Computer simulation of high-current diode: from field emission to space charge limited current flow
The results of numerical simulation of electron beam dynamics in the high-current diode with a blade-like cathode, which operates in the field emission regime, are presented. The computer simulations were performed by particle-in-cell method in the electrostatic approximation. It is shown that there...
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nasplib_isofts_kiev_ua-123456789-1122302025-02-09T20:45:57Z Computer simulation of high-current diode: from field emission to space charge limited current flow Комп’ютерне моделювання потужнострумового діода: від автоелектронної емісії до обмеження струму просторовим зарядом Компьютерное моделирование сильноточного диода: от автоэлектронной эмиссии к ограничению тока пространственным зарядом Manuilenko, O.V. Karas, V.I. Kornilov, E.A. Vinokurov, V.A. Antipov, V.S. Релятивистская электроника The results of numerical simulation of electron beam dynamics in the high-current diode with a blade-like cathode, which operates in the field emission regime, are presented. The computer simulations were performed by particle-in-cell method in the electrostatic approximation. It is shown that there are two modes of diode operation. In the first operation mode the space charge in the diode gap insignificant. In this case the anode current is small, and its value is determined by the Fowler-Nordheim law. This mode, with increasing potential difference between the cath-ode and the anode, passes in a mode of the space charge limited diode current. In this case the anode current is determined by the Child-Langmuir law. It is shown that the electron energy distribution function at the anode is almost monochromatic, with a maximum, which is determined by the potential difference between the cathode and the anode. The spatial electron distribution function is significantly expanded compared with the thickness of the cathode due to the topography of the electric field and space charge forces. Представлено результати чисельного моделювання динаміки електронного пучка в потужнострумовому діоді з ножовим катодом, який працює в режимі автоелектронної емісії. Чисельні моделювання були виконані методом макрочастинок в електростатичному наближенні. Показано, що існують два режими роботи діода. У першому режимі просторовий заряд у діодному проміжку малий. У цьому випадку анодний струм невеликий, а його величина визначається законом Фаулера-Нордгейма. Цей режим при збільшенні різниці потенціалів між катодом і анодом переходить у режим обмеження струму діода просторовим зарядом. У цьому випадку анодний струм визначається законом Чайлда-Ленгмюра. Показано, що функція розподілу електронів за енергією біля анода практично монохроматична, із максимумом, який визначається різницею потенціалів між катодом і анодом. Функція розподілу електронів у просторі значно розширена порівняно з товщиною катода, що пояснюється топографією електричного поля в досліджуваній системі і дією сил об'є-много заряду. Представлены результаты численного моделирования динамики электронного пучка в сильноточном диоде с ножевым катодом, который работает в режиме автоэлектронной эмиссии. Численные моделирования были выполнены методом макрочастиц в электростатическом приближении. Показано, что существуют два режима работы диода. В первом режиме пространственный заряд в диодном промежутке мал. В этом случае анодный ток невелик, а его величина определяется законом Фаулера-Нордгейма. Этот режим при увеличении разности потенциалов между катодом и анодом переходит в режим ограничения тока диода пространственным зарядом. И тогда анодный ток определяется законом Чайлда-Ленгмюра. Показано, что функция распределения электронов по энергии у анода практически монохроматична, с максимумом, определяемым разностью потенциалов между катодом и анодом. Функция распределения электронов в пространстве значительно уширена по сравнению с толщиной катода, что объясняется топографией электрического поля в исследуемой системе и действием сил объемного заряда. 2015 Article Computer simulation of high-current diode: from field emission to space charge limited current flow / O.V. Manuilenko, V.I. Karas, E.A. Kornilov, V.A. Vinokurov, V.S. Antipov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 81-85. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 41.75.-i, 52.40.Mj, 52.58.Hm, 52.59.-f, 52.65.Rr https://nasplib.isofts.kiev.ua/handle/123456789/112230 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Релятивистская электроника Релятивистская электроника |
| spellingShingle |
Релятивистская электроника Релятивистская электроника Manuilenko, O.V. Karas, V.I. Kornilov, E.A. Vinokurov, V.A. Antipov, V.S. Computer simulation of high-current diode: from field emission to space charge limited current flow Вопросы атомной науки и техники |
| description |
