Comparing smoothing techniques for extracting MOSFET threshold voltage

Journal Pre-proofs Comparing smoothing techniques for extracting MOSFET threshold voltage Christopher Stankus, Moinuddin Ahmed PII: DOI: Reference:

S0038-1101(19)30713-0 https://doi.org/10.1016/j.sse.2019.107744 SSE 107744

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Solid-State Electronics

Received Date: Accepted Date:

28 November 2019 2 December 2019

Please cite this article as: Stankus, C., Ahmed, M., Comparing smoothing techniques for extracting MOSFET threshold voltage, Solid-State Electronics (2019), doi: https://doi.org/10.1016/j.sse.2019.107744

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Comparing smoothing techniques for extracting MOSFET threshold voltage Christopher Stankusa , Moinuddin Ahmeda,∗ a Argonne

National Laboratory, Lemont, IL, 60439 USA

Abstract Measurement noise acts as a barrier to the accurate calculation of threshold voltage by derivative-based extraction methods. We examined several smoothing techniques and their effects on measurement noise in threshold voltage extracted by the linear extrapolation method and the gm /Id method. We applied these techniques on a set of SiC power MOSFET devices before and after they had undergone accelerated thermal testing. The smoothing methods examined are least-squares polynomial fitting, low-pass filtering using Butterworth and Chebyshev filters, and differential smoothing. These methods are compared by their shift in threshold voltage compared to unsmoothed data and their standard deviation among the measurements of each device. All methods performed poorly at smoothing when extracting via the gm /Id due to the large amounts of high-magnitude subthreshold noise, mostly tending to extract threshold voltages far below the expected value. Differential smoothing provided results closest to the expected value with an absolute shift of 1.2V on average. The smoothing methods performed better with the LE method. On a failed device, all smoothing methods except polynomial smoothing preserved the large threshold voltage anomaly for four out of five measurements. These results show that choice of smoothing method, if used must be evaluated based on the device and extraction method. Keywords: MOSFETs, Threshold voltage, Linear extrapolation method, Smoothing techniques, Statistical analysis

1. Introduction Threshold voltage (Vth ) is a vital parameter for the modeling and characterization of transistors. It is defined as the gate voltage (Vgs ) where the transistor transitions from weak to strong conduction. There are many methods for extracting threshold voltage from device characteristics, primarily through the Id – Vgs characteristics. Such methods include the constant current method, linear extrapolation method (LE), the second derivative method (SD), and the third derivative method (TD) [1]. However, the methods requiring taking derivatives of the Id – Vgs curve — the LE, SD, and TD ∗ Corresponding

Author Email addresses: [email protected] (Christopher Stankus), [email protected] (Moinuddin Ahmed)

Preprint submitted to Elsevier

December 13, 2019

methods — are susceptible to measurement noise and variability. Unfortunately, these methods are also some of the most popular; the LE and SD methods are particularly popular. Previous work focused on the various methods to find threshold voltage, OrtizConde et al. [1] discussed many of these methods, including the LE method this paper focuses on. However the LE method is sensitive to mobility degradation effects and parasitic series resistances of the source and drain [2]. These effects and the subsequent shift in threshold voltage they cause are used in reliability analysis to determine the health of the device [3]. These effects are less useful in other contexts, so to avoid them several methods of threshold voltage extraction have been developed, including the √ modified Y-function method of Ghibaudo [4], which used extrapolation of the Id / gm function to determine the threshold voltage. A more modern method for dealing with mobility degradation effects and parasitic series resistances was given by Flandre et al. [5], which used the minimum of the d(gm /Id )/dVgs function to determine threshold voltage and showed good results against transistors with high parasitic resistance and mobility degradation effects. This method was found to be much less dependent on low drain voltage than the SD method, reaching a limit quickly rather than monotonically increasing [6]. Other work focused on ways to improve the calculation of threshold voltage using existing methods via numerical smoothing. Yang and Inokawa [7] described a differential smoothing technique applied to the LE method based on a convolutional filter which combines smoothing and differentiation along with a fitness criteria. Gonz´alez et al. [8] described a smoothing technique based on weighted essentially non-oscillatory (WENO) methods, a polynomial interpolation technique, and applied it to the LE and SD methods. Picos et al. [9] described an approach based on low-pass filtering in the frequency domain applied to the SD method. Ib´an˜nez et al. [10] used a technique based on a boolean sum of smoothing splines to smooth data and extract threshold voltage using the SD method. Espi˜nera et al. [11] provided a statistical study of the effects of the LE, SD, TD, and constant current methods on the extracted threshold voltage, and showed a 30% increase in standard deviation for the constant current method versus the other methods. To the authors’ knowledge, no work exists comparing the effects of smoothing methods on extracting threshold voltage. Espi˜nera et al. [11] is similar in its comparative aspect, but is focused on the differences between threshold voltage extraction techniques, not on smoothing methods. This paper presents a comparative statistical study of various smoothing methods which can be easily implemented for computational analysis of threshold voltage extracted from MOS devices [12], as well as for parameters extracted from electronic devices using derivatives in general. In section 2, a variety of smoothing methods are described. In section 3, these methods are applied to extract threshold voltage via the LE and gm /Id methods from a set of SiC power MOSFETs before and after accelerated thermal aging and bias instability testing as described by Ahmed et al. [13], and the statistical results are discussed. Finally, conclusions about the suitability of these methods to threshold voltage extraction and reliability analysis is presented.

