**5.1. The geometry of the proposed technique**

The geometry is shown in **Figure 18**. The laser beam (2) is incident at the target (1). The mask (4) and translationally moving substrate (5) are placed perpendicularly to the axis of the plasma torch. The arrow shows the direction of the substrate displacement.

In order to provide an equal thickness of the film deposited onto the uniformly moving substrate, the following condition should be met: equal amounts of the substance must be deposited per unit time on a unit length section placed along the direction of the substrate displacement and this condition should be satisfied over the entire width of the substrate. Taking into account the peculiarities of angular distribution of the mass‐transfer rate in the plasma torch [6], we propose two configurations of the slit in the mask (**Figure 19**). In both cases, the plasma torch axis passes through the geometrical center of the mask. The displace‐ ment of substrate occurs in the *x*‐direction. The dashed line in **Figure 19a** indicates the line along which the mass‐transfer rate of the deposited matter is constant. A rectangular slit (or slits) curved along dashed line provides, obviously, the uniformity of the film thickness. The second configuration of the mask (**Figure 19b**) takes into account the fact that the angular distribution of the mass‐transfer rate in the plasma plume is given by the function () = cos() [6], that is, the greater is the angle (**Figure 18**), the lower is the mass‐transfer rate; thus, in order to provide the uniformity of the film thickness over the substrate width, the greater should be the width of the slit in the mask.

**Figure 18.** Geometry of deposition onto a moving ribbon.

**Figure 19.** Configurations of slits in the mask.

### **5.2. Calculation of the configuration of slit in the mask**

We now give a method for calculation of the configuration of the slit in the mask (**Figure 19b**) providing the thickness homogeneity of the film deposited onto the substrate moving trans‐ lationally at a constant velocity. The width of the slit *d*(*y*) is determined from the relation ()() = const, where () is the average thickness on the section *d*(*y*) of the film deposited on the resting substrate per unit time. The quantities () are determined from the experi‐ mentally determined function *D*(*x*,*y*). The angular distribution of the ablated material depends on many factors. In our experiments, we tried to change only one parameter, leaving the others unchanged. We chose the laser spot dimensions (*S*) and the laser fluence (*F*) as variable parameters. The values of these parameters and the obtained films relative thickness are listed in **Table 1**. The data of films thickness are fitted by the function *D*(*x*, *y*) = *A*cos*Px + 3*(*θx*) cos*Py + 3*(*θy*). The used designations are clear from **Figure 20**. The obtained values of parameters *A*, *px* and *py* are presented in **Table 2**. Now, it is possible to initiate calculation of the configu‐ ration of the slit.


a Obtained films relative thickness at *θ* = 0.

Taking into account the peculiarities of angular distribution of the mass‐transfer rate in the plasma torch [6], we propose two configurations of the slit in the mask (**Figure 19**). In both cases, the plasma torch axis passes through the geometrical center of the mask. The displace‐ ment of substrate occurs in the *x*‐direction. The dashed line in **Figure 19a** indicates the line along which the mass‐transfer rate of the deposited matter is constant. A rectangular slit (or slits) curved along dashed line provides, obviously, the uniformity of the film thickness. The second configuration of the mask (**Figure 19b**) takes into account the fact that the angular distribution of the mass‐transfer rate in the plasma plume is given by the function () = cos() [6], that is, the greater is the angle (**Figure 18**), the lower is the mass‐transfer rate; thus, in order to provide the uniformity of the film thickness over the substrate width,

166 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

the greater should be the width of the slit in the mask.

**Figure 18.** Geometry of deposition onto a moving ribbon.

**Figure 19.** Configurations of slits in the mask.

**5.2. Calculation of the configuration of slit in the mask**

We now give a method for calculation of the configuration of the slit in the mask (**Figure 19b**) providing the thickness homogeneity of the film deposited onto the substrate moving trans‐ **Table 1.** Variable parameters of a laser deposition.

**Figure 20.** Schematic of deposition geometry. *S* is the laser spot.


a <sup>2</sup> <sup>=</sup> <sup>1</sup> ∑ = 1 ( , − ) 2.

b Volume of a matter deposited on the ribbon through the mask in a unit time.

