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Thin film microstructure and thermal transport simulation using 3D-films
Thin Solid Films 391 Ž2001. 88᎐100
Thin film microstructure and thermal transport simulation using 3D-films T. Smy a,U , D. Walkey a , K.D. Harris b , M.J. Brett b b
a Department of Electronics, Carleton Uni¨ ersity, Ottawa, ON, Canada K1S 5B6 Department of Electrical and Computer Engineering, Uni¨ ersity of Alberta, Edmonton, AB, Canada T6G 2G7
Received 5 July 2000; received in revised form 12 January 2001; accepted 26 March 2001
Abstract Simulations of the flow of heat through porous thin films by the three-dimensional microstructural thin film simulation framework 3D-Films are discussed in this paper. For each simulation, the film structures are generated by the thin film growth model 3D-Films and then used to generate a finite difference based thermal model by the program 3D-FilmsrThermal. This program creates a block based data structure using a 3D quadtree mesh and subsequently solves for the steady state heat flow through the film structure. In this paper the film growth and thermal models are used to analyze and suggest optimization of porous thermal barrier coatings produced by glancing angle deposition techniques. The paper also deals with the determination of the accuracy and efficiency of the thermal model. Studies on the effect of reducing the resolution of the simulated film for less memory intensive thermal simulations are presented, indicating that a reduction of the resolution by a factor of 3 and the number of solution variables by as much as a factor of 27 is feasible. The simulations Of ZrO 2 thermal barriers are compared to experimental results with a relatively close match being obtained. Finally, the simulator is used to analyze the effectiveness of a number of potential thermal barrier structures produced by glancing angle deposition techniques. These results suggest that the most effective thermal barrier film microstructures will be porous films consisting of slanted posts or large pitch helices. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Computer simulation; Evaporation; Thermal properties; Microstructure
1. Introduction The thermal properties of thin films are of significant interest in a number of situations, such as heat flow through the metallization layers of integrated electronic circuitry, micromachined devices including temperature sensors and thin film optical devices, and thermal barrier coatings ŽTBCs. w1x. In thermal barrier coatings, for example, the thermally insulating proper-
U
Corresponding author. Tel.: q1-613-520-3967; fax: q1-613-5205708. E-mail address: [email protected] ŽT. Smy..
ties of a material are exploited to protect surfaces from heat damage. Knowledge of the thermal resistivity dependence on film properties is vital to understanding how it may be improved. In a thin film, thermal properties can vary significantly from that of bulk materials due to microstructure differences. Generally, thin films are deposited using a method, such as sputtering or evaporation, which occurs under conditions far from equilibrium and leads to highly inhomogeneous films. The films are often polycrystalline and, depending on the deposition conditions, may be quite porous. The thermal resistivity of a film is a function of its porosity, connectivity and microstructure. However, due to the size scale involved,
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the experimental determination of the thermal resistivity is challenging, as few techniques provide effective measurements w2x. An alternative method of characterizing and optimizing deposited films is by physically modeling the heat flow through a simulated film structure. Two modeling tools are required: Ž1. a film deposition modeling code; and Ž2. a heat flow solver. This paper presents the use of 3D-Films for modeling the deposition of sputtered and evaporated films and the simulator 3D-FilmsrThermal for modeling the heat flows within the simulated films. As an example, the programs are used to analyze porous films deposited by glancing angle deposition ŽGLAD. w3x for TBC applications. GLAD is physical vapor deposition with carefully controlled substratersource geometry, during which substrate movement may occur about two axes: the first adjusts the angle at which the incident flux arrives at the substrate, while the second rotates the substrate about an axis perpendicular to its surface. At highly oblique deposition angles, extreme shadowing begins to introduce microstructure and porosity into the deposited thin films. During nucleation for growth onto substrates at low temperatures Žrelative to film melting point., any points of the substrate which are shadowed by random three-dimensional variations are prevented from receiving any additional vapor flux. As the film grows, this effect continues, resulting in a microstructure consisting of many, laterally unconnected columns growing outward from the substrate and separated by voids. Since atomic shadowing is the driving mechanism, higher angles of incidence lead to more
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pronounced porosities w5x. For incident flux angles well over 80⬚ from the substrate normal, it has been shown possible to produce porous thin films having densities as little as 15% of the bulk. Without substrate rotation a film grown at glancing incidence consists of many thermally isolated posts or columns. These columns tend to grow in a vapor source seeking manner w3x. However, since it is possible to rotate the substrate throughout the deposition, the direction of column growth may be continuously altered, leading to a variety of interesting microstructures. Using the GLAD technique, structures such as chevron, post and helical have been created w3,4x. A typical GLAD helical SiO 2 film is shown in Fig. 1, illustrating the complex 3D nature of GLAD film microstructure. Traditionally, TBCs are fabricated with a low thermal conductivity material such as ZrO 2 . They are deposited by a method such as plasma spraying which leads to a low density film of high thermal resistance w6x. A significant problem with traditional deposition techniques is thermal expansion stresses causing delamination. Stress problems have been alleviated somewhat by fabrication by electron beam evaporation of thermal barriers with a tight columnar structure, but at the cost of reduced thermal resistance w7x. It has been suggested that GLAD ZrO 2 films could be engineered to provide not only thermally optimized TBC barriers but also films with implicit stress relief mechanisms w8x. The modeling of GLAD thermal barriers is a significant challenge. The inherently complex, 3D nature of GLAD film microstructures and the fine column and grain structure requires a fully three-dimensional simu-
Fig. 1. A helical GLAD SiO 2 film: Ža. SEM Žb. 3D-Films simulation.
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lator. The simulator 3D-Films has been previously used to model both GLAD films and metal deposition over topography w9᎐11x. It is based on a simple cubic data structure in which each cube represents a small amount of film material. Deposition is modeled as a ballistic, Monte Carlo process incorporating nucleation, selfshadowing, adatom mobility and a variety of additional physical effects. 3D-Films provide a completely 3D depiction of the film, its microstructure, and information on local film density and differential and total surface area. The modeling of the heat flow is performed using a new tool, 3D-FilmsrThermal,1 which solves the heat flow equations through the simulated film with a finite difference approach. The heat flow code is based on a 3D thermal simulation tool known as Atar used to solve for the heat flow in integrated device semiconductor devices w12x. 2. Modeling framework
Fig. 2. Basic simulation geometry showing a simple flat substrate, three initial grains and a single particle transport to the surface and then to a growing grain.
