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Zr thin film multilayers
Acta Materialia 51 (2003) 5285–5294 www.actamat-journals.com
Phase stability of bcc Zr in Nb/Zr thin film multilayers G.B. Thompson, R. Banerjee ∗, S.A. Dregia, H.L. Fraser Department of Materials Science and Engineering, The Ohio State University, 477 Watts Hall, 2041 College Road, Columbus, OH 43210, USA Received 5 February 2003; received in revised form 1 July 2003; accepted 3 July 2003
Abstract A change in phase stability from hcp Zr to bcc Zr occurs in Nb/Zr multilayers when the bilayer thickness is reduced to the nanometer scale. This phase stability in these multilayers has been described using a model based on classical thermodynamics. Using a previously reported experimental observation of hcp to bcc transformation in these multilayers, a phase stability diagram (referred to as the biphase diagram) has been proposed. Subsequently, a range of multilayers with varying volume fractions and bilayer thicknesses have been sputter deposited. The crystal structures in these multilayers have been determined using X-ray and electron diffraction. The hcp and bcc Zr phases within the Nb/Zr multilayers are in agreement with the predictions afforded by the proposed biphase diagram. The sequence of the Zr bcc phase stability was accomplished by forming its β-Zr (high temperature bcc phase) prior to forming a bcc coherent interface with Nb. First approximations of the structural and chemical contributions to the interfacial energy accompanying the changes in hcp to bcc phase stability for Zr have been evaluated. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Pseudomorphic; bcc Zr; Multilayers; Nb/Zr; Biphase stability diagram
1. Introduction When the layer thickness in a multilayered thin film is reduced to the nanoscale regime, it often exhibits structural transitions resulting in one or more of the layers adopting a crystal structure different from its bulk equilibrium form [1–3]. Based on the competing influences of the bulk chemical free energy, the interfacial free energy, and the strain energy, it is possible to model and conse-
Corresponding author. Tel.: +1-614-292-6263; fax: +1614-292-7523. E-mail address: [email protected] (R. Banerjee). ∗
quently predict the critical transition thickness in these systems. This paper focuses on the use of a classical thermodynamics model to predict hcp and bcc phase stability in the Zr layer in Nb/Zr multilayers. 1.1. Nb/Zr multilayers The bulk equilibrium structures of Nb and Zr, in the standard condition (room temperature and 1 atm), are bcc and hcp, respectively. Pure elemental Zr exhibits an allotropic transformation from the hcp α-Zr phase to the bcc β-Zr phase at a temperature of 840 °C [4]. Stabilization of a bcc phase in the Zr layers of sputter deposited Nb/Zr multilayers
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00380-X
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at room temperature has been reported by Lowe and Geballe [5]. A series of multilayers of varying equal Nb and Zr layer thicknesses were characterized by the authors [5] using X-ray diffraction (XRD). Their results reveal that when the bilayer thickness, l is reduced below ~3.1 nm (individual layer thickness ~1.55 nm), the bulk bcc Nb/hcp Zr structure transforms to a metastable bcc Nb/bcc Zr structure. Lowe and Geballe [5] did not vary the individual layer thickness within the bilayer pair in their multilayers. In the present study, a detailed investigation of the formation of bcc Zr in Nb/Zr multilayers as a function of varying individual thicknesses has been undertaken. The experimental results have been coupled with modeling efforts. 1.2. The biphase diagram In a recent paper, Dregia et al. [6] proposed a classical thermodynamics model to explain phase stability in multilayered systems. The details of this model have been described elsewhere [6]. Based on this model, a Nb/Zr multilayer can be represented by a reference bilayer consisting of a single layer of Nb, a single layer of Zr, and two Nb/Zr interfaces. The individual layer thickness of this reference bilayer is described in terms of the volume fraction of one of the constituent layers, e.g. fNb, while the bilayer spacing (l), the sum of the individual thickness of a Zr and Nb layer pair, represents the length scale. Thus, fNb and l can be considered to be the two degrees of freedom in this multilayered system. Varying either or both of these degrees of freedom could potentially result in structural transitions in the multilayer. For a Nb/Zr bilayer, the free energy change per unit interfacial area of the multilayer accompanying a structural transition, ⌬g, is given as ⌬g ⫽ 2⌬g ⫹ ⌬GNbfNbl ⫹ ⌬GZr(1⫺fNb)l
