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2023 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Т. 19. Вып. 1
ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ
ИНФОРМАТИКА
UDC 51-73, 537.2 MSC 35J05
Blade-like field cathode with a dielectric coating mathematical modeling
N. V. Egorov, E. M. Vinogradova, G. G. Doronin
St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
For citation: Egorov N. V., Vinogradova E. M., Doronin G. G. Blade-like field cathode with a dielectric coating mathematical modeling. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2023, vol. 19, iss. 1, pp. 65-71. https://doi.org/10.21638/11701/spbul0.2023.106
In this paper the results of a two-dimensional diode emission system based on a blade-like field cathode in a polar coordinate system modeling is presented. The top of the emitter is a circle with a dielectric coating. The anode is a circle coaxial to the top of the emitter. The boundary condition of the first kind is set on the cathode, and the first and second kind on the anode. The problem of calculating the electrostatic potential distribution is reduced to solving a system of linear algebraic equations with constant coefficients. All the geometric dimensions of the system and the values of the potentials on the electrodes are the parameters of the problem.
Keywords: micro- and nanoelectronics, field emitter, mathematical modeling, electrostatic potential distribution, boundary-value problem.
1. Introduction. Over the past few decades, the use and modeling of devices with cathodes based on field emitters has significantly increased in vacuum micro- and nanoelectronics. Such emitters, as a rule, have a small radius of curvature at the apex, which differs by several orders of magnitude in comparison with other geometric parameters of the emission system. Due to such advantages high brightness and low power consumer, the field cathodes are widely used in microscopy, lithography, microwave amplifiers, and X-ray generators [1-3]. To increase the life cycle of field cathodes, the emitter coatings are studied using various materials, including dielectrics [4-9]. To increase the emission current in these systems, so-called blade-like field cathodes are used, which have a much larger emission area compared to single emitters [10].
This work is devoted to the two-dimensional diode emission system with a blade-like field cathode mathematical modeling in a polar coordinate system (r,<f) (Figure). The
© St. Petersburg State University, 2023
Figure. Schematic representation of the diode field emission system
emitter's top is a circle with a dielectric coating. The anode is a circle coaxial to the emitter top.
2. Mathematical model. The problem parameters: r = R1 (p E [—n, n]) — emitter's top surface; r = R3 (p E [p0, n — p0]) — anode surface; r = R3 (p E [—p0, p0]) — emitter foundation surface; r = R2 (p E [—n, n]) — boundary between two dielectrics with the dielectric permittivities e1 and e2\ V1 — emitter's top boundary condition;
dU(r, (p) = Q - emitter foundation boundary condition; V2 — anode boundary
r=Rs
dr
condition.
The electrostatic potential distribution U(r, p) is a solution of the Laplace's equation for the boundary-value problem:
ld_
r dr
dU_ dr
1 d2U _Q r2 d?2
(i)
and boundary conditions
( U(Ri,p) = Vi, U (R3 ,p) = V2,
du (r,?)
dr
r=Rs
= Q,
£1-
du (r,?)
dr
r=R2-0
= £2-
dU(r, ?)
dr
r = R2 +0
? E [-п, , ? E [?0, П - ?0 ] ,
? E [ ?0, ?0] , ? E [-п, n] .
(2)
Due to the fact that the potential distribution function U(r, p) for the entire region (r E [R1, R3], p E [—n, n]) is symmetric with respect to planes p = 0 and p = n, the problem (1), (2) can be solved for (r E [R1, R3], p E [0, n]).
3. Solution of the problem. To solve the boundary-value problem (1), (2) the diode system internal area (r E [R1, R3] , p E [0, n]) can be divide into following subdomains:
(1)-(r e [Ri,R2] , V e [0, vo])
(2)-(r e [R2, R3], v e [0, V0])
(3)-(r e [Ri, R2], v e [vo,n])
(4)-(r e [R2, R3], v e [vo,n]).
Then
Ui(r,V), r e [Ri, R2], V e [0, Vo],
TT(rm)i U2(r,V), r e [R2, R3], V e [0, Vo],
U (r,V) < U3(r,v), r e [Ri, R2], V e [vo, n], w
U4(r, V), r e [R2, R3] , V e [V0, n] •
Let Vi(r) = QR3— In (r/RJ + Vu
£1
V2{r) = QR3 [\n{r/R2) + F^\n{R2/Rl)^ +VU
Vs(r) = £-1£ M-VJMr/Ri) +Vu (4)
£i ^ln(R2/R1)+ln(R3/R2)
£1
V ( \ №~Vi)ln (r/R3)
Vi{r) = TTo--1" V2-
-^ln(R2/R1)+ln(R3/R2)
£1
The functions V1 (r), V2(r), V3(r), V4(r) (4) obey the Laplace's equation and the boundary-conditions (2) for (i)-th subdomain accordingly.
