Approximation of the estimates distribution of parameters for the field emission signal by Johnson and Pearson distribution curves Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

UDC: 519.237

Li Anna Dmitrievna

student

Pakhomova Arina Aleksandrovna Student, Saint Petersburg State University DOI: 10.24411/2520-6990-2019-10517 APPROXIMATION OF THE ESTIMATES DISTRIBUTION OF PARAMETERS FOR THE FIELD EMISSION SIGNAL BY JOHNSON AND PEARSON DISTRIBUTION CURVES

Ли Анна Дмитриевна

студент

Пахомова Арина Александровна

Студент, Санкт-Петербургский государственный университет

АППРОКСИМАЦИЯ РАСПРЕДЕЛЕНИЯ ОЦЕНОК ПАРАМЕТРОВ СИГНАЛА ПОЛЕВОЙ ЭМИССИИ КРИВЫМИ РАСПРЕДЕЛЕНИЯ ДЖОНСОНА И ПИРСОНА

Abstract

A two-parameter model of the field electron emission signal is considered. Within the framework of mathematical modeling, estimates of the parameters of the linearized current dependence on voltage are obtained. For empirical distributions of parameter estimates (not normally distributed), approximations by Johnson and Pearson distribution curves are obtained.

Аннотация

Рассмотрена двухпараметрическая модель сигнала электронной эмиссии поля. В рамках математического моделирования получены оценки параметров зависимости линеаризованного тока от напряжения. Для эмпирических распределений оценок параметров (нормально не распределенных) получены аппроксимации кривыми распределения Джонсона и Пирсона.

Key words: field electron emission, Fowler-Nordheim law, Johnson distribution, Pearson distribution.

Ключевые слова: полевая электронная эмиссия, закон Фаулера-Нордгейма, распределение Джонсона, распределение Пирсона.

Introduction.

The aim of the work is to conduct an approximation of the empirical distributions of the parameters estimates of the signal of the field electron emission.

Autoelectronic emission is caused by electron tunneling into a vacuum [1]. Now this phenomenon is becoming increasingly popular due to its widespread use in microscopy, nanoelectronics and electron holography of atomic resolution. Autoelectronic emission is achieved at high electric field strengths. Thus, a strong electric field contributes to the fact that at the «metal -vacuum» boundary the potential barrier becomes sufficiently thin, which allows electrons to penetrate from a solid into a vacuum.

The Fowler-Nordheim theory describes this process, the purpose of which is to calculate the emission current density as a function of the electric field. In this work, the Fowler-Nordheim formula [2] is used:

j = aF2 exp [— -]

(1)

where j — current density, F — external electric field tensions, a and b — some constants. In a real experiment, instead of j and F, the voltage V between the cathode and the anode and the total current I of electron emission from the cathode surface are measured. Replacing in formula (1) j by the average value of - (here S is the emission area), and F by the average value of pv (here, ft is the geometric field factor depending on the shape of the emitter), which makes it possible to obtain the integral formula FN:

I = AV2 exp[—-], (2)

where A and B — constants, which include geometrical parameters and electron work function.

Approximation methods of distribution.

Earlier, a test was conducted for the normality of estimates of the parameters of the regression model of the signal of a field emission [3]. It was found that in most cases the null hypothesis is not rejected. Nevertheless, the analysis of the results showed that even a large sample size and a small noise level of the field emission signal do not guarantee for the model parameters that their distribution law complies with the normal one, and hence the construction of confidence intervals by the classical method. In this connection, it becomes necessary to select an approximating distribution from experimental data that can satisfactorily describe the results of a computer or full-scale experiment. The most common and frequently used approaches are approximations by Johnson curves and Pearson curves, which were used in this work. The first method is based on replacing the original sample with another one, the distribution of which is standard, and the second is a further development of the moment method.

Johnson curves

Let x be a random variable for which the Johnson distribution is chosen [4]. Johnson's transformation has the form:

Z = y + ^T(X,£,A),^,A > 0, —ro < y < TO,

where y, 77, e and A — Johnson distribution parameters; t(... ) — arbitrary function, z — random variable that has a standard distribution.

As t, Johnson proposed the use of functional transformation forms that have the form [5]: 1. tx(x, = ln—, x > e;

2. t2(x, £,A) = ln

3. t3(x, £,A) = sh-

X

X—£

X+£ —X'

£ < x < £ + A; , —œ < x < œ.

They correspond to the family of curves ., Sg, Sy Johnson, which are expressed respectively through the formulas:

1 /i(x) = v^eXP{—^2[]^^+ln(x — O]2},

X > £, —œ < y* < œ, ^ > 0, —œ < £ < œ, where

i . * M

rç = - and y = — ;

2. /2 (x) =

vX

exp{— 2 ln (X——+£) + y]2},

V2TC(X—£)(X—X+£) 2 VX—X+£

£ < X < £ + A, —œ < y < œ, rç > 0, —œ < £ <

œ,A > 0;

3. /3(x) =

v 1

exp{—l(y +

Cn = —ff2

,+ 1 ,— 1

,c2 = — ô = ■ 6(a4

+ 2

1

- = _ ,

2 , — 1, 2 2,, 1)

3a32 — 2a4 + 6 The type of curve from the Pearson family is determined by the value of the index %:

X =

«32(«4 + 3)2

4(4a4

3a32)(2a4

a

(, + 2)2

6)

16 , — 1

There are 7 main types of Pearson curves (shown schematically in the x-diagram [6, P. 276], which correspond to different values of x, and therefore a3 and a4.

Math modelling.

