- Email: [email protected]
Analysis of the accelerated crucible rotation technique applied to the gradient freeze growth of cadmium zinc telluride
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Energy Storage Materials journal homepage: www.elsevier.com/locate/ensm
Analysis of the accelerated crucible rotation technique applied to the gradient freeze growth of cadmium zinc telluride Mia S. Divecha, Jeffrey J. Derby
⁎
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
A R T I C L E I N F O
A BS T RAC T
Communicated by S. Uda
We employ finite-element modeling to assess the effects of the accelerated crucible rotation technique (ACRT) on cadmium zinc telluride (CZT) crystals grown from a gradient freeze system. Via consideration of tellurium segregation and transport, we show, for the first time, that steady growth from a tellurium-rich melt produces persistent undercooling in front of the growth interface, likely leading to morphological instability. The application of ACRT rearranges melt flows and tellurium transport but, in contrast to conventional wisdom, does not altogether eliminate undercooling of the melt. Rather, a much more complicated picture arises, where spatio-temporal realignment of undercooled melt may act to locally suppress instability. A better understanding of these mechanisms and quantification of their overall effects will allow for future growth optimization.
Keywords: A1. Computer simulation A1. Crystal morphology A1. Convection A1. Heat transfer A1. Mass transfer B1. Semiconducting II-VI materials
1. Introduction
material quality.
Cadmium zinc telluride (CZT) is used for advanced gamma ray detection due to its high resolution and room temperature operability [1,2]. Current growth methods, typically gradient freeze or Bridgman, are plagued by non-homogeneous zinc distribution and deleterious second-phase particle populations [3–5]. Toward improving the quality of CZT grown from the melt, Lynn and coworkers [6] at Washington State University (WSU) have proposed the application the accelerated crucible rotation technique or ACRT [7,8]. The calculations presented here represent initial steps toward understanding and optimizing ACRT growth conditions for CZT. Even though the accelerated crucible rotation technique has long been known to improve mixing in the liquid phase during crystal growth, only general ideas are available to guide the choice of a rotation schedule to be applied to a specific system. Our primary motivation for the use of ACRT during the growth of CZT is to better mix the melt in order to suppress interface instability and thereby favorably affect the population of tellurium-rich, second-phase particles that arise during growth. These second-phase particles are classified into two groups according to their postulated origins. Precipitates are formed during cooling of the crystal via solid-state nucleation and growth and are the inevitable consequence of the retrograde solidus line of CdTe. Much larger and more dangerous to device performance, inclusions are postulated to arise due to the entrapment of tellurium-rich liquid at a morphologically unstable growth interface. The application of ACRT is postulated to stabilize the growth interface, thus eliminating or reducing the trapping of liquid-phase inclusions and improving
2. Model description
⁎
We apply our previously developed, finite element model that describes the conservation of energy, momentum, and mass, along with segregation and solidification in a system assumed to be axisymmetric [11]. For brevity, we do not describe the details of our model here but refer the interested reader to Refs. [9,10] for its application to other ACRT Bridgman systems. For the studies presented here, we choose to ignore the effects of zinc segregation and instead focus on the effects of tellurium transport and segregation. Specifically, we apply front tracking to determine the solid-liquid interface using both the local temperature and the tellurium composition in the melt, according to the phase diagram [12]. These extensions, adapted from our previous efforts to model the growth of CdTe via the traveling heater method [13,14], are needed to assess issue of constitutional supercooling and morphological stability of the growth interface. As will be apparent in forthcoming results, this extended model is able to track a solid-liquid interface with undercooled melt in front of it, in apparent contradiction to the constitutional supercooling criterion of interface stability put forth in the classical analysis of Tiller et al. [15]. In a real system under conditions of morphological instability, the solidification interface evolves into a three-dimensional, cellular form; see, e.g., the recent work by Ma and Plapp [16]. However, such a representation is not possible in our model, which only allows for twodimensional, axisymmetric behavior. The modeling assumption of axisymmetry likely provides artificial stabilization to the computed
Corresponding author.
http://dx.doi.org/10.1016/j.jcrysgro.2016.09.068
Available online xxxx 0022-0248/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Divecha, M.S., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.068
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 1. (a) Steady-state, initial condition showing geometry, streamlines (left), and temperature contours (right). Streamfunction values: ψmin = −0.04306 cm3/s , ψmax = 0.03570 cm3/s ,
Δψ = 7.875e − 04 . Isotherm values: Tmin = 1249 K , Tmax = 1378 K , ΔT = 1.290 K . (b) A single ACRT rotation schedule that is used in the simulations.
Fig. 2. Growth under steady translation without crucible rotation at various times. In each image, regions of melt undercooling are shown on the left by red contour lines, and tellurium concentration fields are shown on the right, with red and blue corresponding to higher and lower concentration values, respectively. Extrema of tellurium mole fraction for all cases shown are cmin=0.53515 and cmax=0.5461. Time in hours: (a) 2.457, (b) 3.084, (c) 3.712, (d) 4.339, (e) 4.966, and (f) 5.593.
