Analytic expression for the electric potential in the plasma sheath
T. E. Sheridan and J. Goree
Department of Physics and Astronomy
The University of Iowa, Iowa City, Iowa 52242
An expression for the spatial dependence of the electric potential in a collisionless and source-free planar plasma sheath is presented. This expression is derived in analogy with Child's law and approaches Child's law asymptotically as the potential drop, [[phi]]w, across the sheath becomes large, |e[[phi]]w/kTe| > 104. Here k is Boltzmann's constant, Te is the electron temperature, and e is the electronic charge. Comparison with numerical solutions of the model equations indicate that the sheath thickness and potential variation predicted by this improved Child's law are accurate for |e[[phi]]w/kTe| > 10. In contrast, we find that Child's law is accurate only when |e[[phi]]w/kTe| > 104.
The plasma sheath is the localized electric field that separates a plasma from a material boundary. The plasma sheath serves to confine the more mobile species in the plasma and to accelerate the less mobile species out of the plasma. For the typical case where the electrons are more mobile than the positively charged ions, the electric field in the sheath points toward the boundary. The sheath thickness is parameterized by the Debye length, [[lambda]]D.
In order for the potential in the sheath to decrease monotonically as we move from the plasma towards the boundary, ions must enter the sheath with a velocity at least as large as the ion acoustic velocity, cs . As a consequence of this Bohm sheath criterion, a presheath is formed in the plasma with a potential drop on the order of |kTe/e|, which accelerates the ions into the sheath. Here k is Boltzmann's constant, Te is the electron temperature, and e is the electronic charge. The scale length of the presheath is set by either the mean free path for ions, or the scale length of the plasma, whichever is shorter .
When the potential drop across the sheath, [[phi]], is large compared to |kTe/e|, we can reasonably ignore the small potential drop across the presheath. The potential variation in the sheath is then approximated by Child's law . However, severe approximations must be made to obtain Child's law, and the agreement between exact numerical solutions of Poisson's equation is within one percent only when |e[[phi]]/kTe| > 104.
In this paper we present an analytic expression for the potential variation in the plasma sheath. It is only slightly more complicated than Child's law, but is in much better agreement with exact numerical solutions. The model we use is appropriate for cathodic sheaths which are planar, collisionless, and source-free, and in steady-state. We further assume that the boundary is perfectly absorbing and that the motion of electrons and ions to the boundary is not impeded by magnetic fields.
In the next section we outline the model of the plasma sheath under consideration, and in section III we discuss approximate solutions to this model. The improved expression we derive provides useful information about the applicability of Child's law and the scaling of the sheath thickness with the potential drop across the sheath.
II. Sheath Model
We consider a widely used time-independent model for the potential in a planar plasma sheath, [[phi]], as a function of position, x . One end of the plasma is terminated by a perfectly absorbing wall held at a negative potential, [[phi]]w. (Here, and throughout this paper, the subscript w will refer to the wall.) We choose the position of the wall to be x = 0 (see Fig. 1). Far from the wall there is a field-free and neutral plasma where [[phi]] = 0. The density of electrons, ne, and ions, ni, are both equal to no in the plasma. At some point x = D, where D is the sheath thickness, there is a transition from the non-neutral sheath to the neutral plasma. We assume that the sheath region is collisionless and source-free. Ions enter the sheath as a monoenergetic beam with a velocity uo. In this model uo must be greater than the ion acoustic velocity, cs, in order that [[phi]] increase monotonically as we move away from the wall.
Since the sheath is source free, the ion density obeys an equation of continuity, niv = nouo, where v is the velocity of the ion beam in the sheath region. Further, since the sheath is collisionless, energy is conserved, Mv2 / 2 = Muo2 / 2 + e[[phi]]. Here M is the ion mass. Combining these two relations we find that the ion density in the sheath is given by
The electrons are assumed to be in thermal equilibrium; therefore, the electron density will follow the Boltzmann relation,
The potential must satisfy Poisson's equation,
where [[epsilon]]o is the permittivity constant. This nonlinear second order ordinary differential equation is autonomous, i.e., d2[[phi]]/dx2 does not depend explicitly on x. To completely specify the problem we need a value for uo and boundary conditions for [[phi]]. Appropriate boundary conditions are [[phi]](0) = [[phi]]w, and [[phi]](x->[[infinity]]) = 0, as shown in Fig. 1.