The potential distribution (3) Ui(r,V) (i = 1, 2, 3, 4) with the variable separation method can be represented for (i)-th subdomain in the form [11, 12]:
~ „_„) 1 + ^^ •
U^r, <p) = Vi(r) + V ane-x^J sin (A„ In (r/i?i)) +
n=1
, ,1 - (r/Ri)-2^m .
+ Y^ Cm (r/R2f- i _ (д1/д2)2мт COS(Mmy),
(5)
1 I p — 2Vkф
U2(r,<p) = V2(r) + ]P bke-^-rt 1 sm К In (г/Д2)) +
k=0
00
(6)
U3(r, v) = V3(r)+J2
(
, 2( —1)" e2 an + —-—
£1
\
(
T2 Tl -дд3||х
^1п(Д2/Д1)+1п(Дз/Д2)
£1
1 + e—2xn(n—V) (7)
x ysm(Anln(r/i?1))+ 1 j + {r/Raf cosM- " ¥>)),
U4(r, V) = V4(r) + Yat ^ sin (& In (r/R2)) +
t=i
1 + e-2it(n-Vo)
+ £ dp (r/R2)
p=0
2Vp
1 - (r/Rs)
1 - (R2/R3)2Vp
cos(r/p(n - ?)),
where
^n —
ln(R2/RiY 2?0
(2p + 1) n
_ (2m + 1) 7Г _ (2k + 1) 7Г
Mm — ~ ' 7 IS к —
2In(R3/R2Y
Пр
2(n - ?0)'
et
nt
(9)
In(Rs/R2)'
The functions (5)-(9) Ui(r, p) (i = 1, 2, 3, 4) as the potential distribution (3) U(r, p), written in the form of series in eigenfunctions, satisfies boundary conditions (2) on the cathode and anode surfaces. These formulas provide the next potential distribution continuity conditions:
Ui(R2,p) = U2R2,p), p E [0, po];
Ui(r,po) = U3(r,po), r E [Ri, R2];
U3(R2,p) = U4(R2,p), p E [po,n];
U2(r,po) = UA(r,po), r E [R2, R3].
To calculate the unknown coefficients an, bk, cm, d,p, gt the additional potential distribution continuity conditions can be used:
U2(r, ?0 ) — U4 (r, ?0 ),
£1
dUi(r, ?)
dr
— £2
dU2(r, ?)
r = R2
dr
r=R2
dUi(r, ?)
dU3(r, ?)
£1
d?
dU(r, ?)
д?
dr dU2(r, ?)
: £2
V=V0
dU4(r, ?)
r = R2
dr
r=R2
d?
dU4(r, ?)
dr
r E [R2, R3], ? E [0, ?0] ,
r E [Ri, R2],
? E [?0, n],
r E [R2, R3] .
(10)
The continuity conditions (10) and the eigenfunctions systems orthogonality lead to the linear algebraic equations system with respect to unknowns coefficients an, bk, cm, dp, gt:
t=i
Ьк-]п(Д3/Д2)-(-1)*Е> 2
2 t-1 Vk Çt
2(-1)n £2
x -J--
£1
T2~Tl -QRsI,
— In (Д2/Rl) + In (Д3/R2)
\ £1
00
(11)
(12)
"к V£l Vk J \-1п(Д2/Д1) + 1п(Д3/Д2) ,
£1
0 ! 1 \ с 0
(-!)" E a™ -WT^r - (-!)m- E Vrr- +
n=1 Xn + Hm £1 k=0 Vk2 + H2m
+ Cmf ^coth (Mm (Д2/Д1)) + ^ coth (Mm (Д3/Д2)) ) = 0,
1 ° (-l)m и
- In (Д2/Д1) ( th (Xnfo) + th (A„(7T - <р0))) + (-1)" E л 2 1 2 +
2 m=0 An + цт
° (-l)P 'П
p=0 лп + 'ip I1«};
/
(-l)f E On + ^ iCOth (Д2/Д1)) + - COth (Дз/Д2))) -
n=i ЛП + Vp 2 \ £1 J
00 . 00
£ £t 1
£2 X —
£1
t=1 « ^ 'IP n=1 'П T 4p
-2-^^--Qifel,
-1п(Д2/Д1) + 1п(Д3/Д2)
£1
a
n
- E Cm 2^7 2 +&fc|ln(fi3/fi2) th(z/fcy>0)
m=0
nm+v2 ' ~k2
+ yd __Lil!___(is)
p=0 P ^ Sh (77р(1П(Д3/Д2))) ^ +
4. Conclusion. This article presents the results of mathematical modeling the two-dimensional diode emission system with a blade-like field cathode system. The emitter top
has a dielectric coating. To solve the boundary problem (1), (2) the variable separation method in polar coordinates was used. The electrostatic potential distribution is presented in terms of eigenfunctions (3)-(9), and the definition of unknown coefficients in potential expansions is reduced to solving a linear algebraic equations system (11)—(15). All the geometric dimensions of the system and the values of the potentials on the electrodes are the parameters of the problem.
