Numerical investigation of the ion temperature effects on magnetized DC plasma sheath

Numerical investigation of the ion temperature effects on magnetized DCplasma sheath

Mansour Khoramabadi,1 Hamid Ghomi,2,a) and Padma Kant Shukla3

1Department of Physics, Boroujerd Branch, Islamic Azad University, Boroujerd, Iran2Laser and Plasma Research Institute, Shahid Beheshti University, Evin 1983963113, Tehran, Iran3Institut fur Theoretische Physik, Fakultat fur Physik und Astronomie, Ruhr-Universitat Bochum, D-44780Bochum, Germany

(Received 18 October 2010; accepted 24 February 2011; published online 6 April 2011)

The properties of a magnetized plasma sheath with finite ion temperature is considered. The effects

of the external magnetic field and the ion temperature on the sheath parameters are examined. It is

found that by increasing the ion temperature and the magnetic field strength there is an increase in

the positive space charge and the ion energy and a decrease in the sheath thickness. Furthermore,

the ion temperature has a direct effect on the ion flux toward the wall. VC 2011 American Institute ofPhysics. [doi:10.1063/1.3569844]

I. INTRODUCTION

The subject of interactions between the plasma and a

solid wall and the formation of a boundary layer around it is

almost as old as plasma physics itself. The problem of a

magnetized plasma boundary has a particular importance in

material processing by the plasma and the plasma confine-

ment in magnetic fusion devices. The importance of the

plasma boundary layer in an external magnetic field has

drawn much attention in recent years. Nonetheless, the

physics of the plasma boundary is still not perfectly under-

stood. In a simple model, a confined plasma is brought into

contact with a conducting wall. Since the mobility of the

electrons is much higher than that of the ions, the wall is

bombarded by the background electrons and attains a nega-

tive potential with respective to the bulk of the plasma. The

negative potential around the plasma attracts the ions toward

the wall and repels the electrons toward the plasma. In the

steady state, a positive space charge forms near the wall to

balance the flow of the ions and electrons into the wall. The

plasma boundary layer is thus divided into two separate

regions: a neutral presheath adjacent to the plasma and a pos-

itive charged sheath adjacent to the wall. The sheath width is

usually a few Debye radius and its task is to shield the

plasma from the wall, while the presheath width is deter-

mined by the size of the plasma container or by the ion mean

free path1 and its function is to fulfill the well known Bohm

criterion. The latter is needed in order to adjust the positive

space charge associated with the ions that are entering into

the sheath.2,3 Usually, the sheath is collisionless and planar

and its width is small in comparison with the presheath.

Since the sheath and presheath regions have different charac-

teristics and functions, it is obvious to use different scales in

studying the plasma boundary layer. According to the two-

scales theory, the sheath region is scaled by the electron

Debye radius, while the presheath is scaled by the size of the

plasma. Thus, we can solve the plasma boundary problem in

these two-scales separately. On the sheath scale the pre-

sheath is infinitely far from the wall where the electric field

is zero. On the presheath scale, the sheath is infinitesimally

thin and the electric there field is infinite. As can be seen the

sheath-presheath interface, the so- called sheath edge, has an

ambiguous behavior and one has to remove the contradicting

behavior while solving the matching problem.4–7 Riemann,8

Franklin,9 and Zimmermann et al.10 investigated the effect

of collisions on the boundary layer of a magnetized plasma

on the presheath scale. They deduced that the magnetic field

along with collisions do not modify the Bohm criterion for

the sheath formation. On the basis of this issue, several

works investigated the plasma boundary layer on the sheath

scale by using the same Bohm criterion as in the unmagne-

tized plasma.11–14 In our previous work, we obtained a gen-

eralized Bohm criterion in a collisional plasma with the ion

temperature effect on the sheath scale.15 Recently, it has

been shown that the ion temperature strongly affects the

sheath properties (such as the ion energy and the density).

Emmert et al.16 presented a formalism for the ion thermal

effect on the electrostatic potential in the bulk of the plasma

and in the sheath near an adsorbing plane wall. They

obtained an integrodifferential equation for the potential in

both the plasma and sheath regions and numerically solved

it. They concluded that an increase of the ion temperature

leads to a slow increase of the sheath width and to a decrease

of the potential at the sheath edge. Here, in a sequel to the

earlier works by Minghao et al.,17 and on the basis of the

two-fluid model, the characteristics of a magnetized DC

plasma sheath are numerically investigated and the effect of

the ion temperature on the sheath properties is examined.