Poisson's equation can be non-dimensionalized by the following transformations:
Here [[eta]] is the dimensionless potential (note the sign of [[eta]] is opposite that of [[phi]]), [[xi]] is distance normalized by the Debye length, and M is the Mach number. The dimensionless Poisson's equation for the potential variation in the sheath is
where [[eta]][[second]] is the second derivative of [[eta]] with respect to [[xi]]. The first term on the right hand side is the dimensionless ion density and the second term is the dimensionless electron density. The boundary conditions are [[eta]](0) = [[eta]]w, and [[eta]]([[xi]]->[[infinity]]) = 0. The Mach number must be specified. For this model the Bohm criterion requires that M > 1.
After multiplying by [[eta]][[minute]], Eq. (5) can be integrated once to give
where [[eta]][[minute]], the dimensionless electric field, is negative since [[eta]] is positive at the wall and falls to zero in the plasma. We have incorporated the conditions that [[eta]]([[xi]]->[[infinity]]) = 0 and [[eta]][[minute]]([[xi]]->[[infinity]]) = 0. Because we must be given [[eta]]w and M in order to completely specify a solution, both [[eta]][[minute]] and [[eta]][[second]], which are given by Eqs. (6) and (5), can be evaluated at the wall. In fact, all derivatives of [[eta]] can be evaluated at the wall. However, derivatives higher than first order all go to zero at the wall as [[eta]]w becomes large.
When might we expect this model to be a good description of the sheath? First, we have assumed the sheath is collisionless. This is a good assumption when the mean free path for ion collisions is much longer the the sheath thickness. Second, we have ignored the presheath. We know from solutions to the plasma equation  that the maximum potential drop across the presheath is [[eta]] = 0.8539. This suggests that the presheath can be treated in a simplified manner for [[eta]] >> 1. The potential in the plasma region is constant (there is no pre-sheath) when there is no source near the sheath , and no error is incurred by neglecting the presheath. Third, it was assumed that the ion distribution entering the sheath is monoenergetic. The distribution of ion energies for ions born with zero energy is known to be very sharply peaked as the ions enter the sheath . Fourth, we have assumed that the electron density obeys the Boltzmann relation. Self  has argued that this assumption has a negligible effect on the results. Fifth, we have assumed that there is no impediment (e.g., a magnetic field parallel to the wall) to the free flow of electrons and ions to the wall. Finally, we have assumed that the wall potential is time independent. The model will still hold for time-varying wall potentials provided the oscillation frequency of the wall potential is less than the ion plasma frequency .
The remainder of this paper considers approximate solutions to the sheath model presented above.
III. Approximate Analytic Solution
We consider the solutions to Poisson's equation, Eq. (5), in the limits of small potential, [[eta]] << 1, and large potential, [[eta]] >> 1. When [[eta]] << 1, the dependence of [[eta]] on [[xi]] is exponential. When [[eta]] >> 1 we find a power law dependence for [[eta]]([[xi]]) .
When [[eta]] << 1 the leading terms in Poisson's equation give
The asymptotic dependence of the potential on position for [[eta]] << 1 is
In the opposite limit, [[eta]] >> 1, Poisson's equation reduces to
[[eta]][[second]] ~ M (2[[eta]])-1/2. (9)
Child  found that the solution to Eq. (9), subject to the boundary conditions at the wall, [[eta]](0) = [[eta]]w, and at the plasma-sheath interface, [[eta]](d) = 0, is
The sheath thickness, d, is given by
and is the thickness of the region where the electron density is negligible. Equations (10) and (11) taken together are called Child's law.
