{K}_{B}T}{q}$, Equation (12) reduces in the following simple form

${I}_{d}\approx {I}_{0}\mathrm{exp}\left(\frac{qV}{n{K}_{B}T}\right)$ (14)

From Equation (13), the SBH at zero bias is expressed as:

${\Phi}_{B0}={K}_{B}T\mathrm{ln}\left(A{A}^{**}{T}^{2}/{I}_{0}\right)$ (15)

where *I*_{0} is the reverse saturation current extrapolated at zero bias. Equation (15) offers a way to determine Φ_{B}_{0} using *AA*^{**} data from Figure 5 and *I _{s}*-values from Figure 4 (
${I}_{0}\approx {I}_{s}$ since extrapolated at zero bias). The results of such a determination are shown in Figure 6.

In accordance with Equation (10), values of Φ_{B}_{0} are lower than those of Φ* _{B}* from Figure 5. Nevertheless, Φ

3.5. *n*, *R _{s}* and Φ

In this method, firstly Equation (6) (with *I _{d}* =

$\mathrm{ln}I=\mathrm{ln}{I}_{s}+\frac{qV}{n{K}_{B}T}-\frac{q{R}_{s}I}{n{K}_{B}T}$ (16)

and thus

$V=\frac{n{K}_{B}T}{q}\mathrm{ln}I+{R}_{s}I-\frac{n{K}_{B}T}{q}\mathrm{ln}{I}_{s}$ (17)

Differentiating Equation (17) provides:

Figure 6. Results of Φ_{B}_{0} vs *T* data for the SBD of this study.

$\text{d}V=\frac{n{K}_{B}T}{q}\text{d}\left(\mathrm{ln}I\right)+{R}_{s}\text{d}I$ (18)

and thus

$\frac{\text{d}V}{\text{d}\left(\mathrm{ln}I\right)}=\frac{n{K}_{B}T}{q}{R}_{s}I$ (19)

Equation (19) shows that, from experimental forward *I-V* data at a given temperature, the curve
$\left[\text{d}V/\text{d}\left(\mathrm{ln}I\right)\right]\text{-}I$ is a straight line from which *R _{s}* and

Secondly, combining Equations (4) and (6) (with *I _{d}* =

$\mathrm{ln}\left(\frac{I}{A\text{}{A}^{**}{T}^{2}}\right)+\frac{{\Phi}_{B}}{{K}_{B}T}=\frac{q}{{K}_{B}nT}V-\frac{q}{{K}_{B}nT}{R}_{s}I$ (20)

Equation (20) is re-arranged to become

$H\left(I\right)={R}_{s}I+n{\Phi}_{B}$ (21)

where

*
$H\left(I\right)=V-\frac{n{K}_{B}T}{q}\mathrm{ln}\left(\frac{I}{A{A}^{**}{T}^{2}}\right)$ * (22)

Equations (19), (21) and (22) are the three auxiliary Cheung’s functions [14]. 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

whereas *H*(*I*) data present a good linear behaviour at bias voltages
$V>\frac{3{K}_{B}T}{q}$ as illustrated in Figure 8.

The values of *n*, *R _{s}* and Φ

It is shown that values of the ideality factor obtained from [d*V*/d(ln*I*)]-*I* plots (average *n* = 1.65) are quite lower than those extracted from ln*I _{d}*-

Figure 7. Plot of [d*V*/d(ln*I*)]-*I* data for the SBD of this study at *T* = 295 K.

Figure 8. Plot of *H*(*I*) data for the SBD of this analysis at *T* = 299.5 K.

Figure 9. Results of *n*, *R _{s}* and Φ

Figure 4, average *n* = 2.04). Moreover, while *n*(*T*) data from Figure 4 show a wavy trend, *n*(*T*) results of the present method ([d*V*/d(*lnI*)]-*I* plots) seem to increase with increasing temperature.

A comparison of *R _{s}*-results of Figure 4 and Figure 5 shows that [d

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

3.6. *V _{bi}* from the Maximum Forward Current Method

At a given temperature, the SBD’s maximum forward current (*I _{d}* =

${I}_{\mathrm{max}}\approx {I}_{0}\mathrm{exp}\left(\frac{q{V}_{bi}}{n{K}_{B}T}\right)$ (23)

or

$\mathrm{ln}{I}_{\mathrm{max}}\approx \mathrm{ln}{I}_{0}+\frac{q{V}_{bi}}{n{K}_{B}}\frac{1}{T}$ (24)

Therefore, in the experimental *I-V* characteristics at different temperatures, accounting only for data corresponding to *I*_{max}, one gets a plot of
$\mathrm{ln}{I}_{\mathrm{max}}\text{-}\frac{1}{T}$. On

the expected resulting straight line, the *V _{bi}* and
${I}_{0}\approx {I}_{s}$ parameters are extracted from the slope and intercept, respectively. Estimates obtained by using that procedure and

3.7. *N _{A}* or

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 $\left|V\right|$. In that case, the reverse current is expressed as [19]

${I}_{R}={I}_{0}\mathrm{exp}\left[\frac{q}{{K}_{B}T}{\left(\frac{qE}{4\pi {\epsilon}_{s}}\right)}^{1/2}\right]$ (25)

where *I*_{0} is the reverse current at zero bias and the *E* quantity is given by

$E={\left[\frac{2qN}{{\epsilon}_{s}}\left(\left|V\right|+{V}_{bi}-\frac{{K}_{B}T}{q}\right)\right]}^{1/2}$ (26)

with *ε _{s}* the semiconductor’s dielectric constant. If
${V}_{eff}=\left|V\right|+{V}_{bi}\gg \frac{{K}_{B}T}{q}$, then Equation (25) becomes

${I}_{R}\approx {I}_{0}\mathrm{exp}\left(\alpha {V}_{eff}^{1/4}\right)$ (27)

where the *α *parameter is expressed as

$\alpha =\frac{q}{{K}_{B}T}{\left(\frac{q}{4\pi {\epsilon}_{s}}\right)}^{1/2}{\left(\frac{2qN}{{\epsilon}_{s}}\right)}^{1/4}$ (28)

Equation (27) may be also written as

$\mathrm{ln}{I}_{R}\approx \mathrm{ln}{I}_{0}+\alpha {V}_{eff}^{1/4}$ (29)

According to Equation (29), by using experimental reverse *I-V* data at a given temperature, together with the
${V}_{bi}$ -value stated in section 3.6,
${\epsilon}_{s}=11.9{\epsilon}_{0}$ for Si [19], and
${\epsilon}_{0}=8.854\times {10}^{-12}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{F}\cdot {\text{m}}^{-1}$ [55], one gets a plot of
$\mathrm{ln}{I}_{R}\text{-}{V}_{eff}^{1/4}$ data, which is expected to yield a straight line, and of which
${I}_{0}\approx {I}_{s}$ and *α* (thus *N* = *N _{A}* or

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

ln*I _{d}*

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}

Figure 10. Plot of
$\mathrm{ln}{I}_{R}$ vs
${V}_{eff}^{1/4}$ data points for the SBD of this study at *T* = 299.

Figure 11. Results of *N _{A}* or

Table 1. Synthesis of the obtained results and the used methods for SBD’s parameters extraction in this analysis.

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].

3.8. 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.

3.9. Device Materials Identification

On one hand, the following data on some SBDs are reported amongst others in literature: 1) Φ* _{B}* = (0.272 ± 0.005) eV and Φ

4. 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 (

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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