**Table 2.** Parameters of a function *D*(*x*, *y*) and corresponding uniform deposition slit.

If the origin of the coordinates is located at the point that corresponds to *θx* = *θy* = 0, and a value *x*o, *y*o is chosen (where *y*o stands for the half‐width of the ribbon and *x*o is determined so that the film thickness is not too small), then the amount of the matter deposited on the segment [(‐*x*o, *y*o), (*x*o, *y*o)] in a unit time will be given by the expression

$$m = \rho dy \int\_{-\chi\_0}^{+\chi\_1} D\left(\mathbf{x}, \boldsymbol{\nu}\right) d\mathbf{x} \tag{9}$$

where *D*(*x*, *y*) is the thickness profile of the film, *ρ* is the matter density, and *dy* is the width of the section. The thickness profile of the film deposited from a point source on a stationary substrate is usually determined by expression *D*(*θ*) = *A*cos*p + 3*(*θ*) [8]. However, when the focal spot has real sizes that differ in different directions, an asymmetry of function *D*(*θ*) is observed. In this case, the profile of the thickness of the film can be described by the expression

$$D\left(\theta\right) = A\cos^{p\_x+3}\left(\theta\_x\right)\cos^{p\_y+3}\left(\theta\_y\right) = A\left(\frac{h}{\sqrt{h^2+x^2}}\right)^{p\_x+3}\left(\frac{h}{\sqrt{h^2+y^2}}\right)^{p\_y+3} \tag{10}$$

Upon substitution of Eq. (10) into Eq. (9), we shall receive an equation for the amount of matter *m*(*y*) deposited in unit time on the segment [(‐*x*, *y*), (*x*, *y*)] for any value of *y*. From the require‐ ment that *m*(*y*) is constant for all *y* from ‐*y*o to *y*o and equals *m*, it is possible to determine the values *x*(*y*) at which this condition is fulfilled:

$$m\_y = m = \rho d\mathbf{v} \int\_{-\mathbf{x}\_0}^{+\mathbf{x}\_0} \left(\frac{h}{\sqrt{h^2 + \mathbf{x}^2}}\right)^{p\_x + 3} \left(\frac{h}{\sqrt{h^2 + \mathbf{y}^2}}\right)^{p\_y + 3} d\mathbf{x} \tag{11}$$

The function *x*(*y*) will determine the profile of a slit that provides uniform deposition of matter on the ribbon. At uniform translation of the ribbon with speed *v*, the thickness of the film *D* will be determined by the expression

$$D = \frac{A}{2\chi\_o \nu} \int\_{-y\_0 - \chi\_0}^{+y\_0 + \chi\_0} \int\_{\gamma} \left(\frac{h}{\sqrt{h^2 + x^2}}\right)^{p\_x + 3} \left(\frac{h}{\sqrt{h^2 + y^2}}\right)^{p\_y + 3} dx dy \tag{12}$$

The computer calculation of the configuration of the slit has been performed. The results are presented in **Figure 21**.

**Figure 21.** The calculated slit configuration for deposition No 5.

**No. A** *px* **+ 3** *py* **+ 3 χ2a Area of a**

a <sup>2</sup> <sup>=</sup> <sup>1</sup>

b

 ∑ = 1 (

, −

) 2.

Volume of a matter deposited on the ribbon through the mask in a unit time.

[(‐*x*o, *y*o), (*x*o, *y*o)] in a unit time will be given by the expression

( ) ( ) ( )

values *x*(*y*) at which this condition is fulfilled:

cos cos

*p p*

*x y*

q

3 3

qq

*x y h h D A <sup>A</sup>*

**Table 2.** Parameters of a function *D*(*x*, *y*) and corresponding uniform deposition slit.

If the origin of the coordinates is located at the point that corresponds to *θx* = *θy* = 0, and a value *x*o, *y*o is chosen (where *y*o stands for the half‐width of the ribbon and *x*o is determined so that the film thickness is not too small), then the amount of the matter deposited on the segment

( )

where *D*(*x*, *y*) is the thickness profile of the film, *ρ* is the matter density, and *dy* is the width of the section. The thickness profile of the film deposited from a point source on a stationary substrate is usually determined by expression *D*(*θ*) = *A*cos*p + 3*(*θ*) [8]. However, when the focal spot has real sizes that differ in different directions, an asymmetry of function *D*(*θ*) is observed.

*m dy D x y dx* (9)

3 3

(10)

+ +

*y x*

*p p*

22 22

ç ÷ ç ÷ è ø + + è ø

*hx hy*

1

*x*

r , +

0

*x*

In this case, the profile of the thickness of the film can be described by the expression

<sup>+</sup> <sup>+</sup> æ ö æ ö ç ÷ = = ç ÷

Upon substitution of Eq. (10) into Eq. (9), we shall receive an equation for the amount of matter *m*(*y*) deposited in unit time on the segment [(‐*x*, *y*), (*x*, *y*)] for any value of *y*. From the require‐ ment that *m*(*y*) is constant for all *y* from ‐*y*o to *y*o and equals *m*, it is possible to determine the


1 66.83 8.45 5.8 2.58 24.76 713 2 80.13 8.2 5.3 4.46 26 925 3 80.28 7.09 4.31 5.7 30.39 1220 4 107 7.8 4.5 6.71 28.11 1403 5 147.33 10.03 5.23 6.67 22.24 1219 6 117.51 11.55 7.69 4.99 18.25 700 7 76.22 12.84 6.86 0.96 16.56 340

168 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

**slit, cm2**

*V***b, a.u.**