Imagine the relationship (2) amperage / with voltage V in the form:

,(„=„„ g) ^[.S]

(3)

V2S V(*-£)2+^2 2

^ln{iZ£+[(£Z£)2+1]l/2})2}>

—œ < x < œ, —œ < y < œ, 77 > 0, —œ < £ < œ,A > 0.

Pearson curves

The Pearson distribution family is given by a differential equation [6] :

dy x + b

y c0 + qx + c2x2' where the constants b, c0, c1, c2 are expressed in terms of the first four moments of the distribution (mathematical expectation, variance, coefficients of

skewness a3 andkurtosis a4):

Construct a linear regression model through transformations:

where 4' = lg4, B' = B/ln10. Herefl = (4,B). Accordingly, the estimates of 44 and B were determined by the formulas 4 = 10-4', B = B' ln(10).

Suppose that in the model of the field emission's signal (3), the voltage values V contain measurement errors, which, by virtue of smallness, are suggested to be measured exactly and operate only with the dimen-sionless quantity e.

In fig. 1 shows a plot of the dependence error values of the regression model from voltage.

3a,2

X

30 20 10 0

n _o

cl ° -10

-20

-30

-40

1 23456789 10

V, b

Fig. 1. The dependence of the values of the error e from the voltage V at N = 100 and S = 15%

Under the assumption of small measurement er- where e' is a standard normally distributed ran-

rors, the expression for the response is: dom variable, the parameter S is responsible for the so-

lg f/+£/° —) « Al - Bl — + called noise level. With a low noise level, it can be ex-

^ 7° ' v ln10' pected that in model (3) the residuals will be distributed

according to the normal law. We introduce the noisy signal designation through Ieps = I + e/0.

Numerical experiment.

In our case, we used the parameters A = 2 and B = 4 [7]. The voltage values Vt are chosen to be equally spaced, while 70 = 1 V and VN =10 V. To find the error e, a built-in MATLAB function is used, which generates a pseudo-random number according to the normal law with zero mathematical expectation

value and the mean square deviation c such experiments were carried out M times.

For various combinations of the parameters M, N and S, an approximation was made of the distribution of linearized estimates of A' and B' by Johnson and Pearson curves was carried out. The families of Johnson and Pearson curves are shown, respectively, in Fig. 2, 3, 4 and fig. 5, 6, 7, 8.

1 1 /■ x

/ \

/ \

/ 7 \

/ / i

/ / \

Fig. 2. SB Johnson family for A' with M = 500, N = 30 and 8 = 5%

o

0.275

\ \ 1 1

f / * * * \ \

\ V *

f

¥ m * \ V

- \ -

\

\

i i * 1 1 1 -----

Fig. 3. SL Johnson family for A' with M = 100, N = 50 and 8 = 10%

Fig. 4. Su Johnson family for B' with M = 1000, N = 100 and 8 = 20%

250 -

200 -

150 -

100 -

Fig. 5. I type of Pearson for B' with M = 100, N = 30 and S = 20%

Fig. 6. II type of Pearson for 41' with M = 100, N = 500 and S = 25%

Fig. 7. VII type of Pearson for B' with M = 500, N = 30 and S = 15%

Fig. 8. The case of the normal Pearson curve for A! with M = 500, N = 30 and S = 150%

In fig. 9 and fig. 10 shows the percentage of the families of the distribution of Johnson curves and the types of Pearson curves in the form of a pie chart. It is worth noting that during the experiment, III and V types of Pearson curves were never obtained.

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Fig. 9. Johnson family of curves

Fig. 10. Pearson family of curves

Conclusion and discussions.

In this paper, using the methods of regression analysis, a signal was simulated based on the two-parameter Fowler-Nordheim model. The possible influence of the error on the response of the system and the parameters of the field emission signal are also analyzed. An approximation was made of the sample distributions of the parameter estimates by the Johnson and Pearson distribution curves, which will make it possible to further carry out a confidential estimation of the parameters of the current-voltage characteristic, both in the case of normal and in other laws of the distribution of estimates.

References:

1. Fowler R. H., Nordheim L. W. Electron Emission in Intense Electric Fields // Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical, physical and Engineering Sciences. London: The Royal Society, 1928, vol. 119, no. 781, pp. 173-181.

2. Fursei G. I. Avtoelektronnaya emissiya [Field emission]. Soros Educational Journal. Moscow, 2000, vol. 6, no. 11, pp. 96-103.

3. Li A. D., Raevskii V. P., Sorokin V. A., Vos-trotina A. V., Omarov R. Z. Proverka normalnosti raspredeleniya ocenok parametrov regressionnoi modeli signala polevoi emissii [Checking the normal of

the estimates distribution of the regression model parameters for the field emission signal]. Young Scientist. Kazan, 2019, no. 263, pp. 3-8.

4. Kobzar A. I. Prikladnaya matematicheskaya statistika. Dlya injenerov i nauchnih rabotnikov [Applied mathematical statistics. For engineers and scientists]. Moscow, 2006. 816 p.

5. Han G. Shapiro S. Statisticheskie modeli v in-jenernih zadachah [Statistical Models in Engineering]. Moscow: World Publ., 1969. 395 p.

6. Mitropolskii A. K. Tehnika statisticheskih vi-chislenii [Statistical Computation Technique]. Moscow: Science Publ., 1971. 576 p.

7. Egorov N. V., Antonov A. Y., Varayun M. I. Analiz emissionnih harakteristik polevogo katoda na osnove regressionnih modelei [Analysis of emission characteristics of field cathods on the basic of regression models]. Journal of Surface Investigation. X-Ray, Synchrotron and Neutron Techniques. Moscow: World Publ., 2018, no. 10, pp. 1-8.