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2. Threshold voltage extraction model We compared the performance of various smoothing methods at providing more stable threshold voltage extraction with the linear extrapolation method and the gm /Id method. The linear extrapolation method consists of finding the Vgs axis intercept (where Id = 0) of the tangent line at the point where the first derivative of the Id – Vgs curve is at maximum. This is the point of maximum transconductance gm = dId /dVgs . The threshold voltage Vth is calculated by subtracting Vds /2 from the resulting Vgs value. The values of gm are sensitive to noise as the derivative acts as a high-pass filter, amplifying noise from variations in the Id – Vgs curve [9]. The gm /Id method extracts threshold voltage by finding the minimum of the first derivative of the gm /Id − Vgs curve. As such, this is similar to the SD method in its sensitivity to interference and noise, but less effected by drain voltage [6]. Noise and interference in this method are particularly large in the subthreshold region. Like the SD method, This method is not effected by parasitic resistance or mobility effects, unlike the LE method, and is thus a more accurate method for extracting threshold voltage. In both cases, we employed numerical smoothing techniques to reduce this measurement noise. 2.1. Smoothing methods We compared the performance of four smoothing methods: Least-squares polynomial fitting, digital Butterworth filters, type I Chebyshev filters, and differential smoothing. These methods were chosen because they are particularly easy to implement to aid computational analysis by being built in to software libraries for scientific computing over other methods such as those given in [8, 9]. These smoothing methods were implemented in Python, using the SciPy library [14]. The remainder of this section gives a brief introduction to each method and describes the parameters that affect it. For all methods, we assume that there are a total of N data points, with Id [i] and Vgs [i] denoting the i-th data point for drain current and gate voltage, respectively. The unsmoothed transconductance is calculated by the direct difference method as gm [i] =

Id [i + 1] − Id [i − 1] Vgs [i + 1] − Vgs [i − 1]

(1)

for 0 ≤ i < N. 2.1.1. Polynomial fitting Polynomial fitting performs a least-squares fit of a polynomial of degree k to the unsmoothed gm curve and Vgs data. The resultant polynomial is then given the measured Vgs data to yield a smooth curve. The resultant curve depends on the degree of the polynomial, with some degrees giving better fit and thus a smoother curve. This is highly dependent upon the structure of the original data. Fig. 1 demonstrates the relationship between extracted threshold voltage and polynomial degree for the LE method. Polynomials of degree 7 and above fluctuate around 4.71V with minor variation in this example. 4

4.76

Vth (V)

4.74 4.72 4.7 4.68 4.66 10 20 15 25 Polynomial Degree

5

30

Figure 1: The threshold voltage given by the LE method, with gm smoothed by a polynomial of the given degree.

4.9 0.50

Vth (V)

4.8

4.7

4.6

f = 0.10

0

10 Order

5

15

20

Figure 2: Threshold voltage extracted with the LE method from gm data smoothed with a Butterworth filter of given order and cutoff frequency. f is stepped by 0.05 half-cycles/sample.