References
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2. Rughoobur G., Ilori О. O., Akinwande A. I. Scanning anode field emission microscopy of a single Si emitter. Journal of Vacuum Science and Technology B, 2022, vol. 40, iss. 4, no. 042803.
3. Sankaran K. J., Kurian J., Sundaravel В., Lin I.-N., Haenen K. Diamond-gold nanohybrids — an enhanced cathode material for field electron emitter applications. Journal of Physics D: Applied Physics, 2021, vol. 54, iss. 5, no. 053002.
4. Bernatskii D., Pavlov V. Creating graphene and graphite coatings on metal field electron emitters to improve their characteristics. Journal of Physics: Conference Series, 2020, vol. 1697, iss. 1, no. 012245.
5. A1 Soud A., A1 Buqain R. N., Mousa M. S. Composite metallic nanoemitters coated with a layer of insulator covered by Au layer. Jordan Journal of Physics, 2020, vol. 13, iss. 3, pp. 253-262.
6. Sominskii G., Tumareva Т., Taradaev E. Protective coatings for tip field emitters operating at high voltages. 2016 Hth International Baltic Conference on Atomic Layer Deposition, BALD 2016 — Proceedings, 2017, no. 7886540, pp. 66-67.
7. Sominskii G. G., Tumareva T. A., Taradaev E. P., Rukavitsina A. A., Givargizov M. E., Stepano-va A. N. Annular multi-tip field emitters with metal-fullerene protective coatings. Technical Physics, 2019, vol. 64, iss. 2, pp. 270-273.
8. Sominskii G. G., Sezonov V. E., Taradaev S. P., Vdovichev S. N. Multilayer field emitters made of contacting hafnium and platinum nanolayers. Technical Physics, 2019, vol. 64, iss. 1, pp. 116-120.
9. 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, no. 104822.
10. Filippov S. V., Dall'Agnol F. F., de Assis T. A., Popov E. O., Kolosko A. G. Properties of blade-like field emitters. Ultramicroscopy, 2022, vol. 233, no. 113462.
11. Vinogradova E. M., Egorov N. V., Starikova A. V., Varayun M. I. Calculating a multipole cylindrical electrostatic system. Technical Physics, 2017, vol. 87, iss. 5, pp. 791-794.
12. Kurbatova G. I., Vinogradova E. M. Analiticheskiye i chislennyye resheniya dvumernykh zadach teploprovodnosti i elektronnoy optiki [Analytical and numerical solutions of two-dimensional problems of heat conduction and electronic optics]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Sciences. Control Processes, 2021, vol. 17, iss. 4, pp. 345-352. https://doi.org/10.21638/11701/spbul0.2021.403 (In Russian)
Received: December 12, 2022.
Accepted: January 19, 2023.
Authors' information:
Nickolay V. Egorov — Dr. Sci. in Physics and Mathematics, Professor; robby7@mail.ru
Ekaterina M. Vinogradova — Dr. Sci. in Physics and Mathematics, Professor; e.m.vinogradova@spbu.ru
Grigoriy G. Doronin - Postgraduate Student; fiuctum@gmail.com
Математическое моделирование полевого катода лезвийной формы с диэлектрическим покрытием
Н. В. Егоров, Е. М. Виноградова, Г. Г. Доронин
Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7-9
Для цитирования: Egorov N. V., Vinogradova Е. М., Doronin G. G. Blade-like field cathode with a dielectric coating mathematical modeling // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2023. Т. 19. Вып. 1. С. 65-71. https://doi.org/10.21638/11701/spbul0.2023.106
В работе представлены результаты моделирования двумерной диодной эмиссионной системы на основе полевого катода лезвийной формы в полярной системе координат. Вершина эмиттера представляет собой окружность, на которую нанесено диэлектрическое покрытие. Анод — окружность, коаксиальная вершине эмиттера. На катоде задано граничное условие первого рода, на аноде — первого и второго рода. Задача вычисления — распределение электростатического потенциала — сведена к решению системы линейных алгебраических уравнений с постоянными коэффициентами. Все геометрические размеры системы и потенциалы на электродах 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.; robby7@mail.ru
Виноградова Екатерина Михайловна — д-р физ.-мат. наук, проф.; e.m.vinogradova@spbu.ru
Доронин Григорий Геннадьевич — аспирант; fluctum@gmail.com
