Field emission system with two emitters mathematical modeling Текст научной статьи по специальности «Физика»

UDC 51-73, 537.2 Вестник СПбГУ. Прикладная математика. Информатика... 2023. Т. 19. Вып. 4 MSC 35J05

Field emission system with two emitters mathematical modeling*

N. V. Egorov, E. M. Vinogradova, G. I. Kurbatova, V. A. Klemeshev

St. Petersburg State University, 7—9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

For citation: Egorov N. V., Vinogradova E. M., Kurbatova G. I., Klemeshev V. A. Field emission system with two emitters mathematical modeling. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2023, vol. 19, iss. 4, pp. 443-448. https://doi.org/10.21638/11701/spbu10.2023.402

This paper presents a mathematical model of a two-dimensional emission diode system with two identical field emitters on a flat substrate; the anode is a plane parallel to the plane of the substrate. According to the proposed method, the real field cathode coincides with the virtual cathode, the shape of which is determined by the zero equipotential surface. The influence of each of the field emitters on the electrostatic potential distribution is replaced by the influence for a finite number of charged filaments. The solution of the boundary value problem for the Poisson equation is found in an analytical form. The potential distribution over the entire region of the emission system under study is presented in the form of expansions in terms of eigenfunctions. The coefficients of the series are calculated explicitly. The graphs presented demonstrate the dependence of the potential distribution on the distance between the emitters. All geometric dimensions of the system are the parameters of the problem. Keywords: micro- and nanoelectronics, field emitter, mathematical modeling, electrostatic potential distribution, boundary-value problem.

1. Introduction. Field cathodes are widely used in a variety of modern vacuum electronic devices [1-3]. The field cathode is structurally a thin emitter with a radius of curvature at its top, on the order of several micro- or nanometers [4-6]. This shape of the emitter makes it possible to obtain intense field electron emission at relatively low voltages [7, 8]. However, one of the main disadvantages of field cathodes is that they give small values of the total emission current for single emitter — on the order of several microamperes. This work is devoted to the mathematical modeling of a two-dimensional emission diode system with two identical field emitters on a flat substrate. The anode is a plane parallel to the plane of the substrate. Two field emitters schematic representation in Cartesian coordinates (x, y) is shown in Figure 1. The influence of each field emitter on the potential distribution can be replaced by the influence of N charged filaments with charge densities т located in the xOy plane with coordinates (x = x0, y = y0i) and (x = x2 — x0, У = y0i), i = 1,N. The surface of the real cathode coincides with the zero equipotential [9-12].

2. Mathematical model. The problem parameters: L — the height of the emitters; y = 0, 0 < x < x2 — the surface of the emitter's substrate; y = yi, 0 < x < x2 — the surface of the anode; x = 0, x = x2, 0 < y < yi — the boundaries of the domain along the x axis; x = xi = x2/2, 0 ^ y ^ yi — the plane of symmetry of the system.

The boundary conditions in an emission diode system with two field emitters: U(x, 0) =0, 0 ^ x ^ x2, — on the substrate surface; U(x,yi) = U0, 0 ^ x ^ x2, —

* This work was funded by the Russian Science Foundation (project N 23-29-00937, https://rscf.ru/project/23-29-00937/).

© St. Petersburg State University, 2023

Figure 1. Schematic representation of the diode field emission system with two identical field emitters

y

on the anode surface; U(0, y) = U(x2,y) = U0—, 0 < y < yi, — the boundaries of the

yi

domain in terms of the variable y.

The potential distribution function U(x,y) for the entire region 0 < x < x2, 0 < y < yi is symmetric with respect to plane x = xi, 0 < y < yi, so the problem can be solved for 0 < x < xi, 0 < y < yi.

Let each charged filament with linear charge density t create a uniformly distributed space charge p in a small volume \x — x0\ < e, \y — y0i \ <8. Thus the electrostatic potential distribution U(x, y) is a solution of the Poisson's equation for the boundary-value problem:

д2U(x,y) + д2U(x,y) _ 1 (x ) dx2 + ду2 _ -^p(x,y)' U(x, 0) _ 0, 0 < x < xi,

U(x,y1 )_ U0, 0 < x < x1, U(0, y)_ Uo 0 < y < yi,

yi

dU (x,y )

( ,y) _0 0 < y < yi.

(1)

dx

_ 0,

The function p(x, y) on the right side of the Poisson equation for the boundary value problem (1) can be represented as

p(x,y) _

pi, \x - xo\ <£ and \y - yoz \ <6, 0, \x - xo \ >£ or \y - y0i \ >6,

(2)

where the relationship between the space charge (2) and the linear charge density of the filament is given by the condition

T = lim 4e8 p., i = 1,N.

