Phase-field simulations of Te-precipitate morphology and evolution kinetics in Te-rich CdTe crystals

ARTICLE IN PRESS Journal of Crystal Growth 311 (2009) 3184–3194

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Phase-field simulations of Te-precipitate morphology and evolution kinetics in Te-rich CdTe crystals Shenyang Hu , Charles H. Henager Jr. Pacific Northwest National Laboratory, Richland, WA 99352, USA

a r t i c l e in f o

a b s t r a c t

Article history: Received 7 October 2008 Received in revised form 25 February 2009 Accepted 25 February 2009 Communicated by M. Skowronski Available online 4 March 2009

Te precipitates are one of principal defects that form during cooling of melt-grown CdTe or CZT crystals when grown Te-rich. Many factors such as the kinetic properties of intrinsic point defects (vacancy, interstitial, and antisite defects); stresses associated with the lattice mismatch between precipitate and matrix; temperature gradients and extended defects (dislocations, twin and grain boundaries); nonstoichiometric composition; thermal treatment history all affect the formation and growth/dissolution of Te precipitates in CdTe. A good understanding of these effects on Te precipitate evolution kinetics is technically important in order to optimize material processing and obtain high-quality crystals. This research develops a phase-field model capable of investigating the evolution of coherent Te precipitates in a Te-rich CdTe crystal undergoing cooling from the melt. Cd vacancies and Te interstitials are assumed to be the dominant diffusing species in the system, which is in two-phase equilibrium (matrix CdTe and liquid Te inclusion) at high temperatures and three-phase equilibrium (matrix CdTe, Te precipitate, and void) at low temperatures. Using available thermodynamic and kinetic data from experimental phase diagrams and thermodynamic calculations, the effects of Te interstitial and Cd vacancy mobility, cooling rates and stresses on Te precipitate, and void evolution kinetics are investigated. Published by Elsevier B.V.

PACS: 61.72.Bb 61.72.Cc Keywords: A1. Growth kinetics A1. Morphology A1. Phase-field model A1. Te precipitate B1. CdTe crystal

1. Introduction CdTe single crystals are promising materials for the applications in optoelectronic devices such as backside illuminated infrared defectors arrays, terrestrial solar cells, X-ray and g-ray detectors because of an optimum band gap and high absorption coefficient [1–3]. CdTe crystals are also used as substrates [4,5] for the epitaxial growth of large area mercury cadmium telluride alloys, which are the most important semiconductors for infrared detection in the 314 mm region. All these applications require high-quality crystals with controlled composition and microstructures. Non-stoichiometric compositions, which are often required for desired properties such as carrier concentration, type of conductivity, diffusivity, absorption behavior, and efficiency of dopant incorporation, enhance the generation of point defects (vacancies, interstitials, and antisites). Furthermore, dislocations and twin structures form easily in CdTe crystals due to intrinsic low stacking fault energy. These defects interact with each other and make the crystal growth complex. For example, in Te-rich CdTe the point defects and dislocations may affect the kinetics of Te precipitation during crystal growth and the dissolution of Te

 Corresponding author. Tel.: +1 509 376 4432; fax: +1509 376 0418.

E-mail address: [email protected] (S. Hu). 0022-0248/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jcrysgro.2009.02.042

precipitates during annealing. Experiments show that the formation of Te precipitates is one of the most pressing materials problems in Te-rich CdTe crystals [6–10]. The reduction in transmittance observed in CdTe crystals corresponds to absorption by Te precipitates and they are also detrimental to a number of optical, electro-optical, and liquid phase epitaxy HgCdTe substrate applications [11–15]. In order to control Te precipitation and eliminate Te precipitates, numerous studies of growth techniques [1,16–21], thermodynamic calculations of phase diagrams [22–25], thermodynamic calculations of kinetic properties of point defects [25–28], and modeling of solidification [29,30] and Te precipitation [31] have been attempted. However, there has been less emphasis in predicting the kinetics of Te precipitation during crystal growth and annealing, where knowledge of Te precipitation kinetics will be helpful in guiding process development. The formation and dissolution of Te precipitates involves the diffusion and ordering of Te and Cd. Therefore, the thermodynamic and kinetic properties of point defects are closely related to the kinetics of Te precipitation. Thermodynamic calculations [25] reveal that different point defects such as vacancies at Cd and Te sublattices, Cd and Te interstitials, Cd and Te antisites, and electrons and holes exist in the crystals with concentrations depending on the temperature and alloy composition. In Te-rich CdTe crystals the dominant point defects that accommodate

