# Quantum Single Theory

### Introduction

Plasma processes are commonly used to deposit thin films on substrates for a variety of applications. The rate of deposition is controlled by the neutral and ion species bombarding the substrate surface. Magnetron sputtering and HiPIMS are two types of deposition plasma process which see widespread use in industry.

HiPIMS processes produce high plasma densities during a short lived pulse of extremely high power density to the target. During the pulse the electron density in the ionization region close to the target surface can be of the order of 1018- 1019 m-3[1, 2]. For magnetron sputtering processes the plasma density is typically much lower with plasma densities up to 1017 m-3. The corresponding ionization mean-free-path for sputtered particles is of the order of 1 cm in HiPIMS compared to 50 cm in magnetron sputtering for typical process conditions. Therefore, the fraction of the sputtered particles that are ionized can vary significantly for different processes. The target material also plays an important role in determining the ionization fraction since the ionization potential is material dependent. Thin film deposition rates are strongly dependent on the ionization fraction and therefore measurement of the ionization fraction and the ionised flux fraction to the substrate is becoming critical for process development.

Measurements of the ionization fraction in deposition processes have been reported by many researchers over the last two decades. Optical emission spectroscopy measurements made by Bohlmark et al [8] indicate the ionization fraction of sputtered Ti in a HiPIMS can be up to 90%. Measurements using a quartz crystal microbalance (QCM) in combination with a retarding field energy analyser (RFEA) show that the ionized flux fraction at the substrate was as low as 4.5% in another HiPIMS process [9].

The Quantum deposition rate monitor was developed specifically to measure the ionised flux fraction and the neutral flux fraction at the substrate location under processing conditions. This system is suitable for most type of deposition processes including magnetron sputtering, HiPIMS, PCVD etc. This system consists of the Semion RFEA with an integrated QCM, with some similarities to the design proposed by Green et al. [14]. The QCM element provides a direct measurement of the deposition rate at the substrate while the RFEA grids can be configured to turn on and off the ion flux to the quartz crystal. In this way the deposition rate can be determined when both neutrals and ions are present and when only neutrals are present. From these two measurements the ionized flux fraction is easily determined.

### 2. Quartz Crystal Microbalance

The use of a piezoelectric quartz crystal resonator to measure mass was first investigated by Sauerbrey in 1959 [ ]. It was found that the change in resonant frequency of the crystal was proportional to the mass of a uniformly deposited layer on the crystal surface. He showed that for small masses of material the shift in resonant frequency is independent of the material properties. Because of this simple relationship, quartz crystal resonators are used extensively to measure thin film deposition rates in plasma processes. The original theory proposed by Sauerbrey has been extended over time to allow measurement of ever greater mass loads deposited on the crystal.

The resonant frequency of the standard A-T cut quartz crystal exhibits a strong temperature dependency and must be kept at a constant temperature to provide reliable results. New quartz crystal materials have been developed to provide a flat frequency response up to several hundred degrees Celsius. This avoids the need for water cooling at the moderated temperatures experienced in many plasma processes. Technical challenges surrounding the accurate measurement of the resonant frequency shift have also been addressed. Technologically advanced quartz crystal microbalance devices are now available from many suppliers.

### 2.1. Quartz Crystal Resonators

A standard QCM measures a mass per unit area by measuring the change in frequency of a quartz crystal resonator. The resonance is altered by the addition/removal of a small mass of material on/from the surface of the resonator. The crystal is comprised of a circular disk of a piezoelectric material sandwiched between two electrodes. The top view of one such crystal is shown below.

Figure 1. Crystal resonator used in the Quantum deposition rate monitor.

Quartz is one member of a family of crystals that experience the piezoelectric effect. The trademarked MQTM material from Tangidyne Corporation is another and preferred in the Quantum product due to its flatter frequency response versus temperature.

The piezoelectric effect causes a mechanical deformation in the crystal when a voltage is applied. The resultant alternating current through the quartz crystal induces oscillations in the crystal structure. With an alternating current between the electrodes of a properly cut crystal, a standing shear wave is generated. The Q factor can be as high as 106. This extremely narrow resonance band results in a highly stable oscillator. Therefore, a high level of accuracy in the determination of the resonance frequency is possible. The standard QCM exploits this ease and precision for sensing. Standard equipment allows frequency resolution down to 1 Hz. Standard crystals designed for QCM products have fundamental resonant frequencies in the 4 – 6 MHz range.

