Extended diode equivalent circuit method (EDM)

The extended diode equivalent circuit method (EDM) aims to improve the IDM and provide more accurate SBHs and other related diode variables. Figure 4-16 shows the equivalent circuit assumed in the IDM compared to that in the EDM. Using the more comprehensive equivalent circuit, two EDMs were created to fit the I-V curves to more accurately measure the saturation current (IS), series resistance (RS), leakage resistance (RL) and ideality factor

(n). The first method was based on an iterative error reduction approach (EDM1) and the second method was based on a gradient descent approach (EDM2), both using the least squares fit method to minimise errors. The codes for both of the EDMs were provided by Dr Rolf Crook, University of Leeds, and can be found in Appendix B and C.

Figure 4-16: Equivalent circuit diagrams for SBH calculation. [Left] IDM assumes a single diode in series, [Right] EDM assumes a diode in series with a resistor and in parallel with a leakage resistor.

Before the EDMs were run the code was analysed to see how the different parameters were affecting the I-V curves. The code allowed different parts of the equivalent circuit in Figure 4-16 to be modelled separately and in combination. This allowed a more thorough analysis of the effect that series and leakage resistance has on a Schottky barrier. Figure 4-17 displays a selection of plots to show how the four main parameters, IS, RS, RL and n effect the I-V characteristics. It is clear that the saturation current affects the shape of the exponential curve, with a smaller saturation current delaying an exponential current response to higher voltages and a larger saturation current producing an exponential curve at low voltages. When n becomes larger, the current reduces in a similar way to IS. When the leakage resistance is much larger than the series resistance, expected for a rectifying device, the leakage resistance sets the gradient at low voltages and the series resistance at high

R

I

V

I

V

R

voltages. If the leakage resistance is small the devices become Ohmic. This is expected since a reduced leakage resistance suggests a short circuit in the barrier.

Figure 4-17: Matlab simulation of parameters which affect the I-V characteristics of a Schottky barrier, based on the extended equivalent circuit, showing variations in [A] IS [B]

n [C] RL [D] RS.

4.4.2.1 EDM 1: Iterative error reduction

EDM 1 uses an iterative error reduction process with least squares fit to fit the I-V data. This technique was selected since the least squares fit method can find an approximate solution for a function in which there are more equations than unknowns, which was the case here. In this method iterative refinement was used to find the best fit, which was found when the iterations minimized the sum of squared residuals. A residual here is the difference between the actual current and the fitted current. Figure 4-19 displays a flow chart to demonstrate the code for EDM 1.

This approach suffered from false minima effects. This was evident during Matlab operation where sometimes the variables did not converge, or very unrealistic values for the four key parameters were obtained. Moreover, this code could produce quite different values for the same fit, especially for leakage resistance. Results from the IDM were used as initial starting parameters for both EDM 1 and 2, to ensure an informed estimate, to avoid the local minima

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 -3 Voltage (V) C u r r e n t ( A ) Is = 1e-5 Is = 1e-10 Is = 1e-15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5x 10 -3 Voltage (V) C u rr e n t (A ) n=2 n=1.3 n=1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5 3 3.5x 10 -3 Voltage (V) C u r r e n t ( A ) Rl = 1e2 Rl = 1e3 Rl = 1e16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5x 10 -3 Voltage (V) C u r r e n t ( A ) Rs=162 Rs=300 Rs=100 Rs=200 A B D C

affect and to ensure consistency between methods. Due to the problems with false minima, a second method, EDM 2, was created to provide more accuracy.

4.4.2.2 EDM 2: Gradient descent

EDM 2 was an alternative least squares method, using a gradient descent approach, to produce a more accurate fit to the I-V data and to overcome the problems of false minima. EDM 2 used the gradient descent algorithm to find the nearest local minimum, using the relationship in Equation 4-6, where 𝑎 is the current parameter from the IDM initially, 𝛾 is the coefficient which dictates the step size and 𝛻 is the gradient operator.

𝑏 = 𝑎 − 𝛾𝛻𝐹 (𝑎)

Equation 4-6

Figure 4-18 presents a schematic for the process in the gradient descent approach. This method involved estimating an initial solution of a function and taking the gradient of the function at that point. Since the function is decreasing, the solution moved in the negative direction (represented by the negative in Equation 4-6) by a predefined step and the gradient at this point is then found. This process is then repeated until convergence, minimising the gradient along the function. A zero gradient corresponds to the local minimum.

The step size is an important parameter which gives this method more stability than the least squares method in EDM 1. It represents the amount that the next point will jump to, to find a solution of the function. Therefore if the step size is too large divergence occurs and the problem of reaching a false local minimum is increased, shown by the green objects in Figure 4-18. However if the step size is too small then convergence takes too long, illustrated by the small red arrows representing step size around the starting red dot in Figure 4-18. Whilst operating this method we used an adaptive approach where the step size was continuously changed to account for divergence and convergence.