ARTICLE IN PRESS S. Hu, C.H. Henager Jr. / Journal of Crystal Growth 311 (2009) 3184–3194

stoichiometric deviations are Cd vacancies, Te interstitials, and Te on Cd antisites. Experiments by Sen et al. [5] demonstrated that Te precipitates in undoped p-type Bridgman growth CdTe crystals are the rhombohedral phase which is stable under hydrostatic pressure above 0.7 GPa. Faceted Te precipitates [6–10] decorated with voids and star-like patterns of dislocation loops are often observed. The {111}-faceted interfaces suggest that Te precipitates have lower interfacial energy on {111} interfaces or were produced via a ledge growth mechanism on the {111} interface. Plus, the presence of voids in the precipitates implies a volume change from liquid phase at high temperature to solid Te at low temperatures. The stress field around Te precipitates was studied by TEM [32] and thermodynamic calculation [31]. Their results show a high compressive stress field in Te precipitates which might explain the formation of Te rhombohedral phase and star-like patterns of dislocation loops observed [4,10,33]. However, the morphology and diffusion are not taken into account in the thermodynamic model [31]. Experiments also show that Te precipitate prefers nucleating on excess volume defects such as grain boundaries and dislocations [7]. Nanosized Te-precipitates are coherent and incoherent in Cd0.9Zn0.1Te (CZT) as studied by high-resolution TEM [34]. The hexagonal and monoclinic phases of Te were coherent with the matrix of CZT, while the high-pressure rhombohedral phase of Te was incoherent with the matrix of CZT. Therefore, Te precipitation is a complex process, as expected. Thermodynamic and kinetic properties of point defects, excess volume defects such as dislocations and grain boundaries, elastic interactions associated with the lattice mismatches of defects and Te precipitates, and interfacial energy anisotropy may all affect the formation and growth of Te precipitates. The phase-field method has been developed and widely used for predicting the phase stability and microstructural evolution kinetics during many materials processes such as solidification, precipitation in alloys, ferroelectric domain evolution in ferroelectric materials, martensitic transformation, dislocation dynamics, and electrochemical processes [35–40]. It describes a microstructure using a set of conserved and nonconserved variables that are continuous across interfacial regions. The temporal and spatial evolution of the variables, i.e. the microstructural evolution, is governed by the Cahn–Hilliard nonlinear diffusion equation and the Allen–Cahn relaxation equation. It uses thermodynamic and kinetic data from atomistic simulation, thermodynamic calculations and experiments, and predicts the kinetics of microstructure evolution. In this work, we develop a phase-field model to simulate the growth of Te precipitates with valid thermodynamic and kinetic data from the literature. The following features are included: (1) Te interstitial and Cd vacancy diffusion, (2) internal stresses associated with the lattice mismatch of point defects and Te precipitates, and (3) interfacial energy anisotropy. The thermodynamic and kinetic properties of the system are summarized; we briefly describe the methodology of the phase-field model, including phase-field variables, total free energy, evolution equations, and model parameters. The effects of Te interstitial and Cd vacancy mobility, cooling rates and stresses on microstructure evolution kinetics, and the morphologies of Te precipitates are investigated using the phase-field model in 1D, 2D, and 3D. Although a complete thermodynamic and kinetic dataset is lacking for CdTe that accounts for point defects, the model developed here is still useful in predicting relative trends in this system and in suggesting avenues of further study.

precipitate with high Te concentration in a Te-rich CdTe matrix during cooling. Fig. 1 shows the experimental phase diagram for Cd–Te and, according to the phase diagram, at high temperatures a Te inclusion should be liquid. The Te inclusion will undergo a phase transition from liquid to solid during cooling. The liquid phase is a Cd–Te solution with equilibrium concentrations that can be calculated from the liquidus line in the phase diagram. The solid phase is the high-pressure rhombohedral Te phase. Voids are often observed inside the Te precipitate or along the boundary between Te precipitate and matrix. Therefore, the system is in two-phase equilibrium (matrix CdTe crystalline phase and liquid Cd–Te solution) at high temperature. We hypothesize that when the temperature is below a critical temperature Tc, the liquid Te inclusion will separate into the rhombohedral Te phase and a void phase, such that below Tc three phases coexist in the system, namely solid CdTe, solid Te precipitates, and voids. Thermodynamic calculation shows that charged defects such as electrons, holes, and vacancies exist and may affect the diffusion of Te, Cd, and vacancies, hence, the growth of Te precipitates. For the sake of simplicity, we only consider uncharged species: Cd vacancies, Te interstitials, and Te and Cd atoms. Te and Cd sublattices are used to describe the crystalline phases and the point defects. The Te sublattice includes Te and interstitial sites in the CdTe crystal while Cd sublattice only includes Cd sites. We treat the Te on Cd antisite as interstitial Vac Te. Thus we use two variables, cTe Te(r,t) and cCd (r,t), to describe time-dependent concentrations of Te atoms and vacancies, where cTe Te(r,t) denotes Te concentration in units of atoms/per Te sublattice site. The variable cVac Cd (r,t) denotes the Cd vacancy concentration in units of atoms/per Cd sublattice site. In our model, we also use an order parameter f(r,t) to distinguish the matrix and precipitate and void phases, such that the function

2. Phase-field model of Te precipitation In Bridgman growth [1], a single CdTe crystal solidifies from the melt. We are interested in the evolution of a small Te

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Fig. 1. CdTe phase diagram reproduced from Ref. [23].

