A method for predicting a selective area growth process based on a kinetic wulff plot

Abstract

The application relates to the technical field of semiconductor micro-nano structure growth regulation and calculation simulation, and discloses a selective area growth process prediction method based on a dynamic Wulff graph, which comprises the following steps: S1, a dynamic Wulff graph (v-plot) of a target material under a preset process window is acquired, and an orientation-related growth rate function corresponding to an interface normal direction is established; S2, a phase field model capable of describing solid-gas interface evolution is constructed; S3, a growth dynamics equation is established; S4, geometric description is established, mask opening geometric parameters are set according to a selective area growth object to be studied; and S5, numerical solution is carried out, and the phase field model is numerically solved. The application does not depend on simple empirical fitting, but directly introduces the material orientation-related growth rate into selective area growth prediction, can explicitly consider the geometric effect of a mask boundary, and can reflect an edge stagnation-bulging-overflow mechanism.

Description

Technical Field

[0001] This invention relates to the field of semiconductor micro / nano structure growth control and computational simulation technology, specifically a method for predicting selective region growth processes based on dynamic Wulff diagrams. Background Technology

[0002] Super-aggregate epitaxial growth (SAG) allows crystals to deposit only within specific regions and extend outwards beyond the opening edges by setting mask openings on the substrate surface, thus enabling the construction of positionally controllable and morphologically tunable micro / nano structures. For quantum dots, nanopillars, and arrayed epitaxial structures, SAG not only determines the final size but also the crystal plane combination, top morphology, lateral expansion path, and consistency between different sites in the array. However, the SAG process differs fundamentally from ordinary free epitaxial growth. Its evolution is typically not a single-stage continuous change but rather involves multiple stages: nucleation at the bottom of the opening, filling within the opening, the leading edge approaching the edge, edge stagnation, bulging, overflow beyond the edge, and lateral overgrowth. These processes are simultaneously controlled by the following factors: 1. intrinsic orientation-dependent growth kinetics of the material; 2. size, shape, and orientation of the mask openings; 3. interface pinning and depinning behavior caused by geometrical abrupt changes at the mask edges; 4. the deposition process and material redistribution near the interface; and 5. uniformity variations caused by differences in geometric parameters within the array of multiple sites.

[0003] In existing technologies, research on SAG often adopts two approaches: one mainly relies on experimental comparison, observing the final morphology by changing the opening size, shape, or growth conditions; the other uses local simplified models or empirical geometric configuration analysis.

[0004] While the above methods can reveal certain patterns in phenomena, they typically have the following shortcomings:

[0005] 1. It is difficult to directly convert the orientation-related dynamic input of materials into calculable interface propulsion laws;

[0006] 2. It is difficult to uniformly describe the entire process of growth within the mask aperture, edge stagnation, bulging, overflow, and lateral overgrowth;

[0007] 3. It is difficult to handle the geometric constraints of 3D masks and their impact on the final state morphology;

[0008] 4. It is difficult to use for mask parameter screening and uniformity prediction at the array scale.

[0009] Therefore, a technical approach is needed that can unify and couple material dynamics, mask geometry, and growth process to predict the complete SAG process and further serve mask parameter design and process optimization. Summary of the Invention

[0010] The purpose of this invention is to provide a method for predicting selective region growth processes based on dynamic Wulff diagrams, in order to solve the problems mentioned in the background art.

[0011] To achieve the above objectives, the present invention provides the following technical solution: a method for predicting selective region growth processes based on dynamic Wulff diagrams, comprising the following steps:

[0012] Step S1: Obtain the kinetic Wulff plot (v-plot) of the target material under the preset process window and establish the orientation-related growth rate function corresponding to the interface normal direction.

[0013] Step S2: Construct a phase-field model that can describe the evolution of the solid-gas interface, and simultaneously introduce the following into the model:

[0014] (1)

[0015] in The local thermodynamic free energy density is expressed using a double potential well function, which is used to stabilize the two-phase state of solid and gas. This is the gradient energy term, used to describe the diffusion interface;

[0016] Step S3: Establish the growth kinetic equation;

[0017] Step S4: Establish a geometric description and set the mask opening geometry parameters according to the selective region growth object to be studied.

