J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

2-D PIC Simulation on Electron Sheath Formation in Magnetized Plasmas
Kazuhito ASANO1, Shin KAJITA2, Dai OIKAWA3 and Noriyasu OHNO3
1

Japan NUS Co., Ltd., 15-9 Kaigan 3, Minato, Tokyo, 108-0022, Japan,

2

EcoTopia Science Institute of Nagoya University, Furo-Chou, Chikusa, Nagoya, 464-8603, Japan,

3

Department of Energy Engineering and Science, Graduate School of Engineering, Nagoya University, Furo-Chou,Chikusa, Nagoya, 464-8603, Japan(Received: 29 October 2009 / Accepted: 5 March 2010)

In order to analyze electron sheath formation we performed two dimensional PIC (Particle In Cell) simulations
in magnetized plasmas. We investigated the influence of the probe geometry on the electron sheath formation interms of the aspect ratio of the plane probe. The dependence of the magnetic field and voltage bias of the probe on
the electron sheath were also taken into consideration. The sheath thickness was also analyzed to compare with thetheoretical formula described by the Child Langmuir (CL) law and the experimental observations. We found thatalteration of the plane probe geometry from rectangular to square decreased sheath thickness, resulting in a close tocylindrical CL sheath, rather than a planar one. These results demonstrate that the behavior of electrons surrounding
the probe is directly affected by the geometry of the probe. Further, it is found that the sheath thickness isdetermined by the ratio between the probe width and the Larmor radius of the electron.
Keywords: electron sheath, Particle In Cell, Child Langmuir law, magnetized plasmas, angular momentum ofelectron, cylindrical effectprediction of electron sheath thickness.

1. Introduction

There have been various studies about the influenceof a magnetic field on ion sheath formation, bothexperimentally and numerically [1]. There are few studies,
however, on the influence of the magnetic field on theelectron sheath formation. The study reported here
provides a detailed analysis of the electron sheathformation around the electrostatic probe biased positively
with respect to plasma potential.
The thickness of the electron sheath that forms
around a planar or cylindrical probe was measured
experimentally in weakly magnetized plasmas, using thelaser photodetachment (LPD) technique [2,3]. Thesestudies show that the electron sheath thickness is sensitive
to the magnetic field as well as the geometry of the probe.
In order to perform a detailed analysis on theinfluence of the magnetic field and the geometry of theprobe on the electron sheath formation, we employed PIC
simulation code, called XOOPIC [4]. XOOPIC supports
two dimensional space, slab and cylindrical geometries
and has electrostatic and electromagnetic solvers. PIC
simulation determines the electric and magnetic fields
self-consistently for various boundary conditions andplays a key role in the analysis and understanding ofelectron sheath formation.
In this paper we report a detailed analysis of the twodimensional electron sheath structure formed around aprobe using a PIC simulation, where the electrostaticsolver is applied. We further evaluate and discuss thevalidity of the Child-Langmuir (CL) formula for accurate

2. Simulation setup

Ly

Figure 1 shows a schematic view of the simulation
setup. The plane probe, biased positively, is located at thecenter of the region along the magnetic field. Eachboundary is fixed at 0 V and hydrogen plasma is injectedfrom boundary with half-Maxwellian velocity distribution.
Electron and ion density, ne and ni are set to be 1017m-3 andLx 䋽 Ly

Ly䋯2yxPlane probet = 2㰱Thickness

LpLength2㰱䌾30㰱

Magnetic field

䌖p
Bias voltage

㰱䋮䋮䋮

B=B x

ne =ni=1017 m-3
Te=1eV,
Ti=0.2eV㰱䋽10㱘m䋮䋮䋮
㰱H plasma

0Lx䋯2

㱗d =23.5㱘m

Lx

Variable ParameterBx

1 䌾 100mT

Vp30 䌾 120V

Lp䋺t

30:2 䌾 2:2

Fig.1 Simulation model and plasma parameters.

author’s e-mail:asano@janus.co.jp563©2010 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchK. Asano et al., 2-D PIC Simulation on Electron Sheath Formation in Magnetized Plasmas

60.00

1/ 4Ly47.95

hp

3H § 2e(V p  Vsp ) ·¸2 0¨¸j ¨©
me
3
¹hc



(2)

2e
(V p  Vsp ) 3 / 2 ,me(3)where Vsp is the space potential, e is the elementarycharge and E f (rp  hc ) / rp , rp is the probe tip radius.