Figure 4-18: Schematic of the gradient descent method, showing the resultant global minimum and false local minimum and maximum. The arrows represent the step size 𝜸 and gradient operator 𝜵. The green arrows and dots illustrate the need to control the step size.

To further enhance the probability of reaching the global minimum rather than false minimum, initial values were taken from the IDM. Figure 4-20 presents a flow chart for the EDM 2 code. -10 -5 0 5 10 0 2 4 6 8 10 Local minimum Global minimum Local maximum Start

Figure 4-19: Flow chart of EDM 1 code for Matlab I-V curve fitting. Load Experimental IV data Select positive variables only Guess at parameters: RS, RL, IS, n Define constants: k, T, q, VT Initiate vectors

I_deviceand V_device

Start Iteration

Select error factor (starting with largest):

5%; 2%; 1%; 0.5%; 0.2%; 0.1%

Find best fit for parameters:

RS, RL, IS, n

Estimate I_device

Find closeness of fit within error range compared to IV data

Best fit obtained for error factor > 0.1% Fit could be improved Error factor reduced Best fit at 0.1% error reached

Display final values for R_S, R_L, I_S, n, R2

Figure 4-20: Flow chart of EDM 2 code for Matlab I-V curve fitting.

Table 4-5 shows the resultant device parameters for RS, RL, IS, and n found from the two EDMs and respective indicator of fit to the experimental data (R2_{). The SBH included in the} table was calculated in all instances from the IS, using Equation 4-5. Despite the local minima problem in EDM 1 where quite different parameters could be achieved for the same device, for example in sample 505leakage resistance of 2e5 _{and 2e}17 _{can be obtained, a good fit was} obtained. Obviously this is more of a concern for large differences in saturation current or ideality factor. Comparing both EDM methods it is clear that the leakage resistances found in EDM 1 fluctuate more and reach higher values.

Table 4-5: RS, RL, IS, n, R2 and SBH for EDM 1 and 2 for I-V characterisation.

Define gradient coefficients- F values for R_S, R_L, I_S, n Define constants: k, T, q, VT Start Iteration Yes No Increase or decrease F values for R_S, R_L, I_S, n Repeat iteration until R2_<0.99

Record best fit values for

RS, RL, IS, n

Best fit found

R2_<0.99

Do resulting

values for R_S, R_L,

I_S, n converge?

Input initial estimated values for:

R_S, R_L, I_S, n

Select positive variables only Load Experimental

Most of the other parameters are comparable between the two methods. The series resistance correlates well between both of the methods, with results in the same order of magnitude, and over half of the samples obtained a resistance within +/- 40 Ω of the other method. The saturation current is also similar between the two methods, but they are generally higher in EDM 2. There is a close correlation in the ideality factor between the methods with a third of the samples only having a difference of +/- 0.1 and 11 out of 17 samples within +/- 0.5 of each other. Individually, both methods produce a very good fit to the I-V data, indicated by the R2 _{values generally exceeding 0.99. The calculated Schottky} barrier heights from the saturated current are also comparable between the two methods. Apart from three, all of the samples have a SBH that is +/- 0.05 eV between EDM 1 and 2. Generally the SBH calculated were slightly higher in EDM 2, especially for the non-ideal I-V curves. Overall, the largest differences between the methods tend to occur for the non- exponential I-V curves.

Comparing these results to the IDM results in Table 4-4, it is evident that the results from the IDM generally differ from the EDMs. This occurs in all parameters apart from the leakage resistance which are generally very similar across all 3 methods, with EDM 1 giving the largest variations and not the IDM as might be anticipated. As explained above, this is probably due to false local minima in EDM 1. The series resistance is generally over estimated in the IDM, apart from a few exceptions. This is probably due to the fact that the EDMs use more of the data to find a more accurate series resistance, rather than the more simplistic method applied in the IDM. The significantly lower series resistance from the EDMs indicates that less tunnelling occurs in the E-A devices and it confirms that hopping transport in the N-A occurs.

The saturation current is generally lower in the IDM which is due to fitting less of the data, since only current above 0.1 V were taken into account, and the inaccuracy of linear fit to the experimental data. Equally the SBHs from the IDM are considerably lower than the SBHs calculated from EDM 1 and 2, expected due to the lower saturation current obtained from the IDM. Significantly lower ideality factors are found in the EDMs which are more comparable with other studies, especially for the Pt devices, discussed previously after Table 4-4. This also suggests that the E-A devices have not deviated much from thermionic emission, as was indicated by the ideality factors found from the IDM. The combination of the above analysis of the results indicate why the data fit (R2_{) is far superior in both of the} EDMs.

Since the EDMs are more accurate, the SBH for the devices can now be updated. Taking n values from EDM 2 the maximum SBH calculated for each device has now increased to 1 eV, 0.97 eV and 0.90 eV for Pt, Ag, and Au devices respectively, which makes them more competitive to SBH found in other studies, shown in Table 4-3.