2.1. Film growth A major concern in a 3D microstructural simulator is the memory requirements of the model. In order to represent the film structure with sufficient accuracy, a linear resolution of 300᎐500 pixels is required in each dimension. This factor combined with the need to represent the internal structure of the film Žessentially a requirement for a ‘volume’ representation rather than a ‘surface’ representation . entails a significant challenge. A resolution of 500 = 500 = 500 pixelsrsimulation using a single 3D array as a data structure would require 1.25= 10 9 cells Žassuming a minimal data structure for each cell of 10 bytes.. This is impractical on typical work stations. In order to minimize memory requirements, a data structure consisting of a two-tiered array is used. The first tier in the structure is a ‘course’ 3D array. Each element of the course array is pointer to a 3D ‘fine’ array structure. Typically, each fine array is 20᎐50 elements on a side. A ‘fine’ array consists of pointers to individual cube data structures, containing information on the block material, surface state, material type, surface normal Žif appropriate. and deposition history. This approach allows for an efficient use of memory, particularly in topographic features such as a vias or in porous films. By this method, resolutions of 500 = 500 = 500 are possible requiring approximately 300 Mbytes of memory. At the beginning of a simulation, an initial substrate is defined and parameters controlling the deposition are provided. The substrate definition is simply an
1
Parties interested in using or acquiring the code should contact T. Srny at [email protected]
initial specification of cubes and can be set at run time to a variety of configurations including a flat substrate, trenches, vias, and steps. During simulation, particles are launched ballistically from slightly above the growing film and follow straight line trajectories until they strike either the substrate or the film. The particles are then permanently incorporated into the film after searching the nearby exposed surface for positions of minimum surface curvature dependent chemical potential. This process is shown schematically at a very low resolution in Fig. 2. The combination of ballistic shadowing and short range surface diffusion successfully accounts for the formation of the columnar thin film microstructure characteristic of high melting point materials at typical deposition temperatures. This is illustrated in Fig. 1 where both a 3D-Films simulation and a scanning electron microscope ŽSEM. image of a helical GLAD structure are shown. The figure clearly shows the ability of the simulator to predict the complex structure of a GLAD film. The boundaries of the simulation must be handled carefully during film growth, diffusion and ballistic transport. Two cases are allowed: periodic or mirrored boundaries. If a boundary is specified as periodic, a particle traveling across the simulation from left to right will re-enter on the left side of the simulation region and the film on the left edge will match that on the right. For a mirrored boundary, particles will ‘bounce’ off the boundary and the film structure will be mirrored at the edge. The primary use of the mirrored boundary condition is to allow more memory efficient simulations. However, the use of a mirrored edge assumes that the initial flux distribution is isotropic across that edge. For incident flux in GLAD, this is not
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usually true and periodic boundary conditions are required. 2.2. Thermal modeling The final product of a film growth simulation is an aggregation of cubes describing a complex 3D structure. Thermal modeling of heat flow through this structure is best done by a numerical solver. The physics involved in solving for the conductive heat flow require the solution of the following partial differential equation w13x: C
⭸T s ⵜ Ž Ž T . ⵜT . q g Ž x, y, z,t . ⭸t
Ž1.
where ŽT . is the temperature-dependant thermal conductivity, C the thermal capacity of the material, the material density, and g Ž x, y, z . describes the heat generation. If there is no heat generated within the film, and we are interested in a steady state solution, this simplifies to: ⵜ Ž Ž T . ⵜT . s 0
Ž2.
Complications in solving this equation within a thin film are caused mainly by the complex geometry of the film and inhomogeneities arising from the use of multiple materials. To produce a numerical solution an appropriately discretized 3D model must be built of the film, boundary conditions must be applied, a mathematical model must be extracted and then the model solved using an appropriate technique. Although the natural data structure of stacked cubes used in the growth model could be exploited without alteration for performing a numerical solution, it generally contains too many blocks. Each block would be a solution variable and the large number of blocks in a typical film
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simulation would be prohibitive for execution on a typical workstation. To alleviate this problem two approaches were taken. The first was to post process the final film structure and reduce its resolution. It was found that although the film growth needs to be simulated with approximately 0.05 m for a block dimension, the thermal properties could be simulated accurately at lower resolutions. Therefore, a code was written that reads the final film structure and agglomerates blocks reducing the film resolution. Typically, a reduction of eight blocks Ž2 = 2 = 2. to 1 or 27 Ž3 = 3 = 3. blocks to 1 is used. A new block is formed if the density in the local region is greater then 0.5. The second method of reducing the number of blocks is to form a three-level 3D quadtree mesh Žalso known as an octree mesh. of the structure from the simulated film. A three-level 2D quadtree mesh is shown in Fig. 3a, illustrating the primary characteristic of the mesh. The mesh is formed by creating blocks in such a manner that the maximum number of smaller neighbors is two on each side in 2D, or four on each face in a 3D mesh. This method of forming the block structure allows for very quick 3D refinement of the mesh structure at interfaces and other regions of significant structure. It effectively and dramatically reduces the number of blocks used to describe the film, while still allowing for a reasonable number of local block topographies. This significantly simplifies coding. The procedure used to create the quadtree mesh was to determine if a cubic area consisted of a single materialrgrain and if the rules for generating the quad tree mesh would not be broken if a single block was formed. When creating the mesh of a porous film a choice needs to made as to if it is needed to mesh the void regions. If the conductivity of the film is significantly larger than that of the voided regions they can be ignored speeding up the simulation due to a smaller
Fig. 3. Ža. Example of a 2D quadtree mesh. Žb. Representative heat flows between 2D blocks which are represented by thermal resistors. Ti and Tn are the temperatures at the nodes i and n, respectively, and rthn r i the thermal resistance between node n and i.
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number of blocks. For the simulations in this paper the conductivity of air is at least 50 times smaller than the conductivity of the film and the voids were left unmeshed.
reformulated by solving for a ‘new’ temperature at each node giving, N
Tnk s Tnky1 q FSOR
is1
2.3. Extraction of a thermal network N
After creation of the 3D quadtree mesh model, a thermal network consisting of resistors is extracted. Each block is visited and, using the topology of the connections with adjacent blocks, a network of resistors to the neighboring nodes is created. As the material properties are, in general, temperature-dependent the resistor values are a function of the node temperature. In Fig. 3 the heat flows for a single 2D block are shown, wherein each heat flow would be represented by a thermal resistor. The value of each resistor is calculated using the common cross-sectional area, the size of the two blocks and the local thermal conductivity. The formulation of the steady state heat flow problem in this manner is basically a finite difference approach, whereby for each block the heat flow must have zero divergence. The solution of this for steady state heat flow therefore requires that the sum of heat flows through each resistor to the node at the center of a block equals zero as given by, N
Ý is1
Ti y Tn q pn s 0 rthn r i
Ž3.
where, Ti and Tn , are the temperatures at nodes i and n, respectively, rthn r i the thermal resistance between node n and i and pn , the thermal power being generated in the block Žusually zero.. A number of boundary conditions can be placed on the model. Regions of the model, such as the top or bottom, can be designated to be at a fixed temperature. Constant power generation can be specified in volumes. The boundary conditions on the vertical sides of the simulation region are either periodic or mirrored to match the previous growth simulation. Once the thermal model has been extracted and the boundary conditions set, the network must be solved. Currently two steady state internal solvers are available. A direct solver can be used which sets up a global sparse array and subsequently solves the linear set of equations representing the thermal resistor network. If the temperature dependence of the thermal conductivity of the film is to be considered, an iterative technique is applied using successive over relaxation ŽSOR. and the repeated solution of the linearized set of equations using the direct solver. A second solver based on explicit SOR of the nodal temperature equations has also been implemented and is particularly useful for large problems. Eq. Ž3. can be
=
žÝ
is1
Ti ky 1 q pn rthn r i
žž / / Ý
1 y Tnky 1 rthn r i
/ Ž4.
where k is the iteration number and FSOR the relaxation parameter Ž1.25 was found to be a good value.. This solver implicitly handles the temperature dependence of the material properties by updating the resistor values at each iteration. Once the solution has been found, the maximum temperature is obtained and a grayscale or color representation of the temperature distribution can be displayed. 2.4. Viewing and post processing The efficient viewing and analysis of the simulated film is of significant importance and is also a considerable challenge. It is obviously necessary to view the film and evaluate its microstructure. In 2D this is fairly simple, however, in 3D a sophisticated approach is required. A 3D film must be rendered in a manner such that coloration and perspective makes the image comprehensible to the viewer. A film produced by 3D-Films contains a significant amount of detail and structure; a typical film consists of millions of cubes. Efficiently rendering such a structure in three dimensions is a challenge. To view the film in 3D an OpenGL ŽSGI, Mountain View, CA. application was written that displays the film on a monitor. The viewer uses OpenGl libraries to handle the creation and rendering of the scene, allows for multiple view points and light sources and provide an interactive interface for moving within the scene. A significant advantage of using OpenGl is the ability to utilize dedicated OpenGl hardware to accelerate the viewer operation. The simulator creates viewing information in the form of simple 3D objects such as triangles and quadrilaterals, each having defined properties such as color and transparency. During the creation of the viewing information, the surface of the film is smoothed to remove the sharp cube edges and then material information or local temperatures are used to ‘color’ the film. The actual film is depicted as a large number Žoften several million. of triangles. The use of the simulator and viewer allows the film to be viewed in a number of ways, including vertical slices, horizontal slices or specified rectangular regions. Once displayed on the monitor, the image may then be captured Žor exported. for future use in an image format such as jpeg.
