Controlling Film Thickness Distribution by Magnetron Sputtering with Rotation and Revolution

Article Controlling Film Thickness Distribution by Magnetron Sputtering with Rotation and Revolution

Handan Huang, Li Jiang, Yiyun Yao, Zhong Zhang, Zhanshan Wang and Runze Qi *

MOE Key Laboratory of Advanced Micro-Structured Materials, Institute of Precision Optical Engineering (IPOE), School of Physics Science and Engineering, Tongji University, Shanghai 200092, China; [email protected] (H.H.); [email protected] (L.J.); [email protected] (Y.Y.); [email protected] (Z.Z.); [email protected] (Z.W.) * Correspondence: [email protected]; Tel.: +86-65984652

Abstract: The laterally graded multilayer collimator is a vital part of a high-precision diffractometer. It is applied as condensing reflectors to convert divergent X-rays from laboratory X-ray sources into a parallel beam. The thickness of the multilayer film varies with the angle of incidence to guarantee every position on the mirror satisfies the Bragg reflection. In principle, the accuracy of the parameters of the sputtering conditions is essential for achieving a reliable result. In this paper, we proposed a precise method for the fabrication of the laterally graded multilayer based on a planetary motion magnetron sputtering system for film thickness control. This method uses the fast and slow particle model to obtain the particle transport process, and then combines it with the planetary motion magnetron sputtering system to establish the film thickness distribution model. Moreover, the parameters of the sputtering conditions in the model are derived from experimental inversion to improve accuracy. The revolution and rotation of the substrate holder during the final deposition process are achieved by the speed curve calculated according to the model. Measurement results from the X-ray reflection test (XRR) show that the thickness error of the laterally graded multilayer Citation: Huang, H.; Jiang, L.; Yao, film, coated on a parabolic cylinder Si substrate, is less than 1%, demonstrating the effectiveness of Y.; Zhang, Z.; Wang, Z.; Qi, R. the optimized method for obtaining accurate film thickness distribution. Controlling Film Thickness Distribution by Magnetron Keywords: film thickness distribution; laterally graded multilayers; magnetron sputter deposition Sputtering with Rotation and Revolution. Coatings 2021, 11, 599. https://doi.org/10.3390/ coatings11050599 1. Introduction Since Gobel successfully applied the laterally graded multilayer collimator in X-ray Received: 26 April 2021 diffraction to modulate the divergence X-ray beam into a parallel X-ray beam [1,2], this type Accepted: 14 May 2021 Published: 19 May 2021 of collimator has had an important role in X-ray diffractometry for realizing high-resolution diffraction, powder diffraction, protein crystallography, and other measurements requiring

Publisher’s Note: MDPI stays neutral high-intensity and parallel X-ray beams. with regard to jurisdictional claims in The collimation is generally attributed to the parabolic shape of the collimator. How- published maps and institutional affil- ever, because of the abnormal shape, in order to realize high efficiency, a laterally graded iations. multilayer was developed whose period depth was customized laterally according to the grazing angle to satisfy Bragg’s law at each point of the optics. Due to the narrow angle bandwidth and the thin layer thickness, the thickness control is very important. The key difficulty in coating laterally graded multilayer films by magnetron sputtering is controlling the deposition rate at different positions on the substrate efficiently and Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. accurately [3–6]. Nagel proposed using the relative geometric position of the target and This article is an open access article the substrate to control the film thickness, and successfully coated a gradient film with a distributed under the terms and periodic film thickness of 2.5–2.9 nm on a 76.2 mm-long substrate [7]. However, in this conditions of the Creative Commons method, the film thickness gradient is limited by geometric conditions. Since the target Attribution (CC BY) license (https:// distance and angle are linearly changed, the film thickness change trend cannot be changed. creativecommons.org/licenses/by/ Villa designed a variety of masks for plane substrates, spherical substrates, and vertical 4.0/). cylindrical substrates [8]. Although this method can be used to deposit multilayer films

Coatings 2021, 11, 599. https://doi.org/10.3390/coatings11050599 https://www.mdpi.com/journal/coatings Coatings 2021, 11, 599 2 of 13

with different gradients in two dimensions, the mask reduces the sputtering efficiency of the target material, and the coating cycle is extended. Morawe designed a coating device with a one-dimensional linear movement of the target along the substrate thickness change direction, in order to coat a one-dimensional laterally graded multilayer film. Film with a periodic thickness of about 3–6 nm was coated on a 240 mm-long substrate, and the film thickness error was less than 1% [6]. However, the multilayer film had poor uniformity in non-gradient directions. We proposed a speed curve design for the substrate based on a planetary motion magnetron sputtering coating system to modulate the speed of the substrate passing through the target. The sputter deposition consists of four processes: (1) the gas discharge process in the cavity [9]; (2) target particle sputtering from the target surface, caused by electron bombardment [10]; (3) the transport of target particles from the target surface to the substrate surface [11]; and (4) the deposition of target particles on the substrate surface. The third process is key to the accuracy of the film thickness [12,13]. Several reports have established models to analyze the transport process. Ecker pointed out that factors such as the initial scattering, model boundaries, and current distribution on the cathode target and mass differences between metal atoms and gas molecules deserved to be analyzed in the distribution of film thickness and proposed a slow particle model that considered the scattering of particles after collision [14]. Petrov established the transport of sputtered particles from a circular target to a substrate parallel to the target based on Ecker’s theory. He found particles sputtered from a target and then deposited on a substrate can be divided into fast particles (direct particles, no collisions) and slow particles (scattered particles, collisions) according to the particle collision probability distribution [15]. Ekpe investigated the effects of the ring width, target sputtering power, operating air pressure, and target distance of circular targets on the thickness distribution of particle deposition on the substrate, based on Petrov’s equations [16]. Synthesizing the research results above, there is no doubt that parameters such as the power of the target material, the sputtering rate, and the sputtering angle distribution all play important roles in the particle transport model. However, the values in the literature above are based on historical experimental data or empirical models. To solve this problem, the parameters in our model were obtained through experimental inversion. In this paper, the entire deposition process is simplified step by step. The particle transport process is illustrated in Section2. Then, based on the planetary motion magnetron sputtering coating system, the film thickness control model, which contains the parameters obtained by experimental inversion to obtain a dependable result, is derived in Section3. Finally, the design of the variable speed profile for the substrate in the cavity and the fabrication process for the laterally graded tungsten/silicon(W/Si) multilayer on a parabolic cylinder Si substrate are described in Section4.

