Czochralski法晶体生长中复杂流动结构及其稳定性研究
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摘要
Czochralski(Cz)熔体晶体生长方法是最常见的单晶生长方法之一，而晶体生长过程中熔体流动稳定性将直接影响晶体材料的制备质量。由于Cz法晶体生长过程中驱动流体流动的力包括浮力、热毛细力、旋转离心力及Coriolis力等，并且各驱动力相互耦合，使得流体流动极其复杂。尽管目前对Cz法晶体生长过程中复杂流动的研究已有不少的报道，但不同学者间研究结果差异大、甚至相互矛盾，其复杂流动的机理仍不清楚。因此，本文以Cz法晶体生长中的复杂流动为研究对象，采用数值模拟和实验观测相结合的方法分别研究了不同驱动力作用下的流动特征，综合分析了各驱动力相互耦合时的流动规律，得到了各驱动力单独或相互耦合作用下流动转变的临界条件，绘制了流动稳定性区域图，获取了流动失稳后各种流型的演变规律，讨论了液池深宽比、半径比、热毛细雷诺数（ReT）以及坩埚和晶体旋转雷诺数（Rec, Res）等对复杂流动的影响，揭示了流动失稳的物理机制，所得结论对丰富和发展复杂流动动力学理论有着重要的学术意义，同时可为完善晶体生长工艺、提高晶体质量提供重要的理论指导。主要研究内容及获得的结果如下：
首先，通过数值模拟研究了Cz结构液池内仅有坩埚和晶体旋转驱动的流动特征，得到了不同旋转条件下的速度场分布，讨论了液池深宽比和半径比对流动结构的影响。结果表明，当旋转雷诺数超过某一临界值时，流动将转变为三维时相关的振荡流动；流动失稳后在流体自由表面上所形成的周向速度振荡波的波数、周向传播方向以及传播速度等都与晶体及坩埚旋转速度的相对大小有关；当晶体和坩埚反向旋转时，流动失稳的物理机制为剪切不稳定性；Cz结构浅液池内当晶体单独旋转或与坩埚同向旋转时，流动失稳的机理为椭圆不稳定性；深液池内当晶体单独旋转时流动的失稳主要是由离心力不稳定性造成的，而当坩埚和晶体同向旋转时流动不稳定性的物理机制为椭圆不稳定性；当液池深宽比一定，在不同的半径比及旋转雷诺数下，发现了从椭圆形到三角形、从四边形至八边形等多种不稳定流动结构。
其次，分析了Cz结构浅液池内低Prandtl数流体热毛细力、旋转离心力和Coriolis力共同作用驱动的流体流动规律。结果表明，当坩埚单独旋转时，流动转变的临界热毛细雷诺数（ReT,c）随着Rec的增加先增加后减小；当晶体和坩埚同向旋转时，在相同Rec下，ReT,c随着Res的增加而减小；另外，晶体和坩埚反向旋转时，流动将经历二次转变；当旋转驱动的强制对流较强时，流动处于不稳定状态，随着热毛细力的增大，旋转对流强度被削弱，流动转变为稳定状态；当ReT进一步增大、热毛细对流占主导地位时，流动再次失稳，转变为不稳定流动；第一次失稳的机理是旋转对流产生的剪切不稳定性，第二次流动转变是由温度和速度振荡不一致激发的。
然后，研究了Cz结构深液池内浮力、热毛细力、旋转离心力和Coriolis力耦合作用驱动的复杂流动机理。研究发现，浮力能促进流动失去稳定性。当ReT较大时，热毛细-浮力对流增强，流动失稳的物理机制为Rayleigh-Bénard流动不稳定性。当各驱动力对流动的影响相当时，其流动失稳的机理为斜压不稳定性。
此外，通过数值模拟讨论了具有液封的Cz结构浅液池内旋转对热毛细对流的影响，获得了流动稳定性区域图。结果发现，当坩埚单独旋转时，ReT,c随坩埚旋转速度的增加而增加；当晶体单独旋转时，ReT,c呈先减小后增大的趋势；当晶体和坩埚反向旋转时，相同Res下坩埚旋转速度越高，ReT,c越小，而当Rec相同时，ReT,c随着Res的增加先减小后增大；当流动失稳后，流场内呈现出多胞振荡结构，且振荡流胞总是在晶体-液封流体边界处产生，然后传向坩埚侧壁，传播过程中振荡逐渐减弱；流动不稳定机理为热流体波不稳定性。另外，研究还发现，坩埚的旋转对液封Cz结构液池内流动的不稳定性有促进作用，而晶体旋转能在一定程度上抑制不稳定性的发生。
最后，采用纹影法对Cz结构液池内硅油失稳后的耗散结构进行了可视化研究。结果表明，当液层较浅时，流动失稳后主要表现为螺纹状的热流体波结构，不同温差条件下，可能出现单列波、双列波甚至多列波相互叠加的结构；当温差增加到一定值时，螺旋状波纹发生变形，特别是对于小半径比结构，两列螺旋波变形叠加形成与Bénard涡胞相似的六边形蜂窝状结构；当液层厚度在58mm时，温度波动呈现出径向条纹结构；随着液池旋转速度的增加，流动逐渐由非稳态振荡结构转变为稳态的轴对称结构，且液层厚度越大，液池旋转对流动不稳定性的抑制作用越大。
The Czochralski (Cz) crystal growth technology is one of the most importantmethods for producing single crystals, where both the crucible containing the melt andthe crystal growing at the melt surface are rotated in opposite directions to smooth theirregular heating. Thus, the forces that can drive the flow include the thermocapillary,buoyancy, centrifugal and Coriolis forces. These forces interact on different scalesmaking the Cz crystal growth process difficult to be controlled and characterized, andseveral flow instabilities may be triggered. These flow instabilities have a