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3. Results 3.1. Simulation accuracy and efficiency The first set of simulations undertaken addressed the issue of model resolution and memory efficiency. To analyze the effect of film depiction resolution on thermal property simulation, a helically structured GLAD film was grown using 3D-Films and then processed to reduce the resolution of the film depiction. The simulated film was 1.2 m= 1.2 m in area and 1.5 m thick. The three investigated film resolutions were a full film done at a resolution of 120 = 120 = 150, a film generated from the full film with 1r2 the resolution Ž60 = 60 = 90., and a film with a resolution of 1r3 Ž40 = 40 = 60.. Fig. 4 shows three different representations of the full resolution and 1r3 resolution films. For each film three views are shown. In Ža. is a depiction of the entire film Žreferred to as ‘untrimmed’.. However, due to the periodic structure of the film and the many helices crossing the edges of the simulation region, this depiction is difficult to interpret. Thus the second depiction in Žb. is a ‘trimmed’ simulation where any helix that crosses a simulation edge is not shown. Only a few helices are shown, but it is much easier to observe the intrinsic structure of the film. Finally in Žc., a depiction of the blocks used to form the thermal simulation is shown Žuntrimmed.. This figure shows two types of blocks, internal blocks and boundary condition blocks Ždarker. at the top and bottom of the film. In the thermal simulation, the boundary condition blocks have an associated fixed temperature. It can be seen from Fig. 4 that although a fine resolution is needed for the simulation of the growth, a post-growth reduction in resolution does not significantly effect even the fine structure. One aspect of reducing the resolution is shown in the trimmed image of the 1r3 resolution film. A helix that did not touch a simulation edge at the higher resolution does so in the 1r3 resolution film. It has been removed, for visualization, by the trimming algorithm. However, this helix is still part of the film depiction data base and will be used for thermal simulations. The thermal simulations for the films of Fig. 4 are presented in Fig. 5. Again, two depictions are presented. A smooth ‘colored’ or ‘grayscale’ depiction of the temperature distribution is shown for the trimmed view in Fig. 5a. The trimmed image is easier to compare than untrimmed images and the temperature distribution can be seen to be very similar for the two resolutions. The second figure Ž5b. for each film makes this distribution much more clear. In Fig. 5b, temperature contours of alternating light and dark regions were imposed on the film surface, enabling a close comparison of the temperature distribution in the two films. It can be seen clearly that the contours are very similar
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displaying a linear thermal flow down through each column. A qualitative analysis of the thermal simulations in Fig. 5 Žincluding 1r2 resolution. is presented in Table 1. By resolution reduction, the number of basic film blocks was reduced dramatically by factors of 8 and 22 for, respectively, 1r2 and 1r3 resolution. The use of a quadtree mesh was more effective for the higher resolution case. A reduction in the number of blocks by 63% was possible in the case of full resolution, whereas only a 33% reduction could be enacted in the 1r3 resolution case. However, in all films, a significant reduction in the overall number of solution variables was possible with a quadtree mesh. The effective thermal resistivity th Žnormalized to the thermal resistivity of a conventional dense thin film. of each of the films was calculated by determining the total heat flow through the film under fixed boundary conditions. Calculating the total heat flow was achieved by determining ŽdTrd n. on the top and bottom boundaries of the film. The film shown in Fig. 5 was found to have a th of approximately 10 times that of the bulk material. As the resolution was altered, the variation in th was on the order of 5%. It was found that much of the variation occurred in the area of the top boundary condition. The close agreement in th is an indication that the reduction in resolution has not fundamentally altered the simulated heat flow as would be expected from Fig. 4. Finally, in this table, the execution time and memory requirements are shown for the three thermal simulations. As can be seen, a dramatic reduction in both computation time and memory usage is observed as the number of blocks is reduced. The simulations were executed on a Sparc ultra 10 workstation having 1 GB of memory. Although simulation of the full resolution film was feasible, it was cumbersome, requiring approximately 10 h of simulation time. This was not a particularly large or detailed simulation, and, as such, the use of model reduction and a quadtree mesh is nearly essential for larger models. 3.2. Application to thermal barriers We use earlier experimental work utilizing the film depicted in Fig. 6a to confirm the accuracy of the simulator. The SEM image shows a GLAD film consisting of alternating layers of porous and dense yttriastabilized zirconia. The composition of the source material for this film was 12% yttria and 88% zirconia by weight. Experimental measurements of the thermal properties of this film were undertaken at the National Institute of Standards and Technology w8x. In this earlier work, two test procedures were employed: the mirage effect and the 3- technique w2x. The mirage technique gave a more reliable value of the effective
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Fig. 4. 3D-Films simulations of a helical film at two resolutions of full Ž120 = 120 = 150. and 1r3 Ž40 = 40 = 50.. Three depictions are shown: Ža. an untrimmed film of the entire film. Žb. A trimmed film of only the helixes that do not cross a simulation boundary. Žc. A depiction of the blocks forming the thermal model ŽBC blocks are shown as darker in color..
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Fig. 5. 3D-FilmsrThermal simulations of the two film resolutions in Fig. 4. Ža. Grayscale temperature distribution plots for a trimmed films. Žb. A temperature contour plot for trimmed films.
thermal diffusivity of 0.0008 cm2rs. A fairly large uncertainty is associated with this result, since measurements were near the edge of sensitivity of the test methods. Comparisons were made to a high density film of the same material deposited using standard evaporation techniques. It was found that the porous
GLAD microstructure made possible a reduction in the effective thermal diffusivity by a factor of 7. If it is assumed that diffusivity is related linearly to thermal conductivity, then the change in diffusivity can be compared directly to the change in effective thermal conductivity or resistivity found by simulation. To verify
Table 1 Structure size, th , execution time and memory for the films in Fig. 5 Resolution
Film blocks
Mesh blocks
Thermal resistivity ŽNormalized.
Execution time Žmin.
Full 1r2 1r3
801 004 93 000 36 306
302 040 55 036 23 622
10.4 9.91 10.2
510 42 8
Memory ŽMB. 557 97 45
96
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Fig. 6. Ža. SEM of zirconia GLAD TBC. Žb. Simulation of a single element of the TBC.
the accuracy of the simulator, a thermal simulation of the film in Fig. 6b was undertaken. The simulation consists of a GLAD film formed by simple oblique deposition at 82⬚ followed by a dense capping layer. The simulated thermal resistivity of this film is 6.8 times that of a bulk film. This is very close to the ratio
of thermal diffusivities measured experimentally. Simulation of only a single element of the TBC may be justified since the heat flow through each film layer is serial. The heat flow in the film of Fig. 6b can be contrasted with the flow in a dense zirconia film. Fig. 7a,b shows a
Fig. 7. 3D-Films simulations. Ža. Dense columnar Zirconia film. Žb. Temperature contours showing heat flow through the dense film. Žc. View of a number of columns forming the film in Fig. 6b. Žd. Heat flow through one of the individual columns in Žc..