[0018] Step S5: Numerical solution, performing numerical solution on the phase field model.

[0019] Preferably, the function in step S1 is used to characterize the relative propulsion capability of different crystal-down interfaces and serves as the kinetic input for subsequent growth evolution calculations.

[0020] Preferably, step S2 employs the smooth boundary method to forcibly incorporate boundary condition constraints into the governing equations through integration by parts, and the chemical potential expression under the SBM framework is:

[0021] (2)

[0022] Wherein, θ is the specified equilibrium contact angle at the interface of the three phases of solid, vapor and mask, which characterizes the wetting behavior on the mask.

[0023] Preferably, step S3 specifically involves first processing the conventional Cahn-Hilliard type equations using the SBM method, then introducing orientation-related growth terms to describe the depositional growth process driven by both chemical potential gradient-driven diffusion and anisotropic deposition. Additionally, an extra three-phase boundary supply term is introduced. Finally, the phase-field governing equations for the SAG process are obtained as follows:

[0024] (3)

[0025] Where the atomic mobility is M s =1,d v is the deposition rate; v(n) is the orientation-dependent growth rate function, and its corresponding visual representation is called the kinetic Wulff plot, which is closely related to material properties. Ensure that deposition occurs only at the interface, d TPB A constant parameter characterizing the strength of the source term. This source term is used to control the application of the source term to only the three-phase interface region.

[0026] Preferably, the mask opening geometry parameters S4 in the step include: opening size; opening shape; opening major and minor axis directions; opening array arrangement; opening center position; and substrate orientation.

[0027] Compared with the prior art, the beneficial effects of this invention are as follows:

[0028] The kinetic Wulff plot is used directly as the growth kinetics input, without relying on simple empirical fitting. Instead, the material orientation-related growth rate is directly introduced into the selective region growth prediction.

[0029] It can handle the complete SAG process, not just the final morphology. It describes the entire process from nucleation to lateral overgrowth, not just the final result.

[0030] It can explicitly consider the geometric effects of mask boundaries, including constraints from the bottom of the hole, sidewalls and top surface, as well as interface stagnation and edge crossing behavior at the edges;

[0031] It can reflect the edge stagnation-bulging-overflow mechanism and is particularly suitable for handling the key transition processes in the selective growth of quantum dots under mask constraints. Attached Figure Description

[0032] Figure 1 This is a flowchart of the method of the present invention;

[0033] Figure 2 This is a 3D surface plot of GaN's v-plot and a cross-sectional view along the critical crystal direction;

[0034] Figure 3 This is a simulation of the SAG process of a two-dimensional GaN single quantum dot under (0001) substrate orientation;

[0035] Figure 4 This is a simulation of the SAG process of a three-dimensional GaN single quantum dot under (0001) substrate orientation;

[0036] Figure 5 This is a simulation of the SAG process of a three-dimensional GaN single quantum dot under (11-20) substrate orientation;

[0037] Figure 6 This is a simulation of the three-dimensional GaN multi-quantum dot SAG process under (11-20) substrate orientation. Detailed Implementation

[0038] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0039] Example

[0040] Please see Figure 1 The illustrated method for predicting selective region growth processes based on dynamic Wulff diagrams includes the following steps:

[0041] Step S1: Obtain the kinetic Wulff plot (v-plot) of the target material under the preset process window and establish the orientation-related growth rate function corresponding to the interface normal direction. The function is used to characterize the relative propulsion capability of different crystals towards the interface and serves as the kinetic input for subsequent growth evolution calculations.

[0042] Step S2: Construct a phase-field model that can describe the evolution of the solid-gas interface, and simultaneously introduce the following into the model:

[0043] (1)

[0044] in The local thermodynamic free energy density is often expressed using the double potential well function, which is used to stabilize the two-phase state of solid and gas. The gradient energy term describes the diffusion interface. In a preferred embodiment, the gradient energy coefficient is... By adopting an isotropic form, it is ensured that anisotropic characteristics mainly originate from growth kinetic factors, i.e. (n).