direct impacton the quality of the growing single crystal, such as the undesired creation of striations.Up to now, it appears that how the rotation influences the thermal and flow fields, thedetails about the complex flow driven by the combination of buoyancy andthermocapillary forces, Coriolis and centrifugal forces have not been investigatedsystematically, and the mechanisms of flow instabilities remain puzzling, although lotsof works have been reported on the flow behaviors during Cz crystal growth process.
In this thesis, both numerical simulations and experiments are performed tocontribute further to the understanding of the complex flow during Czochralski crystalgrowth process. The critical conditions for the onset of flow instabilities are obtained,and the stability diagrams are mapped. Meanwhile, the effects of the aspect ratio, radiusratio, thermocapillary Reynolds number (ReT), as well as the rotation Reynolds number(Rec, Res) are presented. In addition, the mechanisms of flow instabilities are alsodiscussed. The results obtained herein can not only make progress in hydrodynamics butalso provide some new theoretical bases for the optimization of crystal growthtechnology. The main results are as follows:
Firstly, the characteristics of the three-dimensional flow driven by the rotation ofcrucible and crystal are investigated by numerical simulation. Results show that whenthe rotation Reynolds number is small, the basic flow is axisymmetric and steady.However, when the rotation Reynolds number exceeds a critical value, the flow willundergo a transition to a three-dimensional oscillatory flow, which is characterized bythe velocity fluctuation waves travelling in the azimuthal direction. The propagatingdirection and velocity of the waves, as well as the wave number, are dependent on therotation rate and directions of the crucible and crystal. When the crystal counter rotateswith crucible, the mechanism of the flow transition is the shear instability. In the shallow Cz configuration, when the crystal rotates only or co-rotates with crucible, theelliptic instability is responsible for the flow transition. However, in the deepconfiguration, when the crystal co-rotates with crucible, the flow instability is ellipticinstability, and centrifugal instability is the origin of the flow transition for the case ofcrystal rotation only. In addition, the characteristics of flow also show an importantdependence on the radius ratio. Various polygonal flow patterns are presented atdifferent radius ratio.