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simulated film and temperature contour plot of a high density columnar zirconia film. The heat flow can be seen to be nearly one-dimensional and, due to the lack of porosity, vertical. Also shown in Fig. 7c,d are views of a number of the individual columns comprising the film of Fig. 6b and a view of the heat flow through a single one of them. The linear flow downwards along the columns may be clearly seen. It follows that an increase in thermal resistance is not only due to a drop in film density, but also an increase in the length of the thermal flow through the film. 3.3. Alternati¨ e microstructure thermal barriers Due to the flexibility of the GLAD deposition technique, a number of alternative TBC film structures can be imagined. These include chevron structures formed by alternating the side from which the deposition flux arrives w14x, helices with a variety of pitches, and posts which are be formed by rapidly spinning the substrate while maintaining a fixed flux arrival angle Žin essence a helix with a very small pitch. w15x. The thermal simulation tool is ideal for analyzing the relative thermal merits of a variety of microstructures. In Fig. 8 characteristic films and temperature distributions are shown for 1.0-m-thick chevron, three-turn helix and post structures, each with 0.3-m dense capping layers. A plot of the density of these films in as well as the dense film in Fig. 7a is shown in Fig. 9. It can be seen that for all three GLAD films, the density of the porous layer is approximately 0.4 relative to the bulk. The increase in density from 1.0 to 1.3 m is due to the 0.3-m-thick dense capping layer intentionally grown. Table 2 shows simulation results from a number of structures similar to those of Fig. 8. The density of the entire film Žincluding the capping layer. is given as well as the effective thermal resistivity. Although the film density varies by only a few percent, excluding the high density bulk film, th varies by as much as a factor of 2 wbetween the ‘post’ Ž3.76. and 1r2 turn helix films Ž7.22.x. The large variation in th between the films is primarily due to a number of factors: the cross-sectional area of the microstructure carrying the heat flow, the path length of the heat flow, the degree of intercolumn heat flow and heat flow crowding due to the flow ‘turning a corner’. The vertical posts have the shortest path, a vertical one from top to bottom, and are relatively thick. This results in the smallest th . It is interesting to note that even the posts have a th larger by factor of 2 than would be expected due to a straightforward density drop of 2. This can be attributed to thermal spreading resistances at the connection of each post to the cap and substrate. An example of the complex heat flow in these films is presented in Fig. 10. In this figure the temperature contours and heat flow are presented for a thin cross-sectional slice of the film
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in Fig. 8a. These plots clearly show heat flow passing between adjacent columns where they touch and crowding of the heat flow as it flows around a ‘corner’. As the first effect would reduce the thermal resistance of the film and the second increase it, the optimization of the films is a complex matter. A tradeoff appears to be apparent as the more twists in the film the more chance of columns touching and inter-column heat flow. It should also be noted that a direct comparison of the thermal resistivities in Table 2 with the one-turn helical film in Fig. 4 is not possible due to the presence of a capping layer. The dense capping layer has a lower resistance than the porous portion of the film and reduces the overall resistivity. These results indicate that the most effective TBCs Žfrom the perspective of largest thermal resistance per unit thickness . could be fabricated using simple oblique films or large pitch helices. Of course, in practice, there are a number of additional concerns such as film durability and adhesion. 4. Conclusions This paper has presented the simulation of heat through porous films using the three-dimensional microstructural thin film simulation framework 3D-Films. Within this framework, the 3D film growth simulator 3DFilms is used to generate film structures for inhomogeneous films. The simulated film structure, consisting of a cubic data structure, is converted into a finite difference based thermal model by the program 3DFilmsrThermal. This program creates an efficient block based data structure using a 3D quadtree mesh and solves for the steady state heat flow through the film structure. The specific thermal thin film application analyzed is the use of thermal barrier coatings produced by GLAD deposition techniques. Initially, simulations of GLAD helical films were presented showing the feasibility of reducing the problem size by two methods; resolution reduction after the growth simulation, and the use a quadtree mesh. These simulations indicate that a reduction of the resolution by a factor of 3 and the use of quadtree mesh can reduce execution times by a factor of 60 and memory usage by a factor of 5 without sacrificing accuracy in the thermal simulations. To confirm the validity of the thermal simulations, a ZrO 2 thermal barrier was simulated and compared to experimental results from earlier work. A relatively good match was obtained, assuming that thermal diffusivity Žexperimentally measured. and thermal resistivity Žsimulated. are linearly correlated for porous films. Finally, the simulator was used to analyze the effectiveness of a number of alternative thermal barrier microstructures. These results indicate that the most
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Fig. 8. Three GLAD TBCs formed out of a 1.0-m porous film and a 0.3-m capping layer. Structures: Ža. Chevron. Žb. Helix. Žc. Post.
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effective thermal barriers will be simple obliquely deposited films or large pitch helical films. The best thermal barrier was found to be have 1r7 of the thermal conductivity of a bulk homogeneous film.
Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Alberta Microelectronic Corporation, and Micronet.
References Fig. 9. Film density of the films in Fig. 8.
Table 2 Thermal density and thermal resistivity of films structures similar to those in Fig. 8 Type
Density Žnormalized.
Thermal resistivity th Žnormalized.
Bulk Slanted Chevron ŽFig. 8a. Helix 1r2 turns Helix 3r2 turns Helix 3 turn ŽFig. 8b. Posts ŽFig. 8c.
0.98 0.48 0.49 0.46 0.47 0.48 0.50
1.00 6.79 5.51 7.22 5.95 4.91 3.76
w1x B.Z. Janos, E. Lugscheider, P. Remer, Surf. Coat. Technol. 113 Ž1999. 278. w2x D. Josell, E.J. Gonzales, G.S. White, J. Mater. Res 13 Ž1998. 1117. w3x K. Robbie, M. Brett, A. Lakhtakia, Nature 384 Ž1996. 616. w4x R. Messier, T. Gehrke, C. Frankel, V.C. Venugopal, W. Otano, A. Lakhtakia, J. Vac. Sci. Technol. A 15 Ž1997. 2148. w5x D. Vick, L.J. Friedrich, S.K. Dew, M. Brett, K. Robbie, M. Seto, T. Smy, Thin Solid Films 339 Ž1999. 88. w6x S.M. Meier, D.K. Gupta, ASME paper 92-GT-203 Ž1992.. w7x T.E. Strangman, US Patent No. 4,321,204. w8x K.D. Harris, D. Vick, M.J. Brett, K. Robbie, in: M. Pickering, B.W. Sheldon, W.Y. Lee, R.N. Johnson, MRS Symp. Proc., 555 Ž1999. 97. w9x T. Smy, D. Vick, M.J. Brett, S.K. Dew, A.T. Wu, J. Sit, K. Harris, J. Vac. Sci. Technol. A. A18 Ž2000. 2507.
Fig. 10. 3D-FilmsrThermal depictions of the heat flow in a cross-section Ž0.08 m thick. of the structure of Fig. 8a showing a contour plot of the temperature superimposed on the film structure and the heat flow in the film. Ža. Contour plot. Žb. Heat flow.
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w10x T. Smy, D. Vick, M.J. Brett, S.K. Dew, A.T. Wu, J. Sit, K. Harris, in: H.N.G. Wadley, G. Gilmer, W. Barker ŽEds.., MRS Symp. Proc., 616 Ž2000. 112. w11x T. Smy, R.V. Joshi, S.K. Dew, in: M. Gross, T. Gessner, N. Kobayashi, Y. Yasuda ŽEds.., AMC Ž1999. 669.
w12x T. Smy, D. Walkey, S. Dew, IEEE Trans CAD 20 Ž2001. 50. w13x H. Carslaw, Conduction of Heat in Solids, Oxford Press, 1959. w14x K. Robbie, L.J. Friedrich, S.K. Dew, T. Smy, M.J. Brett, J. Vac. Sci. Technol. A 13 Ž1995. 1032. w15x K. Robbie, C. Shafai, M.T. Brett, J. Mater. Res. 14 Ž1999. 3158.