2. Particle Transport Model In this section, a fast and slow particle transport model is established through simulating the actual particle deposition rate distributions. However, a complete simulation would be time-consuming, and errors would accumulate [17–19]. Therefore, we simpliﬁed the former two sputtering processes into the sputtering rate distribution in the target area and sputtering angle distribution of the target particles, respectively. The effective rate of deposition on the substrate was assumed to be the same.

2.1. Assumptions of the Model Before simulating the actual particle deposition rate distributions, the particle transport model was derived based on the following assumptions. CoatingsCoatings.2021 2021, 11, 11, 599, x FOR PEER REVIEW 33 ofof 1314 Coatings. 2021, 11, x FOR PEER REVIEW 3 of 14

When the sputtered particles move from the target to the substrate, collision 2.1.1.When Collision the Probabilitysputtered particles move from the target to the substrate, collision scattering occurs randomly. The probability of this scattering process occurring is given scatteringWhen occurs the sputtered randomly. particles The probability move from theof this target scattering to the substrate, process collisionoccurring scattering is given by the following equation: byoccurs the following randomly. equation: The probability of this scattering process occurring is given by the − d following equation: = − dλ (1) Keλ , (1) Ke=− d , K = e λ , (1) where d is the movement distance, λ is the mean free path of a particle, and K is collision where d is the movement distance, λ is the mean free path of a particle, and K is collision wherescatteringd is probability the movement of sputtered distance, particles.λ is the mean free path of a particle, and K is collision scattering probability of sputtered particles. scattering probability of sputtered particles. 2.1.2. Sputtering Rate for the Target 2.1.2. Sputtering Rate for the Target As shown in Figure 1, the introduction of an electromagnetic field changes the As shown in Figure 1, the introduction of an electromagnetic field changes the distributionAs shown of incident in Figure electrons,1, the introduction and a saddle-shaped of an electromagnetic etch profile field appears changes on the the surface distri- distribution of incident electrons, and a saddle-shaped etch profile appears on the surface butionof the target, of incident which electrons, verifies andthat a the saddle-shaped sputtering rate etch on profile the target appears surface on the is surface not constant of the of the target, which verifies that the sputtering rate on the target surface is not constant target,[20]. Generally, which verifies finite that element the sputtering simulations rate ba onsed the on target the surfacemagnetic is not field constant distribution [20]. Gen- are [20]. Generally, finite element simulations based on the magnetic field distribution are erally,employed finite to element obtain simulationsthe sputtering based rate on distri the magneticbution of field the distributionetching ring, are but employed a series toof employed to obtain the sputtering rate distribution of the etching ring, but a series of obtaininitial parameters the sputtering need rate to distribution be assumed of in the the etching model. ring, In this but study, a series the of rate initial distribution parameters of initial parameters need to be assumed in the model. In this study, the rate distribution of needthe target to be assumedsurface was in thesimply model. assumed In this study,to be proportional the rate distribution to the etching of the target depth surface of the the target surface was simply assumed to be proportional to the etching depth of the wasetching simply ring assumed on the surface. to be proportional In order to toimprove the etching the simulation depth of the efficiency etching ringand onmodel the etchingsurface. ring In order on the to improvesurface. theIn order simulation to improve efficiency the and simulation model accuracy, efficiency the and Gaussian model accuracy, the Gaussian distribution was employed to simulate the etching ring depth accuracy,distribution the was Gaussian employed distribution to simulate was the empl etchingoyed ring to depthsimulate [18 ,21the]: etching ring depth [18,21]: [18,21]: 2x2 −x 2 − x2 Y = Ae −2c2c,2 (2)(2) YAe= 2 , (2) YAe= 2c , where A is the etching depth, c is the etching width coefficient, Y is the etching ring depth where A is the etching depth, c is the etching width coefficient,coefficient, Y is the etching ring depth distribution, and x is the coordinates of each point in the horizontal direction of the etching distribution, and x is the coordinates coordinates of each point in the the horizontal horizontal direction of the etching ring. The change in the distribution of the etching ring is reflected by the fitting parameters ring. The The change change in in the the distribution distribution of the the etch etchinging ring ring is is reflected reflected by the the fitting fitting parameters A and c because the surface shape will change with the coating of the film. The target used A and c because the surface shape will change with the coating of the film.film. The target used in this study was a rectangular shape, and the profile was assumed as shown in Figure 2. in this study was a rectangular shape, and the profileprofile waswas assumedassumed asas shownshown inin FigureFigure2 2..

Figure 1. Etched ring shape. Figure 1. Etched ring shape.

Figure 2. The rectangular target etching ring orbit. Figure 2. The rectangular target etching ring orbit. Coatings.Coatings 20212021,, 1111,, 599x FOR PEER REVIEW 44 of of 14 13

