The classical low aspect ratio DC glow discharges have long time been exploited as sputtering sources and are still used in some applications. These are shown in the figure below

for the case of a planar DC discharge. The discharge is maintained in the usual manner by secondary electrons emitted from the cathode. These energetic secondary electrons accelerated in the cathode fall ensure the necessary ionization and, in this way, maintain the discharge. However, the necessary operating pressures must be sufficiently high, \begin{equation} \label{highP} p \geq 4~Pa \end{equation} in order to ensure that secondary electrons are participating in the ionization process and are not lost immediately to the anode or walls. However, such are pressure is sufficiently high for deposition of sputtered atoms onto the substrates since collisions with plasma gas (typically argon) affects the deposition process. This causes the redeposition of sputtered atom on the cathode, the side walls, and, sometimes, poor substrate adhesion of sputtered films. The condition that sputtered atoms do not experience scattering on the way to the substrate impose the limitation on the sputtered atom mean free path \begin{equation} \lambda = n_g \sigma_{el} \end{equation} where $n_g$ is the density of neutral argon atoms and $\sigma_{el}$ is the scattering cross-section. Namely, it imposes \begin{equation} \lambda \leq l, \end{equation} which for a neutral–neutral scattering cross section $\sigma = 2 \cdot 10^{-16}~cm^2$ means \begin{equation} \label{lowP} p \leq 4~Pa \end{equation} Comparing equations \ref{highP} and \ref{lowP} one can see that the nisha for the stable operation of a dc glow discharge is relatively narrow around $4~Pa$. The conceptual limiting factor here is the poor confinement of secondary electrons. This problem is successfully solved in the hollow cathode discharge. The effective confinement of secondary electrons in the hollow cathode cavity is responsible for the efficiency of these plasma sources. However, the geometry is frequently also a limiting factor.

Magnetron

Operation of sputtering discharge at lower pressure can significantly improve the quality of coating. Driven by this motivation the conventional glow discharge has evolved into the magnetron configuration..

This concept utilizes the DC magnetic field at the cathode to confine the secondary electrons. The conceptual design is shown in the figure below.

The permanent magnets installed behind the cathode target create the magnetic field lines that enter and leave across the cathode plate. The typical cathode voltage to form the discharge is $\gtrsim 200~V$. The discharge forms right above the cathode in the form of a brightly glowing circular plasma ring (typical plasma density $\sim 10^{18} m^{-3}$), as illustrated in the figure. The sputtering takes place at the cathode in a circular track under the ring. The most of the externally applied voltage is deposited across a thin cathode sheath (typical thickness of $1~mm$). The magnetic field in the sheath is not high enough to confine Argon ions. Being accelerated in the high electric field formed in the sheath they strike the cathode at high energy. The ion impact results in the sputtering of the cathode material and produces secondary electron emission necessary for the maintenance of the discharge. The high potential drop in the sheath acts to accelerate secondary electrons back into the plasma, where they are confined by the magnetic field near the cathode. Being confined, they experience sufficient ionizing collisions in order to maintain the discharge, before they get lost at the wall. The typical operation conditions are listed below

Description	Symbol	Value
Working gas		Ar
Working gas pressure	p	$0.3-0.7~Pa$
External magnetic field	$B_0$	$20~mT$
Ion current density at the cathode	$J_i$	$20~\dfrac{mA}{cm^2}$
Applied cathode voltagDischarge Modele	$V_{DC}$	$300-800~V$
Deposition rate		$200~\dfrac{nm}{m}$

$B_0$ corresponds to the magnetic field strength at the radial position where magnetic field lines are parallel to the cathode surface. $J_i$ corresponds to the ion current density at the race track.

Qualitative discharge model

This basic qualitative model reflects some basic properties of the equilibrium magnetron discharge. The following discharge parameters are typically considered: $I_{DC}$, $p$, $B_0$ and radius of the race track $R$.

Voltage $V_{DC}$

The applied cathode voltage $V_{DC}$ is almost completely falls across the cathode sheath. The secondary electron emission coefficient $\gamma_e$ for $200-1000~eV$ argon ions on aluminium is $\gamma_{se}~0.1$. Let's introduce $N$, the number of electron – ion pairs created by each secondary electron before it looses all its energy gained in the sheath. \begin{equation} \label{N} N \approx \dfrac{V_{DC}}{\varepsilon_c}. \end{equation} Here $\varepsilon_c$ corresponds to the energy needed to create one electron-ion pair. For $200-1000~eV$ electrons, $\varepsilon_c \approx 25~eV$. In reality not all secondary electrons dissipate their energy on the ionisation process. Some of them return back to the cathode surface shortly after emission without completely dissipating their energy. As a result, the real effective secondary electron emission coefficient $\gamma_{eff}$ is less than $\gamma_{se}$. \begin{equation} \label{gamma} \gamma_{eff} \approx \dfrac{\gamma_{se}}{2} \end{equation} In steady state the condition should hold \begin{equation} \label{ststate} \gamma_{eff} N = 1 \end{equation} Thus, we obtain \begin{equation} \label{Vdc} V_{DC} \approx \dfrac{2 \varepsilon_c}{\gamma_{se}} \end{equation} For typical $\gamma_{se} \approx 0.1$ and $\varepsilon_c \approx 25~eV$ we obtain $V_{DC} \approx 500~V$.