The results of numerical simulation of electron beam dynamics in the high-current diode with a blade-like cathode, which operates in the field emission regime, are presented. The computer simulations were performed by particle-in-cell method in the electrostatic approximation. It is shown that there are two modes of diode operation. In the first operation mode the space charge in the diode gap insignificant. In this case the anode current is small, and its value is determined by the Fowler-Nordheim law. This mode, with increasing potential difference between the cath-ode and the anode, passes in a mode of the space charge limited diode current. In this case the anode current is determined by the Child-Langmuir law. It is shown that the electron energy distribution function at the anode is almost monochromatic, with a maximum, which is determined by the potential difference between the cathode and the anode. The spatial electron distribution function is significantly expanded compared with the thickness of the cathode due to the topography of the electric field and space charge forces. |
| format |
Article |
| author |
Manuilenko, O.V. Karas, V.I. Kornilov, E.A. Vinokurov, V.A. Antipov, V.S. |
| author_facet |
Manuilenko, O.V. Karas, V.I. Kornilov, E.A. Vinokurov, V.A. Antipov, V.S. |
| author_sort |
Manuilenko, O.V. |
| title |
Computer simulation of high-current diode: from field emission to space charge limited current flow |
| title_short |
Computer simulation of high-current diode: from field emission to space charge limited current flow |
| title_full |
Computer simulation of high-current diode: from field emission to space charge limited current flow |
| title_fullStr |
Computer simulation of high-current diode: from field emission to space charge limited current flow |
| title_full_unstemmed |
Computer simulation of high-current diode: from field emission to space charge limited current flow |
| title_sort |
computer simulation of high-current diode: from field emission to space charge limited current flow |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Релятивистская электроника |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112230 |
| citation_txt |
Computer simulation of high-current diode: from field emission to space charge limited current flow / O.V. Manuilenko, V.I. Karas, E.A. Kornilov, V.A. Vinokurov, V.S. Antipov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 81-85. — Бібліогр.: 15 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 81
COMPUTER SIMULATION OF HIGH-CURRENT DIODE: FROM FIELD
EMISSION TO SPACE CHARGE LIMITED CURRENT FLOW
O.V. Manuilenko, V.I. Karas’, E.A. Kornilov, V.A. Vinokurov, V.S. Antipov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: ovm@kipt.kharkov.ua
The results of numerical simulation of electron beam dynamics in the high-current diode with a blade-like cath-
ode, which operates in the field emission regime, are presented. The computer simulations were performed by parti-
cle-in-cell method in the electrostatic approximation. It is shown that there are two modes of diode operation. In the
first operation mode the space charge in the diode gap insignificant. In this case the anode current is small, and its
value is determined by the Fowler-Nordheim law. This mode, with increasing potential difference between the cath-
ode and the anode, passes in a mode of the space charge limited diode current. In this case the anode current is de-
termined by the Child-Langmuir law. It is shown that the electron energy distribution function at the anode is almost
monochromatic, with a maximum, which is determined by the potential difference between the cathode and the an-
ode. The spatial electron distribution function is significantly expanded compared with the thickness of the cathode
due to the topography of the electric field and space charge forces.
PACS: 41.75.-i, 52.40.Mj, 52.58.Hm, 52.59.-f, 52.65.Rr
INTRODUCTION
The usage of linear induction accelerators (LIA) for
obtaining high-current ion beams (HCIBs) with the pa-
rameters required for different applications – from
heavy-ion fusion up to surface modification and radia-
tion materials science – is perspective, as LIA can oper-
ate at high pulse frequency, and can accelerate HCIBs
of virtually any ions, and perform time compression of
current pulse in the acceleration process, which elimi-
nates the operations related to the increase of current
due to compression rings. Collective focusing tech-
niques can significantly increase the ion beam current.
In such kind of LIA, the ion beam space charge is com-
pensated by electrons, and the electron current is sup-
pressed by magnetic insulation of accelerating gaps [1,
2]. The mechanism for charge and current neutralization
of an HCIB by an electron beam in an axisymmetric
accelerating gap was investigated in [2 - 4]. The trans-
portation, acceleration and stability of compensated ion
beam in the 1-6 cusps were studied in [5 - 7]. The trans-
portation and acceleration of the neutralized high-
current ion bunch with additional compensation of the
accelerated ion bunch space charge by thermal electrons
were studied in [8, 9].