£,5^0

The potential distribution U(x, y) can be represented in the form

U (x,y ) = Uo y + Ui (x,y). yi

(3)

(4)

Х = Х1

Then formulas (2)-(4) for the function U (x, y) make it possible to satisfy all the boundary conditions of the problem (1) with respect to the variables x and y, if the function Ui(x, y) is solution of the Poisson equation with homogeneous boundary conditions in both variables:

д 2U1(x,y) д 2Ui(x,y) ~3x2 + дУ2 Ui(x, 0) = 0, 0 < x < xi,

- ¡F0 P(x,y),

Ui(x, yi) = 0, Ui(0,y) = 0, dUi(x,y)

0 ^ x ^ xi, 0 < y < yi,

(5)

dx

0,

0 < y < yi.

3. Solution of the problem. The function Ui(x,y) is a solution of the boundary value problem (5) for the Poisson equation and with the variable separation method can be represented as an expansion in terms of eigenfunctions in the variable y with functional coefficients vm(x) [9-11]:

x=xi

Ui(x,y)=y^ vm (x) sin amy, am = -. (6)

-, yi

m=i yi

Substituting (6) into the Poisson equation (5), as a result of transformations, it can be obtain that the functions vm(x) are the solution to the system of second-order ordinary differential equations with constant coefficients:

yi

vm(x) - («m)2 Vm(x) = --

and boundary conditions

Vm(0)=0,

£0 yi

dVm(x)

dx

p(x, y) sin amy dy

0.

(7)

The solution of system (7) makes it possible to find the potential distribution in the entire region of the emission system.

The electrostatic potential distribution U(x, y) has the form:

• 0 ^ x ^ xo

N to

y 2 ^ ^ sin (am y0i)cosh(am(xi - xo)) U (x,y) = Uo--1----—-;-x

Ил IT m

y i П£о

i m=i

x sinh (am x) sin (am y) ;

cosh (amxi)

(8)

xo ^ x ^ xi

TU , TT y , 2 ^ ^ sin (am yoi) smh (amxo)

U (x,y) = Uq--1--У Ti >--—-- x

y i n £q m cosh (amxi)

x cosh (am (xi — x)) sin (am y); Вестник СПбГУ. Прикладная математика. Информатика... 2023. Т. 19. Вып. 4

(9)

2

x = xi

x1 < x < x2 - x0

N

U(x,y)_ Uoy + —

yi

i=1 m=1

sin (am y0j ) sinh (amxo) m cosh (amx1 )

x cosh (am (x1 - x2 + x)) sin (am y) ;

x2 - xo ^ x ^ x2

N œ

U (x,y )_ Uo— + —

y1 n£o t=! m=1

^ 2 ^_v^sin(am yoi) cosh (am(x1 - xo))

cosh (am x1 )

x sinh (am (x2 - x)) sin (am y) ;

(10)

(11)

nm

am _

y1

(12)

4. Results of numerical calculations. In accordance with the obtained analytical solution (8)-(12) of the boundary problem (1), the potential distribution was calculated for different values of x0, which specifies the distance between two emitters.

Figures 2, a—d show the equipotential distributions throughout the emission diode systems for the following parameter values: xi = 30, x2 = 60, yi = 10, N = 10, y0i =

y1 2N' t = —40, t0 = Tn (i = 1,N). All geometric parameters and electrode potentials are given in non-dimensional quantities.

Figure 2. Distribution of equipotentials at xo = 15 (a), xo = 25 (b), xo = 27 (c) and xo = 29 (d)

The height of the emitters L can be calculated from condition U(x0,L) = 0. 5. Conclusion. In this article a two-dimensional emission diode system is modeled. The field cathode is a system of two emitters of uniform shape, located on a flat substrate,

x

x

m

and with a flat anode parallel to the substrate. To calculate the electrostatic potential distribution, the influence of emitters on the electric field is replaced by the influence of charged filaments system. The variables separation method in Cartesian coordinates is used to solve the boundary problem (1)-(3). The potential distribution is found in the form of Fourier expansions with known coefficients throughout the domain of the emission diode system — formulas (8)-(12). According to formulas obtained in explicit form, the equipotential distributions were presented in graphs, showing the dependence of electrostatic field on the distance between the emitters. All the geometric dimensions of the system are the parameters of the problem.

References

1. Forbes R. G. Proposal that interpretation of field emission current—voltage characteristics should be treated as a specialized form of electrical engineering. Journal of Vacuum Science and Technology B, 2023, vol. 41, art. no. 028501.

2. Adhikari B. C., Ketan B., Kim J. S., Yoo S. T., Choi E. H., Park K. C. Beam trajectory analysis of vertically aligned carbon nanotube emitters with a microchannel plate. Nanomaterials, 2022, vol. 12, iss. 23, no. 4313.

3. Egorov N. V., Sheshin E. P. Carbon-based field emitters: properties and applications. Topics in Applied Physics, 2020, vol. 135, pp. 449-528.

4. Mousa M. S., Knapek A., Grmela L. Similarities and differences between two researches in field electron emission: a way to develop a more powerful electron source. Jordan Journal of Physics, 2020, vol. 13, iss. 2, pp. 171-179.