Energetic electrons

The extension of the plasma ring over the cathode surface is determined by the gyro-radius of the fast energetic electrons $r_{ce}$ emitted from the cathode surface. These electrons are responsible for the discharge maintenance and confinement of these electrons is decisive for the plasma production. Without going deep into details we state hat the plasma ring has the extension over the cathode surface approximately equal to $r_{ce}$. The energetic electron gyro-radiis is equal to \begin{equation} \label{rce} r_{ce} = \dfrac{\upsilon_e}{\omega_{ce}} = \dfrac{1}{B_0} \sqrt{\dfrac{2mV_{DC}}{e}}, \end{equation} where $\upsilon_e = \sqrt{\dfrac{2eV_{DC}}{m}}$ and $\omega_{ce}$ designate the electron velocity and electron gyro-frequency. For typical $B_0 \approx 20~mT$ and $V_{DC} = 500~V$ we obtain $r_{ce} \approx 5~mm$. The magnetic field lines a the cathode surface are favourable to trap plasma electrons and forcing them to bounce back and forth between radii $r_1$ and $r_2$. The main force reflecting electrons is the electrostatic electric field in the sheath. A electron in a field structure which is convergent at both ends (such as "magnetic mirror" or "magnetic bottle") will be reflected by both mirrors and bounce between them. We introduce the field lines radius of curvature $R_c$. and the ring width $w = r_2 - r_1$. Then the following relations hold \begin{equation} \label{rel1} \dfrac{w}{2} = R_c~sin~\theta \end{equation} and \begin{equation} \label{rel2} r_{ce} + R_c+cos~\theta = R_c \end{equation}

Eliminating $\theta$ from these two equations yields $w$ as a function of $r_{ce}$ and $R_c$. In reality the permanent magnets behind the cathode are positioned such that the magnetic field lines be whenever possible parallel to the surface on the largest possible cathode area. It means practically that $w \ll R_c$, or $sin~\theta \ll 1$. It allows to simplify $sin~\theta \approx \theta$. We obtain thus the following estimate for the ring width: \begin{equation} \label{W} w \approx 2 \sqrt{2 r_{ce} R_c} \end{equation} Taking the typical $r_{ce} \approx 5~mm$ and choosing $R_c \approx 4~cm$, we obtain $w \approx 4~cm$.

Ion Current Density $J_i$ and Sheath Thickness $s$

The magnetic field in the vicinity the cathode surface is not strong enough to magnetise ions, i.e., ions do experience any single gyration between collisions. Also, the neutral gas pressure is not high enough and the cathode sheath is sufficiently thin in order for ions to move across the sheath without collisions. Therefore the ion flow across the sheath (from the plasma ring to the cathode) can be described by the Child law: \begin{equation} \label{Ji} J_i = \dfrac{4}{9} \epsilon_0 \sqrt{ \dfrac{2e}{M} } \dfrac{V_{DC}^{3/2}}{s^2} \end{equation} Assuming for simplicity that the plasma ring is thin $w \ll R$, we can write \begin{equation} \label{Ji_2} J_i = \dfrac{I_{DC}}{2 \pi R w} \end{equation} For typical discharge parameters $I_{DC} = 5~A$, $R = 5~cm$, $w = 4~cm$, $V_{DC}= 500~V$ we obtain from the equation \ref{Ji_2} for the ion current density $J_i \approx 40 \dfrac{mA}{cm^2}$. From the equation \ref{Ji} we obtain for the sheath thickness $s=0.56~mm$.

Plasma density $n_e$

The electron density in the plasma ring can be estimated from the Bohm model of plasma flow in the sheath and pre-sheath. According to the Bohm concept the ion flow a the entrance to the sheath (in our case at the interface between plasma ring and cathode sheath) is given by \begin{equation} \label{Bohm} 0.61 e n_i u_B = J_i \end{equation} where $n_i$ is the ion density ($\approx n_e$) and $u_B$ is the Bohm velocity. For typical conditions in the low pressure discharges $T_e \approx 3~eV$ and with earlier estimated $J_i \approx 40 \dfrac{mA}{cm^2}$ we obtain $n_e \approx 1.5\cdot10^{12}~cm^{-3}$.

Sputtering Rate $R_{spat}$

The sputtering rate is expressed as follows \begin{equation} \label{R_spat} R_{sput} = \gamma_{sput} \dfrac{J_i}{e} \dfrac{1}{n_{Al}}~~cm/s \end{equation} where $\gamma_{sput}$ designate the yield of sputtered atoms per incident ion. It is typically $\gamma_{sput} \sim 1$. $n_{Al}$ is the density of aluminium atoms in the cathode target. $J_i$ has been estimated earlier $J_i \approx 40 \dfrac{mA}{cm^2}$. Thus we obtain $R_{sput} \approx 4.1 \cdot 10^{-6}~cm/s$. A target thickness of 3.6 mm will be sputtered within an 24 hours of operation. It results in the erosion ring at the cathode surface in the vicinity of the plasma ring.

Therefore, the cathode is the consumable part of the magnetron system which must be replaced upon significant erosion. The confining magnetic field in the vicinity of the cathode causes the formation of the dense plasma ring. The plasma density decays very fast with the distance from plasma ring and in the vicinity of the substrate can be an order of magnitude smaller. It can be favourable for some applications and undesirable for others. This issues is addressed in the unbalanced magnetron configuration, where some magnetic field lines connect the cathode and substrate surfaces, enhancing the particle transport towards the substrate.