Secondly, the fundamental characteristics of the three-dimensional flow of lowPrandtl number fluid induced by crucible and/or crystal rotation and the surface tensiongradient during Czochralski crystal growth process are investigated through a series ofunsteady three-dimensional numerical simulations. The results indicate that the criticalthermocapillary Reynolds number varies with the rotation of crucible and crystal. Whenthe crucible rotates only, the critical thermocapillaty Reynolds number, ReT,c, decreasesfirst and then increases with the increase of Rec. If the crystal co-rotates with crucible,the ReT,cincreases with the increase of Rec, and the higher Resresults in a lower ReT,catthe same Rec. In particular, when the crystal counter-rotates with the crucible, threedifferent flow states are observed and mapped with different ReT. If the rotation-inducedflow is strong, the flow field is already located in the unstable state, as the increasinginfluence of the thermocapillary force, the flow strength is weakened, and the3-Dunsteady flow will transit to stable state. If ReTincreases further, the flow driven by thethermocapillary force is dominant and will lose its stability again, and then transit toanother unstable state.
In addition, the flow driven by the coupled buoyancy and thermocapillary forces,Coriolis and centrifugal forces are discussed. It is indicated that the effect of buoyancyforce can instabilize the flow. For a high ReT, the thermocapillary-buoyancy convectionis dominant, thus the origin of the flow instability is Rayleigh-Bénard instability. Whenthe effects of the driving forces are comparable, the mechanism of the flow transitionhas been proved to be the baroclinic instability.
Furthermore, the combined effects of temperature gradient and counter rotation ofcrucible and crystal on the flow instability in a liquid-encapsulated Czochralskiconfiguration are investigated through a series of direct numerical simulations. Resultsshow that when the ReTexceeds a threshold value, the unsteady multi-cellular structuresare developed. The oscillatory flow behaves as fluctuation waves propagating from thecrystal/fluid interface to the crucible sidewall. The amplitudes of the velocity and temperature fluctuations decrease with the increase of crystal rotation rate, but increasewith the crucible rotation rate. The critical conditions for the onset of flow instabilityare obtained. The stability diagram indicates that the rotation of crucible has adestabilizing effect on the flow, but the crystal rotation can depress the flow instabilitywhen the crystal rotation Reynolds number exceeds a certain value.
Finally, the temperature filed of the complex flow in the Cz configuration isinvestigated experimentally by schlieren technique. For the shallow liquid layer, thetemperature disturbance pattern on the free surface is characterized by the curvedspokes. For the different temperature difference, two groups of hydrothermal waveswith different wave numbers are travelling in the opposite directions, especially for thecase of small radius ratio configuration, the typical Bénard cells are observed. When thedepth of liquid layer locates in the range of5~8mm, with the increase of temperaturedifference, the temperature disturbance transits from the straight spoke pattern to the"flower bud typed" pattern. When the depth is greater than8mm, the temperature profileon the surface is represented as striated pattern. In addition, the rotation of crucible hasmore serious inhibition to the flow instability with the higher depth.