phase stability in Nb/Zr multilayers. Thus, in addition to the allotropic free energy differences between the metastable and stable (with reference to the bulk) forms of Nb and Zr, other contributions to the free energy, such as the strain energy, can play a significant role in determining the phase stability. In order to maintain simplicity in the approach, it is assumed that all energetic contributions that scale with the volume are contained within the ⌬Gi terms, and all energetic contributions that scale with the interfacial area are contained in the ⌬g term. A useful way to depict phase stability in Nb/Zr multilayers is to plot the volume fraction fNb vs. 1/l. Such a plot of phase stability has been previously referred to as a biphase diagram [6–8]. An initial approximation to a biphase diagram for Nb/Zr multilayers may be deduced from Lowe and Geballe’s [5] report of a structural transition from hcp Zr to bcc Zr for l ⬍ 3.1 nm at a fNb = 50%. Based on this data point, a single biphase field boundary has been constructed. This is seen in Fig. 1. Thermodynamically, the hcp to bcc phase stability in Zr would be most probable in the Nb-rich regions of the biphase diagram. It has been seen elsewhere [6] that the other constituent phase can exhibit a structural change in regions where it has low volume fraction. The focus of this paper will be on the possible hcp to bcc phase stability of Zr
(1)
where ⌬g is the interfacial energy difference between a Nb/Zr interface in the transformed state and a Nb/Zr interface in the bulk equilibrium state, ⌬Gi is the allotropic volume free energy change for component i, fNb is the volume fraction of Nb, and l is the bilayer spacing. Eq. (1) provides a basic thermodynamic framework for representing
Fig. 1. A proposed, hypothetical biphase stability diagram based on a previously reported [5] structural transition of Zr from hcp to bcc in Nb/Zr multilayers. This point is designated as a hexagonal mark.
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as predicted in the Nb-rich section of the biphase stability diagram. In Eq. (1), by putting ⌬g = 0 and reorganizing the terms, the slope of the biphase boundary separating bcc Nb/hcp Zr and bcc Nb/bcc Zr biphase fields can be evaluated to be [⌬G] / [2(gbcc / bcc⫺ gbcc / hcp)], which has a negative value. Thus, the proposed biphase diagram, shown in Fig. 1, consists of a single biphase boundary with a negative slope separating the bcc Nb/hcp Zr and bcc Nb/bcc Zr biphase fields. The linear boundary assumes that ⌬G and ⌬g are independent of l. The effects of coherency strains on the boundary are considered separately below. The thermodynamics dictates that this biphase boundary line drawn through the transition point at fNb = 0.5 must approach 1 / l = 0 as fNb approaches 1. Thus as the volume fraction of Nb is increased, the critical value of l at which the transition occurs increases. However, it should be noted that since the slope of the biphase boundary, given by [⌬Gzr] / [2(gbcc / bcc⫺gbcc / hcp)], is a constant, the thickness of an individual Zr layer ( = l(1⫺fNb)) remains constant along the boundary. 1.3. Effect of coherency strains on the biphase diagram The biphase diagram shown in Fig. 1 has been constructed using a simplifying assumption that the ⌬Gi and ⌬g are independent of fNb and l. This would not be the case for multilayers that experience coherency strains, where the equilibrium coherency strain in a layer is inversely proportional to its thickness [9]. If coherency strain energy were included in the thermodynamics, ⌬g would depend non-linearly on volume fraction and lead to curved boundaries in 1/l vs. fNb space. For a strain-free unit reference bilayer, the free energy of formation of a metastable biphase, per unit area of interface, is given by Eq. (1). For a coherently strained Nb/Zr bilayer, the strain energy per unit area of interface [8] is given by Wel ⫽ [YNbe2NbfNb ⫹ YZre2Zr(1⫺fNb)]l