We simulate the gradient freeze method after Washington State University's experimental set up [6], which employs temporally changing thermal profiles to directionally solidify the contents of a crucible fixed at one location in the furnace. The following identical growth conditions were used for all simulations: translation velocity of 2 mm/
solidification front and allows the tracking of unstable interfaces without difficulty. Therefore, we instead employ the extent of predicted undercooling in the melt as a proxy for interface instability in the ensuing discussion. We revisit these issues of stability in the conclusions section.
2
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 3. Growth under accelerated crucible rotation at various times within one ACRT cycle. Streamlines (left) and temperature contours (right) are shown. Streamfunction values: ψmin = −0.1072 cm3/s , ψmax = 0.0961 cm3/s , Δψ = 0.002033 cm3/s . Isotherm values: Tmin = 1224 K , Tmax = 13756 , ΔT = 1.515 K . Cases correspond to the 6 distinct regimes in the ACRT cycle (discussed in text) at the following times (hrs): (a) 2.4549, (b) 2.4795, (c) 2.5001, (d) 2.5206, (e) 2.5494, and (f) 2.5617.
⎛ R2 ⎞1/2 τE = ⎜ ⎟ , ⎝ Ων ⎠
h, furnace gradient of approximately 20 K/cm, and crystal diameter of 65 mm. The melt composition is tellurium-rich, as is typically employed for melt growth of CZT for radiation detector applications. For the following calculations, we specify an initial melt with a uniform tellurium mole fraction of c=0.535.
(1)
which is characteristic of the period needed to approach a rotational flow state, where R is the inner radius of the ampoule, Ω is the applied rotation rate, and ν is the kinematic viscosity of the melt. The second consideration is embodied by a Reynolds number,
3. Results
Reω≡
The initial state for the simulations is presented in Fig. 1a and shows the classic, two-cell flow pattern that results from thermal buoyancy. The upper cell circulates with flow upward along the crucible wall and downward along the centerline due to heating from the furnace. The radial temperature gradient reverses near the solid-liquid interface, driving circulation in the opposite direction, with flow upward at the centerline and downward along the crucible wall. The initial tellurium melt composition for all calculations is taken to be constant. To model the effects of ACRT, a time-dependent, azimuthal velocity is imposed on the crucible walls and solid-liquid interface. The rotation schedule used in the calculations is shown in Fig. 1b. This schedule was designed using characteristic measures of time evolution and strength for ACRT flows. Major considerations are a cycle time long enough to produce a well-developed, rotating fluid state, followed by a deceleration from a flow with enough angular momentum to promote the Taylor-Görtler fluid dynamical instability [9,10]. The first of these considerations is addressed by the Ekman time scale,
ΩR2 , ν
(2)
which measures the relative amount inertia imparted to the flow by rotation. For the WSU growth system, τE ≈ 36 , thus the cycle of applied rotation, 150 s before deceleration, corresponds to approximately 4 Ekman units, which should allow enough time for the melt to spin up. The maximum applied rotation gives a value of Reω≈1, 400 , which is sufficient to impart enough angular momentum to the fluid to trigger the nonlinear Taylor-Görtler instability. 3.1. Growth without crucible rotation We first present the case of growth in this system without application of ACRT. Fig. 2 shows contours of melt undercooling on the left and the tellurium concentration field in the melt on the right at successively increasing times. As growth proceeds, the solid-liquid interface moves upward and becomes slightly more concave in shape.. The tellurium concentration field (right) evolves from an initially 3
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 4. Growth under ACRT at the same times as cases shown in Fig. 3. Regions of melt undercooling are shown on the left by red contour lines, and tellurium concentration fields are shown on the right, using the same scale as in Fig. 2. Extrema of tellurium mole fraction for cases shown are cmin=0.5351 and cmax=0.5408.
3.2. Growth with ACRT
homogenous state (not shown) to that seen in Fig. 2a. Due to the rejection from segregation during growth, an enriched layer of tellurium has built up in the melt in front of the solid-liquid interface. The underlying flow, which retains the two-cell character shown in Fig. 1a, sweeps tellurium inward along the solidification interface and upward at the centerline. The upper portion of the melt is isolated from the lower by the shear layer between the two circulation cells, so tellurium from the lower cell can only slowly diffuse into it. Thus, its concentration is much less than the region near the growth interface. As time proceeds, the total amount of tellurium in the melt increases, as indicated by the changing colors. Even though the absolute changes in composition are rather small, the consequences of the build-up of tellurium in the melt are indicated by the regions of undercooled liquid in front of the growth interface, shown as red contours on the left side of the images of Fig. 2. Specifically, we plot contours of ΔT ≡ Tmp (c ) − Tlocal , where Tmp(c) is the constitutionally-dependent melting temperature from the phase diagram and Tlocal is the local melt temperature. Thus, any red indicates undercooled melt that is below its solidification temperature. The existence of these regions indicates the possible onset of morphological instability during crystal growth via the mechanism of constitutional supercooling, as originally put forth by Tiller et al. [15]. As shown by the increased areas enclosed by red contours in Fig. 2, the melt adjacent to the growth interface becomes progressively more unstable as time proceeds. We believe that this is the first calculation showing tellurium-driven constitutional supercooling during the growth of cadmium telluride in a vertical gradient freeze system.
