Gaussian distribution of inhomogeneous nickel–vanadium Schottky interface on silicon (100)

The electrical forward current–voltage ($I$–$V$) measurements are analyzed using the standard TE theory. According to the latter, the theoretical $I$–$V$ relation for a metal–-semiconductor interface is given by [1, 2]

$$\begin{equation}I = {I_{\textrm{s}}}\left[ {\exp \left( {\frac{\beta }{n}\left( {V - {R_{\textrm{s}}}I} \right)} \right) - 1} \right]\end{equation} \tag{ 1 }$$

with the saturation current ${I_\textrm{s}}$ defined by

$$\begin{equation}{I_\textrm{s}} = A{A^*}{T^2}\exp \left( { - \beta {\phi _\textrm{B}}} \right)\end{equation} \tag{ 2 }$$

where $\beta = q/kT$ is the inverse thermal voltage, q is the elementary electric charge, k is the Boltzmann constant, T is the temperature, ${R_\textrm{s}}$ is the series resistance of the diode, V is the applied voltage, n is the ideality factor, A is the diode area, ${A^*}$ is the effective Richardson constant, and ${\phi _{\textrm{B}}}$ is the barrier height.

The obtained experimental $I$–$V$ curves of the NiV/Si diode are shown in figure 1 for various values of the temperature. Our objective then is to find the values of the parameters in equation (1) that best fit the experimental I–V data, i.e. find the values of the ideality factor $n$, the saturation current ${I_\textrm{s}}$, and the series resistance ${R_\textrm{s}}$. The barrier height can then be found by using equation (2) and the theoretical value of Richardson constant $A_{{\text{th}}}^{\text{*}} = 112{\text{ A}}\,{\text{c}}{{\text{m}}^{ - 2}}{\text{ }}{{\text{K}}^{ - 2}}$ of n-type Si [7]. A simple approach that can be used to perform this task is the standard method, which neglects the series resistance and deduces, respectively, $n$ and ${I_\textrm{s}}$ from the slope and intercept of the linear region of the $\ln \left( I \right)$ vs. $V$ curve. The linear fitting of the standard method for the case of the NiV/Si diode is shown in figure 1 (dashed lines). As can be seen from this figure, at each temperature, the linear fitting is only valid for a portion of the I–V curve, which is due to the existence of non-zero series resistance. For the purpose of finding the values of $n$, ${I_\textrm{s}}$, and ${R_\textrm{s}}$ that best fit the $I$–$V$ data in all regions, we use the VOM [49, 50]. Let ${\hat I_k}$ and ${\hat V_k}$ be the ${k^{{\text{th}}}}$ measured current and voltage at a given temperature $T$, respectively, and ${I_k}$ be the corresponding theoretical current obtained by setting $V = {\hat V_k}$ in equation (1). The VOM seeks to find the values of the parameters $n$, ${I_\textrm{s}}$, and ${R_\textrm{s}}$ that minimize the cost function given by

$$\begin{equation}S = \mathop \sum \limits_{k = 1}^N {\left( {\frac{{{I_k} - {{\hat I}_k}}}{{{I_k}}}} \right)^{\!\!2}}{\text{ }}\end{equation} \tag{ 3 }$$

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with $N$ being the total number of measurements. Therefore, the values of $n$, ${I_\textrm{s}}$, and ${R_\textrm{s}}$ can be obtained by solving the system of equations given by

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial \textrm{s}}}{{\partial n}} = 2\mathop \sum \limits_N^{k = 1} \left( {\frac{{{I_k} - {{\hat I}_k}}}{{I_k^3}}} \right){{\hat I}_k}\frac{{\partial {I_k}}}{{\partial n}} = 0} \\ {\frac{{\partial \textrm{s}}}{{\partial {I_\textrm{s}}}} = 2\mathop \sum \limits_N^{k = 1} \left( {\frac{{{I_k} - {{\hat I}_k}}}{{I_k^3}}} \right){{\hat I}_k}\frac{{\partial {I_k}}}{{\partial {I_\textrm{s}}}} = 0} \\ {\frac{{\partial \textrm{s}}}{{\partial {R_\textrm{s}}}} = 2\mathop \sum \limits_N^{k = 1} \left( {\frac{{{I_k} - {{\hat I}_k}}}{{I_k^3}}} \right){{\hat I}_k}\frac{{\partial {I_k}}}{{\partial {R_\textrm{s}}}} = 0} \end{array}} \right.,\end{equation} \tag{ 4 }$$