[0045] One of the most significant characteristics of the SAG process is the constraint of the mask boundary. To characterize this boundary constraint, the Smooth Boundary Method (SBM) is used to forcibly incorporate the boundary condition constraint into the governing equations through integration by parts. In the SBM method, in addition to the field variables used to describe the state of matter, an order parameter also needs to be introduced. Used to distinguish between the boundary region and the solution region, where... =1 indicates a growable region (non-boundary). =0 represents the boundary, and there is a smooth transition between 0 and 1. Therefore, the chemical potential expression under the SBM framework is:

[0046] (2)

[0047] Wherein, θ is the specified equilibrium contact angle at the interface of the three phases of solid, vapor and mask, which characterizes the wetting behavior on the mask.

[0048] Step S3: Establish the growth kinetic equations. First, the conventional Cahn-Hilliard type equations are processed using the SBM method. Then, orientation-dependent growth terms are introduced to describe the deposition growth process driven by both chemical potential gradient-driven diffusion and anisotropic deposition. Furthermore, the contribution of atomic diffusion from the mask surface to the SAG process is not negligible; therefore, an additional three-phase boundary supply term needs to be introduced. Finally, the phase-field governing equations for the SAG process are obtained as follows:

[0049] (3)

[0050] Where the atomic mobility is M s =1,d v It is the deposition rate; v(n) is the orientation-dependent growth rate function, and its corresponding visualization is called the kinetic Wulff plot, also known simply as the v-plot, which is closely related to material properties. Ensure that deposition occurs only at the interface, d TPB A constant parameter characterizing the strength of the source term. Equation (3) describes the SAG process through three complementary terms: mass transport driven by chemical potential gradient, growth driven by deposition, and surface diffusion supply mediated by mask. The first term on the right side of the equation represents the interface diffusion driven by chemical potential during crystal plane evolution. This process redistributes the material between different crystal planes, which is conducive to the continued existence of crystal orientations that grow more slowly but are more kinetically stable. The second term describes the incorporation process of gaseous species at the growth front. Through the growth rate function v(n) related to the crystal orientation, the anisotropic growth kinetics are incorporated into the deposition process, thereby regulating the interface advancement related to the crystal orientation and the overall morphological evolution. The third term describes the diffusion of precursors from the mask surface.

[0051] Step S4: Establish a geometric description. Based on the selective region growth object to be studied, set the mask opening geometric parameters, including: opening size; opening shape; opening major and minor axis directions; opening array arrangement; opening center position; and substrate orientation.

[0052] Step S5, numerical solution, numerical solution of the phase field model, mainly includes spatiotemporal discretization of equation (3). The commonly used spatial discretization scheme is the finite difference method (FDM), or the finite element method (FEM) and finite volume method (FVM). For problems with relatively regular boundary shapes, FDM can be used to approximate the spatial derivative. For microstructure evolution problems with periodic boundary conditions in space, in order to improve computational efficiency, Fourier transform can be used to transform the differential equation in real space into an algebraic equation in reciprocal space, so that the phase field equation becomes simple and easy to solve. When PBC is not satisfied, solving the phase field dynamic equation in reciprocal space becomes very complicated. At this time, FEM is used for solution. The advantage of FEM in solving the evolution equation is that it overcomes the dependence of the reciprocal space solution method on periodic boundary conditions. It can simulate the evolution of material structures with complex geometric shapes and arbitrary boundary conditions, and can be used for secondary development of finite element software to perform large-scale simulation calculations.

[0053] Common time integration schemes are explicit forward Euler method and implicit backward Euler method. The forward Euler method can explicitly obtain the result of the current time step from the result of the previous time step. However, the time step of the forward Euler method has only first-order precision. Therefore, the time step size, the size of the cell mesh, and numerical noise have a certain impact on the computational stability. The implicit backward Euler method can greatly enhance the computational stability, and the time step is much larger than that of the explicit Euler method. In the implicit Euler method, increasing the time step can reduce the number of iterations, but the cost is an increase in the computation time of each iteration step.