The next set of simulations consider the same growth conditions but with the repeated application the ACRT rotation schedule of Fig. 1b to the system. Plots of streamlines (left) and temperature isotherms (right) are shown for ACRT growth in Fig. 3. The system is shown at approximately the same stage of growth as that shown in Fig. 2a, and each case corresponds to a different regime of rotation during one ACRT cycle: (a) acceleration, (b) constant RPM, (c) deceleration, (d) acceleration in the reverse direction, (e) constant RPM in the reverse direction, and (f) deceleration in the reverse direction. Two notable fluid dynamical phenomena are seen during spin up and spin down of the crucible. The characteristic Ekman flow [17], appearing during acceleration, is observed in Figs. 3a and d and is responsible for forcing fluid outwards along the upper and lower interfaces. Taylor-Görtler vortices [7], stacked vertically along the crucible wall and appearing during deceleration, are observed in Fig. 3c and f and are responsible for mixing the bulk. These flow structures are critical to achieving mixing in ACRT and are discussed more extensively in Refs. [9,10]. The isotherms, drawn in the right-side images of these plots, clearly show that the solid-liquid interface does not lie on a single isotherm. With variation of the melt composition caused by segregation and convection, the interface follows instead the solidification temperatures given by the phase diagram. Fig. 4 shows contours of melt undercooling on the left and the tellurium concentration field in the melt on the right at the same times as in the previous figure. Interestingly, the tellurium in the melt is not completely well mixed, even though seven complete ACRT cycles have 4
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
University. A. Yeckel provided support for this work though code development.
been applied to the initial condition for these cases. Although the forced azimuthal flows are strong enough to partially disrupt the underlying thermal-buoyancy driven vortices, there still exists a shear layer that separates two regions of different tellurium levels. Surprisingly, the left images of Fig. 4 show a very complicated spatio-temporal evolution of melt undercooling through the ACRT cycle and demonstrate that, at least for this particular rotation schedule, ACRT does not completely eliminate undercooling. Rather, the regions of undercooled melt sweep across the interface while simultaneously growing or shrinking during different portions of the ACRT cycle. We comment on the significance of this behavior below.
References [1] E. Raiskin, J. Butler, CdTe low level gamma detectors based on a new crystal growth method, IEEE Trans. Nucl. Sci. 35 (1) (1988) 81–84. http://dx.doi.org/ 10.1109/23.12678. [2] J.F. Butler, B.A. Apotovsky, A. Niemela, H. Sipila, Sub-keV resolution detection with Cd1− x Znx Te detectors, in: Proceedings of the SPIE 2009, X-Ray Detector Physics and Applications II, 12, 1, 1993, pp. 121–127. http://dx.doi.org/10.1117/ 12.164731 [3] P. Rudolph, Fundamental studies on Bridgman growth of CdTe, Prog. Cryst. Growth Charact. Mater. 29 (1–4) (1994) 275–381. http://dx.doi.org/10.1016/ 0960-8974(94)90009-4 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0960897494900094〉]. [4] P. Rudolph, Non-stoichiometry related defects at the melt growth of semiconductor compound crystals - a review, Cryst. Res. Technol. 38 (7–8) (2003) 542–554. http://dx.doi.org/10.1002/crat.200310069 [〈http://onlinelibrary.wiley.com.ezp2. lib.umn.edu/doi/10.1002/crat.200310069/abstract〉]. [5] P. Rudolph, M. Mühlberg, Basic problems of vertical bridgman growth of CdTe, Mater. Sci. Eng.: B 16 (1) (1993) 8–16. http://dx.doi.org/10.1016/0921-5107(93) 90005-8 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0921510793900058〉]. [6] A. Datta, S. Swain, Y. Cui, A. Burger, K. Lynn, Correlations of Bridgman-grown Cd 0.9Zn 0.1 Te properties with different ampoule rotation schemes, J. Electron. Mater. 42 (11) (2013) 3041–3053. http://dx.doi.org/10.1007/s11664-013-2782-x. [7] E.O. Schulz-Dubois, Accelerated crucible rotation: hydrodynamics and stirring effect, J. Cryst. Growth 12 (2) (1972) 81–87. http://dx.doi.org/10.1016/00220248(72)90034-6 [〈http://www.sciencedirect.com/science/article/pii/ 0022024872900346〉]. [8] H. Scheel, Accelerated crucible rotation: a novel stirring technique in hightemperature solution growth, J. Cryst. Growth 13–14 (1972) 560–565. http:// dx.doi.org/10.1016/0022-0248(72)90516-7 [URL 〈http://linkinghub.elsevier. com/retrieve/pii/0022024872905167〉]. [9] A. Yeckel, J.J. Derby, Effect of accelerated crucible rotation on melt composition in high-pressure vertical Bridgman growth of cadmium zinc telluride, J. Cryst. Growth 209 (4) (2000) 734–750. http://dx.doi.org/10.1016/S0022-0248(99)00498-4 [URL 〈http://www.sciencedirect.com/science/article/pii/S0022024899004984〉]. [10] A. Yeckel, J.J. Derby, Buoyancy and rotation in small-scale vertical bridgman growth of cadmium zinc telluride using accelerated crucible rotation, J. Cryst. Growth 233 (3) (2001) 599–608. http://dx.doi.org/10.1016/S0022-0248(01) 01601-3 [URL 〈http://www.sciencedirect.com/science/article/pii/ S0022024801016013〉]. [11] A. Yeckel, J.J. Derby, Computer modelling of bulk crystal growth, in: P. Capper, A. Willougby, S. Kasap (Eds.), Crystal Growth of Electronic, Optical, and Optoelectronic Materials, John Wiley & Sons, West Sussex, UK,, 2005, pp. 73–119. [12] J.H. Greenberg, P-T-X phase equilibrium and vapor pressure scanning of nonstoichiometry in CdTe, J. Cryst. Growth 161 (1–4) (1996) 1–11. http://dx.doi.org/ 10.1016/0022-0248(95)00603-6 [URL 〈http://www.sciencedirect.com/science/ article/pii/0022024895006036〉]. [13] J.H. Peterson, A. Yeckel, J.J. Derby, A fundamental limitation on growth rates in the traveling heater method, J. Cryst. Growth (2016) (In press). [14] J.H. Peterson, M. Fiederle, J.J. Derby, Analysis of the traveling heater method for the growth of cadmium telluride, J. Cryst. Growth 454 (2016) 45–58. [15] W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers, The redistribution of solute atoms during the solidification of metals, Acta Metall. 