Child's law relates three quantities: the wall potential, [[eta]]w, the sheath thickness, d, and the Mach number, M . The mach number is related to the current density for ions entering the sheath. Hence, Child's law is often used to determine the current density flowing into a sheath, Ji [[proportional]] e nouo = e nocs M, for given values of [[eta]]w, and d , where d must be determined without using Eq. (11). We, however, are interested in the spatial variation of the potential in the plasma sheath.
Figure 2 compares the spatial variation in the potential predicted by Child's law with an exact numerical solution of Poisson's equation for [[eta]]w <= 800 and M = 1.05. The exact solution is found by integrating [[eta]][[minute]] , given by Eq. (6), from the wall towards the plasma using a Runge-Kutta method . The exact solution's power law nature for [[eta]] >> 1 and exponential nature for [[eta]] << 1 are clearly visible. The exponential solution is dominant for [[eta]] < 0.1. The agreement between Child's law and the exact solution is increasingly poor as the transition from the sheath to the plasma is approached.
We want a more accurate analytic expression for [[eta]]([[xi]]) than Child's law. Instead of looking for an exact solution to an approximate equation, we look for an approximate solution to the exact equation. The power law form of Child's law suggests that we try
Note that [[eta]][[minute]](d) = 0. To find values for the coefficients, a, b, and d, we require [[eta]](0) = [[eta]]w, [[eta]][[minute]](0) = [[eta]][[minute]]w, and [[eta]][[second]](0) = [[eta]][[second]]w. Solving the resulting equations, we find that b, d, and a are given by
Equations (12) and (13a)-(13c) together with Eqs. (5) and (6) for the derivatives make up what we will call the improved Child's law, because of its superior agreement with exact numerical solutions for [[eta]]w < 104.
When the asymptotic values for [[eta]][[minute]]w and [[eta]][[second]]w,
[[eta]][[minute]]wa ~ -23/4 M1/2 [[eta]]w1/4, (14)
[[eta]][[second]]wa ~ M (2[[eta]]w)-1/2 (15)
are substituted into Eqs. (12) and (13), Child's law is recovered. Hence, in the asymptotic limit the improved Child's law approaches Child's law.
Figure 3 shows the difference between the exponent, b, [Eq. (11a)] and the asymptotic value of 4/3, which is used in Child's law. We see that the difference is significant even at [[eta]]w = 100, and that b goes to its asymptotic value as [[eta]]w-1/2. This indicates that Child's law is only accurate for [[eta]]w1/2 >> 1.
The sheath thickness for Child's law, the improved law, and the exact solution are shown plotted against [[eta]]w in Fig. 4. As the wall potential becomes large the improved law exhibits the same 3/4 power scaling of d with [[eta]]w that Child's law shows. There is a minimum in the sheath thickness at [[eta]]w ~ 6 for the improved law (the exact position depends on M). This occurs as the character of the exact solution changes from power law to exponential, and is due to the attempt of the improved law to better fit the exponential by increasing b (see Fig. 3). However, if we ignore this unphysical increase in d for small values of [[eta]]w, the improved Child's law fits the exact numerical solution well even for [[eta]]w on the order of 10.
Figures (5a)-(5c) compare the spatial variation in the electric potential found using Child's law, the improved Child's law, and the exact solution for values of [[eta]]w = 10, 30, and 100 and M = 1.05. We see that even at [[eta]]w = 100 (Fig. 5c) the improved expression is still noticeably different from Child's law, and in much better agreement with the exact solution. Because of the exponential nature of the exact solution for [[eta]] ~ 1, power law solutions, such as Child's law and the improved Child's law, fail for small [[eta]]. However, by allowing the exponent, b, to vary with [[eta]]w, the improved law gives much better agreement with the exact solution than does Child's law. This improved agreement is most apparent, and effective, for smaller values of the wall potential (i.e., [[eta]]w = 10 in Fig. 5a).