[0054] The method provided by the present invention will be simulated and verified below with reference to specific embodiments. (0001) Two-dimensional phase field simulation of GaN quantum dot growth under substrate orientation. First, consider the two-dimensional case of the present invention. The mask thickness of the initial geometric structure is 40nm, the bottom hole diameter is 25nm, and the hole tilt angle is 75°. Figure 2 The two-dimensional growth rate curve shown in b is discretized on a 200×200 grid with a grid spacing of Δx=Δy=1.0nm. Time progression is performed using the Crank-Nicolson scheme with a dimensionless time step of Δt=0.025. All boundaries of the computational domain are subjected to flux-free boundary conditions. The simulated two-dimensional SAG process of the quantum dot is as follows: Figure 3 As shown, simulation results indicate that the crystal first forms an initial nucleus at the bottom of the opening and continues to thicken within the hole. When the growth front reaches the edge of the opening, the three-phase contact line stops at the edge and then forms an upward-bulging front. As the local curvature of the edge increases, the contact line crosses the edge and begins to extend laterally along the top surface of the mask. During epitaxial growth, the kinetically favorable {1-101} crystal plane is gradually retained and expanded, and the overall profile resembles a volcano shape. As growth continues, the (0001) crystal plane with a faster propagation rate gradually decreases, which is manifested as the top concave part gradually becoming smaller. Finally, the two inclined {1-101} crystal planes intersect, and the (0001) crystal plane disappears completely, and the crystal forms a stable isosceles triangular cross-sectional morphology.

[0055] The above implementation steps demonstrate that the present invention can predict the complete evolution path of quantum dots from nucleation to lateral growth in a circular mask opening with inclined sidewalls in a 2D case.

[0056] Next, the morphological evolution prediction of 3D quantum dots under (0001) substrate orientation is performed. Based on the previous implementation step, the model is extended to three dimensions, and the SAG process of 3D quantum dots in the circular opening is also calculated and simulated. The calculation results are as follows: Figure 4As shown, the phenomenon observed in the above embodiments is very consistent with that observed in the above embodiments. After the crystal nucleates in the hole, it continues to thicken. When it crosses the edge of the opening, it undergoes significant lateral expansion and gradually forms six equivalent tilted {1-101} crystal planes and a three-dimensional volcanic structure. The bottom footprint transitions from a near-circular shape to a hexagonal shape. As it continues to grow, the tilted {1-101} crystal planes gradually expand. The (0001) crystal plane corresponding to the top concave region gradually shrinks and disappears with growth, eventually forming a stable hexagonal pyramidal final state morphology.

[0057] This implementation demonstrates that the present invention can not only predict the two-dimensional SAG process, but also effectively simulate the coupling effect in the lateral and height directions and the crystal plane competition evolution process in the three-dimensional case.

[0058] Next, the SAG process on the nonpolar substrate was predicted. Based on the previous step, the substrate orientation was changed to (11-20). Similarly, the SAG process of the 3D quantum dots in the circular opening was calculated and simulated. The calculation results are as follows: Figure 5 As shown, the simulation results indicate that, unlike the previous growth process under polar substrates, the non-polar substrate leads to asymmetry in the in-plane growth rates of different crystal orientations. The growth front on the side closer to the

[0001] direction rises faster, while the one closer to the [000-1] direction rises more slowly. The growth front exhibits an overall morphology of high and low, which causes the two sides of the growth front to reach the opening edge respectively, and then enter the stagnation, bulging and overflow stages successively, thus forming a significant asynchronous epitaxial growth behavior. Upon entering the epitaxial growth stage, the

[0001] direction side... The crystal plane initially evolves into the {1-101} plane, while the [000-1] direction evolves into the {1-100}, (1-120), and (000-1) planes. The overall structure presents a composite morphology composed of half a pyramid and half a prism. As the crystal planes grow further, they gradually shrink due to the rapid growth rate. When the (1-120) plane completely disappears, the final morphology is formed. Subsequent deposition growth will enter an approximately self-similar amplification behavior. Further analysis shows that the lifetime of different transition crystal planes can be used to characterize the time scale required to reach the final morphology.