which we have solved by using Newton–Raphson's method [50, 51]. The obtained results of $n$, ${\phi _{\textrm{B}}}$, ${I_\textrm{s}}$ and ${R_\textrm{s}}$ for various values of the temperature, are shown in figure 2. Moreover, the theoretical I–V curves obtained from equation (1) with the corresponding values of the parameters obtained by the VOM are shown in figure 1 (solid lines). It is evident from figure 1 that the experimental and theoretical results exhibit excellent agreement. Furthermore, it can be seen from the results in figure 2 that, as the temperature increases, the values of the barrier height and the series resistance increase (from ${\phi _\textrm{B}} = {\text{ }}0.461{\text{ eV}}$ and ${R_\textrm{s}} = 15.188{{ \Omega }}$ at $T = 75{\text{ K}}$ to ${\phi _\textrm{B}} = 0.624{\text{ eV}}$ and ${R_\textrm{s}} = 16.145{{ \Omega }}$ at $T = 300{\text{ K}}$), whereas the value of the ideality factor decreases (from $n = 1.453$ at $T = 75{\text{ K}}$ to $n = 1.07$ at $T = 300{\text{ K}}$).

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In figure 3, the experimental reverse bias I–V curves are shown for the temperature range 250 K–300 K. It should be noted, however, that the reverse bias I–V curves for temperatures lower than 250 K are not shown here due to the smallness of their reverse bias current values, which are lower than the precision of our measuring equipment, and thus very noisy and unreliable. From the reverse bias curves shown in figure 3, the value of the reverse saturation current is obtained as $3.32\, \times \,{10^{ - 9}}$, $4.48\, \times \,{10^{ - 8}}$, and $5\, \times \,{10^{ - 7}}$ A for the temperature values of $250$ K, $275$ K, and $300$ K, respectively. These values are very close to their corresponding values obtained from the forward bias curves as can be seen from figure 2(b). From these results and those of the saturation current shown in figure 2(b), it is evident that the saturation current decreases with decreasing temperatures, which is in accordance with the TE theory prediction. Furthermore, the reverse bias I–V curves in figure 3 and their forward counterparts in figure 1 demonstrate clearly that the NiV/Si Schottky diode has a very good forward-to-reverse (rectification) current ratio. For instance, at a voltage of $0.5$ V, the rectification ration is of order ${10^6}$, ${10^5}$, and ${10^4}$ for the temperature values of $250$ K, $275$ K, and $300$ K, respectively.

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The temperature dependence of the ideality factor and the barrier height that are reported above are due to the inhomogeneities of the Schottky barrier. Indeed, the linear relation between the barrier height and ideality factor is demonstrated in figure 4 and it is a signature of inhomogeneities [5, 52, 53].

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As mentioned earlier in the introduction section, the inhomogeneities of the Schottky barrier can be explained by using Werner's model under the assumption of a Gaussian distribution of the barrier height. The probability density function of the barrier height is given by

$$\begin{equation}P\left( {{\phi _\textrm{B}}} \right) = \frac{1}{{\sqrt {2\pi {\sigma ^2}} }}\exp \left( { - \frac{{{{\left( {{\phi _\textrm{B}} - {{\bar \phi }_\textrm{B}}} \right)}^2}}}{{2{\sigma ^2}}}} \right)\end{equation} \tag{ 5 }$$

where ${\phi _{\textrm{B}}}$, ${\bar \phi _{\textrm{B}}}$, and $\sigma$ are the barrier height, its mean, and its standard deviation, respectively. Furthermore, a linear bias of ${\bar \phi _{\textrm{B}}}$ and quadratic bias dependence of $\sigma$ are assumed in Werner's model, i.e.