1 (4) (1953) 428–437. http://dx.doi.org/10.1016/0001-6160(53)90126-6 [URL 〈http://www. sciencedirect.com/science/article/pii/0001616053901266〉]. [16] Y. Ma, M. Plapp, Phase-field simulations and geometrical characterization of cellular solidification fronts, J. Cryst. Growth 385 (2014) 140–147. http:// dx.doi.org/10.1016/j.jcrysgro.2013.03.027 [URL 〈http://www.sciencedirect.com/ science/article/pii/S0022024813002145〉]. [17] P. Capper, J.J.G. Gosney, C.L. Jones, Application of the accelerated crucible rotation technique to the Bridgman growth of CdxHg1- xTe: simulations and crystal growth, J. Cryst. Growth 70 (1) (1984) 356–364. http://dx.doi.org/10.1016/00220248(84)90287-2 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0022024884902872〉].
4. Conclusions We have presented initial studies of the effects of tellurium segregation and transport during the growth of CZT via a gradient freeze crystal growth process representative of that employed at Washington State University [6]. Our simulations demonstrate that steady growth, without rotation, from an initially tellurium-rich melt results in segregation that is sufficient to drive supercooling. We believe that this is the first computation to demonstrate persistent melt supercooling from the effects of excess tellurium for the growth of CZT in a Bridgman orientation. That such conditions exist is consistent with the notion that liquid-phase inclusions may be captured via morphological instabilities of the growth interface. The application of ACRT to this system shows that, despite the formation of Ekman and Taylor-Görtler flows, complete mixing is not attained in the melt. While this outcome calls for the examination of more vigorous rotation schedules for this system, it also raises some interesting questions about how ACRT acts to suppress instability. Conventional wisdom presumes that near-complete mixing driven by ACRT eliminates all supercooling in the melt; however, we observe that undercooled regions of melt shift through time and space in response to ACRT-driven flows. We speculate that the periodic abatement of undercooling may enable local re-stabilization of the interface. In contrast to the case of steady growth, where the persistent presence of undercooling likely forms and propagates a cellular morphology over time, the growth of cells may be inhibited by the short periods of instability over an ACRT cycle or, in effect, healed by periods when the undercooling is momentarily swept away. We intend to clarify the detailed mechanisms by which ACRT acts to stabilize growth in ongoing studies. Toward this general goal, we aim to develop a rigorous, quantitative metric that can account for the temporal and spatial variations in stability along the interface. Such a metric could be applied to not only predict the onset of morphological instability but also optimize rotational schemes, furnace profiles, and geometries. Eventually this will inform our abilities to improve conditions to grow high-quality CZT from the melt. Acknowledgments This work has been supported in part by the U.S. Department of Energy, NNSA Prime Award DE-NA0002565, and Washington State University Sub-award 118717-G003369; no official endorsement should be inferred. We are thankful for significant input of S.K. Swain, J.J. McCoy, S. Kakkireni, and K.G. Lynn of Washington State
5
Contents lists available at ScienceDirect
Energy Storage Materials journal homepage: www.elsevier.com/locate/ensm
Analysis of the accelerated crucible rotation technique applied to the gradient freeze growth of cadmium zinc telluride Mia S. Divecha, Jeffrey J. Derby
⁎
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
A R T I C L E I N F O
A BS T RAC T
Communicated by S. Uda
We employ finite-element modeling to assess the effects of the accelerated crucible rotation technique (ACRT) on cadmium zinc telluride (CZT) crystals grown from a gradient freeze system. Via consideration of tellurium segregation and transport, we show, for the first time, that steady growth from a tellurium-rich melt produces persistent undercooling in front of the growth interface, likely leading to morphological instability. The application of ACRT rearranges melt flows and tellurium transport but, in contrast to conventional wisdom, does not altogether eliminate undercooling of the melt. Rather, a much more complicated picture arises, where spatio-temporal realignment of undercooled melt may act to locally suppress instability. A better understanding of these mechanisms and quantification of their overall effects will allow for future growth optimization.