Finally, by expanding both the exponent, b, and the sheath thickness, d, for [[eta]]w >> 1 we can determine how large [[eta]]w must be for Child's law and the improved law to be in good agreement. We find that
The values of b and d predicted by Eqs. (16) and (17) asymptotically approach those of Child's solution for [[eta]]w >> 1, as they should. However, as seen previously, they only go to the asymptotic limit as [[eta]]w-1/2. The value of b found using Eq. (16) is within one percent of the asymptotic value for [[eta]]w = 2200 (assuming M + 1/M = 2). The sheath thickness predicted from Child's law is within one percent of Eq. (17) for [[eta]]w = 12 000. We can also expand the electric field for [[eta]] >> 1 to give
At the wall this expression also shows the same [[eta]]w-1/2 convergence as Eqs. (16) and (17). For [[eta]]w[[minute]] to be within one percent of its asymptotic value we need [[eta]]w = 5000. We conclude that Child's law only provides a good approximation (within one percent) for the potential variation within the sheath when the potential drop across the sheath, [[eta]]w, is greater than 104.
We have presented an improved Child's law, Eqs.(12) and (13), together with Eqs. (5) and (6), for the time-independent spatial variation of the electric potential for the source-free, collisionless, cathodic plasma sheath. It provides a significant improvement over Child's law when |e[[phi]]/kTe| at the wall is less than 104. The two expressions agree in the asymptotic limit, and the improved expression is no more difficult to evaluate once the coefficients a, b, and d [Eqs. (13a)-(13c)] have been calculated.
The authors wish to acknowledge useful discussions with R. Merlino. This work was supported by the Iowa Department of Economic Development.
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Terrence E. Sheridan, Jr. - received the B.A. in physics from Hiram College, Hiram, Ohio, in 1983 and the Ph.D. in plasma physics from Dartmouth College in 1987 for measurements of the electron distribution function in a magnetized laboratory plasma. He is currently doing post-doctoral research at the University of Iowa. Research topics include experimental measurement and Monte Carlo simulation of transport in magnetron plasmas and particle-in-cell simulation of sheath formation.
John A. Goree - received the B.S. in applied physics from the California Institute of Technology in 1980, and the Ph.D. in plasma physics from Princeton University in 1985 for work on rf waves and far-infrared laser scattering diagnostics of plasmas. He has been an assistant professor of physics at the University of Iowa since 1985, where he has been working on understanding transport processes in glow discharge and magnetron plasmas, laser-induced fluorescence diagnostics, and plasma wave theory. John received an IBM Faculty Development Award in 1986.
FIG 1. Model system for the plasma sheath. The potential, [[phi]], in the plasma sheath is plotted qualitatively as a function of distance from the wall. Ions enter the sheath as a mono-energetic beam with a velocity uo. Dimensionless quantities are shown to the right of their dimensional counterparts. Note that the sign of the dimensionless potential, [[eta]] = - e[[phi]]/kTe, is opposite that of [[phi]].
FIG 2. Semi-log plot for the dimensionless potential, [[eta]], in the sheath as a function of dimensionless distance from the wall, [[xi]], for an exact numerical solution and Child's law with M = 1.05. Note that [[eta]] is a positive quantity, in contrast to the actual potential, [[phi]], which is negative, as exhibited in Fig. 1. Agreement between Child's law and the exact solution is good in the power law regime, [[eta]] >> 1. Agreement is poor in the exponential regime, [[eta]] << 1.
FIG 3. Difference between the exponent in the improved Child's law, b, and the asymptotic value of 4/3 used in Child's law plotted against the wall potential, [[eta]]w, with M = 1.05. Note that this difference goes to zero only as [[eta]]w1/2.
FIG 4. Dimensionless sheath thickness, d, for Child's law, the improved Child's law, and the exact numerical solution plotted against the wall potential, [[eta]]w, with M = 1.05. For the exact solution the sheath thickness is chosen to be the distance from the wall to the point where [[eta]] = 1.
FIG 5. Here the exact numerical solution for [[eta]]([[xi]]), the spatial dependence of the potential in the plasma sheath, is compared to the predictions of the improved Child's law and Child's law. These curves were computed with M = 1.05 and (a) [[eta]]w = 10, (b) [[eta]]w = 30, and (c) [[eta]]w = 100. Note that the improved law is in much better agreement with the exact solution than is Child's law, particularly for [[eta]]w = 10.