Czochralski Crystal Growth Process Research Articles

Study on the Hierarchical Predictive Control of Semiconductor Silicon Single Crystal Quality Based on the Soft Sensor Model

Silicon single crystal (SSC) quality monitoring and control has been a hot research topic in the field of the Czochralski crystal growth process. Considering that the traditional SSC control method ignores the crystal quality factor, this paper proposes a hierarchical predictive control strategy based on a soft sensor model for online control of SSC diameter and crystal quality. First, the proposed control strategy considers the V/G variable (V is the crystal pulling rate, and G is the axial temperature gradient at the solid-liquid interface), a factor related to crystal quality. Aiming at the problem that the V/G variable is difficult to measure directly, a soft sensor model based on SAE-RF is established to realize the online monitoring of the V/G variable and then complete hierarchical prediction control of SSC quality. Second, in the hierarchical control process, PID control of the inner layer is used to quickly stabilize the system. Model predictive control (MPC) of the outer layer is used to handle system constraints and enhance the control performance of the inner layer. In addition, the SAE-RF-based soft sensor model is used to monitor the crystal quality V/G variable online, thereby ensuring that the output of the controlled system meets the desired crystal diameter and V/G requirements. Finally, based on the industrial data of the actual Czochralski SSC growth process, the effectiveness of the proposed crystal quality hierarchical predictive control method is verified.

Development and validation of a thermal simulation for the Czochralski crystal growth process using model experiments

The Czochralski (CZ) process is one of the most important techniques for the production of bulk single crystals. Various numerical models for CZ growth furnaces exist, however, a review of the recent literature demonstrates that their validation is often questionable due to missing in-situ measurement data. In this work, a new 2D open-source thermal CZ growth model is introduced, verified with analytical solutions, and validated using dedicated model experiments with tin and hence easy access for the required measurements. A three-step procedure is applied to identify critical modeling parameters, make adjustments of these parameters, and evaluate the resulting modeling accuracy. Gas and melt convection, modeled with heat transfer coefficients and an effective heat conductivity, are found to be crucial and are adjusted using experimental data. Comparison of measured temperatures in the crucible shows good agreement, with deviations of max. 0.3 / 1.3 K under natural / forced convection. The model is able to predict the influence of crystal pull velocity and melt temperature on crystal diameter qualitatively.

Model experiments for Czochralski crystal growth processes using inductive and resistive heating

The Czochralski (CZ) growth technique is widely applied in crystal growth, using both induction and resistance heaters. In this work, a novel model experiment platform with comprehensive in-situ measurement capability is introduced. Growth experiments with the model material tin applying both heating concepts are performed and analyzed, e.g., in terms of the maximum achievable crystal diameter. Strong asymmetries in the magnetic field of the induction heater are measured and temperature distribution on the resistance heater is found to be non-uniform. Furthermore, significant losses are observed in the power supplies of the resistance heater. The heating efficiency of both concepts is compared considering different insulation geometries. The obtained results show the capability of model experiments for design optimization and will provide valuable input for further validation of numerical simulations.

Minority carrier lifetime in indium doped silicon for photovoltaics

AbstractFor photovoltaics, switching the p‐type dopant in silicon wafers from boron to indium may be advantageous as boron plays an important role in the light‐induced degradation mechanism. With the continuous Czochralski crystal growth process it is now possible to produce indium doped silicon substrates with the required doping levels for solar cells. This study aims to understand factors controlling the minority carrier lifetime in such substrates with a view to enabling the quantification of the possible benefits of indium doped material. Experiments are performed using temperature‐dependent Hall effect and injection‐dependent carrier lifetime measurements. The recombination rate is found to vary linearly with the concentration of un‐ionized indium which exists in the sample at room temperature due to indium's relatively deep acceptor level at 0.15 eV from the valence band. Lifetime in indium doped silicon is also shown to degrade rapidly under illumination, but to a level substantially higher than in equivalent boron doped silicon samples. A window of opportunity exists in which the minority carrier lifetime in degraded indium doped silicon is higher than the equivalent boron doped silicon, indicating it may be suitable as the base material for front contact photovoltaic cells.

Limitations on control performance in the Czochralski crystal growth process using bright ring measurement as a controlled variable

Abstract This paper develops a dynamical model for crystal diameter in the Czochralski process for production of monocrystalline silicon. The model combines simplified crystal growth dynamics with rigorous ray tracing to describe the camera image used for diameter control, and it is demonstrated that the resulting model captures the so-called measurement anomaly that represents a key performance limitation for crystal diameter control.