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f(r,t) is equal to 0 and 1 in the matrix and precipitate, respectively, and it varies smoothly from 0-1 across the interface between matrix and precipitate. The total free energy of the Vac system is written as a function of all the variables (cTe Te(r,t), cCd (r,t), and f(r,t)) and includes chemical free energy, gradient energies, and long-range elastic interaction energy as Z kTe kVac Vac 2 2 E¼ jrcTe jrcVac ðFðcTe Te ; cCd ; fÞ þ wgðfÞ þ Te j þ Cd j 2 2 V 1 Vac þ jgðrfÞj2 þ F elastic ðcTe (1) Te ; cCd ; fÞÞdV 2 Vac Te Vac Te Vac where F(cTe Te, cCd , f, T) ¼ (1h(f))fm(cTe, cCd , T)+h(f)fp(cTe, cCd , T) Te Vac is the total chemical free energy. The functions fm(cTe, cCd , T) Vac and fp(cTe Te, cCd , T) are the chemical free energies of the CdTe matrix phase and the Te precipitate phase, respectively, and h(f) ¼ 3f32f2 is a shape function that is equal to 0 in matrix and 1 in the precipitate, and smoothly changes from 0 to 1 across the interface. The function g(f) ¼ f22f3+f4 is a double-well function and w is a positive constant related to the barrier for the phase transformation. The 3rd through 5th terms in Eq. (1) are gradient energies associated with the contribution of the gradients of compositions and chemical ordering to interfacial energies. kTe and kVac are gradient energy coefficients. The term g(rf) is a function of the order parameter gradient, rf, which describes the interfacial energy anisotropy between matrix and Vac the precipitate phase. The last term Felastic(cTe Te, cCd , f) in Eq. (1) is the elastic energy associated with lattice distortions around defects and from deformation due to temperature gradients and applied stresses.

2.1. Chemical free energy considerations Chemical free energies of the liquid phase [24] and nonstoichiometric CdTe matrix phase [25] were developed by thermodynamic calculations. The compound energy model (CEM) was used to describe the non-stoichiometric CdTe phase and five sublattices and distinct defects (interstitials, antisites, vacancy, charged vacancy, and electron and hole) are considered. For the liquid phase the associated liquid model is used. Since the

proposed thermodynamic model employs different variables, the phase-field model as described here cannot directly use these chemical free energy functions. Furthermore, the developed chemical free energies usually are correct only near equilibrium compositions due to the fact that equilibrium properties are used in developing them. In this work, a model function of chemical free energy is constructed with the equilibrium properties from thermodynamic calculations [24,25] and experimental phase diagram [23] to support the simpler phase-field approach here. The chemical free energy of CdTe matrix phase is designated as e

(2) e cTe m ðTÞ

e cVac m ðTÞ

where and are the equilibrium concentrations of Te and vacancies in the CdTe phase at temperature T. The coefficients Vac ATe m and Am are second derivatives of the chemical free energy at equilibrium concentrations at the temperature T. The chemical Vac free energy of the Te precipitate phase fp(cTe Te, cCd , T) has the form Vac Te 4 Te 3 Te 2 f p ðcTe Te ; c Cd ; TÞ ¼ B1 ðcTe Þ þ B2 ðcTe Þ þ B3 ðc Te Þ e

Vac Vac Vac 2 þ B4 ðcTe Þ Te Þ þ B5 þ Bp =2ðcCd  cp

BVac p ðTÞ,

Initial concentration Equilibrium concentration (solidus line)

M0Te 0 cTe m 0 cVac m Tee cm

4.006  1025(m5/Js) 0.06 0.06 e

3 cTe T þ 1:41  106 T 2 m ¼ 1:30  1:07  10

 8:18  1010 T 3 þ 1:76  1015 T 4 cVac m

e

e

cVac ¼ 0:30  1:07  103 T þ 1:41  106 T 2 m  8:18  1010 T 3 þ 1:76  1015 T 4

Equilibrium concentration (liquidus line, T4Tc)

e

cTe ¼ 4:79  0:0042T þ 1:02  106 T 2 p

e

cTe p

cVac p

e

e

cTe ¼ x0 þ ð1:9  x0 Þð1 þ 0:5ðT 2  TÞ=ðT 2  T c ÞÞ p

e

cTe p

cVac p

e

e

cVac ¼ ðx0  1Þ þ ð2  x0 ÞðT c  TÞ=ðT c  T 2 Þ p x0 ¼ 4:89  4:24  103 T c þ 1:02  106 T 2c

Equilibrium concentration void, TpTc) e

e

cTe v

cTe ¼ 0:5ðT 2  TÞx0 =ðT 2  T c Þ v e

Temperatures

e

cVac ¼ 3:79  0:0042T þ 1:02  106 T 2 p x0 ¼ 4:89  4:24  103 T c þ 1:02  106 T 2c

Equilibrium concentration (precipitate, TpTc)

cVac v T1, Tc, T2

(3) e cVac ðTÞ p

and are where the parameters Bi(T), i ¼ 1, y, 5, determined by the chemical free energy, first and second derivatives at equilibrium concentrations of Te and vacancies at the temperature T in the Te precipitate and void phases, respectively. When the temperature is higher than the critical Vac temperature Tc, the function fp(cTe Te, cCd , T) has one minimum that corresponds to the liquid Te phase to give a two-phase equilibrium (matrix CdTe and liquid Te). When temperature Vac is lower than Tc, the function fp(cTe Te, cCd , T) has two minima corresponding to Te precipitate and void phases and the system contains three-phase equilibria (matrix CdTe, Te precipitate, and void phases). All the thermodynamic data required to determine the parameters in the chemical free energy can be obtained from thermodynamic calculations such as CALPHAD. The parameters in this work are determined partly by the assessed thermodynamic data from experimental phase diagram [23] and thermodynamic calculation [24,25] and are listed in Table 1. The parameters of the