Keywords: A1. Computer simulation A1. Crystal morphology A1. Convection A1. Heat transfer A1. Mass transfer B1. Semiconducting II-VI materials
1. Introduction
material quality.
Cadmium zinc telluride (CZT) is used for advanced gamma ray detection due to its high resolution and room temperature operability [1,2]. Current growth methods, typically gradient freeze or Bridgman, are plagued by non-homogeneous zinc distribution and deleterious second-phase particle populations [3–5]. Toward improving the quality of CZT grown from the melt, Lynn and coworkers [6] at Washington State University (WSU) have proposed the application the accelerated crucible rotation technique or ACRT [7,8]. The calculations presented here represent initial steps toward understanding and optimizing ACRT growth conditions for CZT. Even though the accelerated crucible rotation technique has long been known to improve mixing in the liquid phase during crystal growth, only general ideas are available to guide the choice of a rotation schedule to be applied to a specific system. Our primary motivation for the use of ACRT during the growth of CZT is to better mix the melt in order to suppress interface instability and thereby favorably affect the population of tellurium-rich, second-phase particles that arise during growth. These second-phase particles are classified into two groups according to their postulated origins. Precipitates are formed during cooling of the crystal via solid-state nucleation and growth and are the inevitable consequence of the retrograde solidus line of CdTe. Much larger and more dangerous to device performance, inclusions are postulated to arise due to the entrapment of tellurium-rich liquid at a morphologically unstable growth interface. The application of ACRT is postulated to stabilize the growth interface, thus eliminating or reducing the trapping of liquid-phase inclusions and improving
2. Model description
⁎
We apply our previously developed, finite element model that describes the conservation of energy, momentum, and mass, along with segregation and solidification in a system assumed to be axisymmetric [11]. For brevity, we do not describe the details of our model here but refer the interested reader to Refs. [9,10] for its application to other ACRT Bridgman systems. For the studies presented here, we choose to ignore the effects of zinc segregation and instead focus on the effects of tellurium transport and segregation. Specifically, we apply front tracking to determine the solid-liquid interface using both the local temperature and the tellurium composition in the melt, according to the phase diagram [12]. These extensions, adapted from our previous efforts to model the growth of CdTe via the traveling heater method [13,14], are needed to assess issue of constitutional supercooling and morphological stability of the growth interface. As will be apparent in forthcoming results, this extended model is able to track a solid-liquid interface with undercooled melt in front of it, in apparent contradiction to the constitutional supercooling criterion of interface stability put forth in the classical analysis of Tiller et al. [15]. In a real system under conditions of morphological instability, the solidification interface evolves into a three-dimensional, cellular form; see, e.g., the recent work by Ma and Plapp [16]. However, such a representation is not possible in our model, which only allows for twodimensional, axisymmetric behavior. The modeling assumption of axisymmetry likely provides artificial stabilization to the computed
Corresponding author.
http://dx.doi.org/10.1016/j.jcrysgro.2016.09.068
Available online xxxx 0022-0248/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Divecha, M.S., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.068
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 1. (a) Steady-state, initial condition showing geometry, streamlines (left), and temperature contours (right). Streamfunction values: ψmin = −0.04306 cm3/s , ψmax = 0.03570 cm3/s ,
Δψ = 7.875e − 04 . Isotherm values: Tmin = 1249 K , Tmax = 1378 K , ΔT = 1.290 K . (b) A single ACRT rotation schedule that is used in the simulations.
Fig. 2. Growth under steady translation without crucible rotation at various times. In each image, regions of melt undercooling are shown on the left by red contour lines, and tellurium concentration fields are shown on the right, with red and blue corresponding to higher and lower concentration values, respectively. Extrema of tellurium mole fraction for all cases shown are cmin=0.53515 and cmax=0.5461. Time in hours: (a) 2.457, (b) 3.084, (c) 3.712, (d) 4.339, (e) 4.966, and (f) 5.593.
