Figure 1
Figure 2
The objective of this page is to set some grounds and tools for the measurement of the current-voltage curve of a diode and its analysis. The application of these tools is demonstrated in the experiments regarding a Signal Diode and a Schottky Diode that consisted of measuring such I-V curves for a range of temperatures in order to extract its physically meaningful parameters based on Schockley's model.
The data was measured through a python code that comumunicates with the sourcemeter and the temperature chamber and exports the current-voltage measurements for each temperature as csv files. Furthermore, the code conducts multiple measurements with different current ranges. Another code then choses the most precise current value and exports one final I-V curve for each temperature.
For personal preference, the rest of the analysis is carried out in MATLAB.
The rest of the section is devoted to pointing out problems derived from the experiment codes and solving them a posteriori (i.e. not by fixing the experiment codes) to obtain suitable data for the analysis.
The resulting data contains the absolute value of the current and the I-V points are not sorted by voltage. The following code rearranges the data and rescues the sign of the current:
The voltage sweep to measure the I-V curve was intended to be done in approximately constant steps. However, around certain values, there are many (V, I) points within a very small interval compared to the step size (see figure 1). This phenomenon shows at a zoomed in plot (see figure 2) as a region of accumulation of points, whose difference in current is mainly due to noise from the device (random error). Redundant measurements were performed for points in the sweep whose voltage led to a current exceeding the current limit.
Figure 1 | Figure 2 |
It is enough to take one of them to represent the whole set. In fact, using all of them would affect data fitting by adding weight to arbitrary values. The following code takes the data and substitutes accumulation points for their average, as is shown in figure 2:
A common model for the direct current through a diode is based on Schockley's equation,
$ I = I_S(T)\left(\exp(\frac{e(V-IR)}{\eta k_B T}) - 1\right),\hspace{0.5cm}\text{[A]} \tag{1}$where $I_S(T)$ is the saturation current (with a functional form depending on the type of diode), $\eta$ is the nonideality factor and $R$ is the series resistance of the circuit that reduces the voltage across the diode.
The model fails in predicting a breakdown in which the current drops drastically instead of saturating to $I_S(T)$ and which is triggered in high reverse voltage by the avalanche, tunnel or punch-through effects, depending on the diode characteristics. Also, $\eta$ might vary depending on the voltage and the type of diode. It is expected to be around 2 in low forward bias in p-n diodes due to carrier recombination in the depletion region (neglected by the model). As voltage increases and the region shrinks, it will approach "ideality" ($\eta$ = 1). For further increase, injected minority carriers are no longer negligible compared to majority carriers in the neutral region, which makes $\eta$ approach 2 again.
The numerous limitations of the model can be improved by including more parameters. For example, non-ideality can be better dealt with by using two terms of the Shockley form with different saturation currents, where one is ideal and the other has $\eta = 2$. However, for a certain voltage range, an almost ideal exponential behaviour could be expected, as explained above. Moreover, in this region of not too high and not too low forward bias, the current is presumably small and so is the voltage drop ($V>>IR$) and the equation is dominated by an exponential behaviour ($\exp(\frac{eV}{\eta k_B T})>>1$), resulting in
$ I \approx I_S(T)\exp(\frac{eV}{\eta k_B T}),\hspace{0.5cm}\text{[A]} \tag{2}$with $\eta \approx 1$. Such a simple expression can be easily fitted to the data to obtain the physically meaningful parameters of the model. The existence of this region has been theoretically argued, but it should be experimentally verified along with its size.
This subsection provides a code that finds the exponential region of an experimental curve obtained as explained in the previous section. This problem is equivalent to finding the linear region of the logarithm of the current.
Due to the precision of the measurements and the little noise, an attempt was made by taking the numerical second derivative and searching where it is closest to zero. This would have provided an easy analysis over the whole voltage range. However, the derivatives were found to be very abrupt, which lead to the discovery of the accumulation regions in the data. Even after solving this issue, it was found that numerical differentiation drastically amplifies the error of data, so it is useless for this purpose.
The preferred measure of the linearity of a region was the Pearson coefficient of the variables:
$ r=\frac{cov_{xy}}{\sigma_x\sigma_y} \tag{3}$ where $cov_{xy}$ is the covariance of the variables $x$ and $y$ and $\sigma$ stands for standard deviation.
The code is:
Figure 3 shows an example of how the program works. The number of points of the selected region is usually the imposed minimum (20 in this case), which is arbitrary. An interesting improvement would be to make the size of the region more adaptative, so that there is a balance, based on statistics, between the linearity of the region (perfect for a region of two points) and the sample size.
For higher voltage the plot shows another exponential region with a higher $\eta$, as predicted. The program could be modified to search for more than one exponential region.
Figure 3