5. Fawaeer S. H., Shatnawi M. T. M., Allaham M. M., Mousa M. S. Influence of polystyrene layer on the field electron emission performance of nano-apex carbon fibre emitters. Advances in Materials and Processing Technologies, 2022, vol. 8, iss. 3, pp. 2652-2671.

6. Filippov S. V., Dall'Agnol F. F., Popov E. O., Kolosko A. G., De Assis T. A. Reaching homogeneous field emission current from clusters of emitters with nonuniform heights. Journal of Vacuum Science and Technology B, 2023, vol. 41, iss. 1, art. no. 010601.

7. Sominskii G. G., Sezonov V. E., Taradaev E. P., Tumareva T.A., Taradaev S. P., Rukavitsy-na A. A., Givargizov M. E., Stepanova A. N. Field emitters for miniature high-voltage electronic devices operating in technical vacuum. Radiophysics and Quantum Electronics, 2019, vol. 62, iss. 7-8, pp. 539-546.

8. Allaham M. M., Knapek A., Mousa M. S., Forbes R. G. User-friendly method for testing field electron emission data: Technical report. 34th International Vacuum Nanoelectronics Conference (IVNC). Lyon, France, 2021, pp. 1-2. https://doi.org/10.1109/IVNC52431.2021.9600769

9. Vinogradova E. M., Egorov E. N., Televnyy D. S. Mathematical modeling of field emitter array. Vacuum, 2016, vol. 127, pp. 45-50.

10. Vinogradova E. M., Egorov N. V., Doronin G. G. Mathematical simulation of a 2D diode system with a blade-shaped field emitter. Technical Physics, 2020, vol. 65, iss. 4, pp. 514-518.

11. Vinogradova E. M., Egorov N. V. Effect of dielectrics on the field emission characterisrics in the diode system modelling. Results in Physics, 2021, vol. 30, art. no. 104822.

12. Filip L. D., Carey J. D., Silva S. R. P. Exact equipotential profile mapping: A self-validating method. Journal of Applied Physics, 2011, vol. 109, art. no. 084527.

Received: September 7, 2022.

Accepted: October 12, 2023.

A u t h o r s' i n fo r m a t i o n:

Nickolay V. Egorov — Dr. Sci. in Physics and Mathematics, Professor; n.v.egorov@spbu.ru

Ekaterina M. Vinogradova — Dr. Sci. in Physics and Mathematics, Professor; e.m.vinogradova@spbu.ru

Galina I. Kurbatova — Dr. Sci. in Physics and Mathematics, Professor; g.kurbatova@spbu.ru

Vladimir A. Klemeshev — PhD in Physics and Mathematics, Associate Professor; v.klemeshev@spbu.ru

Математическое моделирование полевой эмиссионной системы с двумя эмиттерами*

Н. В. Егоров, Е. М. Виноградова, Г. И. Курбатова, В. А. Клемешев Санкт-Петербургский государственный университет,

Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7—9

Для цитирования: Egorov N. V., Vinogradova E. M., Kurbatova G. I., Klemeshev V. A. Field émission system with two emitters mathematical modeling // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2023. Т. 19. Вып. 4. С. 443-448. https://doi.org/10.21638/11701/spbu10.2023.402

Рассмотрена математическая модель двумерной эмиссионной диодной системы с двумя полевыми эмиттерами на плоской подложке, анодом служит плоскость, параллельная плоскости подложки. Согласно предложенной методике, реальный полевой катод совпадает с виртуальным, форма которого определяется нулевой эквипотенциальной поверхностью. Влияние каждого из полевых эмиттеров на распределение электростатического потенциала заменяется влиянием конечного числа заряженных нитей. Решение граничной задачи для уравнения Пуассона найдено в аналитическом виде. Распределение потенциала во всей области исследуемой эмиссионной системы представлено в виде разложений по собственным функциям. Коэффициенты рядов вычислены в явном виде. Приведенные графики демонстрируют зависимость распределения потенциала от расстояния между эмиттерами. Все геометрические размеры системы являются параметрами задачи.

Ключевые слова : микро- и наноэлектроника, полевой эмиттер, математическое моделирование, распределение электростатического потенциала, граничная задача.

Контактная информация:

Егоров Николай Васильевич — д-р физ.-мат. наук, проф.; e-mail: n.v.egorov@spbu.ru Виноградова Екатерина Михайловна — д-р физ.-мат. наук, проф.; e.m.vinogradova@spbu.ru Курбатова Галина Ибрагимовна — д-р физ.-мат. наук, проф.; g.kurbatova@spbu.ru Клемешев Владимир Алексеевич — канд. физ.-мат. наук, доц.; v.klemeshev@spbu.ru

* Исследование выполнено за счет гранта Российского научного фонда № 23-29-00937, https://rscf.ru/project/23-29-00937/