Faraday cup

In plasma diagnostics

The Faraday cup uses a physical principle according to which the electrical charges delivered to the inner surface of a hollow conductor are redistributed around its outer surface due to mutual self-repelling of charges of the same sign – a phenomenon discovered by Faraday.

The conventional Faraday cup is applied for measurements of ion (or electron) flows from plasma boundaries and comprises a metallic cylindrical receiver-cup – 1 (Fig. 1) closed with, and insulated from, a washer-type metallic electron-suppressor lid – 2 provided with the round axial through enter-hollow of an aperture with a surface area ${\displaystyle S_F=\pi D_F^2/4}$. Both the receiver cup and the electron-suppressor lid are enveloped in, and insulated from, a grounded cylindrical shield – 3 having an axial round hole coinciding with the hole in the electron-suppressor lid – 2. The electron-suppressor lid is connected by 50 Ω RF cable with the source ${\displaystyle B_{es}}$ of variable DC voltage ${\displaystyle U_{es}}$. The receiver-cup is connected by 50 Ω RF cable through the load resistor ${\displaystyle R_F}$ with a sweep generator producing saw-type pulses ${\displaystyle U_g(t)}$. Electric capacity ${\displaystyle C_F}$ is formed of the capacity of the receiver-cup – 1 to the grounded shield – 3 and the capacity of the RF cable. The signal from ${\displaystyle R_F}$ enables an observer to acquire an I-V characteristic of the Faraday cup by oscilloscope. Proper operating conditions: ${\displaystyle h\ge D_F}$ (due to possible potential sag) and ${\displaystyle h\ll {\lambda }_i}$, where ${\displaystyle {\lambda }_i}$ is the ion free path. Signal from ${\displaystyle R_F}$ is the Faraday cup I-V characteristic which can be observed and memorized by oscilloscope

In Fig. 1: 1 – cup-receiver, metal (stainless steel). 2 – electron-suppressor lid, metal (stainless steel). 3 – grounded shield, metal (stainless steel). 4 – insulator (teflon, ceramic). ${\displaystyle C_F}$ – capacity of Faraday cup. ${\displaystyle R_F}$ – load resistor.

Thus we measure the sum ${\displaystyle i_{\mathrm{\Sigma }}}$ of the electric currents through the load resistor ${\displaystyle R_F}$: ${\displaystyle i_i}$ (Faraday cup current) plus the current ${\displaystyle i_c(U_g)=-C_F(d{U}_g/dt)}$ induced through the capacitor ${\displaystyle C_F}$ by the saw-type voltage ${\displaystyle U_g}$of the sweep-generator: The current component ${\displaystyle i_c(U_g)}$ can be measured at the absence of the ion flow and can be subtracted further from the total current ${\displaystyle i_{\mathrm{\Sigma }}(U_g)}$ measured with plasma to obtain the actual Faraday cup I-V characteristic ${\displaystyle i_i(U_g)}$ for processing. All of the Faraday cup elements and their assembly that interact with plasma are fabricated usually of temperature-resistant materials (often these are stainless steel and teflon or ceramic for insulators). For processing of the Faraday cup I-V characteristic, we are going to assume that the Faraday cup is installed far enough away from an investigated plasma source where the flow of ions could be considered as the flow of particles with parallel velocities directed exactly along the Faraday cup axis. In this case, the elementary particle current ${\displaystyle d{i}_i}$ corresponding to the ion density differential ${\displaystyle dn(v)}$ in the range of velocities between ${\displaystyle v}$ and ${\displaystyle v+dv}$ of ions flowing in through operating aperture ${\displaystyle S_F}$ of the electron-suppressor can be written in the form

where

${\displaystyle e}$ is elementary charge, ${\displaystyle Z_i}$ is the ion charge state, and ${\displaystyle f(v)}$ is the one-dimensional ion velocity distribution function. Therefore, the ion current at the ion-decelerating voltage ${\displaystyle U_g}$ of the Faraday cup can be calculated by integrating Eq. (2) after substituting Eq. (3),

where the lower integration limit is defined from the equation ${\displaystyle M_i{v}_{i,s}^2/2=e{Z}_i{U}_g}$ where ${\displaystyle v_{i,s}}$ is the velocity of the ion stopped by the decelerating potential ${\displaystyle U_g}$, and ${\displaystyle M_i}$ is the ion mass. Thus Eq. (4) represents the I-V characteristic of the Faraday cup. Differentiating Eq. (4) with respect to ${\displaystyle U_g}$, one can obtain the relation

where the value ${\displaystyle -n_i{S}_F(e{Z}_i/M_i)=C_i}$ is an invariable constant for each measurement. Therefore, the average velocity ${\displaystyle ⟨v_i⟩}$ of ions arriving into the Faraday cup and their average energy ${\displaystyle ⟨{\mathcal{E}}_i⟩}$ can be calculated (under the assumption that we operate with a single type of ion) by the expressions

where ${\displaystyle M_A}$ is the ion mass in atomic units. The ion concentration ${\displaystyle n_i}$ in the ion flow at the Faraday cup vicinity can be calculated by the formula

which follows from Eq. (4) at ${\displaystyle U_g=0}$,

and from the conventional condition for distribution function normalizing

Fig. 2 illustrates the I-V characteristic ${\displaystyle i_i(V)}$ and its first derivative ${\displaystyle i_i^{\mathrm{\prime }}(V)}$ of the Faraday cup with ${\displaystyle S_F=0.5c{m}^2}$ installed at output of the Inductively coupled plasma source powered with RF 13.56 MHz and operating at 6 mTorr of H2. The value of the electron-suppressor voltage (accelerating the ions) was set experimentally at ${\displaystyle U_{es}=-170V}$, near the point of suppression of the secondary electron emission from the inner surface of the Faraday cup.

Error sources

The counting of charges collected per unit time is impacted by two error sources: 1) the emission of low-energy secondary electrons from the surface struck by the incident charge and 2) backscattering (~180 degree scattering) of the incident particle, which causes it to leave the collecting surface, at least temporarily. Especially with electrons, it is fundamentally impossible to distinguish between a fresh new incident electron and one that has been backscattered or even a fast secondary electron.