Specifically, our study is concerned with the effects of a con-

stant magnetic field and the ion temperature on the sheath

behavior. In order to understand the main physical features

determining the stationary sheath, the analysis has been re-

stricted to the charged boundary layer (sheath scale), and

processes in the bulk of the plasma and presheath are not

considered. In Sec. I, nonlinear equations in a simple plasma

sheath model are formulated. In Sec. III, numerical results

and the corresponding discussions are presented, displaying

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0021-8979/2011/109(7)/073307/8/$30.00 VC 2011 American Institute of Physics109, 073307-1

JOURNAL OF APPLIED PHYSICS 109, 073307 (2011)

Downloaded 10 Apr 2011 to 217.218.83.61. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

http://dx.doi.org/10.1063/1.3569844

http://dx.doi.org/10.1063/1.3569844

http://dx.doi.org/10.1063/1.3569844

the role of the magnetic field in the formation of the thermal

sheath. Section IV contains a brief summary and conclusions.

II. MODEL AND BASIC EQUATIONS

Here we introduce a magnetized plasma sheath with the

finite ion temperature to examine its properties in the steady

state. If a planar target with a large negative potential is

immersed in this plasma, a planar (one-dimensional) sheath

(dark space) is formed around it. We choose the x-axis nor-

mal to the target surface, and the position of the plasma-

sheath interface at x ¼ 0. In x < 0, there is a quasineutral

plasma with u � 0. The plasma is composed of electrons

and singly charged ions, which are considered as fluids. In

the region x > 0, there is a non-neutral sheath. Indeed, it is

assumed that the ions, which are generated in the bulk of the

plasma (x < 0), are accelerated in a presheath by a weak

electric field toward the sheath, and enter into the sheath

with a nonzero velocity (the Bohm criterion). In the steady

state (@=@t ¼ 0), for a collisionless sheath, the ion fluid

equations are

r � ðniviÞ ¼ 0; (1)

minivi � rvi ¼ eniðEþ vi � BÞ � rpi; (2)

where vi is the average ion flow velocity, e is the magnitude

of the electron charge, E ¼ �ru (with u as the electrostatic

potential), and B defines the electric and magnetic fields,

respectively; mi and ni are the ion mass and the ion number

density, respectively. These equations are identical with

those in Ref. 12. Assuming ion isothermal flow, the ion pres-

sure pi is

pi ¼ kBTini; (3)

where kB is the Boltzmann constant and Ti is the ion temper-

ature. The electrons are in thermal equilibrium and obey the

Boltzmann density distribution

ne ¼ ne0 expeu

kBTe

� �; (4)

where ne0 denotes the electron number density at the sheath

edge and Te is the electron temperature. Finally, by using

Poisson’s equation, which relates the electron and ion num-

ber densities to the electric potential

r2u ¼ � e

e0

ðni � neÞ; (5)

we can close our governing equations. It is assumed that the

ion-neutral collisions indirectly allow the ions to behave like

an isothermal warm fluid. Since there are no variations in the

sheath parameters along the direction parallel to the wall and

the spatial variations are only in the normal to wall direction,

we then consider the geometry of the problem in one-space

dimension and assume that r ! xd=dx. Therefore, from

Eqs. (1) and (5) we have

nitix ¼ ni0tix0 (6)

d2/dx2¼ � e

e0

ðni � neÞ (7)

with ni0 and tix0 as the ion number density and the x-compo-

nent of the average ion velocity at the sheath edge. Accord-

ing to the general criterion for the stationary sheath

formation in the fluid approximation,1,10,15 the isothermal

ions have to enter the sheath region with the normal to wall

initial velocity defined by

tix0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBTe

mi1þ Ti

Te

� �s� cs: (8)

FIG. 1. Schematic configuration of the magnetized sheath in the assumed

model. The magnetic field vector is in the x� z plane and it makes the hangle with the x� axis. The y-axis is pointing into the paper.

FIG. 2. The normalized ion Ni ¼ ni=ne0 (solid line) and electron

Ne ¼ ne=ne0 (dashed line) densities as a function of the normalized distance

from the sheath edge n ¼ x=kD for g0 ¼ 0, g00 ¼ 0:01, Ma ¼ffiffiffiffiffiffiffiffiffiffiffiffi1þ Tp

,

h ¼ p=10; ðaÞ b ¼ 0, and (b) b ¼ 2.