35.90
Ly/2

The f is a function as radius given numerically [5]. Theseequations indicate that the CL sheath in the cylindricalgeometry has stronger dependence on the electron current

h㼠

23.85

11.80

㪈

0Lx/2

㪪㪿㪼㪸㫋㪿㩷㫋㪿㫀㪺㫂㫅㪼㫊㫊㪲㫄㫄㪴-0.24

8SH 09 jE 2, andLx[V]Fig.2 A potential profile and the schematic image of
calculated sheath thickness in B=1 mT,
Vp=60V and Lp:t=30:2. The steady state
potential profile is averaged over 0.7Ǵs㨪1.1
Ǵs or 2000fpe-1㨪3142fpe-1, where fpe is theelectron plasma frequency.

hp㪇㪅㪈㪇㪅㪇㪈hc㪇㪅㪇㪇㪈㪉㪇㪋㪇㪍㪇㪪㪿㪼㪸㫋㪿㩷㫋㪿㫀㪺㫂㫅㪼㫊㫊㪲㫄㫄㪴㪏㪇㪈㪇㪇㪈㪉㪇㪈㪋㪇㪭㫇㪲㪭㪴㪈

temperature, Te and Ti are 1eV and 0.2eV respectively.
Debye length, ǳd is calculated to be 23.5Ǵm. The
length of the simulation region, L, is large enough not toaffect the sheath. The validity of L is confirmed bychecking potential profile, which is approximately 0V in aregion that is far from electrode. This is achieved at a
distance of L > 1mm.
To investigate the effect of the probe geometry on
sheath thickness, various aspect ratios are assessed byadjusting the width of the plane probe as shown in Fig.1.
The static magnetic field applied in the x direction and thebias voltage of the plane probe Vp were varied from 1 mT
to 100 mT and 30 V to 120 V, respectively.
A steady state potential profile can be obtained asshown in Fig.2. The sheath thickness, hӆ, is evaluated byusing this potential profile as follows. The potential of thesheath edge is defined such that, Vedge = Vp / exp, whereexp is an base of natural logarithm. We are able todetermine the position, whose potential corresponds toVedge for vertical direction, in the magnetic field at thecenter of probe as, y(Vedge)x=Lx/2. Subsequently h ӆ is
calculated using following formula.
(1)hA y(Vp )x L / 2  y(Vedge)x L / 2(b) Lp䋺t = 2:2hp

㪇㪅㪈㪇㪅㪇㪈hc㪇㪅㪇㪇㪈㪉㪇㪋㪇㪍㪇

㪏㪇㪈㪇㪇㪈㪉㪇

㪈㪋㪇

㪭㫇㪲㪭㪴

Fig.3 The thickness of the electron sheath formed
around the plane probe. (a) corresponds toLp:t=30:2 (rectangle electrode) and (b)corresponds Lp:t=2:2 (square electrode), in
B=15mT for each case. Lp is length ofelectrode for x direction and t is thickness for ydirection.
㪈㪙㪔㪈㪌㫄㪫㩷㪈㪇㪇㫄㪫㪿㼠䋯㪿㫇㪇㪅㪏㪇㪅㪍㪇㪅㪋㪇㪅㪉㪇㪈㪇㪇㪊㪇䋺㪉(a)h㼠䋯hp㪉㪈䋺㪉㪈㪌䋺㪉㪣㫇䋺㫋㪐䋺㪉㪌䋺㪉㪉䋺㪉㪐䋺㪉㪌䋺㪉㪉䋺㪉

㪏㪇x

㪿㼠䋯㪿㪺

x(a) Lp䋺t = 30:2

Figure 2 shows an example of the calculation ofsheath thickness, where Vedge is calculated to be 60/exp
㨪22V where Vp = 60 V.