The charge and current neutralization of the HCIB
by the electron beam requires electron beams with ener-
gies up to 10 MeV and currents up to 20 kA. The energy
and current of the electron beam should be controlled
independently. These currents can be generated by a
high-current diode with a knife-like cathode which
works in the regime of the field- or explosive emission.
The diode shape for HCIB neutralization in LIA is dif-
ferent from the usual. It consists of a thin ring, which is
a blade-like cathode (Fig. 3). Cylindrical anode having a
radius less than the inner radius of the cathode is insert-
ed into the cathode. This structure is placed in a cylin-
drical vacuum chamber. According to our experiments,
at applying a negative voltage to the cathode
~ 100…200 kV, in this system the electron beam is
generated with a current of about 5…10 kA. Analytical
theory for the high-current diodes of this shape does not
exist at the moment, and the experimental data are lim-
ited.
Fig. 1. The field emission current density as a function
of the electric field for φW=1 eV − barium oxide,
barium on tungsten substrate,
and φW =4 eV − tungsten, iron. 1=FNβ
Fig. 2. The field emission current density as a function
of the electric field for 1=FNβ and 100=FNβ .
4=Wφ eV
Therefore, the diode was investigated numerically
by particles-in-cell method in the electrostatic approxi-
mation with the XOOPIC code [10]. In the numerical
simulations below, we assume that the electron beam is
generated as a result of field emission, which is defined
by the Fowler-Nordheim law [11, 12]:
( )
( ) ( )
⋅−⋅= − y
Ey
EJ
FN
W
W
FN
FN θ
β
φ
ηφ
β 2/3
9
2
2
6 1083.6exp1054.1 ,
where FNJ is the current density, SI units,
WEy φ/1079.3 5−⋅= ; E is the electric field strength
on the cathode; SI units; ( ) 0.1≈yη , it slightly varies
with y ; ( ) 21 yCy V−=θ − Nordheim function [12], in
the simulations below 0=VC , functions ( )yη and
mailto:ovm@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №4(98) 82
( )yθ are tabulated [13, 14], Wφ − work function, FNβ −
field enhancement factor. The current density FNJ de-
pends strongly on the cathode material, i.e. the work
function Wφ (Fig. 1), and the cathode surface quality,
i.e., the field enhancement factor FNβ (Fig. 2). The field
enhancement factor for conventional cathodes ranges
from 10 to 300 [15].
COMPUTER SIMULATION RESULTS
Fig. 3 shows the axially symmetric simulation domain,
where rA is the outer radius of the anode, rA = 0.5 cm,
rL is the outer radius of the cathode, rL = 2.5 cm, zL
is the length of the simulation domain along the longi-
tudinal axis, zL = 4 cm. The cylindrical blade-like
cathode is located in the middle of the computational
domain. The inner radius of the cathode rC = 1.5 cm, its
thickness δ = 0.2 mm. The negative electrical potential
CV is applied to the cathode, the anode is grounded.
Electrons are injected into the computational domain as
a result of field emission in accordance with the Fowler-
Nordheim law both the face of the cathode and its side
surfaces. The outer boundary of the computational do-
main rLr = is a metal, which is under the cathode
potential CV . The left ( z = 0) and right ( z = zL )
boundaries of the computational domain are dielectrics.
The particles that hit the border of the simulation do-
main are absorbed. The numerical simulations, which
are presented below, performed for FNβ = 100 and CV
from -75 up to -300 kV.
Fig. 3. The simulation domain
Figs. 4, 5 show the distributions of the electric po-
tential ( )zrV , and the electric fields ( )zrEr , in the
computational domain for CV = {-75, -150, -300} kV at
a time that is significantly greater than the electron
transit time through the diode gap, i.e. when the system
"electron beam – diode gap" is in equilibrium. As seen
in Figs. 4, 5, with the growth of the cathode potential,
the space charge is accumulated in the diode gap, which
shields the "vacuum" electric field of the cathode.