Table 1 The thermodynamic properties and parameters used in the simulations. Mobility Initial concentration

e

Te Vac 2 2 Vac Te Te Vac Vac f m ðcTe Te ; cCd ; TÞ ¼ Am =2ðcTe  cm ðTÞÞ þ Am =2ðc Cd  c m ðTÞÞ

e

cVac ¼ ðx0  1Þ þ ð2  x0 ÞðT c  TÞ=ðT c  T 2 Þ v 1000 K, 950 K, 900 K

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Table 2 The dimensionless parameters used in the simulations. Time Characteristic length Mobility coefficient Gradient coefficient Gradient coefficient Gradient coefficient Double well height Chemical free energy (matrix) Chemical free energy (precipitate, T4Tc) Chemical free energy (precipitate, TpTc)

w*

0.4 4 nm 1.6 0.002 0.002 g1 ¼ 1.0, g2 ¼ 8.0, g3 ¼ 8.0 0.011

n Te ATe m ¼ Am /C44

n ATe m ¼ 6

n ¼ Avac Avac m m /C44 Bi* ¼ Bi/C44

n Avac ¼ 20 m B1 ¼ B2 ¼ 0, B3 ¼ 0.30

n Bvac ¼ Bvac p /C44 p

n B4 ¼ 1.00, B5 ¼ 0.84, Bvac ¼ 2.0 p B1 ¼ 2=ðx1  x2 Þ2

Dt* r0 L*

kTe kVac g i*

Bi* ¼ Bi/C44

B2 ¼ 4ðx1 þ x2 Þ=ðx1  x2 Þ2 B3 ¼ ð2x21 þ 6x1 x2 þ 2x22 Þ=ðx1  x2 Þ2 B4 ¼ ð8x30  12x20 x1 þ 4x0 ðx21 þ 4x1 x2 þ x22 Þ=ðx1  x2 Þ2 B5 ¼ ð6x40  8x30 ðx1 þ x2 Þ þ 2x20 ðx21 þ 4x1 x2 þ x22 Þ þ 0:1ðx21 þ 2x1 x2 þ x22 ÞÞ=ðx1  x2 Þ2

Elastic constants Elastic constants

e

e

n Bvac ¼ Bvac p /C44 p

n Bvac ¼ 10; p

n Cm ij DC pijn

m m Cm 11* ¼ 2.73, C12* ¼ 1.91, C44* ¼ 1.0

x1 ¼ cTe p ;

x2 ¼ cTe v

DCp11* ¼ 1.17, DCp12* ¼ 1.11, DCp44* ¼ 0.019

chemical free energies are listed in Table 2. It is our intent, with the development of thermodynamic databases and interface between phase-field model and thermodynamic software, which the thermodynamic data can be directly employed in the phasefield model [41], but this is left for future work.

and the elastic constants CdTe, given as C11 ¼ 56.2, C12 ¼ 39.4, and C44 ¼ 20.6 GPa; Te (precipitate): C11 ¼ 32, C12 ¼ 16.5, and C44 ¼ 21 GPa [44] and void: C11 ¼ 0, C12 ¼ 0, and C44 ¼ 0 GPa are used to p determine C m ij and DC ij in Eq. (5). The elastic solution in an elastic inhomogeneous solid is solved by an iteration method [39].

2.2. Elastic energy considerations

2.3. Interfacial energy considerations

The average atomic volumes in CdTe, liquid, and Te precipitate are OTe(CdTe) ¼ 34.0 A˚3, OTe(Liquid) ¼ 37 A˚3, and OTe(Precipitate) ¼ 39 A˚3 [31,42,43], respectively. Consider N atoms in a small region in the matrix CdTe crystal imagine that if the N atoms transfer to liquid or Te precipitate phase, the volume increases because the atomic volume in liquid or Te precipitate phase is larger than that in the crystal. Similarly, Te interstitials cause volume expansion while Cd vacancies cause volume contraction. Therefore, the formation of liquid and precipitate phases and diffusion of Te interstitials and Cd vacancies will cause a redistribution of elastic energies or internal stresses in the system. Assume that the interface between the precipitate and matrix is coherent, which is true for small Te precipitate [34]. We also assume that the variation of stress-free lattice parameter of the matrix phase with vacancies and interstitials obeys Vegard’s law. Then, the stress-free strain associated with the formation of Te precipitates and the distributions of Te interstitials and vacancies can be described by,