(2)
where ei is the coherency strain and, for each metal, Y is a biaxial elastic modulus defined as Y⫽
E 1⫺n
(3)
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where E is Young’s modulus and n is Poisson’s ratio. The natural misfit, mo, is partitioned such that 兩eNb兩 + 兩eZr兩 = mo, and for force balance, (Yefl)Nb + (Yefl)Zr = 0. Thus, both eZr and eNb can be expressed in terms of mo, and Eq. (2) becomes Wel ⫽ Ylm2o,
(4)
with Y⫽
YNbYZrfNb(1⫺fNb) YNbfNb ⫹ YZr(1⫺fNb)
(5)
Inserting Wel into Eq. (1), the total free energy of formation of a unit bilayer in a coherent metastable Nb/Zr multilayer is ⌬gc ⫽ 2⌬gc ⫹ [⌬GNbfNb ⫹ ⌬GZr(1⫺fNb)
(6)
⫹ Ym ]l 2 o
At the nanometer size scale, in addition to the coherency stresses, the surface and interface stresses can make a contribution to the total strain energy of the multilayer. However, as discussed by Banerjee et al. [8], it is difficult to quantify the interfacial stress, especially in the context of the energetics of a multilayer. Furthermore, very little is known about the magnitude or sense of the interfacial stresses involved in the Nb/Zr system. Therefore, following along the lines of classical treatments of coherency strains in thin films [20], it has been assumed that the contribution of the interfacial stresses is significantly lower than the contribution from coherency strains.
2. Experimental procedure The Nb/Zr films were grown by DC magnetron sputtering in a custom-built stainless steel UHV chamber. Each target was of commercial grade purity. The sputtering gas, Ar, was flowed though a Ti gettering furnace in order to reach a purity of 99.999% before entering the chamber. Sputtering pressures were 2–4 mTorr. The sputtering chamber has a base pressure in the low 10⫺9 Torr regime and is evacuated by molecular sorption and cryogenic pumps. The sputtering rate was power regulated and the deposition runs are automated through LabVIEW software [10]. Each film was
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grown on a Si wafer that had a 200 nm thick amorphous oxide layer. The amorphous layer provided a smooth, non-crystallographic surface from which the film could grow. All films were deposited at ambient temperatures (~40 °C). Structural characterization of the multilayers was carried out using XRD and transmission electron microscopy (TEM). The XRD scans were performed on a Scintag PAD V diffractometer using Cu Kα radiation at 45 kV and 20 mA. Scan rates used were 0.5–1°/min. TEM samples were prepared in the plan-view geometry for characterization using electron diffraction. The TEM samples were analyzed in a FEI/Philips CM200 microscope at an accelerating voltage of 200 keV. The volume fractions were confirmed by compositional analysis of the Zr and Nb in each of the multilayered thin films using energy dispersive spectroscopy (EDS) in a FEI/Philips XL30FEG scanning electron microscope (SEM). Analysis of the XRD and TEM results allowed for the determination of the film’s growth texture, crystal structures, and lattice parameters of the Nb and Zr layers in each of the multilayers. The bilayer spacing, l, for the multilayers was determined from the separation of satellite peaks in the XRD scans [11].
3. Results Using the proposed Nb/Zr biphase diagram as a guide, a series of multilayered films have been deposited. These data points are plotted in Fig. 2(a). Results of both XRD and plan-view electron diffraction studies indicate that the multilayers exhibit crystal structures commensurate with their location in the phase space described by the proposed biphase diagram. For example, the TEM diffraction patterns in Fig. 2(b),(c), correspond to those multilayers (marked #1 and #2) on either sides of the biphase boundary between bcc Nb/hcp Zr and bcc Nb/bcc Zr. It is clearly evident that hcp Zr’s characteristic rings are no longer present in Fig. 2(c) as compared to Fig. 2(b). Furthermore, all the rings of Fig. 2(c) could be consistently indexed to the bcc structure. Thus, Zr has adopted the bcc phase as predicted by the thermodynamics outlined in the biphase diagram construction.