We simulate the gradient freeze method after Washington State University's experimental set up [6], which employs temporally changing thermal profiles to directionally solidify the contents of a crucible fixed at one location in the furnace. The following identical growth conditions were used for all simulations: translation velocity of 2 mm/
solidification front and allows the tracking of unstable interfaces without difficulty. Therefore, we instead employ the extent of predicted undercooling in the melt as a proxy for interface instability in the ensuing discussion. We revisit these issues of stability in the conclusions section.
2
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 3. Growth under accelerated crucible rotation at various times within one ACRT cycle. Streamlines (left) and temperature contours (right) are shown. Streamfunction values: ψmin = −0.1072 cm3/s , ψmax = 0.0961 cm3/s , Δψ = 0.002033 cm3/s . Isotherm values: Tmin = 1224 K , Tmax = 13756 , ΔT = 1.515 K . Cases correspond to the 6 distinct regimes in the ACRT cycle (discussed in text) at the following times (hrs): (a) 2.4549, (b) 2.4795, (c) 2.5001, (d) 2.5206, (e) 2.5494, and (f) 2.5617.
⎛ R2 ⎞1/2 τE = ⎜ ⎟ , ⎝ Ων ⎠
h, furnace gradient of approximately 20 K/cm, and crystal diameter of 65 mm. The melt composition is tellurium-rich, as is typically employed for melt growth of CZT for radiation detector applications. For the following calculations, we specify an initial melt with a uniform tellurium mole fraction of c=0.535.
(1)
which is characteristic of the period needed to approach a rotational flow state, where R is the inner radius of the ampoule, Ω is the applied rotation rate, and ν is the kinematic viscosity of the melt. The second consideration is embodied by a Reynolds number,
3. Results
Reω≡
The initial state for the simulations is presented in Fig. 1a and shows the classic, two-cell flow pattern that results from thermal buoyancy. The upper cell circulates with flow upward along the crucible wall and downward along the centerline due to heating from the furnace. The radial temperature gradient reverses near the solid-liquid interface, driving circulation in the opposite direction, with flow upward at the centerline and downward along the crucible wall. The initial tellurium melt composition for all calculations is taken to be constant. To model the effects of ACRT, a time-dependent, azimuthal velocity is imposed on the crucible walls and solid-liquid interface. The rotation schedule used in the calculations is shown in Fig. 1b. This schedule was designed using characteristic measures of time evolution and strength for ACRT flows. Major considerations are a cycle time long enough to produce a well-developed, rotating fluid state, followed by a deceleration from a flow with enough angular momentum to promote the Taylor-Görtler fluid dynamical instability [9,10]. The first of these considerations is addressed by the Ekman time scale,
ΩR2 , ν
(2)
which measures the relative amount inertia imparted to the flow by rotation. For the WSU growth system, τE ≈ 36 , thus the cycle of applied rotation, 150 s before deceleration, corresponds to approximately 4 Ekman units, which should allow enough time for the melt to spin up. The maximum applied rotation gives a value of Reω≈1, 400 , which is sufficient to impart enough angular momentum to the fluid to trigger the nonlinear Taylor-Görtler instability. 3.1. Growth without crucible rotation We first present the case of growth in this system without application of ACRT. Fig. 2 shows contours of melt undercooling on the left and the tellurium concentration field in the melt on the right at successively increasing times. As growth proceeds, the solid-liquid interface moves upward and becomes slightly more concave in shape.. The tellurium concentration field (right) evolves from an initially 3
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
Fig. 4. Growth under ACRT at the same times as cases shown in Fig. 3. Regions of melt undercooling are shown on the left by red contour lines, and tellurium concentration fields are shown on the right, using the same scale as in Fig. 2. Extrema of tellurium mole fraction for cases shown are cmin=0.5351 and cmax=0.5408.
3.2. Growth with ACRT
homogenous state (not shown) to that seen in Fig. 2a. Due to the rejection from segregation during growth, an enriched layer of tellurium has built up in the melt in front of the solid-liquid interface. The underlying flow, which retains the two-cell character shown in Fig. 1a, sweeps tellurium inward along the solidification interface and upward at the centerline. The upper portion of the melt is isolated from the lower by the shear layer between the two circulation cells, so tellurium from the lower cell can only slowly diffuse into it. Thus, its concentration is much less than the region near the growth interface. As time proceeds, the total amount of tellurium in the melt increases, as indicated by the changing colors. Even though the absolute changes in composition are rather small, the consequences of the build-up of tellurium in the melt are indicated by the regions of undercooled liquid in front of the growth interface, shown as red contours on the left side of the images of Fig. 2. Specifically, we plot contours of ΔT ≡ Tmp (c ) − Tlocal , where Tmp(c) is the constitutionally-dependent melting temperature from the phase diagram and Tlocal is the local melt temperature. Thus, any red indicates undercooled melt that is below its solidification temperature. The existence of these regions indicates the possible onset of morphological instability during crystal growth via the mechanism of constitutional supercooling, as originally put forth by Tiller et al. [15]. As shown by the increased areas enclosed by red contours in Fig. 2, the melt adjacent to the growth interface becomes progressively more unstable as time proceeds. We believe that this is the first calculation showing tellurium-driven constitutional supercooling during the growth of cadmium telluride in a vertical gradient freeze system.
