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

## 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.

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 | $200~\dfrac{nm}{m}$ |

## 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}

#### 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.
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