Theoretical analysis of field electron emission must consider many parameters, one of the most critical being the field enhancement factor (FEF). In a single tip form, the FEF can vary several orders of magnitude and depends only on the system geometry, when the gap length between the emitter and counter-electrode is much greater than the height of the emitter. In this work, we determine very accurate analytical expressions for the FEF of five emitters with various shapes, which are often considered in the literature: Ellipsoidal, Hemisphere-on-Cylindrical post, Hemisphere-on-Orthogonal cone, Paraboloidal, and Hyperboloidal. We map the FEF as a function of the aspect ratio with an error smaller than 2% to serve as a quick reference database. Additionally, we calculate the electric field distribution over the emitters, which can give an insight into the effective notional emission area and the influence of the emitter’s base.
REFERENCES
1.
H.
Dai
, J.
Hafner
, A.
Rinzler
, D. T.
Colbert
, and R. E.
Smalley
, Nature
384
, 147
(1996
). 2.
S. K.
Kanth
, A.
Sharma
, B. Ch.
Park
, and H. S.
Kim
, J. Vac. Sci. Technol., B
34
, 011805
(2016
). 3.
X.
Shao
, A.
Srinivasan
, W. K.
Ang
, and A.
Khursheed
, Nat. Commun.
9
, 1288
(2018
). 4.
M.
Croci
, I.
Arfaoui
, T.
Stockli
, A.
Chatelain
, and J.-M.
Bonard
, Microelectron. J.
35
, 329
(2004
). 5.
A. N.
Obraztsov
and V. I.
Kleshch
, J. Nanoelectron. Optoelectron.
4
, 207
(2009
). 6.
Y.
Ohkawa
, T.
Okumura
, K.
Iki
, H.
Okamoto
, and S.
Kawamoto
, J. Vac. Sci. Technol., B
37
, 022203
(2019
). 7.
N.
Yamamoto
, T.
Morita
, Y.
Ohkawa
, M.
Nakano
, and I.
Funaki
, J. Propul. Power
35
, 490
(2019
). 8.
D.
Li
, Y.
Cheng
, Y.
Wang
, H.
Zhang
, C.
Dong
, and D.
Li
, Appl. Surf. Sci.
365
, 10
(2016
). 9.
N. N.
Rudyk
, O. I.
Il’in
, M. V.
Il’ina
, A. A.
Fedotov
, V. S.
Klimin
, and O. A.
Ageev
, J. Phys. Conf. Ser.
917
, 082008
(2017
). 10.
G. S.
Bocharov
and A. V.
Eletskii
, Tech. Phys.
50
, 944
(2005
). 11.
J.
Bieker
, R. G.
Forbes
, S.
Wilfert
, and H. F.
Schlaak
, IEEE J. Electron Devices Soc.
7
, 997
(2019
). 12.
G. E.
Vibrans
, J. Appl. Phys.
35
, 2855
(1964
). 13.
R. G.
Forbes
, J. Appl. Phys.
120
, 054302
(2016
). 14.
T.
Utsumi
, IEEE Trans. Electron Devices
38
, 2276
(1991
). 15.
G.
Kokkorakis
, C.
Kokkorakis
, A.
Modinos
, and J. P.
Xanthakis
, J. Appl. Phys.
91
, 4580
(2002
). 16.
R. G.
Forbes
, C. J.
Edgcombe
, and U.
Valdrè
, Ultramicroscopy
95
, 57
(2003
). 17.
J.
Wei
, G.
Zhang
, Y.
Xing
, J.
Yang
, and W.
Zhou
, NANO Brief Rep. Rev.
14
, 1950105
(2019
). 18.
S.
Podenok
, M.
Sveningsson
, K.
Hansen
, and E. E. B.
Campbell
, NANO Brief Rep. Rev.
1
, 87
(2006
). 19.
W.
Zeng
, G.
Fang
, N.
Liu
, L.
Yuan
, X.
Yang
, Sh.
Guo
, D.
Wang
, Zh.
Liu
, and X.
Zhao
, Diam. Relat. Mater.
18
, 1381
(2009
). 20.
F. H.
Read
and N. J.
Bowring
, Nucl. Instrum. Methods Phys. Res., Sect. A
519
, 305
(2004
). 21.
A.
Zhbanov
and S.
Yang
, 25th International Vacuum Nanoelectronics Conference
, Jeju, Korea, 9–13 July 2012 (IEEE
, Piscataway, NJ, 2012
), pp. 1
–2
.22.
X. Q.
Wang
, M.
Wang
, Z. H.
Li
, Y. B.
Xu
, and P. M.
He
, Ultramicroscopy
102
, 181
(2005
). 23.
X. Q.
Wang
, M.
Wang
, P. M.
He
, Y. B.
Xu
, and Z. H.
Li
, J. Appl. Phys.
96
, 6752
(2004
). 24.
C. J.
Edgcombe
and U.
Valdrè
, J. Microsc.
203
, 188
(2001
). 25.
C. A.
Spindt
, I.
Brodie
, L.