㪍㪇㪋㪇㪉㪇㪇㪊㪇䋺㪉

3. Simulation results

Theoretically, the electron sheath thickness obeys theCL law such that the thickness of the CL sheath formed in
front of a planar electrode, hp, or that of a the cylindricalelectrode, hc, can be written as follows [5],(b)h㼠䋯hc㪉㪈䋺㪉㪈㪌䋺㪉㪣㫇䋺㫋Fig.4 Ratio of sheath thickness to (a) plane CLsheath and (b) cylindrical CL sheath atdifferent aspect ratio. Vp is set to be 60V.

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K. Asano et al., 2-D PIC Simulation on Electron Sheath Formation in Magnetized Plasmas

density j than CL sheath in plane-parallel one depicted in
Fig.1. This means that a cylindrical probe has a greater
capability for collection of an electron current.
Figure 3 shows the sheath thickness at different bias
voltages of the probe in B=15mT. It can be seen, from
Fig.3 (a), that the sheath thickness calculated bysimulation is approximately equal to hp under a highaspect ratio, where Lp:t=30:2. On the other hand, as
shown in Fig.3(b), the sheath thickness is close to hc undera low aspect ratio where, Lp:t=2:2. This clearly indicatesthat the geometry of the plane probe significantly
influences the behavior of electrons around the probe
resulting in a change in the potential profile around theprobe which leads to a decrease in the sheath thickness.
This also indicates that a low aspect ratio has more
electron collective effect than rectangle one as well as
cylindrical geometry.
The ratio of sheath thickness to CL sheath, atdifferent aspect ratios is analyzed as shown in Fig.4. This
figure shows that sheath thickness in weakly magnetizedplasma has stronger dependence on the aspect ratio thanstrongly magnetized. The sheath thickness decreases as
the aspect ratio decreases in weakly magnetized plasmaand close to CL sheath in the cylindrical geometry.

(a) High aspect ratio-㪉㪚㫌㫉㫉㪼㫅㫋㩷㪻㪼㫅㫊㫀㫋㫐㩷㪲㫂㪘㪆㫄 㪴㪇㪄㪈㪇㪄㪉㪇㪄㪊㪇㪄㪋㪇㪚㫌㫉㫉㪼㫅㫋㩷㪻㪼㫅㫊㫀㫋㫐㩷㪲㫂㪘㪆㫄㪉 㪴㪇㪄㪈㪇Ly/2㪊㪇㪋㪇㪌㪇㪍㪇㪙㪲㫄㪫㪴㪎㪇㪏㪇㪐㪇㪈㪇㪇㪋㪇㪌㪇㪍㪇㪙㪲㫄㪫㪴㪎㪇㪏㪇㪐㪇㪈㪇㪇(b) Lp䋺t = 2:2㪈㪇㪉㪇㪊㪇angular momentum of the electron results in an increase inelectron density around the probe, which in turn shields
the electric field, due to a decreased incidence ofcollisions between electron and the probe. In order tocharacterize the effect of the shielding effect due toangular momentum of the electron, the current density j,with different magnetic fields and bias voltage wereplotted in Fig.7. The aspect ratio corresponds to Lp:t=30:2
in Fig.7(a) and 2:2 in Fig.7(b), respectively. It wasrevealed that the current density increases negatively asthe aspect ratio decreases. This indicates that the collectiveability of the electron current or the cylindrical effect isdominant in a low aspect ratio. This cylindrical effectmakes the sheath thickness smaller than the CL sheath inplane parallel geometry as predicted by CL sheath incylindrical one, shown in Eq. (2). As can be seen, acylindrical effect, due to the effect of angular momentum