2.1.3.2.1.3. Sputtering Sputtering Angle Angle Distribution Distribution of of Target Target Particles Particles AtAt present, thethe cascadecascade collision collision model model based based on momentumon momentum transfer transfer theory theory can better can betterexplain explain the phenomenon the phenomenon of high-speed of high-speed particle particle bombardment bombardment causing causing sputtering sputtering of target ofparticles target particles [22,23]. When[22,23]. the When energy the of energy incident of particlesincident isparticles high, nearly is high, all nearly the emissions all the emissionsare normal are to normal the target to surface,the target and surface, practically and nonepractically are sidewise. none are On sidewise. the contrary, On the the contrary,numberof the particles number sputtered of particles near sputtered the target surfacenear the increases target surface when the increases energy ofwhen incident the energyparticles of isincident low. Broadway particles is assumed low. Broadway that the assumed angular that distribution the angular curve distribution is an elliptical curve isdistribution, an elliptical as distribution, shown in Figure as shown3[24]. in The Figure expression 3 [24]. The for thisexpression is: for this is: 2 mcos (θ) S(θ) = 2m cos(θ) , S(θ) = 222+− ,(3) (3) αα2 + ((11 −αα2)) coscos 2 (θθ)) where θ is the angle between the sputtering direction of the particles and the normal where θ is the angle between the sputtering direction of the particles and the normal direction (Z direction) of the target surface, α = m/n is the ellipticity coefficient, m is the direction (Z direction) of the target surface, α = m/n is the ellipticity coefficient, m is the length of the ellipse in the normal direction (Z direction) of the target surface, and n is the length of the ellipse in the normal direction (Z direction) of the target surface, and n is the minor axis of the ellipse in the tangential direction (X direction) of the target surface. This minor axis of the ellipse in the tangential direction (X direction) of the target surface. This study quotes this formula, where the major axis of the ellipse and the ellipse coefficient α study quotes this formula, where the major axis of the ellipse and the ellipse coefficient α areare set set as as the the parameters parameters to to be be fitted. fitted.

FigureFigure 3. 3. AngularAngular distribution distribution of of particle particle sp sputteringuttering on on the the target target surface. surface.

InIn addition, addition, this this study study did did not not discuss discuss the the growth growth process process of of the the film, film, assuming assuming that that thethe effective effective deposition coefficientcoefficient of of all all particles particles on on the the substrate substrate surface surface is theis the same. same. In theIn case of the barrier of the sleeve to the particles, we simply assumed it was a linear barrier; the case of the barrier of the sleeve to the particles, we simply assumed it was a linear this means when the substrate and the target are blocked by the sleeve, the contribution of barrier; this means when the substrate and the target are blocked by the sleeve, the the sputtered particles to the film deposition is not considered. contribution of the sputtered particles to the film deposition is not considered. 2.2. Target Particle Sputtering Model 2.2. Target Particle Sputtering Model In magnetron sputtering, due to the influence of gas pressure, deposited particles can be roughlyIn magnetron divided sputtering, into fast particlesdue to the (which influence do of not gas collide) pressure, and deposited slow particles particles (which can becollide). roughly Fast divided particles into representfast particles the particles(which do that not directly collide) reachand slow the substrateparticles surface(which collide).without anyFast collisionsparticles afterrepresent sputtering the particles from the targetthat directly surface, reach and slow the particlessubstrate represent surface withoutthe particles any thatcollisions reach the after substrate sputtering surface from by diffusive the target motion, surface, with collisionaland slow scattering.particles representZhang the presents particles the that number reach the of substrate fast particles surface and by number diffusive ofslow motion, particles with collisional deposited scattering.per unit area on the substrate by integrating all the positions on the target [25]: Zhang presents the number of fast particles and number of slow particles deposited RR per unit area on the substrate by integratingNf(xf, yf) =all the dpositionsNf on the target [25]: − d , (4) RR 2m cos(θ) Yt cos θ(e λ ) Nxy=(,)= d N dx dy fffα2+(1−α2) fcos2(θ) d2 t t − d 1 RR , (4) Ns(xs, ys) = dNs λ 2m cos (θ) 4π Yet cos θ() = D d RR 2m ts +2 − d ddxytt = 2222+−1 1 κY d λ e λ dx dy , (5) α 4π(1λ αt ) cos (θ)D2 d4d t t α2+(1−α2) ts d2 Coatings. 2021, 11, x FOR PEER REVIEW 5 of 14

1 Nxy(,)= d N sss4π  s Dts d + 22m − d , (5) = 11κ d λ λ Yexytttdd Coatings 2021, 11, 599 4πλ Dd24 5 of 13 α22+−(1 α ) ts d2

where Yt is the number of sputtered particles per unit area of the target, (,xytt ) is the where Yt is the number of sputtered particles per unit area of the target, (xt, yt) is the point point coordinates on the target, and κ is the diffusion coefficient (m2 / s) . coordinates on the target, and κ is the diffusion coefficient (m2/s). TheThe totaltotal thickness thickness distributiondistribution needsneeds toto considerconsider bothboth thethe fastfast andand slowslow particles.particles. FromFrom Equations Equations (4)(4) and and (5), (5), we we can can see see that that the the film film thickness thickness distribution distribution is is under under the the influenceinfluence of of geometric geometric parameters: parameters: the target–substrate the target–substrate distance, distance, temperature, temperature, gas pressure, gas electromagneticpressure, electromagnetic force, and otherforce, factors.and other factors.

3.3. Inversion Inversion 3.1.3.1. Planetary Planetary Motion Motion Magnetron Magnetron Sputtering Sputtering Coating Coating System System DepositionDeposition was was carried carried out out using using a customized a customized direct currentdirect current (DC) magnetron (DC) magnetron sputter- ingsputtering coating systemcoating withsystem substrate with substrate planetary planetary motion, motion, shown schematically shown schematically in Figure in4 Figure. The diameter4. The diameter of the deposition of the deposition chamber chamber is 1.4 m. Theis 1.4 system m. The was system equipped was withequipped two rectangu- with two larrectangular targets (127 targets mm × (127381 mm mm) × with381 mm) an adjustable with an adjustable height; this height; means this the means target–substrate the target– distancesubstrate was distance adjustable. was adjustable. In order to avoidIn order the to atomic avoid shadow the atomic effect, shadow which deteriorateseffect, which thedeteriorates quality of the the quality deposited of the ﬁlm, deposited a rectangular film, sleevea rectangular was used sleeve around was each used target around [26 ,each27]. Whentarget depositing[26,27]. When multilayer depositing ﬁlms, themultilay two targetser films, were the placed two symmetrically, targets were and placed the substratesymmetrically, performed and high-speedthe substrat rotatione performed at a speed high-speed of 100 rpm rota whiletion at revolving a speed aroundof 100 rpm the mainwhile axis revolving of the system. around Thethe rotationmain axis of of the the substrate system. holderThe rotation is controlled of the substrate by an indepen- holder dentis controlled rotation switch, by an independent while the revolution rotation is switch, controlled while by athe computer revolution program. is controlled During by the a revolution, the positioning accuracy is better than 1mm. The vacuum system is composed computer program. During the revolution, the positioning accuracy is better than 1mm. of two sets of mechanical pump and molecular pump; the pumping speed of the molecular The vacuum system is composed of two sets of mechanical pump and molecular pump; pump is 2000 L/s. the pumping speed of the molecular pump is 2000 L/s.