The next set of simulations consider the same growth conditions but with the repeated application the ACRT rotation schedule of Fig. 1b to the system. Plots of streamlines (left) and temperature isotherms (right) are shown for ACRT growth in Fig. 3. The system is shown at approximately the same stage of growth as that shown in Fig. 2a, and each case corresponds to a different regime of rotation during one ACRT cycle: (a) acceleration, (b) constant RPM, (c) deceleration, (d) acceleration in the reverse direction, (e) constant RPM in the reverse direction, and (f) deceleration in the reverse direction. Two notable fluid dynamical phenomena are seen during spin up and spin down of the crucible. The characteristic Ekman flow [17], appearing during acceleration, is observed in Figs. 3a and d and is responsible for forcing fluid outwards along the upper and lower interfaces. Taylor-Görtler vortices [7], stacked vertically along the crucible wall and appearing during deceleration, are observed in Fig. 3c and f and are responsible for mixing the bulk. These flow structures are critical to achieving mixing in ACRT and are discussed more extensively in Refs. [9,10]. The isotherms, drawn in the right-side images of these plots, clearly show that the solid-liquid interface does not lie on a single isotherm. With variation of the melt composition caused by segregation and convection, the interface follows instead the solidification temperatures given by the phase diagram. Fig. 4 shows contours of melt undercooling on the left and the tellurium concentration field in the melt on the right at the same times as in the previous figure. Interestingly, the tellurium in the melt is not completely well mixed, even though seven complete ACRT cycles have 4
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
M.S. Divecha, J.J. Derby
University. A. Yeckel provided support for this work though code development.
been applied to the initial condition for these cases. Although the forced azimuthal flows are strong enough to partially disrupt the underlying thermal-buoyancy driven vortices, there still exists a shear layer that separates two regions of different tellurium levels. Surprisingly, the left images of Fig. 4 show a very complicated spatio-temporal evolution of melt undercooling through the ACRT cycle and demonstrate that, at least for this particular rotation schedule, ACRT does not completely eliminate undercooling. Rather, the regions of undercooled melt sweep across the interface while simultaneously growing or shrinking during different portions of the ACRT cycle. We comment on the significance of this behavior below.
References [1] E. Raiskin, J. Butler, CdTe low level gamma detectors based on a new crystal growth method, IEEE Trans. Nucl. Sci. 35 (1) (1988) 81–84. http://dx.doi.org/ 10.1109/23.12678. [2] J.F. Butler, B.A. Apotovsky, A. Niemela, H. Sipila, Sub-keV resolution detection with Cd1− x Znx Te detectors, in: Proceedings of the SPIE 2009, X-Ray Detector Physics and Applications II, 12, 1, 1993, pp. 121–127. http://dx.doi.org/10.1117/ 12.164731 [3] P. Rudolph, Fundamental studies on Bridgman growth of CdTe, Prog. Cryst. Growth Charact. Mater. 29 (1–4) (1994) 275–381. http://dx.doi.org/10.1016/ 0960-8974(94)90009-4 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0960897494900094〉]. [4] P. Rudolph, Non-stoichiometry related defects at the melt growth of semiconductor compound crystals - a review, Cryst. Res. Technol. 38 (7–8) (2003) 542–554. http://dx.doi.org/10.1002/crat.200310069 [〈http://onlinelibrary.wiley.com.ezp2. lib.umn.edu/doi/10.1002/crat.200310069/abstract〉]. [5] P. Rudolph, M. Mühlberg, Basic problems of vertical bridgman growth of CdTe, Mater. Sci. Eng.: B 16 (1) (1993) 8–16. http://dx.doi.org/10.1016/0921-5107(93) 90005-8 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0921510793900058〉]. [6] A. Datta, S. Swain, Y. Cui, A. Burger, K. Lynn, Correlations of Bridgman-grown Cd 0.9Zn 0.1 Te properties with different ampoule rotation schemes, J. Electron. Mater. 42 (11) (2013) 3041–3053. http://dx.doi.org/10.1007/s11664-013-2782-x. [7] E.O. Schulz-Dubois, Accelerated crucible rotation: hydrodynamics and stirring effect, J. Cryst. Growth 12 (2) (1972) 81–87. http://dx.doi.org/10.1016/00220248(72)90034-6 [〈http://www.sciencedirect.com/science/article/pii/ 0022024872900346〉]. [8] H. Scheel, Accelerated crucible rotation: a novel stirring technique in hightemperature solution growth, J. Cryst. Growth 13–14 (1972) 560–565. http:// dx.doi.org/10.1016/0022-0248(72)90516-7 [URL 〈http://linkinghub.elsevier. com/retrieve/pii/0022024872905167〉]. [9] A. Yeckel, J.J. Derby, Effect of accelerated crucible rotation on melt composition in high-pressure vertical Bridgman growth of cadmium zinc telluride, J. Cryst. Growth 