0ion

Ly/200㪉㪇Fig.7 Current density of electron to the probe atdifferent aspect ratio, (a) high aspect ratio and (b)low aspect ratio.

electronLy㪈㪇㪄㪉㪇㪄㪊㪇㪄㪋㪇㪄㪌㪇㪄㪍㪇㪄㪎㪇㪄㪏㪇㪇(b) Lp:t = 2:2

㪭㫇㪔㪊㪇㪭㩷㩷㩷㩷㩷㪍㪇㪭㩷㩷㩷㩷㩷㪐㪇㪭
㩷㩷㩷㩷㩷㪈㪉㪇㪭(a) Lp䋺t = 30:2

㪄㪌㪇㪄㪍㪇㪄㪎㪇㪄㪏㪇㪇

4.1 Geometric effect of plane probe
Figure 5 shows the distribution of electron and ion.
This figure suggests that the electrons are condensed, incase of low aspect ratio and shield the electric field. In Fig.
5(b), the turn of electron around probe is observed due toits angular momentum. Fig.6 shows a schematic drawingof the effect of the angular momentum of electron. TheLy-Fig.6 Schematic view of geometric effect. (a): electron
vanishes after incidence of collision withelectrode. (b): electron turns around electrodedue to its angular momentum rather thanincidence of collision. This is caused by electricfield around electrode and angular momentumof electron.

4. Discussion(a) Lp:t = 30:2(b) Low aspect ratio

Lx/2

Lx 0

Lx/2

Lx

Fig.5 Distribution of electron and ion to show theeffect of aspect ratio of plane probe, Lp:t is (a)30:2 and (b) 2:2, respectively. The magnetic
field is set to be 1mT.

565

K. Asano et al., 2-D PIC Simulation on Electron Sheath Formation in Magnetized Plasmasof electrons, is produced on the plane probe by decreasing
the aspect ratio.

B
Stronglymagnetized

4.2 Effect of magnetic field on sheath formation
Figure 8 shows the ratio of sheath thickness to CL
sheath in plane parallel geometry at different magnetic
fields and bias voltages. In Fig.8 (a), using a high aspectratio, at the sheath thickness peaks around B=30mT.
Conversely, as in the case of a low aspect ratio, no peak isobserved as shown in Fig.8 (b).
The physical reason for these differential
dependencies of the magnetic field on the sheath thickness
could be effected by the considering the relationshipbetween the probe width and the Larmor radius of theelectron. In order to understand this phenomenon, we
define the following index, Le.
(4)Le  v ! TC / 2(a) <v>㬍Tc(B)䋯2 䋼 Lp-(b) <v>㬍Tc(B)䋯2 䌾 Lp-

Weaklymagnetized

2Sme ,eB

8kTe,TSme C

v!

where kTe is the electron temperature in J. <v> is arandom velocity of the electron and , Tc-1 is a cyclotron
frequency. Le means the distance an electron moves in halfa Larmor motion. The relation between Le and Lp, oraspect ratio, characterizes sheath formation.
Figure 9 shows the schematic view of the magnetic㪿㼠䋯㪿㫇(c) <v>㬍Tc(B)䋯2 㻦 Lp(b) B=100mT(a) B=15mT㪭㫇㪔㪊㪇㪭
㩷㩷㩷㩷㩷㪍㪇㪭
㩷㩷㩷㩷㩷㪐㪇㪭
㩷㩷㩷㩷㩷㪈㪉㪇㪭㪇㪅㪏

Lp

Fig.9 The schematic view of the magnetic field effectwhen the Larmor radius is (a) very small, (b)comparable with the probe width and (c) largeenough.