Thin film microstructure and thermal transport simulation using 3D-films T. Smy a,U , D. Walkey a , K.D. Harris b , M.J. Brett b b
a Department of Electronics, Carleton Uni¨ ersity, Ottawa, ON, Canada K1S 5B6 Department of Electrical and Computer Engineering, Uni¨ ersity of Alberta, Edmonton, AB, Canada T6G 2G7
Received 5 July 2000; received in revised form 12 January 2001; accepted 26 March 2001
Abstract Simulations of the flow of heat through porous thin films by the three-dimensional microstructural thin film simulation framework 3D-Films are discussed in this paper. For each simulation, the film structures are generated by the thin film growth model 3D-Films and then used to generate a finite difference based thermal model by the program 3D-FilmsrThermal. This program creates a block based data structure using a 3D quadtree mesh and subsequently solves for the steady state heat flow through the film structure. In this paper the film growth and thermal models are used to analyze and suggest optimization of porous thermal barrier coatings produced by glancing angle deposition techniques. The paper also deals with the determination of the accuracy and efficiency of the thermal model. Studies on the effect of reducing the resolution of the simulated film for less memory intensive thermal simulations are presented, indicating that a reduction of the resolution by a factor of 3 and the number of solution variables by as much as a factor of 27 is feasible. The simulations Of ZrO 2 thermal barriers are compared to experimental results with a relatively close match being obtained. Finally, the simulator is used to analyze the effectiveness of a number of potential thermal barrier structures produced by glancing angle deposition techniques. These results suggest that the most effective thermal barrier film microstructures will be porous films consisting of slanted posts or large pitch helices. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Computer simulation; Evaporation; Thermal properties; Microstructure
1. Introduction The thermal properties of thin films are of significant interest in a number of situations, such as heat flow through the metallization layers of integrated electronic circuitry, micromachined devices including temperature sensors and thin film optical devices, and thermal barrier coatings ŽTBCs. w1x. In thermal barrier coatings, for example, the thermally insulating proper-
U
Corresponding author. Tel.: q1-613-520-3967; fax: q1-613-5205708. E-mail address: [email protected] ŽT. Smy..
ties of a material are exploited to protect surfaces from heat damage. Knowledge of the thermal resistivity dependence on film properties is vital to understanding how it may be improved. In a thin film, thermal properties can vary significantly from that of bulk materials due to microstructure differences. Generally, thin films are deposited using a method, such as sputtering or evaporation, which occurs under conditions far from equilibrium and leads to highly inhomogeneous films. The films are often polycrystalline and, depending on the deposition conditions, may be quite porous. The thermal resistivity of a film is a function of its porosity, connectivity and microstructure. However, due to the size scale involved,
0040-6090r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 0 9 7 4 - 9
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the experimental determination of the thermal resistivity is challenging, as few techniques provide effective measurements w2x. An alternative method of characterizing and optimizing deposited films is by physically modeling the heat flow through a simulated film structure. Two modeling tools are required: Ž1. a film deposition modeling code; and Ž2. a heat flow solver. This paper presents the use of 3D-Films for modeling the deposition of sputtered and evaporated films and the simulator 3D-FilmsrThermal for modeling the heat flows within the simulated films. As an example, the programs are used to analyze porous films deposited by glancing angle deposition ŽGLAD. w3x for TBC applications. GLAD is physical vapor deposition with carefully controlled substratersource geometry, during which substrate movement may occur about two axes: the first adjusts the angle at which the incident flux arrives at the substrate, while the second rotates the substrate about an axis perpendicular to its surface. At highly oblique deposition angles, extreme shadowing begins to introduce microstructure and porosity into the deposited thin films. During nucleation for growth onto substrates at low temperatures Žrelative to film melting point., any points of the substrate which are shadowed by random three-dimensional variations are prevented from receiving any additional vapor flux. As the film grows, this effect continues, resulting in a microstructure consisting of many, laterally unconnected columns growing outward from the substrate and separated by voids. Since atomic shadowing is the driving mechanism, higher angles of incidence lead to more
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pronounced porosities w5x. For incident flux angles well over 80⬚ from the substrate normal, it has been shown possible to produce porous thin films having densities as little as 15% of the bulk. Without substrate rotation a film grown at glancing incidence consists of many thermally isolated posts or columns. These columns tend to grow in a vapor source seeking manner w3x. However, since it is possible to rotate the substrate throughout the deposition, the direction of column growth may be continuously altered, leading to a variety of interesting microstructures. Using the GLAD technique, structures such as chevron, post and helical have been created w3,4x. A typical GLAD helical SiO 2 film is shown in Fig. 1, illustrating the complex 3D nature of GLAD film microstructure. Traditionally, TBCs are fabricated with a low thermal conductivity material such as ZrO 2 . They are deposited by a method such as plasma spraying which leads to a low density film of high thermal resistance w6x. A significant problem with traditional deposition techniques is thermal expansion stresses causing delamination. Stress problems have been alleviated somewhat by fabrication by electron beam evaporation of thermal barriers with a tight columnar structure, but at the cost of reduced thermal resistance w7x. It has been suggested that GLAD ZrO 2 films could be engineered to provide not only thermally optimized TBC barriers but also films with implicit stress relief mechanisms w8x. The modeling of GLAD thermal barriers is a significant challenge. The inherently complex, 3D nature of GLAD film microstructures and the fine column and grain structure requires a fully three-dimensional simu-
Fig. 1. A helical GLAD SiO 2 film: Ža. SEM Žb. 3D-Films simulation.
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lator. The simulator 3D-Films has been previously used to model both GLAD films and metal deposition over topography w9᎐11x. It is based on a simple cubic data structure in which each cube represents a small amount of film material. Deposition is modeled as a ballistic, Monte Carlo process incorporating nucleation, selfshadowing, adatom mobility and a variety of additional physical effects. 3D-Films provide a completely 3D depiction of the film, its microstructure, and information on local film density and differential and total surface area. The modeling of the heat flow is performed using a new tool, 3D-FilmsrThermal,1 which solves the heat flow equations through the simulated film with a finite difference approach. The heat flow code is based on a 3D thermal simulation tool known as Atar used to solve for the heat flow in integrated device semiconductor devices w12x. 2. Modeling framework
Fig. 2. Basic simulation geometry showing a simple flat substrate, three initial grains and a single particle transport to the surface and then to a growing grain.