FigureFigure 4. 4.Planetary Planetary motion motion magnetron magnetron sputtering sputtering coating coating system. system.

Here,Here, thethe directdirect coordinatecoordinate systemsystem isis usedused toto describe describe thethe trajectories trajectories ofof the the unit unit pointspoints at at different different rotational rotational radii radii of of the the substrate substrate holder, holder, as as shown shown in in Figure Figure4. 4. Then, Then, the the

trajectoriestrajectories of of the the unit unit pointspoints atat differentdifferent rotationalrotational radiiradii ririof of the the substrate substrate holder holder can can be be expressed as: expressed as: xn = R cos θr + ri cos(θs(θr)) xR=+ cos θ rcos (θ (θ )), (6) yn =nrisrR sin θr + ri sin(θs(θr))  =+ , (6) yRnrisr sin θ r sin (θ (θ )) where R is the rotational radius of the substrate holder, θr is the revolution angle of the substrate holder, and θs(θ r) is the rotational angle of the substrate holder. The rotational speed of the substrate holder of this experimental device is adjustable 0 by setting the rotational angle change node θr , and this rotational angle at the rotation 0 angle rate ωr can adjust the rate curve, where the rotational speed between nodes and the rotational angle changes linearly. When calculating the running trajectory, we ﬁrst segment according to the rotation angle nodes; if there are n nodes, the number of segments is n − 1. 0 0 Therefore, with the start rotation angle θr (n), end rotation angle θr (n + 1), start rotation Coatings. 2021, 11, x FOR PEER REVIEW 6 of 14

where R is the rotational radius of the substrate holder, θr is the revolution angle of the

substrate holder, and θsr(θ ) is the rotational angle of the substrate holder. The rotational speed of the substrate holder of this experimental device is adjustable

by setting the rotational angle change node θr' , and this rotational angle at the rotation

angle rate ωr' can adjust the rate curve, where the rotational speed between nodes and the rotational angle changes linearly. When calculating the running trajectory, we first Coatings 2021, 11, 599 segment according to the rotation angle nodes; if there are n nodes, the number6 of 13of

segments is n−1. Therefore, with the start rotation angle θr'(n ) , end rotation angle + + θr '(n 1) , start rotation angle rate ωr '(n ) , and end rotation angle rate ωr '(n 1) of each 0 0 anglesegment, rate ωther (rotationn), and endrate rotationof each segment angle rate mayωr (ben +expressed1) of each as: segment, the rotation rate of each segment may be expressed as: =+ ωθr kb (7)

−+ ωr = kθ + b (7) = ωωrr'(nn ) '( 1) =− where k , bnknωθrr'( ) '( ) . ωθθ0 '(nn )ω− 0 '(+ 1) = r rr(n)− r (n+1) = ω 0( ) − θ 0( ) where k θ 0( )−θ 0( + ) , b r n k r n r ndθ r n 1 SinceSinceω ω== dθ ,, the the time time to to turn turn to toθ θis: is: dtt r r θ 0 Z r θr ( θ + ++) ( θ ( ) + ) =+=−+1 1 ln lnk (kbrθrrb )ln ln k( knbθr '(n )b ) t(θr) =ttntn(θrr)ddθr +θ t(n)() = − + ()t(n,) , (8)(8) θ 0 θ '(θn ) + + r (n) kr r kbθbr k kk wherewheret (tnn()) ==0 0is is the the start start time time of of this this segment segment and and the the end end time time of of thethe previousprevious segment,segment, whichwhich can can be be calculated calculated from from the the previous previous segment segment (for (for the the ﬁrst first segment segmentt (ntn()) ==0). 0). Since the rotation speed ωs is constant, the rotation angle: Since the rotation speed ωs is constant, the rotation angle:

θs(θ ) = ω t(θr), (9) θrsr(θ )=ωs st(θ r) , (9) Thus,Thus, thethe trajectoriestrajectories ofof thethe unitunit pointspoints atat differentdifferent rotationrotation radiiradii onon thethe substratesubstrate holderholder cancan bebe obtained,obtained, as as shown shown in in Figure Figure5 .5.

FigureFigure 5. 5.The The trajectorytrajectory ofof unitunit pointspoints atat different different rotation rotation radii radii on on the the substrate substrate holder. holder.

3.2.3.2. ExperimentalExperimental InversionInversion CombiningCombining thethe substratesubstrate motionmotion trajectorytrajectory ofof 3.13.1 andandthe thetarget target particle particle sputtering sputtering modelmodel ofof 2.2,2.2, thethe depositiondeposition thicknessthickness perper unitunit areaarea atat differentdifferent rotationrotation radiiradii onon thethe substrate can be simulated after the following parameters are fitted: substrate can be simulated after the following parameters are fitted: 1. W target sputtering coefficient K ; 1. W target sputtering coefficient WKW ; 2. W target etching ring etching width coefficient cW; 2. W target etching ring etching width coefficient cW ; 3. W target sputtering angular distribution coefficient αW; 4.3. W W particletarget sputtering momentum angu scatteringlar distribution cross-sectional coefficient area ασWW;; 5. W particle diffusion coefficient NW; 6. Si target sputtering coefficient KSi; 7. Si target etching ring etching width coefficient cSi; 8. Si target sputtering angular distribution coefficient αSi; 9. Si particle momentum scattering cross-sectional area σSi; 10. Si particle diffusion coefficient NSi. The above parameters are fitted by the following evaluation function:

Fn =|dmn − den|−den × ξ < 0, (10) Coatings. 2021, 11, x FOR PEER REVIEW 7 of 14

4. W particle momentum scattering cross-sectional area σW ;

5. W particle diffusion coefficient NW ;

6. Si target sputtering coefficient KSi ;

7. Si target etching ring etching width coefficient cSi ;

8. Si target sputtering angular distribution coefficient αSi ;

9. Si particle momentum scattering cross-sectional area σSi ;

10. Si particle diffusion coefficient NSi . The above parameters are fitted by the following evaluation function: Coatings 2021, 11, 599 =−−×< 7 of 13 Fdnmnenen|| d dξ 0 , (10)

where dmn and den are the model simulation period thickness and calibration sample

periodwhere d mnthicknessand den atare the the rotation model simulation radius r periodn , respectively. thickness andξ calibrationis the tolerance sample of period film thickness.thickness atIn the this rotation study, radiuswe set rnξ, respectively.equal to 0.2%ξ isto the make tolerance the film of thickness film thickness. close Into thisthe idealstudy, value. we set ξ equal to 0.2% to make the film thickness close to the ideal value. In order order to to reduce reduce the the difficulty difficulty of fitting, of fitting, the therevolution revolution rate rateis set is to set a constant to a constant speed whenspeed calibrating when calibrating the multilayer the multilayer film. At film. the At same the sametime, time,to ensure to ensure the accuracy the accuracy of the of experimentalthe experimental parameter parameter inversion, inversion, the thetwo two samples, samples, A Aand and B, B, are are coated coated at at different constant revolution speeds. For For the substrate, the revolution speeds passing over the W target were 0.200 and 0.211 rpm, and the revolution speeds passing over the Si target were 0.240 and 0.176 rpm, respectively. The depositiondeposition conditionsconditions areare shownshown inin TableTable1 .1.

Table 1.1. Deposition conditions. Target- Background Working Working Argon Target Sputtering Target- Revolution Background Working Working Argon Target Sputtering Voltage Substrate Revolution Pressure Gas Pressure Flow Material Power Voltage Substrate Radius Pressure Gas Pressure Flow Material Power Distance Radius Distance −4 5050 W W(W), (W), 150 327 V (W), 2.62.6× ×10 10−4 PaPa 99.999% 99.999% Ar Ar 2 2 mTorr mTorr 20 sccm W, W, Si Si 120.5120.5 mm mm 340 340 mm mm 150W W(Si) (Si) 436 V (Si)

Figure6 6 isis thethe normalizednormalized curvecurve ofof thethe thicknessthickness ofof SampleSample AA andand SampleSample BB fromfrom thethe XRR. (The test range is within a radius of 15 to 90 mm, and the film film thickness at the radius of 15 mm is the maximum value for normalization. The normalization processing rules in the following text are the same as here.) In th thee figure, figure, the gradient of Sample B is greater than that of Sample A. When Sample B is coated, the time of coating W is longer than that of Si, whilewhile forfor SampleSample A A it it is is vice vice versa. versa. It It is is inferred inferred that that the the film film thickness thickness gradient gradient of the of Wthe layer W layer is greater is greater than than that that of the of Sithe layer. Si layer.

Figure 6. Comparison of periodic thickness gradient curves of calibration Samples A-1# and B. The final fitting results are shown in Table2, and the film thickness distribution calculated by the model is compared with the experimentally obtained film thickness distribution, as shown in Figure7. The sputtering distribution is the undercosine type because the bombarding particles are argon ions, and their bombardment energy is too small to provide sustained recoil. As a result, the local spike projectile effect is obvious.

Table 2. Fitting parameters.

Sputtering of Target Momentum Sputtering Etching Width Diffusion Particles’ Angular Scattering Target Coefficient of the Coefficient of the Coefficient of Distribution Cross-Sectional Area Target Etched Ring Target Particles Coefficient of the Target Particle W 1.3 10.3 0.7 5.1 × 10−22 mm2 0.8 Si 1 19.2 0.6 6.2 × 10−22 mm2 0.7 Coatings. 2021, 11, x FOR PEER REVIEW 8 of 14

The final fitting results are shown in Table 2, and the film thickness distribution calculated by the model is compared with the experimentally obtained film thickness distribution, as shown in Figure 7. The sputtering distribution is the undercosine type because the bombarding particles are argon ions, and their bombardment energy is too small to provide sustained recoil. As a result, the local spike projectile effect is obvious.

Table 2. Fitting parameters.

Sputtering of Target Momentum Scattering Sputtering Coefficient of the Etching Width Coefficient of Particles’ Angular Diffusion Coefficient Target Cross-Sectional Area of Target the Etched Ring Distribution of Target Particles the Target Particle Coatings 2021, 11, 599 Coefficient 8 of 13 W 1.3 10.3 0.7 5.1 × 10−22 mm2 0.8 Si 1 19.2 0.6 6.2 × 10−22 mm2 0.7

FigureFigure 7. 7. ComparisonComparison of of the the film film thickness thickness distribu distributiontion calculated calculated by by the the fitting fitting model model and and experi- experimentalmental film thickness film thickness distribution. distribution.