[0059] This implementation procedure illustrates that the present invention is applicable to the prediction of complex SAG behavior under nonpolar or semipolar orientations.

[0060] Next, the SAG process prediction of multiple quantum dots on a nonpolar substrate is performed. Based on the previous step, a 3×3 staggered close-packed mask array is constructed by increasing the number of openings and changing the mask aperture size and shape. The shape, size parameters, and positions of the mask are shown in Table 1. Figure 6The simulation results show that each quantum dot undergoes a similar growth evolution process according to the phenomenon mentioned in the third implementation step, but the different masks lead to significant differences in their respective evolution processes. Specifically, for circular openings, increasing the size mainly causes an approximately isotropic amplification of the final morphology; for elliptical openings, increasing the opening size along a certain in-plane direction will promote growth in that direction, thereby producing a directional scaling effect on the crystal.

[0061] This implementation process illustrates that the present invention can be used for parameter selection and uniformity optimization at the array scale.

[0062] Table 1. Summary of the position and geometric parameters of each mask opening in the 3×3 mask array

[0063]

[0064] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0065] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for predicting selective region growth processes based on dynamic Wulff diagrams, characterized in that, Includes the following steps: Step S1: Obtain the kinetic Wulff plot (v-plot) of the target material under the preset process window and establish the orientation-related growth rate function corresponding to the interface normal direction. Step S2: Construct a phase-field model that can describe the evolution of the solid-gas interface, and simultaneously introduce the following into the model: (1)， in The local thermodynamic free energy density is expressed using a double potential well function, which is used to stabilize the two-phase state of solid and gas. This is the gradient energy term, used to describe the diffusion interface; Step S3: Establish the growth kinetic equation; Step S4: Establish a geometric description and set the mask opening geometry parameters according to the selective region growth object to be studied. Step S5: Numerical solution, performing numerical solution on the phase field model.

2. The method for predicting selective region growth processes based on dynamic Wulff diagrams according to claim 1, characterized in that: The function in step S1 is used to characterize the relative propulsion capability of different crystal-down interfaces and serves as the kinetic input for subsequent growth evolution calculations.

3. The method for predicting selective region growth processes based on dynamic Wulff diagrams according to claim 2, characterized in that: Step S2 employs the smooth boundary method to forcibly incorporate boundary condition constraints into the governing equations through integration by parts, and the chemical potential expression under the SBM framework is as follows: (2)， Wherein, θ is the specified equilibrium contact angle at the interface of the three phases of solid, vapor and mask, which characterizes the wetting behavior on the mask.

4. The method for predicting selective region growth processes based on dynamic Wulff diagrams according to claim 3, characterized in that: The specific process of step S3 is as follows: First, based on the conventional Cahn-Hilliard type equation, the SBM method is used for processing. Then, orientation-related growth terms are introduced to describe the depositional growth process under the combined effects of chemical potential gradient-driven diffusion and anisotropic deposition. In addition, an extra three-phase boundary supply term is introduced. Finally, the phase-field governing equations of the SAG process are obtained as follows: (3)， Where the atomic mobility is M s =1,d v is the deposition rate; v(n) is the orientation-dependent growth rate function, and its corresponding visual representation is called the kinetic Wulff plot, which is closely related to material properties. Ensure that deposition occurs only at the interface, d TPB A constant parameter characterizing the strength of the source term. This source term is used to control the application of the source term to only the three-phase interface region.

5. The method for predicting selective region growth processes based on dynamic Wulff diagrams according to claim 4, characterized in that: The mask opening geometry parameters S4 in the step include: opening size; opening shape; opening major and minor axis directions; opening array arrangement; opening center position; and substrate orientation.

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