Experimental investigation on the buoyancy-induced flow in a model of the Czochralski crystal growth process

Within this paper we present a model experiment focusing on investigations of the flow field in a Czochralski puller. Low melting point liquid metals as GaInSn are an important tool to investigate the flow structure for such industrial processes. The topology of the prevailing thermally-driven convection might be very complex and is mainly determined by the aspect ratio of the liquid volume and the strength of the convection described by the characteristic dimensionless Grashof number. The measurements of the fluid flow have been conducted by means of the ultrasound Doppler velocimetry (UDV) with and without the influence of external magnetic fields. Two kinds of sensor configurations were used to investigate the flow. Firstly, measurements of the radial velocity component by means of single UDV transducers were carried out shortly below the melt surface across the entire diameter of the cylindrical liquid column at various azimuthal angles. Secondly, a vertically arranged UDV array was applied at the side of the cylinder to obtain more detailed information about the radial velocities in the covered meridional plane. The results reveal the complex flow structure of natural convection in a Czochralski crucible which gains in complexity with applied external magnetic fields.

Physical modelling of Czochralski crystal growth in horizontal magnetic field

This study addresses experimentally the heat transfer, the temperature azimuthal non-uniformity and the onset of oscillations in a low temperature physical model of a medium-sized Czochralski crystal growth process with a strong horizontal magnetic field (HMF). It is observed that under certain conditions the integral heat flux may decrease with increasing magnetic field strength at the same time as the flow velocity increases. The azimuthal non-uniformity of the temperature field in the melt near the crystal model rim is only little influenced by its rotation rate outside of a narrow range where the centrifugal force balances the buoyant one. The flow oscillation onset has been observed for two values of the HMF strength. Conditions of this onset are little influenced by the crystal rotation. The critical temperature difference of the oscillation onset considerably exceeds that of the Rayleigh–Bénard (RB) cell in a strong HMF.

Stability analysis of the first-order steady-state solution in the czochralski crystal growth process using perturbation techniques

Stability analysis of the first-order steady-state solution in the czochralski crystal growth process using perturbation techniques

Analysis of the effect of symmetric/asymmetric CUSP magnetic fields on melt/crystal interface during Czochralski silicon growth

With the use of 300mm silicon wafers for industrial semiconductor device manufacturing, the Czochralski (Cz) crystal growth process has to be optimized to achieve higher quality and productivity. Numerical studies based on 2D global thermal models combined with 3D simulation of melt convection are widely used today to save time and money in the process development. Melt convection in large scale Cz Si growth is controlled by a CUSP or transversal magnetic field (MF) to suppress the melt turbulence. MF can be optimized to meet necessary characteristics of the growing crystal, in terms of point defects, as MF affects the melt/crystal interface geometry and allows adjustment of the pulling rate.Among the different knobs associated with the CUSP magnetic field, the nature of its configuration, going from symmetric to asymmetric, is also reported to be an important tool for the control of crystallization front. Using a 3D unsteady model of the CGSim software, we have studied these effects and compared with several experimental results. In addition, physical mechanisms behind these observations are explored through a detailed modeling analysis of the effect of an asymmetric CUSP MF on convection features governing the heat transport in the silicon melt.

Physical modelling of temperature fluctuations in a high aspect ratio model of the Czochralski crystal growth

A low temperature liquid metal model of the Czochralski (CZ) crystal growth process is considered experimentally for a high aspect ratio. Temperature fluctuations close to the edge of the model crystal are studied under the action of a rotating magnetic field (RMF) and/or rotation of the model crystal. A rotation of thermal structures is observed which loses its periodicity at sufficiently high strengths of the RMF. This finding is in qualitative agreement with previous findings in Rayleigh–Bénard (RB) cells. Opposing to that more generic case, the remaining amplitude of the temperature fluctuations stays significantly higher. I.e., the suppression of the fluctuations, which are detrimental to the growth of a mono-crystal, is weaker in the model under investigation.