5

4.8

4.8

Vth (V)

Vth (V)

5

4.6 4.4

4.6 4.4 4.2

r p = 3dB

4

5 Order

5 r p (dB)

10

(a)

f = 0.1 f = 0.15

10

(b)

f = 0.2 f = 0.25

f = 0.3 f = 0.35

f = 0.4 f = 0.45

f = 0.5

Figure 3: Threshold voltage extracted using the LE method from gm data smoothed with Chebyshev filters with a) varying order and cutoff frequency with r p = 3dB b) varying r p and cutoff frequency with 3rd-order filters.

2.1.2. Butterworth filters Butterworth filters have a flat response before their cutoff frequency, rolling off at a rate dependent on their order. Applied to smoothing, the parameters of interest are the order and the cutoff frequency. For the digital filters we employ, the cutoff frequency is given as half-cycles per sample on the interval (0, 1). Higher cutoff frequencies preserve more of the original gm curve, and higher filter orders cause much faster rolloff after the cutoff frequency. Fig. 2 shows how order and cutoff frequency effect extracted threshold voltage. In this case for all but the highest and lowest cutoff frequencies, the filters with order greater than or equal to five have converged to a single value. 2.1.3. Chebyshev filters Chebyshev filters have a steeper cutoff than Butterworth filters for a given order, but have ripple in their frequency response. In type I Chebyshev filters, this ripple is before the cutoff frequency, attenuating and amplifying low-frequency signals. In type II Chebyshev filters, the ripple is after the cutoff frequency and effects higher-frequency signals. We focus on Type I filters for smoothing. In a Type I Chebyshev filter, the order of the filter has a larger effect on the results than with a Butterworth filter due to ripple. This ripple manifests as a series of maxima and minima in the filter’s gain response towards the cutoff. Odd order filters begin at a maximum, so any ripple causes attenuation. Even order filters begin at a minimum, so ripple causes amplification [15]. With both even and odd order filters scaled such that their initial gain is unity, applying the Chebyshev filter will result in smooth data with slight attenuation or amplification, respectively, proportional to the order and the ripple level which is independent of the cutoff frequency. The cutoff frequency, like with the Butterworth filter, controls the overall amount of smoothing. Fig. 3 shows the relationship between these three parameters on extracted threshold voltage. 6

Both Chebyshev and Butterworth filters have been implemented to filter the data twice, once in the forwards direction and once in the reverse, as these methods introduce a “lag” if data is simply filtered in one direction, causing undesired shifting of the peaks in the data. A consequence of filtering twice is that the order of these filters is effectively doubled compared to their given order [16]. 2.1.4. Differential smoothing The differential smoothing (DS) technique is described by Yang and Inokawa [7] using a window convolution filter given by 1 P j=r j=−r Id [i + j] · j NC gm [i] = (2) ∆Vgs for r ≤ i < N − r, with ∆Vgs assumed constant. r is the smoothing rank of the filter, and NC is a normalization coefficient given by r(r + 1)(2r + 1) . (3) 3 The smoothing rank r effects the smoothness of the curve, and Yang and Inokawa [7] give a method for finding the most optimal smoothing rank as the minimum of the “mismatch rate” between two ranks Pn=m+5 |gmr [n] − gmp [n]| nP n=m−5 o × 100%. η= (4) P n=m+5 0.5 · n=m−5 |gmr [n]| + n=m+5 n=m−5 |gmp [n]| NC =

We use this method when applying the smoothing, as well as the local-average smoothing described below. Unlike the previous methods, this method does not smooth the unsmoothed gm curve given by (1) but rather computes a smoothed gm curve directly from the Id – Vgs curve. The assumption of constant ∆Vgs given by Yang and Inokawa [7] may not be valid for experimental data, as measurement and testing equipment have some variation resulting in a non-constant ∆Vgs . In that case, we take the average of the differences in adjacent Vgs data points, given by N−1

∆Vgs =

1 X Vgs [i + 1] − Vgs [i] N − 1 i=0

(5)

but this means that there will inevitably be error in the case where ∆Vgs is not constant. A more locally accurate average can be given by ∆r Vgs [i] =

r−1 1 X Vgs [i + j + 1] − Vgs [i + j] 2r j=−r

for r ≤ i < N − r − 1. This gives a modified version of (2) using this average 1 P j=r j=−r Id [i + j] · j NC gm [i] = . ∆r Vgs We refer to this modified equation as local average differential smoothing (LADS). 7

(6)

(7)

6

Unsmoothed Polynomial Butterworth Chebyshev DS LADS

gm (A/V)

4

2

5.4 5.2

0

5 13 0

5

10 Vgs (V)

14 15

15

16 20

Figure 4: The effect of each of the five smoothing techniques on an unsmoothed gm curve. Polynomial is degree 7. Butterworth is order 4, f = 0.25. Chebyshev is order 3, f = 0.25 with r p = 3dB.