### 2.2. Deposition Rate Determination

Sauerbrey [7 ] was the first to realize that a quartz crystal resonator could be used to measure an extremely small mass of a substance deposited on the crystal surface. His theory is still used extensively in modern deposition rate monitoring systems.

### Sauerbrey’s Equation

Sauerbrey derived the following relationship

$\LARGE&space;\Delta&space;f&space;=&space;-C_f\Delta&space;m$

where Δf is the change in resonant frequency of the crystal in Hz, Cfis the sensitivity factor of the crystal in Hz.ng-1.cm2 and Δm is the change in mass per unit area in gcm2. This relationship assumes that the additional material has the same electro-acoustic properties as that of the underlying crystal. In this case the sensitivity factor is given by

where f0 is the unloaded resonant frequency of the crystal in Hz, is the density of the crystal material in g.cm-3 and crystal in g.cm-1.s-2. μx is the shear modulus of the Since the deposited material is assumed to have the same properties as that of the crystal, a number of criteria must be satisfied for the Sauerbrey equation to apply:

• The deposited mass must form a rigid layer
• The deposited mass must be distributed evenly on the crystal surface
• The frequency change Δf0 / f0 ≤ 0.02 [ ]

Z-Match Equation

If Δf0 / f0 ≤ 0.02 then the so-called Z-match equation, derived by Lu and Lewis, should be applied. These authors analyzed the loaded crystal by modeling a resonator split into two components i.e. the main crystal and the deposited film. This analysis led to the following equation:

where Nx is the frequency constant for the crystal in Hz.cm, Rz is the Z-factor of the material deposited such that

$\LARGE&space;R_z=\sqrt{(\rho_x\mu_x)/(\rho_f\mu_f)}$

is the density of the deposited material, μ is the shear modulus of the deposited material and f is the resonant frequency of the loaded crystal. The parameter Rz introduces the ratio of acoustic impedance of the crystal to that of the deposited film. It can be shown that if the ratio of acoustic impedances is one i.e. the crystal and deposited material are the same, then the Z-match equation reduces to the Sauerbrey equation.

### Film thickness

The growth rate of the deposited film is often the parameter of most interest to researchers. The thickness Tk f of the deposited film can be calculated from equation (3):

The Z-match equation has been shown to be in good agreement with experiments for frequency changes up to 40% [ ]. Certain types of films, such as organic polymers, have viscoelastic properties that are not accounted for in these simple models and significant departures from equations 3 and 4 are to be expected. Recent advances have led to the extension of these models to incorporate liquids and viscoelastic materials. However, the equations presented above cover the majority of applications for which the Semion deposition rate monitor is applicable.

### RFEA with Integrated QCM

The idea of using an RFEA with integrated QCM to measure the ionised flux fraction has been around for a number of decades. Rossnagel and Hopwood [ ] used one such device to measure metal ionisation fractions in ICP plasmas. The use of mesh components with large orifices limited its use to very low density to avoid plasma forming inside the device.

Green et al improved on the original design considerably by incorporating mesh components with much smaller apertures. This design was also capable of being DC biased at the substrate potential.

### Semion Deposition Rate Monitor

A schematic of the Quantum deposition rate monitor is shown in figure 2.

Figure 3.1-1: Schematic of the Quantum deposition rate monitor.

Unlike other devices, the crystal is embedded in the RFEA. Other research devices have been developed by simply building a grid stack around the off- the-shelf deposition rate monitor, including the large metal water cooled support structure. This approach eliminates the possibility of miniaturizing the device and reduces the operating pressure range that can be achieved.

The Quantum deposition rate monitor incorporates only the crystal which is approximately 250µm thick. When combined with the stack of grids, insulators and mechanical housing the overall thickness of the device is 5 mm. The total depth, from orifice to crystal, is approximately 1 mm which allows the sensor to be operated to at least 50 mTorr in Argon without ion collisions inside. Other designs reported on in the literature [ ] have depths which are at least 10 times that of the Quantum deposition rate monitor.

A major advantage of the Semion product over other devices is that the sensor can be placed on dc, pDC or rf biased substrates without the need for any modification of the substrate holder. This allows the user to measure ionised flux fractions directly at the substrate location under real processing conditions.

### Theory of operation

The entrance orifice is 5 mm in diameter and allows a sample of ions, neutrals and electrons into the analyser for detection. A first grid G0 is placed over the entrance orifice to reduce the open diameter exposed to the plasma, to a size equivalent to the grid aperture size. In this device the grid aperture size is a 25 µm x 25 µm square. For proper operation the aperture size should be less than the Debye length to prevent plasma forming inside the analyser. The aperture size used is less than the Debye length, near the substrate, for most applications encountered.