(a) Lp䋺t = 30:2,Lp=0.3mm
㪈㪅㪇-LyBelectron㪇㪅㪍㪇㪅㪋<v>㬍Tc(B)䋯2 䋾 Lp㪇㪅㪉

Ly/2<v>㬍Tc(B)䋯2 䋼 Lp㪇㪅㪇㪇㪈㪇㪉㪇㪊㪇㪋㪇㪌㪇㪍㪇㪎㪇㪏㪇㪐㪇㪈㪇㪇㪙㪲㫄㪫㪴(b) Lp䋺t = 2:2,Lp=0.02mm

Ly㪈㪅㪇ion㪿㼠䋯㪿㫇

㪇㪅㪏<v>㬍Tc(B)䋯2 䋾 Lp㪇㪅㪍

Ly/2㪇㪅㪋㪇㪅㪉㪇㪅㪇㪇㪈㪇㪉㪇㪊㪇㪋㪇㪌㪇

㪍㪇

㪎㪇

㪏㪇

㪐㪇

㪈㪇㪇

0㪙㪲㫄㪫㪴

Lx/2

Lx 0

Lx/2

Lx

Fig.8 The ratio of sheath thickness to CL sheath inplane parallel geometry at different magnetic
field, (a) high aspect ratio and (b) low aspectratio. <v> is a random velocity of the electron, Tcis a cyclotron period.

Fig.10 Distribution of electron and ion to show theeffect of the magnetic field. In (a) B=15mT, andin (b) B=100mT. The bias voltage and the aspectratio are set to be 60V and Lp:t=30:2,
respectively.

field effect on the Larmor motion of electrons todemonstrate the dependency of sheath formation on theLarmor motion of an electron. Three cases are considered.
Case (a) where Larmor radius is small (Le < Lp) and Case(b) Le㨪Lp and Case (c) Le > Lp. In Case (c), an electron

could likely cross the probe due to the larger Larmor
radius. In Case (b), Le is comparable to Lp, thus electrons
would have a tendency to collide before crossing theprobe. These behaviors are slightly different from eachother but an impact on the sheath formation is comparable

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K. Asano et al., 2-D PIC Simulation on Electron Sheath Formation in Magnetized Plasmas

in both cases. Finally, in Case (a), an electron can moveclose to the probe resulting in a potential profile aroundthe probe that decreases as the Larmor radius decreases, orthe magnetic field increases. In case of the high aspectratio shown in Fig.8 (a), Le is comparable with Lp in B㨪
40mT. On the other hand, Le is always greater than Lp incase of the low aspect ratio as shown in Fig.8 (b). These
results agree with the relation between Larmor radius andprobe width shown in Fig.9.
Figure 10 illustrates the distribution of electrons andions at different magnetic fields. This clearly shows thatthe magnetic field influences the spatial distribution ofelectrons and ion.

4. Conclusion

We performed two dimensional PIC simulations inmagnetized plasmas to analyze electron sheath formation.
The effect of the geometry of the plane probe as well as
the magnetic field of the probe on the electron sheathformation was analyzed.
The results revealed that changing the geometry of
the plane probe from rectangular to square resulted in adecreases in sheath thickness and a close to CL sheath in
the cylindrical geometry. Furthermore, the simulation
results showed that the angular momentum of an electron,in case of low aspect plane probe, contributes to shield
the electric field and decrease sheath thickness. This
indicates that a cylindrical effect is dominates in a low
aspect ratio plane probe.
Moreover, as illustrated in Fig.8, the effect ofmagnetic field on sheath formation was shown to dependon the relative relationship between the probe width andthe Larmor radius of the electron.

Acknowledgements

One of the authors (K.A) wishes to thank to Mr. D. Atkinsof Japan NUS for useful comments.

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For example, P.C Stangeby, The Plasma Boundary of
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S. Kajita, S. Kado, A. Okamoto, and S. Tanaka, Phys. Rev.
E 70, 066403 (2004).
S. Kajita, S. Kado, and S. Tanaka, Contrib. Plasma Phys.
46, 373 (2006).
J.P. Verboncoeur, A.B. Langdon, N.T. Gladd, Compt. Phys.
Commun. 87, 199 (1995).
I. Langmuir and K.B. Blodgett, Phys. Rev. 22, 347 (1925).

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