073307-2 Khoramabadi, Ghomi, and Kant Shukla J. Appl. Phys. 109, 073307 (2011)


Conversely, referring to the geometry sketched in Fig. 1 and

assuming that B ¼ B0ðx cos hþ z sin hÞ, one can obtain from

Eq. (2)

tixdvix

dx¼ � e

mi

d/dxþ eB0

mitiy sin h� kBTi

mi

1

ni

dni

dx; (9)

tixdtiy

dx¼ eB0

miðtiz cos h� tix sin hÞ; (10)

tixdtiz

dx¼ � eB0

mitiy cos h: (11)

For convenience, we introduce the dimensionless variables,

Ni ¼ ni=ne0, Ne ¼ ne=ne0, T ¼ Ti=Te, n ¼ x=kD, qi ¼ css,

xc ¼ eB0=mi ¼ 1=s, uj ¼ tij=cs ðj ¼ x; y; and zÞ,kD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie0kBTe=ne0e2

p, b ¼ kD=qi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie0=mine0

pB0, cs

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBTe=mi

p, g ¼ �eu=kBTe. Hence, with Eqs. (4) and (6)–

(11), together with the charge neutrality condition in the

sheath edge ne0 ¼ ni0, we have

Ni ¼Ma

ux; (12)

Ne ¼ expð�gÞ; (13)

ux0 ¼dux

dn¼ ux

u2x � T

dgdnþ buy sin h

� �; (14)

uy0 ¼duy

dn¼ b

uz

uxcos h� sin h

� �; (15)

uz0 ¼duz

dn¼ �b

uy

uxcos h; (16)

g00 ¼ d2g

dn2¼ Ni � Ne; (17)

where cs is the ion-acoustic velocity, Ma ¼ tix0=cs is the ion

Mach number, kD is the electron Debye length at the sheath

edge, s is the ion gyroperiod, qi is the ion gyroradius, and nis the distance from the sheath edge normalized by kD. b is

the parameter that indicates the magnetic field magnitude.

III. NUMERICAL RESULTS AND DISCUSSION

To solve the equations of the ion motion and Poisson’s

equation, we use gð0Þ ¼ 0, g0ð0Þ ¼ 0:01, uxð0Þ ¼ Ma

¼ffiffiffiffiffiffiffiffiffiffiffiffi1þ Tp

, uyð0Þ ¼ uzð0Þ ¼ 0, and Nið0Þ ¼ Neð0Þ ¼ 1 as

boundary conditions at the plasma-sheath interface (n ¼ 0).

As shown, the Bohm velocity at the sheath edge is a function

of T,8–10,15 and is independent of B in weak magnetic fields.

To examine the sheath structure, it is appropriate to investi-

gate the profiles of the particle density, the electric potential,

FIG. 3. The normalized ion Ni (solid line) and electron Ne (dashed line)

densities as a function of the normalized distance from the sheath edge n for

g0 ¼ 0, g00 ¼ 0:01, Ma ¼ffiffiffiffiffiffiffiffiffiffiffiffi1þ Tp

, b ¼ 1, (a) h ¼ 0, and (b) h ¼ p=12.

FIG. 4. The normalized space charge Ni � Ne across the sheath as a function

of the normalized distance from the sheath edge n. The plasma parameters

and initial conditions are the same as in Fig. 2; (a) b ¼ 0 and (b) b ¼ 2.



and the ion velocity. For this purpose, by using a Runge-

Kutta method of the fourth-order, Eqs. (12)–(17) are solved

and the numerical results are discussed in the next

subsections.

A. Ion temperature effect on particle density

The numerical solutions of Eqs. (12)–(17) are displayed

in Figs. 2–11. In Figs. 2 and 3 the spatial variations of the

normalized ion and electron densities have been illustrated

for different values of b (b ¼ 0; 2), h (h ¼ 0; p=12), and TðT ¼ 0; 0:2; 0:4; and 0:6Þ across the sheath. These figures

exhibit that increasing the magnitude of an oblique magnetic

field ðbÞ for constant h (Fig. 2), or increasing the magnetic

field angle with the normal to the wall ðhÞ for constant b(Fig. 3), increases the aggregation of both kinds of particles

near the sheath edge and increasing the ion temperature

slightly intensifies this effect. All electron densities (dashed

line) decrease smoothly according to the Boltzmann relation.