A second grid G1, insulated from G0 (all grids are insulated from each other to allow independent biasing), is biased with a negative potential relative to G0 to repel any electrons that may enter the sensor.

A third grid G2 is biased with a positive potential sweep to generate a gradually increasing retarding field to discriminate incoming ions based on their kinetic energy [ ] - when the sensor is being operated in RFEA mode. When the sensor is being operated in deposition rate monitor mode then G2 can be biased with a positive potential to repel all incoming ions or it can be biased such that all incoming ions pass through unperturbed. The deposition rate at the crystal can thus be determined with and without the ion fraction.

A fourth grid G3 is biased negative 10 volts with respect to the fifth grid G4, when the device is operated in RFEA mode. G4 collects the ion current passing G3 to generate the current –voltage characteristic from which the ion energy distribution is calculated. G3 suppresses secondary electron emission from G4. When operated in deposition rate mode, G3 and G4 are biased to the same potential as G2

The crystal terminates the stack of components. It is held at the same DC potential as G4 so that ions travelling from G4 to the crystal do not experience an additional electric field which might prevent them from being detected. The crystal is also excited with a radio frequency bias to cause it to resonate at its natural frequency. The shift in resonant frequency due to deposition on the crystal surface is monitored and enables the deposition rate to be calculated.

### Calibration

The deposition rate measured by the crystal at the bottom of the gridded structure will deviate from the deposition rate at the sensor surface. The ions arrive at normal incidence to the sensor surface because they are accelerated in the sheath electric field adjacent to the surface. Therefore, the ions entering the sensor orifice travel straight through to the crystal surface. The neutral species, on the other hand, arrive at the sensor surface with an isotropic distribution. The neutrals entering the orifice do so with a spread of trajectories – some will be lost to the side walls and some will make it to the crystal for detection. This ‘shadowing’ effect has been addressed for specific discharge geometry by Green et al [ ]. The deposition rate seen by the crystal at bottom of the grid stack can be determined from the following relationship

where Rsurf is the deposition rate at the sensor surface (that seen by the substrate), Rcrystal is the deposition rate at the crystal, G accounts for the reduction in neutral flux reaching the crystal due to the shadowing effect and Tg is the transmission factor of each grid where n is the number of grids used. The 3 grids used by Green et al had 52.7% transmission each giving Tgn = 14.6%. The aspect ratio of their device (depth/width) was approximately 15/18 as shown in the schematic below [ ]. It was shown that this ratio can be used, with reasonable accuracy, to estimate the percentage of neutrals reaching the crystal as a result of side wall shadowing. Therefore, the aspect ratio of 15/18 gives G =16.6% and Rcrystal = 2.4%× Rsurf which compared favourably with their experimentally determined value of 1.6%.

Green et al also give a plot of theoretically calculated values for G as a function of sensor aspect ratio (see figure 5 in reference [ ]). This will be used as a guide for estimating G for the Semion QC deposition rate monitor.
The Quantum deposition rate monitor utilizes 5 grids, each with 50% transmission giving a value for Tn of approximately 3%. This is significantly less than that of Green et al due to incorporation of 2 additional grids to ensure correct operation of the RFEA element of the device. However, the shadowing effect is much less due to the small internal depth of the sensor. The aspect ratio is approximately 1/5 from which the total flux reaching the crystal is estimated to be about 70%. Therefore, Rcrystal = 2.1% × Rsurf, which is crystal surf very similar to that of the device designed by Green et al.Given the fact that the neutral flux is reduced by approximately 98% and the ion flux, due to the directional nature, is reduced by 97% an accurate experimental calibration is required since small errors in the estimation of Tn and G can cause the calculated deposition rates to deviate significantly from reality. To provide the best possible calibration of the device a second quartz crystal is incorporated into the sensor design, close the gridded element. The reference crystal is located at the surface of the sensor housing and so there is no neutral flux loss due to the aspect ratio shadowing effect. There are no grids used either, which eliminates the transmission effect described above. At the beginning of each experimental run a calibration can be performed in which the deposition rates of both crystals are compared to find a scaling factor for the crystal at the bottom of the grid stack.

Figure 3: Schematic of the Quantum deposition rate monitor showing the gridded element and the reference crystal side by side.

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