Notes on plasma sheaths

I follow Bittencourt’s and Chabert’s books

Why sheaths form


The simplest sheath model assumes:

bg right:45% 80%

Basic equations and sheath solution

Neglecting electron inertia, their momentum equation reads \(0 = -\frac{1}{n_e}\frac{d (n_eT_e)}{d x} + e\frac{d\phi}{dx}\) If most electrons are repelled by the sheath, to a good approximation they satisfy the Boltzmann relation with constant $T_e$: \(n_e(x) = n_s \exp\left[ \frac{e \phi(x)}{T_e}\right]\)

Cold ions momentum equation can be integrated into the conservation of ion mechanical energy: \(m_i u_i \frac{du_i}{dx} = - e\frac{d\phi}{dx} \quad\Rightarrow\quad \frac{1}{2}m_i u^2_i(x) + e\phi(x) = \frac{1}{2}m_i u^2_s\)

The 1D continuity equation of ions states simply \(n_i(x) = \frac{n_su_{is}}{u_i(x)} =n_s\left[1-\frac{2e\phi(x)}{m_iu_{is}^2}\right]^{-1/2}\)

Sheath equation

Poisson’s equation for $\phi$ reads: \(\frac{d^2\phi}{dx^2} = \frac{e}{\varepsilon_0} [n_e(x)-n_i(x)]\) Substituting, we obtain the sheath equation: \(\frac{d^2\phi}{dx^2} = \frac{en_s}{\varepsilon_0} \left\{\exp\left[ \frac{e \phi(x)}{T_e}\right]-\left[1-\frac{2e\phi(x)}{m_iu_{is}^2}\right]^{-1/2}\right\}\)

This is a nonlinear equation that must be solved with the condition $j_i = j_e$ at $x= 0$.

Observe that $u_{is}$ is still unknown.

We can find a condition on $u_{is}$ if we linearize the equation about $x=s$, \(\frac{d^2\phi}{dx^2} = \frac{en_s}{\varepsilon_0} \left\{\left[1 + \frac{e \phi}{T_e}\right]-\left[1+\frac{e\phi}{m_iu_{is}^2}\right]\right\} \\ = \frac{e^2n_s}{\varepsilon_0} \left\{\frac{1}{T_e}-\frac{1}{m_iu_{is}^2}\right\}\phi = \frac{1}{\lambda_D^2} \left\{1-\frac{T_e}{m_iu_{is}^2}\right\}\phi\)

Bohm’s criterion states that $u_{is} \geq \sqrt{T_e/m_i}$ (Bohm’s velocity) at the sheath edge. I.e., ions must arrive at the sheath edge with sonic velocity.

Observe that if $n_i=n_e=n_s$, $u_i=u_{is}=\sqrt{T_e/m_i}$ at the sheath edge $x=s$, the density $n_0$ far into the plasma is higher: \(m_i u_i \frac{du_i}{dx} = - T_e\frac{d \ln n}{d x} \quad\Rightarrow\quad \frac{1}{2} T_e = -T_e \ln n_s + T_e \ln n_0\)

\[\frac{n_s}{n_0} = \exp(-1/2) \simeq 0.6\]

Sheath potential

More complex models

RF sheaths

Imagine now that we have the set up of the figure, and we bias a section of the wall with a sinusoidal AC voltage at angular frequency $\omega$. The capacitor stills prevent any DC current to the wall, but within each AC cycle instantaneous current may be collected.

Alternatively, the plasma potential may vary at a frequency $\omega$, giving rise to similar effects.

Read more: Chabert’s chapter 4

bg right:50% 90%

Plasma frequency of ions and electrons

The characteristic times in which a typical ion or traverses a Debye length are \(\tau_i = \frac{\lambda_D}{\sqrt{T_e/m_i}} = \omega_{pi}^{-1}\) \(\tau_e = \frac{\lambda_D}{\sqrt{T_e/m_e}} = \omega_{pe}^{-1}\) Here, $\omega_{pi}$, $\omega_{pe}$ are the ion and electron plasma frequency, respectively.

  1. If $\omega \ll \omega_{pi}$, the AC frequency is so slow that ions and electrons do not see any major changes during their passage through the sheath. The solution is essentially a collection of quasi-static solutions, based on the stationary I-V characteristic.
  2. If $\omega_{pi} \sim \omega \ll \omega_{pe}$, the slower ions see the change while in the sheath, but the solution of the faster electrons is essentially quasi-static. Ions gain a different amount of energy depending at which instant they enter the sheath, complicating their dynamics
  3. If $\omega_{pi} \ll \omega \ll \omega_{pe}$, oscillations are too fast to have an impact on the ions, so they barely disturb them. Electrons are able to redestribute quickly in response to $\omega$.
  4. If $\omega_{pi} \ll \omega \sim \omega_{pe}$, The effect on electrons is large. Their dynamics are complex.
  5. If $\omega_{pe} \ll \omega$, oscillations are too fast for either ions or electrons to follow them. In essence, we have the DC solution again.

When the potential difference increases, so does the sheath width.These figures exemplify the motion of ions in the varying sheath.The greatest effect on ion motion happens when their transit time in the sheath is similar to the RF time.

Regardless, the average ion flux continues to be \(\left<\Gamma_i\right>= n_s\sqrt{\frac{T_e}{m_i}}\)

bg vertical right:40% 90% bg right:40% 90%

Electron response

As long as $\omega \ll \omega_{pe}$, electron response to the varying potential is instantaneous. The electron current collected by the wall is: \(n_s \exp\left(\frac{e\phi_w(t)}{T_e}\right) \sqrt{\frac{T_e}{2\pi m_e}}\)

Electron response is exponential:

RF floating potential

The floating potential is different in this case due to the rectifying effect of electron motion. Averaging over an RF period:

\[\left<n_s \exp\left(\frac{e[\phi_0 + \phi_1\sin(\omega t)]}{T_e}\right) \sqrt{\frac{T_e}{2\pi m_e}}\right> = n_s\sqrt{\frac{T_e}{m_i}}\]

Integrating we find that the floating potential becomes more negative, due to the so-called RF self bias: \(\phi_w = \frac{T_e}{2e}\left[ -\ln \left(\frac{m_i}{2\pi m_e}\right) -\ln \left(\mathrm{I}_0\left(\frac{e\phi_1}{T_e}\right)\right) \right]\)

bg 60%