首先，通过数值模拟研究了Cz结构液池内仅有坩埚和晶体旋转驱动的流动特征，得到了不同旋转条件下的速度场分布，讨论了液池深宽比和半径比对流动结构的影响。结果表明，当旋转雷诺数超过某一临界值时，流动将转变为三维时相关的振荡流动；流动失稳后在流体自由表面上所形成的周向速度振荡波的波数、周向传播方向以及传播速度等都与晶体及坩埚旋转速度的相对大小有关；当晶体和坩埚反向旋转时，流动失稳的物理机制为剪切不稳定性；Cz结构浅液池内当晶体单独旋转或与坩埚同向旋转时，流动失稳的机理为椭圆不稳定性；深液池内当晶体单独旋转时流动的失稳主要是由离心力不稳定性造成的，而当坩埚和晶体同向旋转时流动不稳定性的物理机制为椭圆不稳定性；当液池深宽比一定，在不同的半径比及旋转雷诺数下，发现了从椭圆形到三角形、从四边形至八边形等多种不稳定流动结构。
其次，分析了Cz结构浅液池内低Prandtl数流体热毛细力、旋转离心力和Coriolis力共同作用驱动的流体流动规律。结果表明，当坩埚单独旋转时，流动转变的临界热毛细雷诺数（ReT,c）随着Rec的增加先增加后减小；当晶体和坩埚同向旋转时，在相同Rec下，ReT,c随着Res的增加而减小；另外，晶体和坩埚反向旋转时，流动将经历二次转变；当旋转驱动的强制对流较强时，流动处于不稳定状态，随着热毛细力的增大，旋转对流强度被削弱，流动转变为稳定状态；当ReT进一步增大、热毛细对流占主导地位时，流动再次失稳，转变为不稳定流动；第一次失稳的机理是旋转对流产生的剪切不稳定性，第二次流动转变是由温度和速度振荡不一致激发的。
然后，研究了Cz结构深液池内浮力、热毛细力、旋转离心力和Coriolis力耦合作用驱动的复杂流动机理。研究发现，浮力能促进流动失去稳定性。当ReT较大时，热毛细-浮力对流增强，流动失稳的物理机制为Rayleigh-Bénard流动不稳定性。当各驱动力对流动的影响相当时，其流动失稳的机理为斜压不稳定性。
此外，通过数值模拟讨论了具有液封的Cz结构浅液池内旋转对热毛细对流的影响，获得了流动稳定性区域图。结果发现，当坩埚单独旋转时，ReT,c随坩埚旋转速度的增加而增加；当晶体单独旋转时，ReT,c呈先减小后增大的趋势；当晶体和坩埚反向旋转时，相同Res下坩埚旋转速度越高，ReT,c越小，而当Rec相同时，ReT,c随着Res的增加先减小后增大；当流动失稳后，流场内呈现出多胞振荡结构，且振荡流胞总是在晶体-液封流体边界处产生，然后传向坩埚侧壁，传播过程中振荡逐渐减弱；流动不稳定机理为热流体波不稳定性。另外，研究还发现，坩埚的旋转对液封Cz结构液池内流动的不稳定性有促进作用，而晶体旋转能在一定程度上抑制不稳定性的发生。
最后，采用纹影法对Cz结构液池内硅油失稳后的耗散结构进行了可视化研究。结果表明，当液层较浅时，流动失稳后主要表现为螺纹状的热流体波结构，不同温差条件下，可能出现单列波、双列波甚至多列波相互叠加的结构；当温差增加到一定值时，螺旋状波纹发生变形，特别是对于小半径比结构，两列螺旋波变形叠加形成与Bénard涡胞相似的六边形蜂窝状结构；当液层厚度在5
The Czochralski (Cz) crystal growth technology is one of the most importantmethods for producing single crystals, where both the crucible containing the melt andthe crystal growing at the melt surface are rotated in opposite directions to smooth theirregular heating. Thus, the forces that can drive the flow include the thermocapillary,buoyancy, centrifugal and Coriolis forces. These forces interact on different scalesmaking the Cz crystal growth process difficult to be controlled and characterized, andseveral flow instabilities may be triggered. These flow instabilities have a direct impacton the quality of the growing single crystal, such as the undesired creation of striations.Up to now, it appears that how the rotation influences the thermal and flow fields, thedetails about the complex flow driven by the combination of buoyancy andthermocapillary forces, Coriolis and centrifugal forces have not been investigatedsystematically, and the mechanisms of flow instabilities remain puzzling, although lotsof works have been reported on the flow behaviors during Cz crystal growth process.