$$\begin{equation}{\bar \phi _\textrm{B}} - {\bar \phi _{\textrm{B}0}} = {\rho _2}V,\end{equation} \tag{ 6 }$$

and

$$\begin{equation}{\sigma ^2} - \sigma _0^2 = {\rho _3}V\end{equation} \tag{ 7 }$$

where ${\rho _2}$ and ${\rho _3}$ are coefficients that are temperature independent. Hence, the total current across the Schottky contact can be obtained as

$$\begin{equation}I = \mathop \int \limits_{ - \infty }^\infty i\left( {{\phi _\textrm{B}},V} \right)P\left( {{\phi _\textrm{B}}} \right){\text{ d}}{\phi _\textrm{B}}\end{equation} \tag{ 8 }$$

with $i\left( {{\phi _\textrm{B}},V} \right)$ being the current at a given barrier height ${\phi _{\textrm{B}}}$ and voltage $V$. By substituting $i\left( {{\phi _\textrm{B}},V} \right)$ and $P\left( {{\phi _\textrm{B}}} \right)$ from equations (1) and (5), respectively, equation (8) yields

$$\begin{equation} I = {I_{0}}\left[ {\exp \left( {\frac{\beta }{{{n_{{\text{ap}}}}}}\left( {V - {R_\textrm{s}}I} \right)} \right) - 1} \right] \end{equation} \tag{ 9 }$$

where

$$\begin{equation}{I_0} = A{A^*}{T^2}\exp \left( { - \beta {\phi _{{\text{ap}}}}} \right),\end{equation} \tag{ 10 }$$

${n_{{\text{ap}}}}$ and ${\phi _{{\text{ap}}}}$ are, respectively, the apparent barrier height and the apparent ideality factor, which are given in Werner's model [4, 22, 23, 54] by

$$\begin{equation}{\phi _{{\text{ap}}}} = {\bar \phi _{\textrm{B}0}} - \frac{{\beta \sigma _0^2}}{2}\end{equation} \tag{ 11 }$$

and

$$\begin{equation}\frac{1}{{{n_{{\text{ap}}}}}} = 1 - {\rho _2} + \frac{{\beta {\rho _3}}}{2}.\end{equation} \tag{ 12 }$$

According to equation (11), the plot of ${\phi _{{\text{ap}}}}$ versus $q/2kT$ should be a straight line. The plot of ${\phi _{{\text{ap}}}}$ versus $q/2kT$ for the NiV/Si diode is given in in figure 5. As shown in this figure, the results are in good agreement with equation (11), and thus the values ${\bar \phi _{\textrm{B}0}} = {\text{ }}0.68{\text{ eV}}$ and ${\sigma _0} = {\text{ }}53.665{\text{ mV }}$ are obtained from this plot. Figure 6 shows the ($n_{{\text{ap}}}^{ - 1} - 1$) versus $q/2kT$ plot of the NiV/Si diode. As seen, the obtained results are in accordance with equation (12). Therefore, the values of the voltage coefficients ${\rho _2}$ and ${\rho _3}$ can be obtained, respectively, from the intercept and the slope of the curve in figure 6 as ${\rho _2} = - 0.0188$ and ${\rho _3} = - 4.322{\text{ mV}}$

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The conventional Richardson plot of $\ln \left( {{I_\textrm{s}}/{T^2}} \right) = \ln \left( {A{A^*}} \right) - \beta {\phi _{\textrm{B}}}$ versus $q/kT$ is usually used to determine the Richardson constant ${A^*}$ and the barrier height ${\phi _{\textrm{B}}}$. Figure 7 shows the conventional Richardson plot for the NiV/Si diode from which the values of the barrier height and the Richardson constant are obtained as ${\phi _\textrm{B}} = {\text{ }}0.410{\text{ eV}}$ and ${A^*} = {\text{ }}3.072\, \times \,{10^{ - 3}}{\text{ }}$ A cm⁻² K⁻², respectively. However, the obtained value of the Richardson constant is much smaller than the theoretical value of 112 A cm⁻² K⁻² of n-type Si [7]. The small value of the Richardson constant suggests that the effective area is much smaller than the diode area [18]. Moreover, it is apparent from figure 7 that the conventional Richardson plot deviates from linearity, which is due to the inhomogeneity of the barrier height [54, 55]. This anomalous behavior can be explained by taking into account the inhomogeneity of the barrier height, which can be achieved by combining equation (10) and equation (11) as

$$\begin{equation}\ln \left( {\frac{{{I_0}}}{{{T^2}}}} \right) - \frac{{{\beta ^2}{\sigma ^2}}}{2} = \ln \left( {A{A^*}} \right) - \beta {\bar \phi _{\textrm{B}0}}.\end{equation} \tag{ 13 }$$

