Laser Interferometry A. Principles The method of measurement of thin film deposition or etch rate is based on the interference principle of light waves. Interference is the phenomenon that can be observed when two or more light beams from the same source, reach the same point at the same time but they have travel different optical paths.  The existence of this phenomenon is due to the coherence property of waves. It is easily observed when a monochromatic light source is used. This happens because two monochromatic light beams that reach the same point at the same time, have a coherence coefficient equal to 1.  The physical result of waves overlapping is the observation of dark and bright regions that are called fringes.  Bright regions are observed when a number of light waves interact resulting to a light wave of maximum intensity. On the other hand, dark regions are observed when a number of light waves interact, resulting to a light wave of minimum, very often null, intensity.

The observation of minima and maxima of light intensity, when a light beam is reflected on a thin film surface, corresponds to conditions of minimum and maximum coherence of light waves that are reflected on the  thin film-air interface (0-1), of the light waves that are reflected on the  thin film-substrate interface (1-2) and finally of the waves that are reflected on substrate - substrate holder interface (2-3). The variation of the thin film thickness, either increase (deposition) or decrease (etching),  results to a continuous  change of the optical path of the light wave that transmits the 0-1 interface or/and the 1-2 interface. These light waves that are finally reflected either on the 1-2  or on the 2-3 interface, interfere with the light beam that is initially reflected on the 0-1 interface, leading to the observation of fringes.

The mathematical description of the interference phenomenon that can lead to the calculation of the deposition and the etch rate is based on the Freshnel formulas. According to this method the electric field and consequently the intensity of a light wave of general polarization that is reflected on a planar surface can be analyzed to two components - one parallel (from now one denoted as p) and one vertical (denoted as s) to the surface. The ratio of the  reflected and the incident electric field components are called Freshnel complex-amplitude reflection coefficients and can be related to the angles of light incidence and the dielectric indexes of the materials. For the system that is presented on the above figure that includes three interfaces (0-1, 1-2 and 2-3) and four materials (air, thin film, substrate and substrate holder) there are three parallel and and three vertical reflection coefficients that corresponds to each of the three interfaces (0-1, 1-2 and 2-3) These coefficients can be written as: In the above relations the unknown angles φ1 and φ2 has been expressed as a function of the angle of incidence φo and the refraction indexes no, n1, n2  according to Snell' s law: This allows the expression of the Freshnel coefficients as a function of dielectric constants  and the angle of incidence φο which are all known. Moreover, the resultant reflected wave in medium 0, which is the measured quantity, will be the sum of the wave that is initially reflected on 0-1 boundary and of the planar waves that  after multiplex reflections in all of the interfaces they finally reach again the 0-1 interface. The total reflectance R of the parallel and the vertical components of the incident light wave can be expressed as geometric series of the Freshnel coefficients by using the relations: where In the last two formulas, L is the wavelength of the incident beam in Å, d1 the film thickness and d2 the thickness of the substrate. In fact β1 and β2 express the phase change that the multiply-reflected wave experiences inside the film (β1) or inside the substrate (β2). They are called phase angles and are the most important parameters in the method.

Finally, the intensity of the reflected light wave that result from the interference of the multiple reflected light waves can be expressed in terms of the total p and s reflectance and the phase angles  β1 and β2, as: In the figures a, b is presented the variation of the reflected intensity of the light beam in terms of thin film thickness as calculated from the above formula. The theoretical calculations have been performed for a system with three interfaces and the materials: air (0), microcrystalline silicon thin film (1, n1=3.65), glass Corning 7059 (2, substrate) and stainless steel (3, substrate holder). In figure a, one can observe a periodic variation of the reflected intensity with film thickness, the thickness that corresponds to one period being about 895 Å. The interference of the light waves result to the appearance of two maxima and one minimum  during this period. The first maximum results from the interference of light waves that are reflected from the 0-1 and the 1-2 interface and take place when β1 becomes equal to: while the second peak result from the interference of light waves that are reflected from the 0-1 and the 2-3 interface and take place when β1 becomes equal to: It has to be mentioned that these values of β1 stand only for the specific system (three interfaces and n1=3.65).