In this thesis, both numerical simulations and experiments are performed tocontribute further to the understanding of the complex flow during Czochralski crystalgrowth process. The critical conditions for the onset of flow instabilities are obtained,and the stability diagrams are mapped. Meanwhile, the effects of the aspect ratio, radiusratio, thermocapillary Reynolds number (ReT), as well as the rotation Reynolds number(Rec, Res) are presented. In addition, the mechanisms of flow instabilities are alsodiscussed. The results obtained herein can not only make progress in hydrodynamics butalso provide some new theoretical bases for the optimization of crystal growthtechnology. The main results are as follows:
Firstly, the characteristics of the three-dimensional flow driven by the rotation ofcrucible and crystal are investigated by numerical simulation. Results show that whenthe rotation Reynolds number is small, the basic flow is axisymmetric and steady.However, when the rotation Reynolds number exceeds a critical value, the flow willundergo a transition to a three-dimensional oscillatory flow, which is characterized bythe velocity fluctuation waves travelling in the azimuthal direction. The propagatingdirection and velocity of the waves, as well as the wave number, are dependent on therotation rate and directions of the crucible and crystal. When the crystal counter rotateswith crucible, the mechanism of the flow transition is the shear instability. In the shallow Cz configuration, when the crystal rotates only or co-rotates with crucible, theelliptic instability is responsible for the flow transition. However, in the deepconfiguration, when the crystal co-rotates with crucible, the flow instability is ellipticinstability, and centrifugal instability is the origin of the flow transition for the case ofcrystal rotation only. In addition, the characteristics of flow also show an importantdependence on the radius ratio. Various polygonal flow patterns are presented atdifferent radius ratio.
Secondly, the fundamental characteristics of the three-dimensional flow of lowPrandtl number fluid induced by crucible and/or crystal rotation and the surface tensiongradient during Czochralski crystal growth process are investigated through a series ofunsteady three-dimensional numerical simulations. The results indicate that the criticalthermocapillary Reynolds number varies with the rotation of crucible and crystal. Whenthe crucible rotates only, the critical thermocapillaty Reynolds number, ReT,c, decreasesfirst and then increases with the increase of Rec. If the crystal co-rotates with crucible,the ReT,cincreases with the increase of Rec, and the higher Resresults in a lower ReT,catthe same Rec. In particular, when the crystal counter-rotates with the crucible, threedifferent flow states are observed and mapped with different ReT. If the rotation-inducedflow is strong, the flow field is already located in the unstable state, as the increasinginfluence of the thermocapillary force, the flow strength is weakened, and the3-Dunsteady flow will transit to stable state. If ReTincreases further, the flow driven by thethermocapillary force is dominant and will lose its stability again, and then transit toanother unstable state.
In addition, the flow driven by the coupled buoyancy and thermocapillary forces,Coriolis and centrifugal forces are discussed. It is indicated that the effect of buoyancyforce can instabilize the flow. For a high ReT, the thermocapillary-buoyancy convectionis dominant, thus the origin of the flow instability is Rayleigh-Bénard instability. Whenthe effects of the driving forces are comparable, the mechanism of the flow transitionhas been proved to be the baroclinic instability.
Furthermore, the combined effects of temperature gradient and counter rotation ofcrucible and crystal on the flow instability in a liquid-encapsulated Czochralskiconfiguration are investigated through a series of direct numerical simulations. Resultsshow that when the ReTexceeds a threshold value, the unsteady multi-cellular structuresare developed. The oscillatory flow behaves as fluctuation waves propagating from thecrystal/fluid interface to the crucible sidewall. The amplitudes of the velocity and temperature fluctuations decrease with the increase of crystal rotation rate, but increasewith the crucible rotation rate. The critical conditions for the onset of flow instabilityare obtained. The stability diagram indicates that the rotation of crucible has adestabilizing effect on the flow, but the crystal rotation can depress the flow instabilitywhen the crystal rotation Reynolds number exceeds a certain value.
Finally, the temperature filed of the complex flow in the Cz configuration isinvestigated experimentally by schlieren technique. For the shallow liquid layer, thetemperature disturbance pattern on the free surface is characterized by the curvedspokes. For the different temperature difference, two groups of hydrothermal waveswith different wave numbers are travelling in the opposite directions, especially for thecase of small radius ratio configuration, the typical Bénard cells are observed. When thedepth of liquid layer locates in the range of5~8mm, with the increase of temperaturedifference, the temperature disturbance transits from the straight spoke pattern to the"flower bud typed" pattern. When the depth is greater than8mm, the temperature profileon the surface is represented as striated pattern. In addition, the rotation of crucible hasmore serious inhibition to the flow instability with the higher depth.
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