The interfacial energy includes the excess energy across the interface due to the change of composition and due to ordering. From the total free energy formula (1), it can be seen that all the terms contribute to the interfacial energy. In this paper, we assume that the compositional gradient is isotropic, but that the order parameter gradient is anisotropic and makes the dominant contribution to the interfacial energy. From the symmetries of the matrix and precipitate phases, a four-fold symmetry of the interfacial energy anisotropy is constructed as [45]

0 Vac 0 ij ðr; tÞ ¼ 0Te ðcTe Te  1:0Þdij þ Cd cCd dij þ f fdij

(4)

The coefficients e0Te ¼ 0.04, e0Cd ¼ 0.02, and 0f ¼ 0.06 are chosen from the formation volumes of Te interstitials and Cd vacancies, the atomic volumes, and equilibrium concentration of different phases. Other terms will be required for consideration of semi-coherent or incoherent interfaces. The stress-free strain is positive when it describes a lattice expansion in the Te precipitate, and is zero in the void. The elastic properties are different in Te precipitate, matrix, and void phases. We let the elastic constants vary with the concentrations of Te and vacancy as p p Vac m C ij ðr; tÞ ¼ C m ij þ ðDC ij  ðC ij þ DC ij ÞcCd ðr; tÞÞ

 ð1  tanhð20  ð0:5  fðr; tÞÞ=2

(5)

gðrfÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 ðf2x þ f2y þ f2z Þ þ g2 ðf2y f2z þ f2x f2z þ f2x f2y Þ þ g3 ðf4x þ f4y þ f4z Þ

(6) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The interfacial energy is proportional to g ¼ g1 þ g3 in {1 0 0} pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi interfaces, g ¼ g1 þ g2 =4 þ g3 =2 in {11 0} interfaces, and g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 þ g2 =3 þ g3 =3 in {111} interfaces, such that the interfacial energy is anisotropic if g2a0 and/or g3a0. 2.4. Evolution equations The migration of Te interstitials and Cd vacancies, and microstructure evolution are described by the Cahn–Hilliard [46] and Allen–Cahn equations [47] Vac dEðcTe @cTe Te ; c Cd ; f; TÞ Te ðr; tÞ ¼ r  M Te r Te @t dcTe ðr; tÞ

(7)

Vac @cVac dEðcTe Te ; cCd ; f; TÞ Cd ðr; tÞ ¼ r  M Vac r Vac @t dcCd ðr; tÞ

(8)

Vac dEðcTe @fðr; tÞ Te ; cCd ; f; TÞ ¼ LðrfÞ dfðr; tÞ @t

(9)

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Vac Te Vac where dE(cTe Te, cCd , f, T)/dx, x ¼ cTe, cCd , and f is the differential of the total free energy respect to the phase-field variables, MTe and MVac are the mobility of the Te interstitial and Cd vacancy, respectively, and L(rf) is the interface mobility. This mobility is assumed be isotropic and set to be a constant, L0. The Eqs. (7)–(9) are efficiently solved using a semi-implicit method [48] with dimensionless units

r i ¼ 

fm ¼

kTe ¼ gi ¼

ri ; r0

t ¼

fm ; C 44

kTe

r 20 C 44

gi r 20 C 44



M 0Te C 44 t ; r 20

fp ¼ ; ;

fp ; C 44

kVac ¼

L ¼

E ¼

kVac

L0 r 20 M 0Te

E C 44

r 20 C 44 C w ij C ij ¼ ; w ¼ C 44 C 44

(10)

where r0 is a characteristic length, C44 is the shear modulus of the matrix, and M0Te is the mobility of Te atoms at temperature T1. Tables 1 and 2 list all the model parameters used in the simulations.

3. Results and discussion To demonstrate the capability of this phase-field model, the growth kinetics and morphology of Te precipitates in 1D, 2D, and 3D are simulated using various parameter combinations. The simulations start in a simulation cell with an initial Te precipitate at temperature, T1. The radius of the Te precipitate is R0 ¼ 15r0. e Vace ðT 1 Þ in the The initial concentrations are cTe p ðT 1 Þ and cp Te0 Vac0 in the matrix. The temperature precipitate, and cm and cm decreases from T1 to T2 with constant cooling rate, T˙. During cooling the Te phase will separate into Te solid precipitate and void, but the model does not stipulate how the phase separation occurs. The void and Te precipitate formation could form through homogeneous nucleation, such as spinodal decomposition, or via inhomogeneous nucleation at excess volume structural defects such as dislocations, twin boundaries and grain boundaries. The nucleation of voids and Te precipitates needs to consider statistical events at atomic level in order to capture a mechanistic understanding of the nucleation, and although we are pursuing a multi-scale model to include nucleation into a phase-field model, it is not considered here. Rather, in these simulations, a void nucleus is introduced into the Cd–Te phase when the temperature is equal to Tc. The volume fraction of the void in the precipitate is 0.2 and determines the radius of the void. The concentrations of Te e interstitials and Cd vacancies in the void are calculated by cTe v ðT c Þ Vace and cv ðT c Þ. The concentrations of Te and vacancy in the precipitate are scaled by mass and lattice conservation when a void nucleus is introduced. The mobility of Te is assumed to be MTe ¼ 3.40  1022 exp(6743.8/T)(m5/Js). The mobility ratio of Te atom and Cd vacancy M0 ¼ MTe/MVac, the elastic interaction energy, and the cooling rate are chosen as simulation parameters to determine their relative effects on growth kinetics of Te precipitates and voids. 3.1. 1D simulations First we examine the equilibrium composition in a Te precipitate in 1D with a simulation cell of size 512r0. Fig. 2 shows the temporal evolution of Te and vacancy concentration in 1D. We can see that the concentrations of Te and vacancies in the precipitate change with temperature. But since the system does not reach equilibrium during cooling the concentration in the Te precipitate is not equal to the equilibrium concentration described