(Invited) Fabrication of High-Quality Ge-on-Insulator Structures by Lateral Liquid Phase Epitaxy

Germanium (Ge) has been considered as a key material for next-generation electronic–photonic integration in large-scale integrated circuits on Si substrates. Ge-based electronic device is beneficial for high-performance complementary metal-oxide semiconductor application compatible with Si technology because Ge is a Group IV element and the electron and hole mobilities are 2.5-4 times higher than those of Si. Ge metal-oxide-semiconductor field-effect transistors (MOSFETs) might suffer from the large off-leakage current caused by the small bandgap, however Ge-on-insulator (GOI) structure has a great potential to overcome this obstacle. Ge is also expected as light emission and absorption material in the photonic devices. Although Ge has the electronic band structure of indirect band gap, the direct bandgap at Γ valley is 0.8 eV, corresponding to 1550-nm wavelength, same as used in optical communication, and the energy difference between Γ and L valley is only 136 meV. Furthermore, inducing lattice strain and Tin (Sn) incorporation into Ge enhance the carrier mobility and modulate the band structure from the indirect to the direct band structure, resulting in the enhancement of electronic device performance and the increase of light emission and absorption efficiency in photonic devices, respectively. Therefore, a lot of studies have been extensively carried out in order to realize Ge-based electronic–photonic integration. One of the critical challenges for the realization is to fabricate GOI structures with high-crystalline-quality Ge layers. Among various methods of GOI fabrication, layer transfer and Ge condensation are the ones most widely used. However, the former requires complex processing and the latter has difficulties in achieving high-quality GOI layers. Recently, rapid melt growth, in other words, the lateral liquid-phase epitaxy (LLPE) method, was developed for producing local GOI structures on Si substrates [1,2]. In this method amorphous Ge wires connected to the Si substrate at seed area are annealed above the melting point of Ge and lateral epitaxial growth from the seed area along the Ge wires occurs during cooling. This method has the remarkable advantage for heteroepitaxial growth containing large lattice mismatch, where misfit dislocations frequently generate at the interface between the epitaxial layer and the substrate. In the fabrication of GOI structures by this method crystallographic defects generate at the Si seed region because the lattice mismatch between Ge and Si is 4.2%. However, they quickly move out of the thin wires, similar to necking in the Czochralski crystal-growth process, resulting in single-crystalline Ge wires containing no dislocations or stacking faults. This method has another advantage to induce tensile strain in GOI layers. The lattice spacing of Ge is larger than that of Si, so that compressive strain is typically induced in the Ge layer epitaxially grown on the Si substrate. Insertion of low-temperature-grown buffer layer and following high temperature growth allow 0.2% tensile strain in the Ge layer originating from the difference in the thermal-expansion coefficients of the Ge layer with respect to the Si substrate [3]. In the case of the LLPE method, tensile strain of 0.35% is expected because Ge wires are stress free at the melting point (938°C) and cooled down to room temperature, where the difference in the thermal-expansion coefficients is most effectively utilized to induce tensile strain. In this paper, we report the electrical and photonic properties of the LLPE-grown Ge wires. We have investigated the electrical properties of the Ge wires by the back-gate GOI MOSFETs [4,5]. We confirmed a high on/off source-drain current ratio of 106 and high hole mobility of 480 cm2/Vs, which is 2.8 times higher than that of the reference SOI device. We have also examined direct band gap shrinkage in the LLPE-grown Ge wires by means of micro (μ)-photoluminescence (PL) and μ-Raman spectroscopy [6]. Strain analysis and temperature dependent PL spectra indicated that a direct band gap shrinkage amounting to 45 meV was mainly due to a tensile strain of about 0.4% induced by rapid crystallization from the Ge melting point during the LLPE process. [1] Y. Liu et al., Appl. Phys. Lett. 84, 2563 (2004). [2] T. Hashimoto et al., Appl. Phys. Express 2, 066502 (2009). [3] Y. Ishikawa et al., Appl. Phys. Lett. 82, 2044 (2003). [4] Y. Suzuki et al., Appl. Phys. Lett. 101, 202105 (2012). [5] T. Hosoi et al., Appl. Phys. Lett. 105, 173502 (2014). [6] M. Matsue et al., Appl. Phys. Lett. 104, 031106 (2014).

PDE backstepping control of one-dimensional heat equation with time-varying domain

In this work a PDE backstepping-based control law for one-dimensional unstable heat equation with time-varying spatial domain is developed. The underlying parabolic partial differential equation (PDE) with time-varying domain is a model emerging from process control applications such as crystal growth. The use of backstepping control methodology yields the inherent feature of a time-varying PDE describing the kernel of the associated Volterra integral. The well-posedness of PDE kernel is proven and a numerical method to compute the solution of PDE kernel augmented with the error analysis to establish the accuracy of the proposed numerical method is demonstrated. Finally, the explicit form of the full state-feedback control law is given and appropriate simulation is provided for the application of temperature regulation in the Czochralski crystal growth process.