Fig. 6 shows the distribution of the electrons in the
configuration space {r, z} for CV . = {-75, -150, -300} kV
at the time when the "electron beam – diode gap" sys-
tem is in equilibrium. The increasing cathode potential
results in an increase of the longitudinal dimension of
the beam at the anode. Fig. 6 also shows that at steady
state electron emission occurs, mainly, with the end of
the cathode, while in the initial time interval, when in
the diode are not a lot of electrons, emission takes place
from the side surfaces of the anode which are close to
the end face of the cathode.
a
b
c
Fig. 4. The electric potential ( )zrV , in the diode:
CV = -75 kV (a); CV = -150 kV(b); CV = -300 kV(c)
a
b
c
Fig. 5. The electric field ( )zrEr , in the diode:
CV = -75 kV (a); CV = -150 kV (b); CV = -300 kV (c)
The bunching of the electron beam, which can be
seen in the Fig. 6,c, can be attributed to over-voltage of
the diode gap, which leads to the explosive growth of
ISSN 1562-6016. ВАНТ. 2015. №4(98) 83
the electron emission from the cathode surface and rapid
screening of the initial "vacuum" electric field on the
surface of the cathode by the space charge of the elec-
tron cloud, further rapid displacement of electrons to-
ward the anode, which is leads to the unscreening of the
cathode electric field, which in turn again leads to an
explosive growth of emission from the cathode surface
and the process repeated.
a
b
c
Fig. 6. The distribution of the electrons
in the configuration space {r, z}: CV = -75 kV (a);
CV = -150 kV (b); CV = -300 kV (c)
Fig. 7, as an example, shows the electron distribu-
tion on the anode surface as function of energy and lon-
gitudinal coordinate at CV = -75 kV. It is seen that the
electron energy distribution function is almost mono-
chromatic, with a maximum which is determined by the
cathode potential CV , in this case ~ 75 keV. This behav-
ior of the electron energy distribution function, depend-
ing on the cathode potential, is maintained for all stud-
ied potentials. The distribution function in the longitu-
dinal direction significantly broadened in comparison
with the thickness of the cathode. This is due to the to-
pography of the electric field at the cathode, and the
influence of the space charge.
Fig. 7. The electron distribution on the anode surface
as function of energy and longitudinal coordinate
Fig. 8 shows, as a function of time, the anode current
)(tI A (left column) and the cathode electric field in the
its center )(tEr (right column) at different applied cath-
ode voltages.
a
b
c
d
e
f
g
h
Fig. 8. The anode current )(tI A (a, c, e, g) and the
cathode electric field )(tEr (b, d, f, h) vs time:
CV = -75 kV (a, b); CV = -102 kV (c, d);
CV = -120 kV (e, f); CV = -300 kV (g, h)
As seen in Fig. 8, regardless of the cathode potential,
after the start of the field emission, electric field at the
cathode surface is reduced due to the occurrence of a
negative space charge in the diode gap, which leads to
the suppression of electron emission. If the applied volt-
age VC = {-75, -102} kV, the decreasing of the electric
field is insignificantly. When the front of emitted elec-
trons reaches the anode, the electric field at the cathode
takes its self consistent steady-state value. The station-
ary value of the cathode electric field is slightly differ-
ent from the minimal field at the cathode. The field at
the cathode is minimized when the front of emitted elec-
trons crosses the plane of the anode. This mode of diode
operation corresponds to the case, when the space
charge in the diode gap is insignificant. The current at
the anode is determined by the Fowler-Nordheim law. If
the applied potential CV = {-120, -300} kV, then after
the front of emitted electrons reaches the anode, the
electric field at the cathode significantly increases as
compared with the minimal electric field at the cathode,
but remains lower than the initial “vacuum” field. This
increase in the cathode electric field, compared to the
minimal cathode field, is due to the fact that the flow of
electrons through the anode is greater than the flow of
ISSN 1562-6016. ВАНТ. 2015. №4(98) 84
injection of electrons into the diode gap due to field
emission. The effect of the space charge on the anode
current is significant in this mode of diode operation.
The diode current in this operation mode is limited by
the space charge.
Fig. 9 shows the maximal current ( )CVImax and the
steady-state/averaged ( )Cave VI current at the anode de-
pending on the applied cathode voltage. The data for
Fig. 9 are taken from the Fig. 8. It can be seen that when
the space charge is not significant ( || CV < 120 kV), the
maximal and steady-state currents are practically the
same. In diode operation mode, when its current is lim-
ited by the space charge ( || CV > 120 kV)
( ) ( )CaveC VIVI >max .