209 (4) (2000) 734–750. http://dx.doi.org/10.1016/S0022-0248(99)00498-4 [URL 〈http://www.sciencedirect.com/science/article/pii/S0022024899004984〉]. [10] A. Yeckel, J.J. Derby, Buoyancy and rotation in small-scale vertical bridgman growth of cadmium zinc telluride using accelerated crucible rotation, J. Cryst. Growth 233 (3) (2001) 599–608. http://dx.doi.org/10.1016/S0022-0248(01) 01601-3 [URL 〈http://www.sciencedirect.com/science/article/pii/ S0022024801016013〉]. [11] A. Yeckel, J.J. Derby, Computer modelling of bulk crystal growth, in: P. Capper, A. Willougby, S. Kasap (Eds.), Crystal Growth of Electronic, Optical, and Optoelectronic Materials, John Wiley & Sons, West Sussex, UK,, 2005, pp. 73–119. [12] J.H. Greenberg, P-T-X phase equilibrium and vapor pressure scanning of nonstoichiometry in CdTe, J. Cryst. Growth 161 (1–4) (1996) 1–11. http://dx.doi.org/ 10.1016/0022-0248(95)00603-6 [URL 〈http://www.sciencedirect.com/science/ article/pii/0022024895006036〉]. [13] J.H. Peterson, A. Yeckel, J.J. Derby, A fundamental limitation on growth rates in the traveling heater method, J. Cryst. Growth (2016) (In press). [14] J.H. Peterson, M. Fiederle, J.J. Derby, Analysis of the traveling heater method for the growth of cadmium telluride, J. Cryst. Growth 454 (2016) 45–58. [15] W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers, The redistribution of solute atoms during the solidification of metals, Acta Metall. 1 (4) (1953) 428–437. http://dx.doi.org/10.1016/0001-6160(53)90126-6 [URL 〈http://www. sciencedirect.com/science/article/pii/0001616053901266〉]. [16] Y. Ma, M. Plapp, Phase-field simulations and geometrical characterization of cellular solidification fronts, J. Cryst. Growth 385 (2014) 140–147. http:// dx.doi.org/10.1016/j.jcrysgro.2013.03.027 [URL 〈http://www.sciencedirect.com/ science/article/pii/S0022024813002145〉]. [17] P. Capper, J.J.G. Gosney, C.L. Jones, Application of the accelerated crucible rotation technique to the Bridgman growth of CdxHg1- xTe: simulations and crystal growth, J. Cryst. Growth 70 (1) (1984) 356–364. http://dx.doi.org/10.1016/00220248(84)90287-2 [URL 〈http://www.sciencedirect.com/science/article/pii/ 0022024884902872〉].
4. Conclusions We have presented initial studies of the effects of tellurium segregation and transport during the growth of CZT via a gradient freeze crystal growth process representative of that employed at Washington State University [6]. Our simulations demonstrate that steady growth, without rotation, from an initially tellurium-rich melt results in segregation that is sufficient to drive supercooling. We believe that this is the first computation to demonstrate persistent melt supercooling from the effects of excess tellurium for the growth of CZT in a Bridgman orientation. That such conditions exist is consistent with the notion that liquid-phase inclusions may be captured via morphological instabilities of the growth interface. The application of ACRT to this system shows that, despite the formation of Ekman and Taylor-Görtler flows, complete mixing is not attained in the melt. While this outcome calls for the examination of more vigorous rotation schedules for this system, it also raises some interesting questions about how ACRT acts to suppress instability. Conventional wisdom presumes that near-complete mixing driven by ACRT eliminates all supercooling in the melt; however, we observe that undercooled regions of melt shift through time and space in response to ACRT-driven flows. We speculate that the periodic abatement of undercooling may enable local re-stabilization of the interface. In contrast to the case of steady growth, where the persistent presence of undercooling likely forms and propagates a cellular morphology over time, the growth of cells may be inhibited by the short periods of instability over an ACRT cycle or, in effect, healed by periods when the undercooling is momentarily swept away. We intend to clarify the detailed mechanisms by which ACRT acts to stabilize growth in ongoing studies. Toward this general goal, we aim to develop a rigorous, quantitative metric that can account for the temporal and spatial variations in stability along the interface. Such a metric could be applied to not only predict the onset of morphological instability but also optimize rotational schemes, furnace profiles, and geometries. Eventually this will inform our abilities to improve conditions to grow high-quality CZT from the melt. Acknowledgments This work has been supported in part by the U.S. Department of Energy, NNSA Prime Award DE-NA0002565, and Washington State University Sub-award 118717-G003369; no official endorsement should be inferred. We are thankful for significant input of S.K. Swain, J.J. McCoy, S. Kakkireni, and K.G. Lynn of Washington State
5