The ion distribution, however, strongly depends upon the

magnetic field strength and for a critical value it begins to

fluctuate, but the sheath formation condition ðNi � NeÞ is

compliant across the sheath. Also the spatial variations of the

difference between the ion and electron densities are

depicted in Figs. 4 and 5. From these figures, we see that

ðNi � NeÞ � 0 across the sheath and a positive space charge

is formed near the wall. According to these figures, it is evi-

dent that an increase in b and h values gives rise to an

enhancement of the fluctuation rate of the positive space

charge and shifts the location of the charge accumulation to

the sheath edge. Both Figs. 4 and 5 exhibit that an increase

in the ion temperature leads to a growth of the positive space

charge and a transmission of the positive space charge to-

ward the sheath edge. From Figs. 4 and 5 we see that there is

no difference between the space charge profile in the absence

of a magnetic field b ¼ 0 and the space charge profile in the

presence of a normal to wall magnetic field ðh ¼ 0Þ. We con-

clude that only the transverse component of the magnetic

field ðbT ¼ b sin hÞ plays the original role and the normal to

wall magnetic field component ðbN ¼ b cos hÞ does not have

any influence on the spatial variation of the sheath

parameters.

B. Electric potential and ion velocity

From the ion continuity equation we know that the xcomponent of the ion velocity is inversely proportional to

FIG. 5. The normalized space charge Ni � Ne across the sheath as a function

of the distance from the sheath edge n. The plasma parameters and initial

conditions are the same as in Fig. 3; (a) h ¼ 0 and (b) h ¼ p=12.

FIG. 6. The normalized x-component of the ion velocity ux ¼ vix=cs across

the sheath as a function of the distance from the sheath edge n. The plasma

parameters and initial conditions are the same as in Fig. 2; (a) b ¼ 0 and

(b) b ¼ 2.



the ion density. Therefore, From Figs. 2 and 3, we expect

that ux increases across the sheath and oscillates with an

increase in the transverse component of the magnetic field

ðbTÞ. In Fig. 6, the spatial variations of the x component of

the normalized ion velocity is shown for different values of

bðb ¼ 0; 0:2Þ and TðT ¼ 0; 0:2; 0:4; and 0:6Þ across the

sheath. The x component of the initial velocity of the ions is

a function of T (according to the boundary conditions). From

Poisson’s equation it is well known that the space charge cre-

ates the electric field. The positive space charge in the sheath

region creates an electric field toward the wall, which is

associated with a decreasing electric potential (increasing

normalized electric potential) from the sheath edge toward

the wall. According to Figs. 4 and 5, an increase in bT or Tgives rise to an increase in the space charge and shifts it to-

ward the sheath edge. Thus, both effects increase the normal-

ized electric potential ðgÞ across the sheath. Figure 7

apparently displays the increase of the normalized electric

potential with the increase of b (or h) or T. It shows that the

sheath width (the distance between the sheath edge and the

wall with a constant electric potential) descends with an

increase in the ion temperature or the transverse component

of the magnetic field in a constant wall potential. Note that

the wall location is characterized by its electric potential.

From this figure, it is clear that in contrast with the space

charge distribution, there is no fluctuation in the electric

potential distribution. Indeed, the fluctuation on the space

charge distribution results in the fluctuation in the electric

field and then in the x-component of the ion velocity distri-

bution. Alternatively, on the basis of the energy conservation

law, the electric potential energy of the ions in the sheath is

equal totheir kinetic energy (the ion speed) in the absence of

collisions. So the electric potential and the ion speed distri-

butions across the sheath are proportional to each other.

Figure 8 illustrates the ion speed distribution u (magnitude

of the ion velocity) across the sheath for different values of band T. As expected, the distribution of the ion speed and the

electric potential are smoothly increasing functions. Accord-

ing to Figs. 2–8, the presence of an oblique magnetic field

strongly affects the ion dynamics in a collisionless magne-

tized sheath. Indeed the transverse component (parallel to

the wall) of the magnetic field allows the ions to gyrate and

FIG. 7. The normalized electric potential g ¼ �eu=kBTe across

the sheath as a function of the distance from the sheath edge n.