InIn addition, addition, due due to to the the existence existence of of the the sleeve sleeve,, the the particle particle deposition deposition rate rate distribution distribution isis not not a a regular regular ellipse ellipse but but a a rectangular rectangular compr compression.ession. The The figure figure also also shows shows the the trajectory trajectory ofof the the unit unit points at different rotation radii. The The unit unit point point with with a larger rotation radius entersenters the effective depositiondeposition areaarea before before the the unit unit point point with with a smaller a smaller rotation rotation radius, radius, but butthe the rate rate of particle of particle deposition deposition is slower is slower at the at largerthe larger radius radius of rotation. of rotation. 4. Deposition Experiment 4. Deposition Experiment 4.1. Design Basis of the Speed Curve 4.1. Design Basis of the Speed Curve After obtaining the particle deposition rate distribution at the moving position of the substrateAfter obtaining holder, the the particle substrate deposition holder moving rate distribution speed curve at the can moving be designed position to change of the substratethe film thickness holder, the distribution substrate holder at different moving rotation speed radii.curve Atcan the be beginningdesigned to of change designing the filmthe variablethickness speed distribution curve, the at different variable speedrotation node radii. that At has the the beginning greatest of influence designing on the variabledeposition speed thickness curve, per the unit variable area at speed different no rotationde that radiihas the on thegreatest substrate influence holder on should the depositionbe determined. thickness per unit area at different rotation radii on the substrate holder should be determined.Figure8 shows the curve of the deposition thickness (deposition angular rate) per unitFigure revolution 8 shows angle the per curve unit of area the at deposition different rotation thickness radii (deposition on the substrate angular holderrate) per as unita function revolution of the angle revolution per unit angle, area at which different is faster rotation at small radii angleson the becausesubstrate it holder enters as the a functiondeposition of areathe firstrevolution at larger angle, rotation which radii. is However,faster at small when angles the smaller because radius it ofenters rotation the Coatings. 2021, 11, x FOR PEER REVIEWdepositionenters the area deposition first at area,larger the rotation deposition radii. However, angle rate when increases the smaller more rapidly radius of at rotation a small9 of 14 entersradius the than deposition at a large area, radius, the becausedeposition the an smallergle rate the increases radius of more rotation, rapidly the at closer a small the radiustrajectory than is at to thea large center radius, of the because target. the smaller the radius of rotation, the closer the trajectory is to the center of the target.

Figure 8. FilmFilm thickness deposition thickness distributiondistribution perper unitunit areaarea andand unitunit revolution revolution angle angle on onthe the substrate substrate holder holder at differentat different rotation rotation radii. radii.

During the revolution, increasing the revolution speed of the section with an angle

of θk to θjk effectively reduces the difference between the thickness at rk and the θ θ thickness at rj . For example, in the figure, 70 and 90 are the angles at which effective θ deposition starts at the rotation radii of 70 and 90 mm, respectively. 7090 is the angle at which the angular deposition rate is the same under the radii of rotation of 70 and 90 mm. θ θ Therefore, changing the revolution speed from 70 to 7090 can efficiently change the relative thickness between the rotation radius of 70 mm and the rotation radius of 90 mm. The thickness at a 90 mm radius of rotation decreases relative to the thickness at a 70 mm radius of rotation as the revolution speed of this section increases.

In order to find this key revolution angle θjk , a third-order polynomial was applied to fit the thickness growth curve: 4 ==+++i 23 Daaaaa irθθθθ01r2r3r, (11) i=0 Then, let the polynomial coefficient of the thickness growth fitting equation at the

larger radius be Ki and the polynomial coefficient of the thickness growth fitting

equation at the smaller radius be ki . When the following relation is satisfied, the rate of increase in thickness at the smaller radius begins to exceed that at the larger radius. ++23 + = + + 2 + 3 kkk123θθ k 4 KK 1 2 θθθ K 3 K 4; (12) KkbKkcKkdKk−=−=−=−,,, Let 44 33 22 11, then: 1 bbac2(3 −+2 3 ) θ =− 3a 1 3(ab−+ 23223322 9 abcad − 27 +−+ 4( b 3 ac ) +−+ ( 2 b 9 abcad − 27 ) )3 1 (13) 32× 3 a + , 1 (2−+babcadbacbabcad3223322 9 − 27 +−+ 4( 3 ) +−+ (2 9 − 27 ))3 With the above cubic polynomial, the key revolution angle for this study could be calculated according to Formula (13). Meanwhile, the revolution speed curve based on the

key angle nodes θk and θjk , which controls the motion of our magnetron sputtering coating system, can be derived as shown in Figure 9. Coatings 2021, 11, 599 9 of 13

During the revolution, increasing the revolution speed of the section with an angle of θk to θjk effectively reduces the difference between the thickness at rk and the thickness at rj. For example, in the figure, θ70 and θ90 are the angles at which effective deposition starts at the rotation radii of 70 and 90 mm, respectively. θ7090 is the angle at which the angular deposition rate is the same under the radii of rotation of 70 and 90 mm. Therefore, changing the revolution speed from θ70 to θ7090 can efficiently change the relative thickness between the rotation radius of 70 mm and the rotation radius of 90 mm. The thickness at a 90 mm radius of rotation decreases relative to the thickness at a 70 mm radius of rotation as the revolution speed of this section increases. In order to find this key revolution angle θjk, a third-order polynomial was applied to fit the thickness growth curve:

4 i 2 3 D = ∑ aiθr = a0 + a1θr + a2θr + a3θr , (11) i=0

Then, let the polynomial coefficient of the thickness growth fitting equation at the larger radius be Ki and the polynomial coefficient of the thickness growth fitting equation at the smaller radius be ki. When the following relation is satisfied, the rate of increase in thickness at the smaller radius begins to exceed that at the larger radius.

2 3 2 3 k1 + k2 + k3θ + k4θ = K1 + K2θ + K3θ + K4θ ; (12)

Let K4 − k4, b = K3 − k3, c = K2 − k2, d = K1 − k1, then:

1 b 2 3 (−b2+3ac) θ = − 3a 1 q 3 3a(−2b3+9abc−27a2d+ 4(−b2+3ac)3+(−2b3+9abc−27a2d)2) 1 (13) 3×2 3 a + 1 , q 3 (−2b3+9abc−27a2d+ 4(−b2+3ac)3+(−2b3+9abc−27a2d)2)

With the above cubic polynomial, the key revolution angle for this study could be Coatings. 2021, 11, x FOR PEER REVIEWcalculated according to Formula (13). Meanwhile, the revolution speed curve based10 on of the14

key angle nodes θk and θjk, which controls the motion of our magnetron sputtering coating system, can be derived as shown in Figure9.