Fig. 2. Temporal and spatial evolution of (a) Te concentration and (b) vacancy concentration on cooling from 1000 to 900 K with cooling rate T˙ ¼ 2.5 K/min and mobility ratio M0 ¼ MTe/MVac ¼ 1.0. At T ¼ 900 K, a void nucleus with 20% volume fraction of the Te precipitate is generated randomly in the Te precipitate. The insert plot in (a) shows the temperature profile during cooling.

by the phase diagram. The cooling rate, mobility, and concentration of Te and vacancies in the matrix all affect the Te concentration in the precipitate. It is interesting to find that Te precipitate grows fast at high temperature and then shrinks. The fast growth rate at high temperatures can be explained by initial high Te and vacancy concentrations in the matrix, high mobility of diffusive species, and low equilibrium concentration for the Te phase. The Te precipitate shrinks due to Te condensation and phase separation of Te precipitate, and void at low temperatures. Observing the evolution of the concentrations in Fig. 2, we can see that the concentrations in the Te precipitate, void, and matrix reach their equilibrium concentrations after long time annealing e Vace at temperature T ¼ 900 K, cTe m ð900 KÞ ¼ 1:00002 and cm ð900 KÞ ¼ e Vace ð900 KÞ ¼ 1:899 and c ð900 KÞ ¼ 1:0 0:00002 in the matrix; cTe p p e Vace ð900 KÞ ¼ 0:0 and c ð900 KÞ ¼ 1:0 in the in the Te precipitate; cTe v v void.

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Fig. 3. The effect of cooling rates on growth kinetics is shown as the system cools from 1000 to 900 K with three cooling rates. Shown in (a) are the volume fractions of the phases as a function of time. The lines show the volume fraction of Te precipitate and the symbols show the volume fraction of void phase. The Te precipitate is defined by Te Vac ˙ ¼ 6.4 K/min. The colors indicate Te cTe TeX1.2 while the voids are defined by cTep0.5 and cCd X0.5. In (b) the evolution of Te precipitate and void is shown for cooling rate T Te concentration with red having the maximum value cTe Te ¼ 2.0 and blue the minimum value cTe ¼ 0.0, such that the red region is precipitate while the blue region is void. Shown in (c) are the final morphologies of precipitate and voids at t ¼ 892 s for three cooling rates.

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With the equilibrium concentration and order parameter profiles the interfacial energies are calculated by

g¼

Z

0:98

n

kTe

Vac 2 jrcTe FðcTe Te ; c Cd ; fÞ þ wgðfÞ þ Te j 2  kTe 1 2 2 df=l jrcVac þ Cd j þ jgðr fÞj 2 2

f¼0:02

(11)

where l is the thickness of the interface. For the interfaces: CdTe matrix/Te precipitate, Te precipitate/void, and void/CdTe matrix, the calculated interfacial energies are 0.66, 0.40, and 0.68 J/m2, respectively. In summary, the 1D simulation results demonstrate the phase-field model developed in this study can capture the temporal and spatial evolution of concentrations of Te and vacancies during cooling, and the equilibrium concentrations when the system reaches equilibrium. The chosen model parameters also produce reasonable interfacial energies. Although 1D simulations can reach reasonable cooling rates, relative long cooling times (tens of hours) and large length scales (tens of microns), we have to perform simulations in 2D and 3D to take curvature and elastic energy into account. 3.2. 2D simulations 2D simulations consider the effects of interfacial energy anisotropy and elastic interactions, cooling rate, and the relative mobilities of Te and vacancy on the microstructural evolution. The same model parameters such as initial concentrations in the matrix, the initial sizes and concentrations in the Te precipitate and voids, and temperature ranges in 1D are used in the 2D simulations, which are carried out in a simulation cell 256r0  256r0. First the effect of cooling rate on the growth is simulated and shown in Fig. 3. Fig. 3a plots the volume fraction evolution of Te precipitates and voids for three cooling rates: T˙ ¼ 25.2, 12.8, and 6.4 K/min. In the calculation of volume fraction the Te precipitate is defined by cTe TeX1.2, while the voids are Vac defined by cTe Tep0.5 and cCd X0.5. A larger cooling rate decreases the particle growth rate, since a faster cooling rate brings the system to a lower temperature earlier and the mobility of diffusive species at lower temperatures is reduced. The inflections in the Te precipitate volume fraction curves correspond to the introduction of voids into the precipitate at the phase-separation temperature Tc ¼ 950 K. The introduced void nuclei could be stable or unstable, which depends on the void size. If the void is larger than the critical nucleus size, it will grow, otherwise, it will shrink. Compared with the evolution of Te precipitate and void volume fraction in Fig. 2a, the time lag seen in the void growth curve indicates the time to relax the introduced void to a stable void. Since the void has 20% volume fraction of the precipitate, the decrease of precipitate volume fraction at Tc is proportional to the precipitate size. The time evolution of the microstructure is shown in Fig. 3b. The colors indicate Te concentrations with red having the maximum value cTe Te ¼ 2.0 and blue the minimum value cTe Te ¼ 0.0, such that the red region is precipitate while the blue region is void. The Te concentration in the precipitate gradually increases and reaches equilibrium at the end. Fig. 3c shows the final morphologies of the Te precipitate and void. The faceted interfaces are due to the lower interfacial energy on the {11 0} interface based on Eq. (6). Because of the lack of chemical free energy of the system there are some uncertainties in total free energy calculation. The chemical free energy used in the model only presents the phase equilibrium information, but not the absolute chemical energy. Therefore, the elastic energy is relative. In addition, accurate thermodynamic properties such as the dependence of lattice mismatch and elastic constants on temperature and concentration