Backstepping output-feedback control of moving boundary parabolic PDEs

This paper extends the backstepping-based observer design to the state estimation of parabolic PDEs with time-dependent spatial domain. The design is developed for the stabilization of a collocated boundary measurement and actuation of an unstable 1D heat equation with the application to the temperature distribution regulation in Czochralski crystal growth process. The PDE system that describes the estimation error dynamics is transformed to an exponentially stable target system through invertible transformations to obtain the time-varying kernel PDE defined on the time-varying triangular-shape domain. The exponential stability of the closed-loop system with an observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system.

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015

Model predictive temperature tracking in crystal growth processes

The temperature gradients and distribution evolution within the crystal domain in Czochralski crystal growth process have important role in produced crystal's quality. Precise and tight regulation of temperature distribution and gradients is the most promising approach to ensure the crystal quality. In this work, the coupled crystal pulling dynamics and heat transfer models in Czochralski crystal growth with a time-varying boundary is considered. The moving boundary finite element method is used to obtain the optimal reference temperature trajectory associated with the reference crystal shape taking into account constraints on input and the temperature gradients. The obtained reference trajectory is used to implement a model predictive control strategy to track the reference temperature despite uncertainties in crystal domain geometry evolution and disturbances. Finally, the proposed method is implemented on a high fidelity finite element process model with non-planar interface and results are presented to validate the success of the proposed methodology.

Nonlinear model-based control of the Czochralski process IV: Feedforward control and its interpretation from the crystal grower׳s view

The paper proposes a model-based feedforward control for the Czochralski crystal growth process. It completes the system for nonlinear model-based control presented by the authors in previous publications. From a system theoretical point of view such a feedforward control requires a dynamic lumped parameter model of the hydrodynamical–geometrical as well as of the thermal subsystem of the process. For the heater feedforward control one lacks the latter one with the necessary accuracy. A static approach circumventing this problem is presented here, based on a mathematical description regarding the most important qualitative dependencies between relative changes of manipulated and controlled variables during growth. In combination with feedback control, high precision tracking of the most important system variables, crystal diameter and crystal growth rate, is ensured. Additionally, a comprehensive discussion of the dependencies of the heater feedforward control on technological parameters, growth regimes and crystal shapes is given.

Temperature distribution reconstruction in Czochralski crystal growth process

A mechanical geometric crystal growth model is developed to describe the crystal length and radius evolution. The crystal radius regulation is achieved by feedback linearization and accounts for parametric uncertainty in the crystal growth rate. The associated parabolic partial differential equation (PDE) model of heat conduction is considered over the time‐varying crystal domain and coupled with crystal growth dynamics. An appropriately defined infinite‐dimensional representation of the thermal evolution is derived considering slow time‐varying process effects. The computational framework of the Galerkin's method is used for parabolic PDE order reduction and observer synthesis for temperature distribution reconstruction over the entire crystal domain. It is shown that the proposed observer can be utilized to reconstruct temperature distribution from boundary temperature measurements. The developed observer is implemented on the finite‐element model of the process and demonstrates that despite parametric and geometric uncertainties present in the model, the temperature distribution is reconstructed with the high accuracy. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2839–2852, 2014

Anisotropic study of thermal stresses induced by diameter fluctuation during Czochralski silicon single crystal growth

Instabilities in the Czochralski crystal growth process result in deviation from the desired crystal diameter i.e. uneven crystal surface. The excessive thermoelastic stress induced by diameter perturbation is studied by mean of numerical simulations. A set of 3D simulation has been performed for an axisymmetric crystal. The crystal anisotropy is taken into account. The influence of crystal diameter fluctuation on stress field inside the crystal has been studied. The resolved shear stresses have been calculated for 12 slip systems for three crystal orientations. Accumulated excess stress from its critical value is calculated. Simulation results suggest that crystal surface undulation affects both thermal field and stress distribution inside the crystal. The crystal region with high risk of dislocation generation is discussed for three crystal orientation. The stress level in [111] crystal orientation is found to be lower than other orientations.

Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

A Review of the Automation of the Czochralski Crystal Growth Process

On the occasion of the centennial of the invention of the Czochralski crystal growth process by the Polish scientist Jan Czochralski, a review of selected strategies for the automatic control of this process is given. This review provides a sketch of the fundamental challenges of controlling the Czochralski process and the basic concepts of feedback control. Both early and modern approaches to the control of the Czochralski process are described. The discussion focuses on questions related to feed-forward control, feedback control, and state estimation. The presented methods rely on simple mathematical process models in contrast to the nite element model-based approaches typically used in crystal growth process design and analysis. Such mathematical models motivate both the structure and parameters of the chosen controller. A comprehensive list of references to background literature on this topic completes this survey.