2.1. Film growth A major concern in a 3D microstructural simulator is the memory requirements of the model. In order to represent the film structure with sufficient accuracy, a linear resolution of 300᎐500 pixels is required in each dimension. This factor combined with the need to represent the internal structure of the film Žessentially a requirement for a ‘volume’ representation rather than a ‘surface’ representation . entails a significant challenge. A resolution of 500 = 500 = 500 pixelsrsimulation using a single 3D array as a data structure would require 1.25= 10 9 cells Žassuming a minimal data structure for each cell of 10 bytes.. This is impractical on typical work stations. In order to minimize memory requirements, a data structure consisting of a two-tiered array is used. The first tier in the structure is a ‘course’ 3D array. Each element of the course array is pointer to a 3D ‘fine’ array structure. Typically, each fine array is 20᎐50 elements on a side. A ‘fine’ array consists of pointers to individual cube data structures, containing information on the block material, surface state, material type, surface normal Žif appropriate. and deposition history. This approach allows for an efficient use of memory, particularly in topographic features such as a vias or in porous films. By this method, resolutions of 500 = 500 = 500 are possible requiring approximately 300 Mbytes of memory. At the beginning of a simulation, an initial substrate is defined and parameters controlling the deposition are provided. The substrate definition is simply an
1
Parties interested in using or acquiring the code should contact T. Srny at [email protected]
initial specification of cubes and can be set at run time to a variety of configurations including a flat substrate, trenches, vias, and steps. During simulation, particles are launched ballistically from slightly above the growing film and follow straight line trajectories until they strike either the substrate or the film. The particles are then permanently incorporated into the film after searching the nearby exposed surface for positions of minimum surface curvature dependent chemical potential. This process is shown schematically at a very low resolution in Fig. 2. The combination of ballistic shadowing and short range surface diffusion successfully accounts for the formation of the columnar thin film microstructure characteristic of high melting point materials at typical deposition temperatures. This is illustrated in Fig. 1 where both a 3D-Films simulation and a scanning electron microscope ŽSEM. image of a helical GLAD structure are shown. The figure clearly shows the ability of the simulator to predict the complex structure of a GLAD film. The boundaries of the simulation must be handled carefully during film growth, diffusion and ballistic transport. Two cases are allowed: periodic or mirrored boundaries. If a boundary is specified as periodic, a particle traveling across the simulation from left to right will re-enter on the left side of the simulation region and the film on the left edge will match that on the right. For a mirrored boundary, particles will ‘bounce’ off the boundary and the film structure will be mirrored at the edge. The primary use of the mirrored boundary condition is to allow more memory efficient simulations. However, the use of a mirrored edge assumes that the initial flux distribution is isotropic across that edge. For incident flux in GLAD, this is not
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usually true and periodic boundary conditions are required. 2.2. Thermal modeling The final product of a film growth simulation is an aggregation of cubes describing a complex 3D structure. Thermal modeling of heat flow through this structure is best done by a numerical solver. The physics involved in solving for the conductive heat flow require the solution of the following partial differential equation w13x: C
⭸T s ⵜ Ž Ž T . ⵜT . q g Ž x, y, z,t . ⭸t
Ž1.
where ŽT . is the temperature-dependant thermal conductivity, C the thermal capacity of the material, the material density, and g Ž x, y, z . describes the heat generation. If there is no heat generated within the film, and we are interested in a steady state solution, this simplifies to: ⵜ Ž Ž T . ⵜT . s 0
Ž2.
Complications in solving this equation within a thin film are caused mainly by the complex geometry of the film and inhomogeneities arising from the use of multiple materials. To produce a numerical solution an appropriately discretized 3D model must be built of the film, boundary conditions must be applied, a mathematical model must be extracted and then the model solved using an appropriate technique. Although the natural data structure of stacked cubes used in the growth model could be exploited without alteration for performing a numerical solution, it generally contains too many blocks. Each block would be a solution variable and the large number of blocks in a typical film
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simulation would be prohibitive for execution on a typical workstation. To alleviate this problem two approaches were taken. The first was to post process the final film structure and reduce its resolution. It was found that although the film growth needs to be simulated with approximately 0.05 m for a block dimension, the thermal properties could be simulated accurately at lower resolutions. Therefore, a code was written that reads the final film structure and agglomerates blocks reducing the film resolution. Typically, a reduction of eight blocks Ž2 = 2 = 2. to 1 or 27 Ž3 = 3 = 3. blocks to 1 is used. A new block is formed if the density in the local region is greater then 0.5. The second method of reducing the number of blocks is to form a three-level 3D quadtree mesh Žalso known as an octree mesh. of the structure from the simulated film. A three-level 2D quadtree mesh is shown in Fig. 3a, illustrating the primary characteristic of the mesh. The mesh is formed by creating blocks in such a manner that the maximum number of smaller neighbors is two on each side in 2D, or four on each face in a 3D mesh. This method of forming the block structure allows for very quick 3D refinement of the mesh structure at interfaces and other regions of significant structure. It effectively and dramatically reduces the number of blocks used to describe the film, while still allowing for a reasonable number of local block topographies. This significantly simplifies coding. The procedure used to create the quadtree mesh was to determine if a cubic area consisted of a single materialrgrain and if the rules for generating the quad tree mesh would not be broken if a single block was formed. When creating the mesh of a porous film a choice needs to made as to if it is needed to mesh the void regions. If the conductivity of the film is significantly larger than that of the voided regions they can be ignored speeding up the simulation due to a smaller
Fig. 3. Ža. Example of a 2D quadtree mesh. Žb. Representative heat flows between 2D blocks which are represented by thermal resistors. Ti and Tn are the temperatures at the nodes i and n, respectively, and rthn r i the thermal resistance between node n and i.
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number of blocks. For the simulations in this paper the conductivity of air is at least 50 times smaller than the conductivity of the film and the voids were left unmeshed.
reformulated by solving for a ‘new’ temperature at each node giving, N
Tnk s Tnky1 q FSOR
is1
2.3. Extraction of a thermal network N
After creation of the 3D quadtree mesh model, a thermal network consisting of resistors is extracted. Each block is visited and, using the topology of the connections with adjacent blocks, a network of resistors to the neighboring nodes is created. As the material properties are, in general, temperature-dependent the resistor values are a function of the node temperature. In Fig. 3 the heat flows for a single 2D block are shown, wherein each heat flow would be represented by a thermal resistor. The value of each resistor is calculated using the common cross-sectional area, the size of the two blocks and the local thermal conductivity. The formulation of the steady state heat flow problem in this manner is basically a finite difference approach, whereby for each block the heat flow must have zero divergence. The solution of this for steady state heat flow therefore requires that the sum of heat flows through each resistor to the node at the center of a block equals zero as given by, N
Ý is1
Ti y Tn q pn s 0 rthn r i
Ž3.
where, Ti and Tn , are the temperatures at nodes i and n, respectively, rthn r i the thermal resistance between node n and i and pn , the thermal power being generated in the block Žusually zero.. A number of boundary conditions can be placed on the model. Regions of the model, such as the top or bottom, can be designated to be at a fixed temperature. Constant power generation can be specified in volumes. The boundary conditions on the vertical sides of the simulation region are either periodic or mirrored to match the previous growth simulation. Once the thermal model has been extracted and the boundary conditions set, the network must be solved. Currently two steady state internal solvers are available. A direct solver can be used which sets up a global sparse array and subsequently solves the linear set of equations representing the thermal resistor network. If the temperature dependence of the thermal conductivity of the film is to be considered, an iterative technique is applied using successive over relaxation ŽSOR. and the repeated solution of the linearized set of equations using the direct solver. A second solver based on explicit SOR of the nodal temperature equations has also been implemented and is particularly useful for large problems. Eq. Ž3. can be
=
žÝ
is1
Ti ky 1 q pn rthn r i
žž / / Ý
1 y Tnky 1 rthn r i
/ Ž4.
where k is the iteration number and FSOR the relaxation parameter Ž1.25 was found to be a good value.. This solver implicitly handles the temperature dependence of the material properties by updating the resistor values at each iteration. Once the solution has been found, the maximum temperature is obtained and a grayscale or color representation of the temperature distribution can be displayed. 2.4. Viewing and post processing The efficient viewing and analysis of the simulated film is of significant importance and is also a considerable challenge. It is obviously necessary to view the film and evaluate its microstructure. In 2D this is fairly simple, however, in 3D a sophisticated approach is required. A 3D film must be rendered in a manner such that coloration and perspective makes the image comprehensible to the viewer. A film produced by 3D-Films contains a significant amount of detail and structure; a typical film consists of millions of cubes. Efficiently rendering such a structure in three dimensions is a challenge. To view the film in 3D an OpenGL ŽSGI, Mountain View, CA. application was written that displays the film on a monitor. The viewer uses OpenGl libraries to handle the creation and rendering of the scene, allows for multiple view points and light sources and provide an interactive interface for moving within the scene. A significant advantage of using OpenGl is the ability to utilize dedicated OpenGl hardware to accelerate the viewer operation. The simulator creates viewing information in the form of simple 3D objects such as triangles and quadrilaterals, each having defined properties such as color and transparency. During the creation of the viewing information, the surface of the film is smoothed to remove the sharp cube edges and then material information or local temperatures are used to ‘color’ the film. The actual film is depicted as a large number Žoften several million. of triangles. The use of the simulator and viewer allows the film to be viewed in a number of ways, including vertical slices, horizontal slices or specified rectangular regions. Once displayed on the monitor, the image may then be captured Žor exported. for future use in an image format such as jpeg.