3. Results and discussion The smoothing methods described above have been used to smooth the gm – Vgs curves of a set of 60 SiC power MOSFETs. 30 of the devices were exposed to accelerated thermal testing at 120°C for 200 hours, with the devices measured before and after testing, and the other 30 were exposed to bias temperature instability testing at 120°C for 200 hours, with the devices measured before testing, every 50 hours during testing, and after testing. This was done by sweeping Vgs from 0V to 20V, and Vds from 0V to 20V (Vds = Vgs ) and gathering Ids measurements using a Vgs step size of 0.1V. These measurements were performed five times for each device using an Agilent 1505A power semiconductor parameter analyzer during each measuring of the device. We considered each of these measurement runs a discrete sample for comparison, giving a total of 210 samples. Variation among the measurements in each device comes mainly from the uneven distribution of dopants and oxide thickness in the MOSFET channel as noted in [13]. This results in the presence of charge traps along the source and drain ends of the channel—reflected in the LE threshold voltage increase and decrease, respectively —and resultant 1/ f noise [17] which tend towards an average with continued measurement as the trap density distributes more evenly across the channel. The smoothing techniques from section 2 have been applied to this data along with an unsmoothed control. For the LE method, the exact parameters used for each method are as follows: A polynomial of degree 7, Butterworth filtering with order 5 and cutoff frequency of 0.25 half-cycles/sample, Chebyshev filtering with order 3, r p = 3dB and cutoff frequency of 0.25 half-cycles/sample, and differential smoothing described by Eq. (2) and Eq. (7), with rank determined by the mismatch criteria described by Eq. (4). Fig. 4 demonstrates each of these methods with a single device.

8

LE Average Smoothing Method

σ (V)

Unsmoothed Polynomial Butterworth Chebyshev DS LADS

0.0922 0.0689 0.0813 0.0780 0.0795 0.0783

|∆Vth | (V)

— 0.1012 0.0586 0.0877 0.0762 0.0777

gm /Id Average σ (V) 1.0440 2.4073 1.0891 1.1434 1.8068 1.8556

|∆Vth | (V)

3.0055 2.1666 3.3894 3.2054 1.2002 1.2157

Table 1: The per-device standard deviation (σ of threshold voltages and per-device mean absolute shift in threshold voltage (|)∆Vth |) after smoothing for the LE method and the gm /Id method. For the gm /Id method the shift is relative to the unsmoothed voltage extracted via the LE method.

We compared each smoothing method by looking at the standard deviation of the extracted threshold voltage for the measurements in each sample, as well as the mean of the absolute differences of the threshold voltages extracted after smoothing against the unsmoothed values for the measurements in each sample. The standard deviation was chosen as the threshold voltage measurements in each sample are from the same device, so this value should be small. Threshold voltage will always vary slightly for any given device but a decrease in the variation among measurements reflects an overall damping of channel effects on the extracted threshold voltage. The mean absolute difference between the smoothed and unsmoothed results gives an indication of how the smoothing has shifted the extracted threshold voltage. The LE method is sensitive to both the position and magnitude of the peak gm curve, and smoothing affects both of these. A significantly large difference would indicate that important information was lost by the smoothing. Also tested was the gm /Id method. Due to the significantly increased amount of sub-threshold interference caused by this method, This includes interference with much larger magnitudes than that at the threshold voltage on the d(gm /Id )/dVgs curve. These magnitudes are so large that smoothing just the d(gm /Id )/dVgs curve is ineffective. Thus for each smoothing technique we have smoothed both the gm curve and the d(gm /Id )dVgs curve. As the unsmoothed threshold voltage attained by this method is not useful, we cannot look at the shift in threshold voltage from that to judge any effect. We instead compared with the results from the LE method, as both methods should be in rough agreement. Table 1 shows these measures for the smoothing techniques we have described for the LE and gm /Id methods. For the LE method, all smoothing methods give lower standard deviation over a device’s runs than the unsmoothed data, with a maximum 25% decrease from polynomial smoothing. Additionally, all smoothing methods have shifted the extracted threshold voltage by more than 50mV on average, with polynomial smoothing giving the largest shift of 101.2mV. The average unsmoothed threshold voltage of our data was 5.4238V, so this represents a percent shift of 1.86%. Both versions of differential smoothing perform almost identically, with only a 1% difference in standard deviation and threshold voltage shift. For the gm /Id method, only the differential