Figure b presents a case that is quite closer to conditions of the deposition of a thin film from plasma i.e the thickness of a thin film increases and simultaneously the refractive index (real part of n1) and the absorption coefficient (imaginary part of n1) of the film increases. This scheme result to a variation of the reflected intensity that is not periodic and to a continuous dumping of the reflected intensity. The film thickness that corresponds to the appearance of two successive maxima is reduced due to the increase of the refractive index, while the second peak, which has been attributed to the interference of light waves that are reflected from the 0-1 and the 2-3 interfaces, is step by step disappeared, due to the increase of the film absorption. Figures c, d present the variation of the reflected intensity with the film thickness for the simpler case of two interfaces (0-1 and 1-2). In figure c, are presented results that correspond to a film refractive index of 3.65. In this case the variation of the reflected intensity is periodic and the film thickness that corresponds to the appearance of two successive maxima or minima is 895 Å i.e equal to the thickness presented in figure a. This is not a surprising result because the appearance of the maxima and minima are based on the value of β1 that in turn does not depends on the existence of two or three interfaces but only on the value of the film thickness. On the other hand, in this simple approach the second peak that corresponds to the interference of light waves that are reflected from the air-film and the substrate-substrate holder interfaces is not observed as the 2-3 interface has been ignored. Finally, figure d presents a case where the refractive index of the thin film has a quite low value (n1=1.5). In this situation, the thickness that corresponds to the appearance of two minima is high (2800 Å) and the intensity of the reflected wave is low, more than an order of magnitude lower compared to the case where n1=3.65. In both cases (n1=3.65 and n1=1.5) the same value of the incident light wave has been used (Iin=2 x 10-5 mW).

B. Setup

The setup used for measuring the deposition and etch rates is presented in the figure below:

#### It usually consists of a CW He-Ne or diode laser with power of a few mW, focusing lenses and a photodiode with appropriate wavelength response. The laser beam is focused on the specific point of interest at the surface of the growing film or the material to be treated. The angle of incidence is our case is usually in the range 77–82ο  and depends on the specific reactor configuration, the distance between the two electrodes and the distance to the optical windows. The reflected beam is collected from the opposite  side of the reactor and it is focused on the photodiode. The signal is then recorded, analyzed and stored in a personal computer using a simple A/D card.

C. Deposition rate measurements An example of thin film deposition rate measurements is presented in the above figure. The plasma conditions are typical of microcrystalline silicon thin film growth i.e high pressure, high power high dilution of silane in hydrogen. The reflected intensity of the light wave is recorded as a function of the deposition time by applying the configuration that has been presented in the setup section. According to the interference concept that has been presented to the principles section the appearance of two successively maxima or minima will take place in specific thin film thickness and when the relative phase difference have a value β1i1(i+1)=π. When this condition is satisfied the variation of the film thickness will be equal to: As a matter of fact, the record in time of the appearance of two successive maxima or minima corresponds to a specific thin film thickness and lead to the calculation of the deposition rate, according to the relation: where ΔΤ is the time that required for the appearance of two successive maxima or minima.

Concerning the shape of the recorded intensity and according to the discussion in the principle section, one can say that the initial film growth (first two fringes) is better described from the three interface configuration and the continuous increase of the thin film refractive index and the absorption coefficient. On the other hand, as the film thickness increases the simpler scheme of the two interfaces and the constant refractive index describes better the recorded intensity. Thus, for the accuracy of the deposition rate measurement the first two fringes have to be ignored as the refractive index in this stage is also unknown. The usefulness of the method can be understood from the above figure. In cases where the synergetic effect of different combinations of plasma parameters on the deposition rate is searched, a quite large number of depositions are required. For example, the investigation of the combined effect of the total gas pressure and the interelectrode space on the microcrystalline silicon deposition rate and the determination of the optimal deposition rate conditions in terms of these two parameters requires about 100 measurements. If for the measurement of the film thickness an ex-situ method was applied this investigation would take about 200 working days (including dead times for reactor pumping to the base vacuum). On the other hand, the application of an in - situ method as Laser Reflectance Interferometry significantly reduces the investigation time to less than 50 working days as 3 to 5 conditions can be checked at each deposition.

D. Etch rate measurements Another application of the method is the measurement of the etch rate during the plasma treatment of different materials. In the figure above, is presented the variation of the reflected intensity as a function of treatment time of  Polyethylene Terephthalate (PET) films from He-O2 plasma. The refractive index of the material has the quite low value of 1.2 and this lead to the observation of broad maxima and sharp minima of the reflected intensity. In addition, due to the low value of n1 the thickness of the layer that corresponds to the time that is required for the appearance of two minimum has the quite high value of 2150 Å. The application of the Laser Reflectance Interferometry method for the investigation of the effect of external plasma parameters on the etch rate of PET from He discharges is presented in the figure above. The application of the method in etch systems as in the case of deposition systems gives the capability of quite large number of etch rate measurements,  thus reducing significantly the time that is required for an optimization of a process.

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