Fig. 4. The effect of elastic interaction on growth kinetics is shown. The system cools from 1000 to 900 K with a cooling rate T˙ ¼ 6.4 K/min, mobility ratio M0 ¼ 1.0, and different escale parameters. Shown in (a) is the evolution of the volume fractions of Te precipitate and void phase. Shown in (b) is the microstructural evolution of Te precipitate and voids for escale ¼ 1.5, and in (c) the final morphologies of precipitate and voids at t ¼ 892 s are shown.

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Fig. 5. The effects of elastic interaction and mobility of Te and vacancy on growth kinetics are shown. The system cools from 1000 to 900 K with a cooling rate T˙ ¼ 6.4 K/ min, escale ¼ 1, and three mobility ratios M0 ¼ MTe/MVac. Shown in (a) is the temporal evolution of the volume fractions of Te precipitate and void phase with different mobility ratios. Shown in (b) is the temporal evolution of Te precipitate and voids for M0 ¼ 0.2, and in (c) is the distribution of pressure at t ¼ 892 s. The yellow, red, and blue colors denote the pressure defined by P ¼ (sxx+syy)/2. For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.

are unavailable. To study the elastic energy effect of elastic interaction on precipitate growth, we artificially scale the stress free strain  ij ¼ escaleij by a factor escale and retain the chemical

free energy and model parameters to investigate how elastic interactions affect the precipitate growth kinetics. Fig. 4 shows the evolution of volume fraction and microstructure, and final

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Fig. 6. Shown in (a) is the morphological evolution of a Te precipitate in 3D starting with a spherical shape at t ¼ 0. The faceting occurs along {111} planes in the CdTe system according to Eq. (6) as the particle evolves. The 3D octahedron is rotated in (b) so that the view is along a /111S-type direction such that the emerging hexagonal shape of the faceted particle can be seen. Shown in (c) is an experimental TEM image of a large, faceted Te precipitate in CdTe [8] and in (d) is an experimental SEM image of a Te precipitate taken at PNNL. (Note: SEM image from CZT sample discussed in C.H. Henager, Jr. et al., ‘‘Preferential orientation of Te precipitates in melt-grown CZT,’’ submitted to J. Cryst. Growth (2008). See also E.A. Miller, M. Toloczko, C.E. Seifert, A. Seifert, W. Liu, and M. Bliss, ‘‘Differential aperture X-ray microscopy near Te precipitates in CdZnTe,’’ in Proceedings of SPIE—The International Society for Optical Engineering, 2007, Vol. 6706, 670609, SPIE, Bellingham WA, WA 98227-0010, United States).