(a) (b)

FigureFigure 9. 9. RevolutionRevolution speed speed dist distributionribution curve. curve. (a ()a )W W target; target; (b (b) )Si Si target. target.

4.2.4.2. Results Results TheThe film ﬁlm was was deposited deposited at at a a background background pressure ofof 2.62.6 ×× 1010−−4 4PaPa and and in in the the presence presence ofof Ar, Ar, 99.999% 99.999% purity. The workingworking pressurepressure was was controlled controlled at at 2 mTorr2 mTorr (1 mTorr(1 mTorr = 0.133 = 0.133 Pa). Pa).All All the the depositions depositions were were performed performed while while keeping keeping the the substratessubstrates at room room temperature temperature andand facing facing the the target target at at the the distance distance of of 120. 120.55 mm, mm, while while the the W W and and Si Si targets targets were were placed placed diagonally and adjusted to the same height. The sputtering power supply was 50 and 150 W, and the voltage was 327 and 436 V, respectively. The substrate of the finished product was a parabolic cylinder Si substrate produced by Zeiss. The size of it was 40 mm × 20 mm × 20 mm, and the surface roughness was 0.127– 0.155 nm (root mean square roughness). Since the substrate of the finished product was thick and the shape was parabolic, the surface reflectance could not be measured directly. In this regard, the method of placing a silicon wafer at a symmetrical position in the center of the finished product was applied, as shown in Figure 10, to determine the reflectance and film thickness distribution of the finished wafer by testing the silicon wafer. In order to confirm whether this method was reliable, we placed two calibrated silicon wafers symmetrically and tested the film thickness distribution curve. As shown in Figure 11, the maximum deviation of the overall film thickness was 0.005 nm, so we believe this method is reliable.

Figure 10. The placement of the finished product and the calibrated silicon wafer. Coatings. 2021, 11, x FOR PEER REVIEW 10 of 14

(a) (b) Figure 9. Revolution speed distribution curve. (a) W target; (b) Si target.

4.2. Results The film was deposited at a background pressure of 2.6 × 10−4 Pa and in the presence Coatings 2021, 11, 599 of Ar, 99.999% purity. The working pressure was controlled at 2 mTorr (1 mTorr =10 0.133 of 13 Pa). All the depositions were performed while keeping the substrates at room temperature and facing the target at the distance of 120.5 mm, while the W and Si targets were placed diagonally and adjusted adjusted to to the the same same height. height. The The sputtering sputtering power power supply supply was was 50 and 50 and 150 150W, and W, and the thevoltage voltage was was 327 327and and 436 436V, respectively. V, respectively. The substrate of the finishedfinished productproduct waswas aa parabolicparabolic cylindercylinder SiSi substratesubstrate producedproduced by Zeiss. The The size size of of it itwas was 40 40 mm mm × 20× mm20 mm × 20× mm,20 mm,and the and surface the surface roughness roughness was 0.127– was 0.127–0.1550.155 nm (root nm mean (root meansquare square roughness). roughness). Since thethe substratesubstrate of of the the finished finished product product was was thick thick and and the the shape shape was was parabolic, parabolic, the surfacethe surface reflectance reflectance could could not be not measured be measured directly. directly. In this regard,In this theregard, method the of method placing of a siliconplacing wafer a silicon at a wafer symmetrical at a symmetrical position inposition the center in the of center the finished of the productfinished wasproduct applied, was asapplied, shown as in shown Figure 10in, toFigure determine 10, to the dete reflectancermine the and reflectance film thickness and distributionfilm thickness of thedistribution finished of wafer the finished by testing wafer the siliconby testing wafer. the silicon In order wafer. to confirm In order whether to confirm this whether method wasthis reliable,method wewas placed reliable, two we calibrated placed two silicon calibrated wafers symmetricallysilicon wafers andsymmetrically tested the filmand thicknesstested the distribution film thickness curve. distribution As shown in curve. Figure As 11 , theshown maximum in Figure deviation 11, the of themaximum overall filmdeviation thickness of the was overall 0.005 film nm, thickness so we believe was 0.005 this method nm, so we is reliable. believe this method is reliable.

Coatings. 2021, 11, x FOR PEER REVIEW 11 of 14

Figure 10. The placement of the ﬁnishedfinished productproduct andand thethe calibratedcalibrated siliconsilicon wafer.wafer.

Figure 11. Comparison chart of the d-spacing ofof calibrationcalibration SamplesSamples 1#1# andand 2#.2#. Finally, a laterally graded multilayer film was coated with the revolution speed curve Finally, a laterally graded multilayer film was coated with the revolution speed curve in Figure9. The thickness distribution test curve is shown in the brown line in Figure 12, in Figure 9. The thickness distribution test curve is shown in the brown line in Figure 12, with a thickness error of less than 1%. Figure 13 shows the XRR test curve of the finished with a thickness error of less than 1%. Figure 13 shows the XRR test curve of the finished product with a test interval of 5 mm. The x value is the abscissa value of the parabola product with a test interval of 5 mm. The x value is the abscissa value of the parabola where the substrate is located. The larger the x is, the closer the substrate is to the center of where the substrate is located. The larger the x is, the closer the substrate is to the center rotation on the substrate holder, the better the film quality is, and the higher the reflectivity, of rotation on the substrate holder, the better the film quality is, and the higher the owing to fewer obliquely incident particles. reflectivity, owing to fewer obliquely incident particles.

Figure 12. Film thickness distribution curve (red line: target thickness curve; blue line: simulated thickness curve; brown line: actual coating thickness distribution curve).

Figure 13. XRR test curve.

In practical applications, what really affects the working efficiency of the mirror is the shift of the first-order peak position. As shown in Figure 14, the reflected light intensity at the working grazing angle of incidence is always greater than 95% of the peak-to-peak value. Coatings. 2021,, 11,, xx FORFOR PEERPEER REVIEWREVIEW 11 of 14

Figure 11. Comparison chart of the d-spacing of calibration Samples 1# and 2#.