The plasma parameters and initial conditions are the same as in

Fig. 2; (a) b ¼ 0 and (b) b ¼ 2.



consequently, the ion density and all components of the ion

velocity fluctuate.

C. Wall net current

In the sheath region, there is an increasing normalized

electric potential, a decreasing electron number density

according to the Boltzmann relation, along with a descend-

ing electron current to the wall. One can then define the net

current to the wall as

jn ¼ ji � je ¼ evixni �1

4ecene; (18)

where ce ¼ ð8kBTe=pmeÞ1=2is the electron thermal speed

and me the electron mass. Normalizing the net current by

j0 ¼ ene0cs one obtains from Eq. (18)

Jn ¼ Ji � Je ¼ uxNi �mi

2pme

� �1=2

Ne: (19)

Figure 9 displays the normalized net current of the wall

across the sheath for mi ¼ 40mH (considering argon atoms as

ionized gas) with mH as the hydrogen atom mass. This figure

shows that by increasing the wall electric potential and wid-

ening the sheath width, the net current to the wall rises from

a large negative value in the sheath edge and is saturated to a

positive value. Additionally, in this figure the characteristic

curve, I � V, has been illustrated on a smaller scale to clarify

the saturation current and floating potential (a wall electric

potential in which the wall net current becomes zero). From

Fig. 9 it is apparent that increasing the transverse component

of the magnetic field decreases the sheath width in the float-

ing potential and it does not have any effect on the floating

potential and saturation net current. We deduce from this fig-

ure that increasing the ion temperature decreases the floating

potential and raises the saturation net current to the wall.

D. Transverse drift of ions

In contrast to the presheath,9 in the sheath both the mag-

netic and electric forces play an important role. Figures 6

and 8 show that the ion speed ðuÞ is a smoothly increasing

function despite the fact that the x� component of the

FIG. 8. The normalized ion speed u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

x þ u2y þ u2

z

qacross the sheath as a

function of the distance from the sheath edge n. The plasma parameters and

initial conditions are the same as in Fig. 2; (a) b ¼ 0 and (b) b ¼ 2.

FIG. 9. The normalized net current of the wall across the sheath as a func-

tion of the distance from the sheath edge n. The plasma parameters and ini-

tial conditions are the same as in Fig. 2; (a) b ¼ 0 and (b) b ¼ 2.



ion velocity fluctuates across the sheath. It means that the

y� and z� components of the ion velocity have to fluctuate

as well. Indeed, the transverse component of the ion velocity

ðuT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

y þ u2z

qÞ fluctuates inversely with its longitudinal

component ðuxÞ. The z� component of the ion velocity is

shown in Fig. 10 as a function of ux for different values of hand T. This figure is of particular significance, since it shows

the net direction of the ion motion in the x� z plane. This

direction is close to the magnetic field direction, but the tur-

bulent motion of the ion fluid has another drifting motion

along the E� B direction. The y�component of the ion ve-

locity as a function of ux has been plotted in Fig. 11, showing

the net direction of the ion motion in the x� y plane. The

motion of the ions along the y axis is the well-known drifting

motion of E� B in the plasma and increases with the

increase of h (the angle between E and B). Figures 10 and 11

show that an increase in the ion temperature only increases

the longitudinal component of the ion velocity and does not

have any effect on its transverse component. It is clear from

these figures that the ion temperature, in contrast to the trans-

verse component of the magnetic field, increases the ion lon-

gitudinal motion (normal to the wall).

IV. SUMMARY AND CONCLUSION

In this paper, we have presented a general description of

a magnetized plasma sheath composed of warm positive ions

and Boltzmann distributed inertia-less electrons. Based on

the two-fluid model, the ion and electron densities, the ion

velocity, the wall net current, and the electric potential have

been investigated across the sheath for different values of the

ion temperature and the transverse component of the mag-

netic field. It is found that the normal component of the mag-

netic field does not have any effect on the sheath parameters.

However, the transverse component of the magnetic field

causes the positive charge accumulation toward the sheath

edge and the ion temperature further enhances this effect. It

is found that by increasing the ion temperature and the

strength of the oblique magnetic field in the sheath, the posi-

tive space charge increases. Accordingly, the ion tempera-

ture and the transverse component of the magnetic field

decrease the sheath width. Furthermore, our numerical

results reveal that the higher plasma temperature increases

the net current to the wall. We note that the external oblique

magnetic field does not have any effect on the wall saturation

current and the floating potential. Moreover, the magnetic

FIG. 10. The normalized z component of the ion velocity uz as a function of

the normalized x-component of the ion velocity ux. The plasma parameters

and initial conditions are the same as in Fig. 3; (a) h ¼ p=12 and

(b) h ¼ p=8.