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morphologies of Te precipitate and voids. We can see trends in Fig. 4a that increasing the elastic interaction (increasing the value of escale) slightly increases the Te precipitate growth rate (T4Tc). Since Te and vacancy have opposite stress-free strains, they attract each other to minimize the pressure in the Te precipitate. As a consequence, the elastic interaction increases the Te and vacancy flux from matrix to the precipitate, hence, the growth kinetics. The volume fraction drop of Te precipitate at phase separation temperature and subsequent volume fraction evolution of Te precipitate and voids in Fig. 4a shows that more vacancies will be pulled out the Te precipitate with increasing elastic interaction energy. As a result, the volume fraction of voids increases while the volume fraction of the Te precipitates decreases as the contribution of elastic energy to the total free energy is increased (by increasing the value of escale). The relationship of relative volume fractions of Te precipitate and void is the inverse of that observed in Fig. 3c. This implies that elastic interactions also enhance the separation of Te precipitate and void phases. Examining the effect of elastic interaction on the evolution of Te and vacancy concentrations, we find that the elastic interaction strengthens the segregation of vacancies in interfaces as shown in Fig. 4b, and results in the formation of voids on interfaces having a large tensile stress. Fig. 4c shows that the elastic interaction will considerably affect the void formation and their morphologies. Lastly we consider the effect of mobility of Te and vacancies on the precipitate growth. The results are presented in Fig. 5. The mobility of Te and vacancies dramatically affects the growth kinetics, volume fractions, and morphologies of Te precipitates and voids. Changing the vacancy mobility affects the growth process for both Te diffusion controlled growth or vacancy controlled growth. As the vacancy mobility increases, the growth becomes Te diffusion controlled. In contrast, the growth is vacancy diffusion controlled at low vacancy mobility. The Te concentration in the precipitate in Te diffusion controlled growth is lower than that in vacancy controlled growth. The Te precipitate with lower Te concentration is unstable and tends to separate into Te precipitate and voids at transition temperature Tc. Fig. 5b shows the microstructure evolution with high vacancy mobility M0 ¼ 0.2 and we observe that the Te precipitate is unstable at Tc. A spinodal decomposition occurs in the Te precipitate. However, in the case of M0 ¼ 5.0, the Te precipitate has high Te concentration such that a void introduced at Tc is unstable. It shrinks and disappears as inferred from the volume fraction evolution in Fig. 5a and the final microstructures in Fig. 5c. The pressure distributions in the final microstructures are plotted in Fig. 5c. The yellow, red, and blue colors denote zero stress, tensile, and compressive stresses, respectively. The stress is zero in voids and compressive in Te precipitates. Outside of the Te precipitate there exist larger tensile stresses, which may cause dislocation loop punching, observed in some experiments. Based on thermodynamic calculation [31] and experiment [49], the equilibrium pressure in the Te precipitate should be about 0.7 GPa. Our simulations show that the pressure in the Te precipitate is about 0.2 GPa. Of course, Te concentration in the precipitate, the size of the precipitate, the morphology of precipitate and voids, and relaxation of the interface all affect the pressure.

3.3. 3D simulations In 3D simulations, the focus is on the Te precipitate morphology. A spherical Te precipitate with equilibrium Te and vacancy concentrations at T ¼ 1000 K is placed in the center of a simulation cell of size 96r0  96r0  96r0. The radius of the precipitate is R0 ¼ 20r0. The simulation is carried out at temperature T ¼ 1000 K. The Te and vacancy have the same

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mobility, M0 ¼ 1. Fig. 6 shows the Te precipitate morphology evolution in 3D predicted using the phase-field model. The isosurface shows that the Te concentration is equal to 1.25. The initial spherical precipitate grows and becomes a slightly faceted particle, such that the flat interfaces are the {111} planes of CdTe, which have the lowest interfacial energy in 3D according to Eq. (6). The effect of elastic energy on the morphology is also included. The results show that including the elastic energy slightly increases the rate of evolution, but does not affect the final morphology, which is dominated by the minimization of the interfacial energy. Similarly, the faceted particle may be caused by a ledge growth mechanism, which could be integrated into phasefield models by introducing a ledge nucleation process. Fig. 6c and d shows the morphology of Te precipitates observed in experiments and the morphology from our simulations is in agreement with that. The octahedral solid has the same underlying symmetry as a cube and contains a 3-fold rotation axis along the /111S-type direction such that triangular and hexagonal shaped particles are predicted for Te precipitates viewed in CZT crystals cut along (111) planes. In general, these predicted shapes agree with observations and one of these hexagonal shapes are shown in Fig. 6d. The lack of sharp facets in the 3D model is due to the diffuse nature of the phase-field interfaces but the particle is definitely faceted as shown in Fig. 6a. Others have suggested that the Te precipitates begin to form within a negative crystal of CZT corresponding to a truncated tetrahedron but this is not consistent with the observations of hexagonal Te precipitates.

4. Summary and conclusions In this work, Te precipitation is modeled through migration of two dominant point defects: Te interstitials and Cd vacancies in Te-rich Cd–Te alloys using a phase-field model to simulate the growth of Te precipitates and voids during cooling from the melt as in a Bridgman furnace. The effect of interfacial energy anisotropy, elastic energy, elastic inhomogeneity, mobility of diffusive species, and cooling rates are considered; and the model parameters are determined by existing thermodynamic and kinetic data for the CdTe system. The simulations demonstrate that the phase-field model is able to correctly capture the Te precipitate morphology and find the equilibrium concentrations in a three-phase equilibrium consisting of CdTe matrix, Te precipitate, and void phase within the precipitate. It is found that the cooling rates, elastic interaction, and diffusing species mobility dramatically affect final Te concentration in Te precipitates and precipitate growth rates, as well as affecting the phase separation occurring in the Te precipitate to form voids. For quantitative simulations, there is a need to establish a better thermodynamic and kinetic database of the CdTe system.

Acknowledgments PNNL is operated for the US Department of Energy by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. This work was funded at PNNL by the Office of Defense Nuclear Nonproliferation, Office of Nonproliferation Research and Development (NA-22). References [1] P. Rudolph, Fundamental-studies on Bridgman growth of CdTe, Progress in Crystal Growth and Characterization of Materials 29 (1–4) (1994) 275–381. [2] Y. Nemirovsky, et al., Study of the charge collection efficiency of CdZnTe radiation detectors, Journal of Electronic Materials 25 (8) (1996) 1221–1231.

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