Humphrey
, and E. R.
Westerberg
, J. Appl. Phys.
47
, 5248
(1976
). 26.
C. A.
Spindt
, J. Appl. Phys.
39
, 3504
(1968
). 27.
R. B.
Marcus
, K. K.
Chin
, Y.
Yuan
, H.
Wang
, and W. N.
Carr
, IEEE Trans. Electron Devices
37
, 1545
(1990
). 28.
W. D.
Keslling
and C. E.
Hunt
, J. Vac. Sci. Technol., B
11
, 518
(1993
). 29.
D. W.
Jenkins
, IEEE Trans. Electron Devices
40
, 3
(1993
). 30.
W. P.
Dyke
and W. W.
Dolan
, Adv. Electron. Electron Phys.
8
, 89
(1956
). 31.
Zh.
Zhang
, G.
Meng
, Q.
Wu
, Zh.
Hu
, J.
Chen
, Q.
Xu
, and F.
Zho
, Sci. Rep.
4
, 4676
(2014
). 32.
F.
Rohrbach
, CERN Report 71-5/TC-L, 1971.33.
R.
Latham
, High Voltage Vacuum Insulation: Basic Concepts and Technological Practice
(Academic
, London
, 1995
).34.
E. G.
Pogorelov
, A. I.
Zhbanov
, and Y. C.
Chang
, Ultramicroscopy
109
, 373
(2009
). 35.
H. G.
Kosmahl
, IEEE Trans. Electron Devices
38
, 1534
(1991
). 36.
D.
Biswas
, Phys. Plasmas
25
, 043113
(2018
). 37.
D.
Biswas
, Phys. Plasmas
26
, 073106
(2019
). 38.
Sh.
Sarkar
and D.
Biswas
, J. Vac. Sci. Technol., B
37
, 062203
(2019
). 39.
S.
Lenk
, C.
Lenk
, and I. W.
Rangelow
, J. Vac. Sci. Technol., B
36
, 06JL01
(2018
). 40.
A.
Knápek
, J.
Sýkora
, J.
Chlumská
, and D.
Sobola
, Microelectron. Eng.
173
, 42
(2017
). 41.
T.
Yamaguchi
, E.
Inami
, Y.
Goto
, Y.
Sakai
, S.
Sasaki
, T.
Ohno
, and T. K.
Yamada
, Rev. Sci. Instrum.
90
, 063701
(2019
). 42.
S.
Li
and R. R.
Jones
, Nat. Commun.
7
, 13405
(2016
). 43.
K. F.
Hii
, R. R.
Vallance
, S. B.
Chikkamaranahalli
, M. P.
Menguc
, and A. M.
Rao
, J. Vac. Sci. Technol., B
24
, 1081
(2006
). 44.
H. S.
Sim
, S. P.
Lau
, L. K.
Ang
, G. F.
You
, M.
Tanemura
, K.
Yamaguchi
, M.
Zamri
, and M.
Yusop
, Appl. Phys. Lett.
93
, 023131
(2008
). 45.
M.
Passacantando
, F.
Bussolotti
, S.
Santucci
, A.
Di Bartolomeo
, F.
Giubileo
, L.
Iemmo
, and A. M.
Cucolo
, Nanotechnology
19
, 395701
(2008
). 46.
J.-M.
Bonard
, K. A.
Dean
, B. F.
Coll
, and C.
Klinke
, Phys. Rev. Lett.
89
, 197602
(2002
). 47.
S.
Sun
and L. K.
Ang
, Phys. Plasmas
19
, 033107
(2012
). 48.
T. A.
de Assis
and F. F.
Dall’Agnol
, J. Vac. Sci. Technol., B
37
, 022902
(2019
). 49.
T. A.
de Assis
and F. F.
Dall’Agnol
, Nanotechnology
27
, 44LT01
(2016
). 50.
M.
Rezeq
, Appl. Surf. Sci.
258
, 1750
(2011
). 51.
G. S.
Bocharov
, A. V.
Eletskii
, D. G.
Kvashnin
, and L. A.
Chernozatonskii
, J. Vac. Sci. Technol., B
33
, 041801
(2015
). 52.
I. L.
Jityaev
and A. M.
Svetlichnyi
, J. Vac. Sci. Technol., B
37
, 012201
(2019
). 53.
G. C.
Kokkorakis
, J. A.
Roumeliotis
, and J. P.
Xanthakis
, J. Appl. Phys.
95
, 1468
(2004
). 54.
Y.
Liu
and Y. Y.
Lau
, J. Vac. Sci. Technol., B
14
, 2126
(1996
). 55.
E. O.
Popov
, A. G.
Kolosko
, M. A.
Chumak
, and S. V.
Filippov
, Tech. Phys.
64
, 1530
(2019
). 56.
T. E.
Stern
, B. S.
Gossling
, and R. H.
Fowler
, Proc. R. Soc. A
124
, 699
(1929
). 57.
See supplementary material at https://doi.org/10.1116/6.0000949 for the raw simulation data of the Field Enhancement Factor.
© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