FIG. 11. The normalized y component of the ion velocity uy as a function of

the normalized x component of the ion velocity ux. The plasma parameters

and initial conditions at the sheath edge are the same as in Fig. 3;

(a) h ¼ p=12 and (b) h ¼ p=8.



field increases the transverse drifting motion of the ions and

the ion temperature enhances their longitudinal drifting

motion. The present results have applications in plasma proc-

essing; for example, in ion implantation using magnetized

electropositive plasmas. Specifically, we note that an

increase of the ion temperature would increase the penetra-

tion depth into the target by increasing the ion’s energy and

their longitudinal velocity. Also, in magnetron sputtering,

the ion temperature increases the rate of the coating by rais-

ing the ion flux to the target.

1K. U. Riemann, J. Phys. D: Appl. Phys. 24, 493 (1991).2L. Tonks and I. Langmuir, Phys. Rev. 34, 876 (1929).3D. Bohm, Characteristics of Electrical Discharges in Magnetic Fields(McGraw-Hill, New York, 1949).

4K. U. Riemann, Phys. Plasmas 4, 4158 (1997).

5K. U. Riemann, Plasma Phys. Controlled Fusion 47, 1949 (2005).6R. N. Franklin, J. Phys. D: Appl. Phys. 31, 2532 (1998).7R. N. Franklin, J. Phys. D: Appl. Phys. 36, R309 (2003).8K. U. Riemann, Phys. Plasmas 1, 552 (1994).9R. N. Franklin, J. Phys. D: Appl. Phys. 38, 3412 (2005).

10T. M. G. Zimmermann, M. Coppins, and J. E. Allen, Phys. Plasmas 15,

072301 (2008).11S. F. Masoudi, J. Phys. D: Appl. Phys. 40, 6641 (2007).12Z. Xiu, L. Y. Jin, G. Ye, W. X. Zheng, L. Yue., and W. G. Xiao, Vacuum

73, 681 (2004).13K. Yasserian, M Aslaninejad, M. Ghoranneviss, and F. M. Aghamir,

J. Phys. D: Appl. Phys. 41, 105215 (7pp) (2008).14M. M. Hatami, B. Shokri, and A. R. Niknam, J. Phys. D: Appl. Phys. 42,

025204 (2009).15H. Ghomi, and M. Khoramabadi, J Plasma Phys. 76, 247 (2010).16G. A. Emmert, R. M. Wieland, A. T. Mense, and J. N. Davidson, Phys.

Fluids 23, 803 (1980).17L. Minghao, Z. Yu, D. Wanyu, L. Jinyuan, and W. Xiaogang, Plasma Sci.

Technol. 8, 544 (2006).



http://dx.doi.org/10.1088/0022-3727/24/4/001

http://dx.doi.org/10.1103/PhysRev.34.876

http://dx.doi.org/10.1063/1.872536

http://dx.doi.org/10.1088/0741-3335/47/11/006

http://dx.doi.org/10.1088/0022-3727/31/19/026

http://dx.doi.org/10.1088/0022-3727/36/22/R01

http://dx.doi.org/10.1063/1.870800

http://dx.doi.org/10.1088/0022-3727/38/18/014

http://dx.doi.org/10.1063/1.2946436

http://dx.doi.org/10.1088/0022-3727/40/21/024

http://dx.doi.org/10.1016/j.vacuum.2003.12.099

http://dx.doi.org/10.1088/0022-3727/41/10/105215

http://dx.doi.org/10.1088/0022-3727/42/2/025204

http://dx.doi.org/10.1017/S0022377809990304

http://dx.doi.org/10.1063/1.863062

http://dx.doi.org/10.1063/1.863062

http://dx.doi.org/10.1088/1009-0630/8/5/10

http://dx.doi.org/10.1088/1009-0630/8/5/10

Numerical investigation of the ion temperature effects on magnetized DC plasma sheath

Documents

Transcript of Numerical investigation of the ion temperature effects on magnetized DC plasma sheath