9

d(gm /Id )/dVgs (V −1 )

2

1

0

−1 −2

0

5

10 Vgs (V)

15

20

Figure 5: A d(gm /Id )/dVgs curve, with minimum near 5V obscured by noise.

smoothing methods are even close to useful, though still with large variance between runs on each device due to the interference. These smoothing methods struggled with the gm /Id method because The noise is much higher magnitude than the critical value. Fig. 5 shows a d(gm /Id )/dVgs curve; we can see a minimum is reached near 5V, but this is obscured by the noise prior to that point. The smoothing methods we have tested do a poor job in this situation, as they tend to preserve large peaks. Smoothing the Id curve would eliminate these peaks, but also any large changes in current indicative of failure. We also compared the performance of each smoothing method with a failed device using the LE method to extract threshold voltage. As the LE method is sensitive to parasitic resistances and mobility effects, degradation of the MOSFET channel will have a large effect on it. Table 6 shows the results for each smoothing technique with this failed device. All smoothing methods shift the extracted threshold voltage by significant amounts each run, with the second run having its anomalous peak completely smoothed away, as the peak there lasts for only two data points. Save for polynomial smoothing, the other runs have their peaks preserved, though damped, and the resulting threshold voltage is still much higher than the average of 4.5–5V. Polynomial smoothing, however, completely eliminates the peaks in all cases and is useless here for detecting the device failure based on the change in threshold voltage. The performance of the smoothing methods on this device demonstrates that they should be checked against the unsmoothed results to ensure that they do not mask any measurements that would be indicative of anomalous behavior.

10

Unsmoothed Polynomial Butterworth Chebyshev DS LADS

Vth (V)

10

8

6

4

1

2

3 Run

4

5

Figure 6: Extracted threshold voltage with the LE method for the failed device when each smoothing method is used.

4. Conclusion Extracting the threshold voltage of a MOSFET is complicated by noise due to environmental sources and measurement error. Smoothing methods such as low-pass filtering and polynomial fitting are effective techniques to smooth this noise and allow automated analysis of MOSFET threshold voltage. These smoothing methods effect the extracted value by shifting threshold voltage compared to unsmoothed data and reducing variation between measurements. Using the LE method Butterworth and Chebyshev filters and differential smoothing perform similarly, with polynomial failing to preserve high magnitude variations that may be predictive of failure. All methods struggled with the gm /Id method due to the difficulty in eliminating high-magnitude subthreshold noise. The performance of smoothing methods is dependent upon the data being smoothed, and this must be taken into account during analysis. Acknowledgement This work was partially funded by Department of Energy—Power-America (Grant PA100215CFP51). This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships Program (SULI). [1] Ortiz-Conde A, Garc´ıa-S´anchez FJ, Muci J, Barrios AT, Liou JJ, Ho CS. Revisiting MOSFET threshold voltage extraction methods. Microelectron Reliab 2013;53:90–104. doi:10.1016/j.microrel.2012.09.015. [2] Schroeder DK. Semiconductor material and device characterization. 3rd ed.; Wiley; 2006. 11

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[15] Shenoi BA. Introduction to digital signal processing and filter design. Wiley; 2006. [16] Gustafsson F. Determining the initial states in forward-backward filtering. IEEE Trans Signal Process 1996;44:988 –92. doi:10.1109/78.492552. [17] Sikula J, Levinshtein M. Advanced experimental methods for noise research in nanoscale electronic devices. New York: Kluwer Academic Publishers; 2004,doi:10.1007/1-4020-2170-4.

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