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3. Results 3.1. Simulation accuracy and efficiency The first set of simulations undertaken addressed the issue of model resolution and memory efficiency. To analyze the effect of film depiction resolution on thermal property simulation, a helically structured GLAD film was grown using 3D-Films and then processed to reduce the resolution of the film depiction. The simulated film was 1.2 m= 1.2 m in area and 1.5 m thick. The three investigated film resolutions were a full film done at a resolution of 120 = 120 = 150, a film generated from the full film with 1r2 the resolution Ž60 = 60 = 90., and a film with a resolution of 1r3 Ž40 = 40 = 60.. Fig. 4 shows three different representations of the full resolution and 1r3 resolution films. For each film three views are shown. In Ža. is a depiction of the entire film Žreferred to as ‘untrimmed’.. However, due to the periodic structure of the film and the many helices crossing the edges of the simulation region, this depiction is difficult to interpret. Thus the second depiction in Žb. is a ‘trimmed’ simulation where any helix that crosses a simulation edge is not shown. Only a few helices are shown, but it is much easier to observe the intrinsic structure of the film. Finally in Žc., a depiction of the blocks used to form the thermal simulation is shown Žuntrimmed.. This figure shows two types of blocks, internal blocks and boundary condition blocks Ždarker. at the top and bottom of the film. In the thermal simulation, the boundary condition blocks have an associated fixed temperature. It can be seen from Fig. 4 that although a fine resolution is needed for the simulation of the growth, a post-growth reduction in resolution does not significantly effect even the fine structure. One aspect of reducing the resolution is shown in the trimmed image of the 1r3 resolution film. A helix that did not touch a simulation edge at the higher resolution does so in the 1r3 resolution film. It has been removed, for visualization, by the trimming algorithm. However, this helix is still part of the film depiction data base and will be used for thermal simulations. The thermal simulations for the films of Fig. 4 are presented in Fig. 5. Again, two depictions are presented. A smooth ‘colored’ or ‘grayscale’ depiction of the temperature distribution is shown for the trimmed view in Fig. 5a. The trimmed image is easier to compare than untrimmed images and the temperature distribution can be seen to be very similar for the two resolutions. The second figure Ž5b. for each film makes this distribution much more clear. In Fig. 5b, temperature contours of alternating light and dark regions were imposed on the film surface, enabling a close comparison of the temperature distribution in the two films. It can be seen clearly that the contours are very similar
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displaying a linear thermal flow down through each column. A qualitative analysis of the thermal simulations in Fig. 5 Žincluding 1r2 resolution. is presented in Table 1. By resolution reduction, the number of basic film blocks was reduced dramatically by factors of 8 and 22 for, respectively, 1r2 and 1r3 resolution. The use of a quadtree mesh was more effective for the higher resolution case. A reduction in the number of blocks by 63% was possible in the case of full resolution, whereas only a 33% reduction could be enacted in the 1r3 resolution case. However, in all films, a significant reduction in the overall number of solution variables was possible with a quadtree mesh. The effective thermal resistivity th Žnormalized to the thermal resistivity of a conventional dense thin film. of each of the films was calculated by determining the total heat flow through the film under fixed boundary conditions. Calculating the total heat flow was achieved by determining ŽdTrd n. on the top and bottom boundaries of the film. The film shown in Fig. 5 was found to have a th of approximately 10 times that of the bulk material. As the resolution was altered, the variation in th was on the order of 5%. It was found that much of the variation occurred in the area of the top boundary condition. The close agreement in th is an indication that the reduction in resolution has not fundamentally altered the simulated heat flow as would be expected from Fig. 4. Finally, in this table, the execution time and memory requirements are shown for the three thermal simulations. As can be seen, a dramatic reduction in both computation time and memory usage is observed as the number of blocks is reduced. The simulations were executed on a Sparc ultra 10 workstation having 1 GB of memory. Although simulation of the full resolution film was feasible, it was cumbersome, requiring approximately 10 h of simulation time. This was not a particularly large or detailed simulation, and, as such, the use of model reduction and a quadtree mesh is nearly essential for larger models. 3.2. Application to thermal barriers We use earlier experimental work utilizing the film depicted in Fig. 6a to confirm the accuracy of the simulator. The SEM image shows a GLAD film consisting of alternating layers of porous and dense yttriastabilized zirconia. The composition of the source material for this film was 12% yttria and 88% zirconia by weight. Experimental measurements of the thermal properties of this film were undertaken at the National Institute of Standards and Technology w8x. In this earlier work, two test procedures were employed: the mirage effect and the 3- technique w2x. The mirage technique gave a more reliable value of the effective
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Fig. 4. 3D-Films simulations of a helical film at two resolutions of full Ž120 = 120 = 150. and 1r3 Ž40 = 40 = 50.. Three depictions are shown: Ža. an untrimmed film of the entire film. Žb. A trimmed film of only the helixes that do not cross a simulation boundary. Žc. A depiction of the blocks forming the thermal model ŽBC blocks are shown as darker in color..
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Fig. 5. 3D-FilmsrThermal simulations of the two film resolutions in Fig. 4. Ža. Grayscale temperature distribution plots for a trimmed films. Žb. A temperature contour plot for trimmed films.
thermal diffusivity of 0.0008 cm2rs. A fairly large uncertainty is associated with this result, since measurements were near the edge of sensitivity of the test methods. Comparisons were made to a high density film of the same material deposited using standard evaporation techniques. It was found that the porous
GLAD microstructure made possible a reduction in the effective thermal diffusivity by a factor of 7. If it is assumed that diffusivity is related linearly to thermal conductivity, then the change in diffusivity can be compared directly to the change in effective thermal conductivity or resistivity found by simulation. To verify
Table 1 Structure size, th , execution time and memory for the films in Fig. 5 Resolution
Film blocks
Mesh blocks
Thermal resistivity ŽNormalized.
Execution time Žmin.
Full 1r2 1r3
801 004 93 000 36 306
302 040 55 036 23 622
10.4 9.91 10.2
510 42 8
Memory ŽMB. 557 97 45
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Fig. 6. Ža. SEM of zirconia GLAD TBC. Žb. Simulation of a single element of the TBC.
the accuracy of the simulator, a thermal simulation of the film in Fig. 6b was undertaken. The simulation consists of a GLAD film formed by simple oblique deposition at 82⬚ followed by a dense capping layer. The simulated thermal resistivity of this film is 6.8 times that of a bulk film. This is very close to the ratio
of thermal diffusivities measured experimentally. Simulation of only a single element of the TBC may be justified since the heat flow through each film layer is serial. The heat flow in the film of Fig. 6b can be contrasted with the flow in a dense zirconia film. Fig. 7a,b shows a
Fig. 7. 3D-Films simulations. Ža. Dense columnar Zirconia film. Žb. Temperature contours showing heat flow through the dense film. Žc. View of a number of columns forming the film in Fig. 6b. Žd. Heat flow through one of the individual columns in Žc..