Finally, a laterally graded multilayer film was coated with the revolution speed curve inin FigureFigure 9.9. TheThe thicknessthickness distributiondistribution testtest curvecurve isis shownshown inin thethe brownbrown lineline inin FigureFigure 12,12, with a thickness error of less than 1%. Figure 1313 showsshows thethe XRRXRR testtest curvecurve ofof thethe finishedfinished product with a test interval of 5 mm. The x value is the abscissa value of the parabola Coatings 2021, 11, 599 where the substrate is located. The larger the x is, the closer thethe substratesubstrate isis toto thethe center11center of 13 of rotation on the substrate holder, the better the film quality is, and the higher the reflectivity, owing to fewer obliquely incident particles.

Figure 12. Film thickness distribution curve (red line: target thickness curve; blue line: simulated Figure 12. Film thickness distribution curve (red line: targettarget thicknessthickness curve;curve; blueblue line:line: simulatedsimulated thickness curve; brown line: actual coating thickness distribution curve). thicknessthickness curve;curve; brownbrown line:line: actualactual coatingcoating thicknessthickness distributiondistribution curve).curve).

FigureFigure 13. 13. XRRXRR test test curve. curve.

InInIn practicalpractical practical applications,applications, applications, whatwhat what reallyreally really affectsaffects affects thethe the workingworking working efficiencyefficiency efficiency ofof ofthethe the mirrormirror mirror isis thetheis the shiftshift shift ofof thethe of first-orderfirst-order the first-order peakpeak peak position.position. position. AsAs shshown As shownin Figure in 14, Figure the reflected 14, the reflectedlight intensity light Coatings. 2021, 11, x FOR PEER REVIEWatintensity the working at the grazing working angle grazing of incidence angle of is incidencealways greater is always than greater95% of the than peak-to-peak 95%12 of of the 14 at the working grazing angle of incidence is always greater than 95% of the peak-to-peak value.peak-to-peak value.

FigureFigure 14. 14. TheThe first-order first-order peak of thethe XRRXRR testtest curvecurve (the (the straight straight line line is is the the working working grazing grazing incident incidentangle of angle the position). of the position). 5. Conclusions 5. Conclusions In this paper, we proposed an effective method for controlling the film thickness In this paper, we proposed an effective method for controlling the film thickness distribution. The aim of the study was to find a technology for fabricating laterally graded distribution. The aim of the study was to find a technology for fabricating laterally graded multilayers required for parabolic collimators, applied in a high-precision diffractometer. multilayers required for parabolic collimators, applied in a high-precision diffractometer. A fast and slow particle transport model was established based on Petrov’s theory, which considered the collisions that occur during the deposition. In order to improve the simulation efficiency and model accuracy, the Gaussian distribution was employed to simulate the etching ring depth, and the sputtering distribution was assumed to be the undercosine type. A film thickness control model based on the planetary motion magnetron sputtering coating system is presented. The parameters in the model were set as objects to be fitted, and experimental results were introduced to invert the sputtering parameters, which improved the accuracy of the model while reducing the difficulty of model from a practical point of view. We coated two W/Si multilayer film samples, A and B, at different constant revolution speeds, to ensure the accuracy of the experimental parameter inversion. We also analyzed the film thickness at different radii on the substrate holder under different revolution angles. A third-order polynomial was applied to fit the thickness growth curve to find the key revolution angle, in order to derive the speed curve that controls the motion of our magnetron sputtering coating system and builds the required film thickness distribution. Finally, we studied the thickness error of the W/Si ML deposited onto the parabolic cylinder Si substrate, and found it was less than 1%. It was shown that the reflected light intensity at the working grazing angle of incidence is always greater than 95% of the peak- to-peak value from the XRR test curve. In conclusion, it should be noted that these experiments indicate the prospect of using the planetary motion magnetron sputtering coating system and correcting for the fabrication of the laterally graded multilayer.

A fast and slow particle transport model was established based on Petrov’s theory, which considered the collisions that occur during the deposition. In order to improve the simulation efficiency and model accuracy, the Gaussian distribution was employed to simulate the etching ring depth, and the sputtering distribution was assumed to be the undercosine type. A film thickness control model based on the planetary motion magnetron sputtering coating system is presented. The parameters in the model were set as objects to be fitted, and experimental results were introduced to invert the sputtering parameters, which improved the accuracy of the model while reducing the difficulty of model from a practical point of view. We coated two W/Si multilayer film samples, A and B, at different constant revolution speeds, to ensure the accuracy of the experimental parameter inversion. We also analyzed the film thickness at different radii on the substrate holder under different revolution angles. A third-order polynomial was applied to fit the thickness growth curve to find the key revolution angle, in order to derive the speed curve that controls the motion of our magnetron sputtering coating system and builds the required film thickness distribution. Finally, we studied the thickness error of the W/Si ML deposited onto the parabolic cylinder Si substrate, and found it was less than 1%. It was shown that the reflected light intensity at the working grazing angle of incidence is always greater than 95% of the peak-to-peak value from the XRR test curve. In conclusion, it should be noted that these experiments indicate the prospect of using the planetary motion magnetron sputtering coating system and correcting for the fabrication of the laterally graded multilayer.

Author Contributions: Conceptualization, H.H., R.Q. and Z.W.; methodology, R.Q.; software, H.H. and L.J.; validation, R.Q. and Z.W.; formal analysis, H.H.; investigation, H.H., R.Q., L.J. and Z.Z.; resources, R.Q., Z.Z. and Z.W.; data curation, H.H. and Y.Y.; writing original draft preparation, H.H.; writing review and editing, R.Q.; visualization, H.H.; supervision, R.Q., Z.Z. and Z.W.; project administration, R.Q.; funding acquisition, R.Q., Z.Z. and Z.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Research Program of Science and Technology Commission of Shanghai Municipality (20142200500), National Natural Science Foundation of China (12027810, 11805212, 12003016). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conﬂicts of Interest: All authors have read and approved this version of the article, and due care has been taken to ensure the integrity of the work. Neither the entire paper nor any part of its content has been published or has been accepted elsewhere.

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