Fig. 9. The ( )CVImax and the ( )Cave VI at the anode
depending on the applied cathode voltage
Fig. 10 shows the steady-state/averaged current den-
sity ( )Cave VJ as a function of the cathode applied volt-
age CV . In addition, Fig. 10 shows an approximation of
the ( )Cave VJ by the Fowler-Nordheim law and the
Child-Langmuir law. As can be seen from the Fig. 9,
when || CV < 120 kV, the ( )Cave VJ is quite well ap-
proximated by the Fowler-Nordheim law, otherwise
( || CV > 120 kV) the ( )Cave VJ is quite well approximat-
ed by the Child-Langmuir law.
Fig. 10. The averaged current density ( )Cave VJ
as a function of the cathode applied voltage CV
and its approximation by the Fowler-Nordheim
the Child-Langmuir laws
CONCLUSIONS
In this paper we have studied numerically the elec-
tron beam dynamics in the high-current diode with a
blade-like cathode, which operates in the field emission
regime. The computer simulations were performed by
particle-in-cell method in the electrostatic approxima-
tion. It is shown that there are two modes of diode oper-
ation. In the first operation mode the space charge in the
diode gap is insignificant. In this case the anode current
is small, and its value is determined by the Fowler-
Nordheim law. This mode, with increasing potential
difference between the cathode and the anode, passes
into a mode of the space charge limited diode current. In
this case the anode current is determined by the Child-
Langmuir law. It is shown that the electron energy dis-
tribution function at the anode is almost monochromatic,
with a maximum, which is determined by the potential
difference between the cathode and the anode. Spatial
electron distribution function is significantly expanded
compared with the thickness of the cathode due to the
topography of the electric field and space charge forces.
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Article received 25.05.2015
КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ СИЛЬНОТОЧНОГО ДИОДА: ОТ АВТОЭЛЕКТРОННОЙ
ЭМИССИИ К ОГРАНИЧЕНИЮ ТОКА ПРОСТРАНСТВЕННЫМ ЗАРЯДОМ
О.В. Мануйленко, В.И. Карась, Е.А. Корнилов, В.А. Винокуров, В.С. Антипов
Представлены результаты численного моделирования динамики электронного пучка в сильноточном ди-
оде с ножевым катодом, который работает в режиме автоэлектронной эмиссии. Численные моделирования
были выполнены методом макрочастиц в электростатическом приближении. Показано, что существуют два
режима работы диода. В первом режиме пространственный заряд в диодном промежутке мал. В этом случае
анодный ток невелик, а его величина определяется законом Фаулера-Нордгейма. Этот режим при увеличе-
нии разности потенциалов между катодом и анодом переходит в режим ограничения тока диода простран-
ственным зарядом. И тогда анодный ток определяется законом Чайлда-Ленгмюра. Показано, что функция
распределения электронов по энергии у анода практически монохроматична, с максимумом, определяемым
разностью потенциалов между катодом и анодом. Функция распределения электронов в пространстве зна-
чительно уширена по сравнению с толщиной катода, что объясняется топографией электрического поля в
исследуемой системе и действием сил объемного заряда.
КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ ПОТУЖНОСТРУМОВОГО ДІОДА: ВІД АВТОЕЛЕКТРОННОЇ
ЕМІСІЇ ДО ОБМЕЖЕННЯ СТРУМУ ПРОСТОРОВИМ ЗАРЯДОМ
О.В. Мануйленко, В.І. Карась, Є.О. Корнілов, В.О. Вінокуров, В.С. Антіпов
Представлено результати чисельного моделювання динаміки електронного пучка в потужнострумовому
діоді з ножовим катодом, який працює в режимі автоелектронної емісії. Чисельні моделювання були вико-
нані методом макрочастинок в електростатичному наближенні. Показано, що існують два режими роботи
діода. У першому режимі просторовий заряд у діодному проміжку малий. У цьому випадку анодний струм
невеликий, а його величина визначається законом Фаулера-Нордгейма. Цей режим при збільшенні різниці
потенціалів між катодом і анодом переходить у режим обмеження струму діода просторовим зарядом. У
цьому випадку анодний струм визначається законом Чайлда-Ленгмюра. Показано, що функція розподілу
електронів за енергією біля анода практично монохроматична, із максимумом, який визначається різницею
потенціалів між катодом і анодом. Функція розподілу електронів у просторі значно розширена порівняно з
товщиною катода, що пояснюється топографією електричного поля в досліджуваній системі і дією сил об'є-
много заряду.
COMPUTER SIMULATION RESULTS
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