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The modified Richardson plot of $\ln(I_0/T^2)-\beta^2\sigma^2/2$ versus $q/kT$ for the NiV/Si Schottky diode according to equation (13) is shown in figure 8. As seen, the plot is well-fitted by a straight line that gives $\bar\phi_{B0}=0.68$ eV and ${A^*} = 111.56$ A cm^-2 K^-2 from the intercept and slope, respectively. The obtained value of $\bar\phi_{B0}$ matches perfectly the previously calculated mean barrier height that has been obtained from the plot of figure 5 (see section 3.2). Furthermore, the obtained value of the Richardson constant is very close to the theoretical value of 112 A cm^-2 K^-2 of n-type Si. These results clearly indicate the usefulness and validity of the modified Richardson plot.

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The variation of the ideality factor with temperature in Schottky diodes is known as the T0 effect, or alternatively, as the T0 anomaly [18, 56]. According to Saxena [57], the temperature dependence of the ideality factor is usually given by

$$\begin{equation}n\left( T \right) = 1 + \frac{{{T_0}}}{T}\end{equation} \tag{ 14 }$$

where ${T_0}$ is a positive constant that is referred to as the excess temperature. On the other hand, according to Werner, the temperature dependence of the ideality factor can be obtained from equation (15) as

$$\begin{equation}n\left( T \right) = \frac{1}{{1 - {\rho _2} + \frac{{q{\rho _3}}}{{2kT}}}}.\end{equation} \tag{ 15 }$$

In the case of $\left| {{\rho _2}} \right| \ll \left| {q{\rho _3}/2kT} \right| \ll 1$, which is valid for most cases of Schottky diodes [23], $n\left( T \right)$ in equation (15) can be approximated as [23]

$$\begin{equation}n\left( T \right) \approx 1 - \frac{{q{\rho _3}}}{{2kT}},\end{equation} \tag{ 16 }$$

which is of the same form as equation (14). Therefore, by comparing equation (14) and equation (16), a theoretical approximation of ${T_0}$ can be obtained as

$$\begin{equation}{T_0} \approx - \frac{{q{\rho _3}}}{{2k}}.\end{equation} \tag{ 17 }$$

By using the calculated value of ${\rho _3} = - 4.322{\text{ mV}}$, we obtain from equation (17), ${T_0} \approx 25.07{\text{ K}}$. Figure 9 shows the experimental $nT$ versus $T$ plot together with its linear fitting and illustrates the temperature dependence of the ideality factor for the NiV/Si Schottky diode. In the same figure, the $nT$ versus $T$ plot corresponding to equation (16) with the calculated value of ${T_0}$ is shown as well. As can be seen from figure 9, equation (16) well approximates the temperature dependence of the ideality factor. Nevertheless, it is evident that, in contrast to equation (16), the slope of the experimental data deviates slightly from unity. Such deviation of the slope has been observed in various Schottky diodes [23] and it is mainly due to the temperature dependence of T0 [49, 56]. In addition, equation (16) is derived from an approximation that may not always be accurate [56]. Indeed, in our case, the condition of $\left| {q{\rho _3}/2kT} \right| \ll 1$ is weakly valid in the considered temperature range. Therefore, as has been reported in several works [18, 23, 56, 58, 59], the temperature dependence of the ideality factor, in this case, is given by

$$\begin{equation}n\left( T \right)\, = \,{n_0}\, + \,\frac{{{{\tilde T}_0}}}{T}\end{equation} \tag{ 18 }$$

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with ${n_0}$ and $\,{\tilde T_0}$ can be obtained for the NiV/Si diode from the slope and intercept of the linear fitting of the experimental $nT$ versus $T$ plot as ${n_0} = 0.95$ and ${\tilde T_0}\, = \,36\,{\text{K}}$. As demonstrated in figure 9, with the obtained values of ${n_0}$ and $\,{\tilde T_0}$, equation (18) describes the temperature dependence of the ideality factor more accurately than equation (16).