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simulated film and temperature contour plot of a high density columnar zirconia film. The heat flow can be seen to be nearly one-dimensional and, due to the lack of porosity, vertical. Also shown in Fig. 7c,d are views of a number of the individual columns comprising the film of Fig. 6b and a view of the heat flow through a single one of them. The linear flow downwards along the columns may be clearly seen. It follows that an increase in thermal resistance is not only due to a drop in film density, but also an increase in the length of the thermal flow through the film. 3.3. Alternati¨ e microstructure thermal barriers Due to the flexibility of the GLAD deposition technique, a number of alternative TBC film structures can be imagined. These include chevron structures formed by alternating the side from which the deposition flux arrives w14x, helices with a variety of pitches, and posts which are be formed by rapidly spinning the substrate while maintaining a fixed flux arrival angle Žin essence a helix with a very small pitch. w15x. The thermal simulation tool is ideal for analyzing the relative thermal merits of a variety of microstructures. In Fig. 8 characteristic films and temperature distributions are shown for 1.0-m-thick chevron, three-turn helix and post structures, each with 0.3-m dense capping layers. A plot of the density of these films in as well as the dense film in Fig. 7a is shown in Fig. 9. It can be seen that for all three GLAD films, the density of the porous layer is approximately 0.4 relative to the bulk. The increase in density from 1.0 to 1.3 m is due to the 0.3-m-thick dense capping layer intentionally grown. Table 2 shows simulation results from a number of structures similar to those of Fig. 8. The density of the entire film Žincluding the capping layer. is given as well as the effective thermal resistivity. Although the film density varies by only a few percent, excluding the high density bulk film, th varies by as much as a factor of 2 wbetween the ‘post’ Ž3.76. and 1r2 turn helix films Ž7.22.x. The large variation in th between the films is primarily due to a number of factors: the cross-sectional area of the microstructure carrying the heat flow, the path length of the heat flow, the degree of intercolumn heat flow and heat flow crowding due to the flow ‘turning a corner’. The vertical posts have the shortest path, a vertical one from top to bottom, and are relatively thick. This results in the smallest th . It is interesting to note that even the posts have a th larger by factor of 2 than would be expected due to a straightforward density drop of 2. This can be attributed to thermal spreading resistances at the connection of each post to the cap and substrate. An example of the complex heat flow in these films is presented in Fig. 10. In this figure the temperature contours and heat flow are presented for a thin cross-sectional slice of the film
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in Fig. 8a. These plots clearly show heat flow passing between adjacent columns where they touch and crowding of the heat flow as it flows around a ‘corner’. As the first effect would reduce the thermal resistance of the film and the second increase it, the optimization of the films is a complex matter. A tradeoff appears to be apparent as the more twists in the film the more chance of columns touching and inter-column heat flow. It should also be noted that a direct comparison of the thermal resistivities in Table 2 with the one-turn helical film in Fig. 4 is not possible due to the presence of a capping layer. The dense capping layer has a lower resistance than the porous portion of the film and reduces the overall resistivity. These results indicate that the most effective TBCs Žfrom the perspective of largest thermal resistance per unit thickness . could be fabricated using simple oblique films or large pitch helices. Of course, in practice, there are a number of additional concerns such as film durability and adhesion. 4. Conclusions This paper has presented the simulation of heat through porous films using the three-dimensional microstructural thin film simulation framework 3D-Films. Within this framework, the 3D film growth simulator 3DFilms is used to generate film structures for inhomogeneous films. The simulated film structure, consisting of a cubic data structure, is converted into a finite difference based thermal model by the program 3DFilmsrThermal. This program creates an efficient block based data structure using a 3D quadtree mesh and solves for the steady state heat flow through the film structure. The specific thermal thin film application analyzed is the use of thermal barrier coatings produced by GLAD deposition techniques. Initially, simulations of GLAD helical films were presented showing the feasibility of reducing the problem size by two methods; resolution reduction after the growth simulation, and the use a quadtree mesh. These simulations indicate that a reduction of the resolution by a factor of 3 and the use of quadtree mesh can reduce execution times by a factor of 60 and memory usage by a factor of 5 without sacrificing accuracy in the thermal simulations. To confirm the validity of the thermal simulations, a ZrO 2 thermal barrier was simulated and compared to experimental results from earlier work. A relatively good match was obtained, assuming that thermal diffusivity Žexperimentally measured. and thermal resistivity Žsimulated. are linearly correlated for porous films. Finally, the simulator was used to analyze the effectiveness of a number of alternative thermal barrier microstructures. These results indicate that the most
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Fig. 8. Three GLAD TBCs formed out of a 1.0-m porous film and a 0.3-m capping layer. Structures: Ža. Chevron. Žb. Helix. Žc. Post.
T. Smy et al. r Thin Solid Films 391 (2001) 88᎐100
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effective thermal barriers will be simple obliquely deposited films or large pitch helical films. The best thermal barrier was found to be have 1r7 of the thermal conductivity of a bulk homogeneous film.
Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Alberta Microelectronic Corporation, and Micronet.
References Fig. 9. Film density of the films in Fig. 8.
Table 2 Thermal density and thermal resistivity of films structures similar to those in Fig. 8 Type
Density Žnormalized.
Thermal resistivity th Žnormalized.
Bulk Slanted Chevron ŽFig. 8a. Helix 1r2 turns Helix 3r2 turns Helix 3 turn ŽFig. 8b. Posts ŽFig. 8c.
0.98 0.48 0.49 0.46 0.47 0.48 0.50
1.00 6.79 5.51 7.22 5.95 4.91 3.76
w1x B.Z. Janos, E. Lugscheider, P. Remer, Surf. Coat. Technol. 113 Ž1999. 278. w2x D. Josell, E.J. Gonzales, G.S. White, J. Mater. Res 13 Ž1998. 1117. w3x K. Robbie, M. Brett, A. Lakhtakia, Nature 384 Ž1996. 616. w4x R. Messier, T. Gehrke, C. Frankel, V.C. Venugopal, W. Otano, A. Lakhtakia, J. Vac. Sci. Technol. A 15 Ž1997. 2148. w5x D. Vick, L.J. Friedrich, S.K. Dew, M. Brett, K. Robbie, M. Seto, T. Smy, Thin Solid Films 339 Ž1999. 88. w6x S.M. Meier, D.K. Gupta, ASME paper 92-GT-203 Ž1992.. w7x T.E. Strangman, US Patent No. 4,321,204. w8x K.D. Harris, D. Vick, M.J. Brett, K. Robbie, in: M. Pickering, B.W. Sheldon, W.Y. Lee, R.N. Johnson, MRS Symp. Proc., 555 Ž1999. 97. w9x T. Smy, D. Vick, M.J. Brett, S.K. Dew, A.T. Wu, J. Sit, K. Harris, J. Vac. Sci. Technol. A. A18 Ž2000. 2507.
Fig. 10. 3D-FilmsrThermal depictions of the heat flow in a cross-section Ž0.08 m thick. of the structure of Fig. 8a showing a contour plot of the temperature superimposed on the film structure and the heat flow in the film. Ža. Contour plot. Žb. Heat flow.
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w10x T. Smy, D. Vick, M.J. Brett, S.K. Dew, A.T. Wu, J. Sit, K. Harris, in: H.N.G. Wadley, G. Gilmer, W. Barker ŽEds.., MRS Symp. Proc., 616 Ž2000. 112. w11x T. Smy, R.V. Joshi, S.K. Dew, in: M. Gross, T. Gessner, N. Kobayashi, Y. Yasuda ŽEds.., AMC Ž1999. 669.
w12x T. Smy, D. Walkey, S. Dew, IEEE Trans CAD 20 Ž2001. 50. w13x H. Carslaw, Conduction of Heat in Solids, Oxford Press, 1959. w14x K. Robbie, L.J. Friedrich, S.K. Dew, T. Smy, M.J. Brett, J. Vac. Sci. Technol. A 13 Ž1995. 1032. w15x K. Robbie, C. Shafai, M.T. Brett, J. Mater. Res. 14 Ž1999. 3158.