A further analysis of these data points near the biphase boundary at f Nb~0.85 has shown the sequence of the bcc transition. The #1 data point in Fig. 3(a) corresponds to a Nb/Zr multilayer in which the layers exhibit the bulk equilibrium structures. This is confirmed by the plan-view electron diffraction pattern shown in Fig. 3(b) (marked #1). This pattern can be indexed consistently as bcc Nb/hcp Zr. As the bilayer spacing (l) is reduced, just above the biphase stability boundary, the Zr transforms to a bcc structure. Indicative of the transformation to bcc Zr is the disappearance of the hcp Zr {101¯ 0} reflection. Interestingly, the diffraction pattern shown in Fig. 3(b) (#2) shows a smearing of the {1 1 0} bcc intensity. The line scan of this ring indicates an asymmetric intensity profile. Using Gatan Digital Micrograph imaging process software, this type of intensity profile in this diffraction pattern can be interpreted on the basis of two {1 1 0} reflections, one from bcc Nb and the other from bcc Zr. The interplanar spacings for these two rings indicated that Zr’s bcc lattice parameter value was 0.354 nm; this lattice parameter is within 1.2% of the lattice parameter of β-Zr extrapolated from high temperatures to room temperature [12]. Nb retained its bulk lattice parameter value of 0.330 nm. The bcc Zr phase formed a semi-coherent bcc/bcc interface with Nb that is essentially strain-free. Upon further reduction of l, the Nb/Zr interfaces became coherent as indicated by a more symmetric intensity distribution in the {1 1 0} diffracted ring (Fig. 3(b) #3). Both layers at this data point have adopted a bcc lattice parameter of 0.338 nm. A similar result of a semi-coherent to coherent transformation of bcc Zr was seen for a set of multilayers grown at fNb = 0.5. These data points are labeled #4, #5, and #6 in Fig. 2. XRD scans indicated that the bulk equilibrium phases of Nb and Zr grew in a texture orientation associated with their closest pack direction, e.g. {0 1 1} and {0 0 0 2}, respectively. For extremely thick bilayers of 500 nm bcc Nb/500 nm hcp Zr, the Zr showed the presence of a {101¯ 0} peak along with the {0 0 0 2}. For the multilayers studied in this paper (l ⬍ 15 nm), only the {0 1 1} and {0 0 0 2} textures were observed. Coupling the XRD results for the films studied
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Fig. 2. (a) Nb-rich section of the Nb/Zr biphase diagram. The solid markers correspond to multilayers deposited and characterized in the present study. The hexagonal marker corresponds to the result reported by Lowe and Geballe [5]. (b) TEM plan-view of a multilayer marked #1 in (a) indicating a bcc Nb/hcp Zr structure. (c) TEM plan-view diffraction corresponding to the multilayer marked #2 in (a) indicating an bcc Zr/bcc Nb structure. Note the loss of the {101¯ 0} ring.
in this paper along with the electron diffraction results from plan-view TEM specimens, a Burgers or near-Burgers orientation relationship between the bcc Nb and hcp Zr phases is reported. This orientation relationship being defined as {0 1 1}bcc / / {0 0 0 1}hcp 具1 1 1典bcc / / {112¯ 0}hcp Upon transformation to bcc Zr, the orientation relationship with bcc Nb was {0 1 1}bcc / / {0 1 1}bcc 具1 1 1典bcc / / {1 1 1}bcc
4. Discussion Using a previously reported experimental data point for the change in phase stability of hcp Zr to bcc Zr in Nb/Zr multilayers, and a classical thermodynamics model, a biphase diagram has been constructed. Subsequently, a whole series of Nb/Zr multilayers with different values of fNb and l have been deposited on the Nb-rich side of the biphase diagram. The results of structural characterization of these multilayers are in agreement with the predictions afforded by the proposed biphase diagram. The experimental results clearly indicate that on
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4.1. Biphase boundaries and associated interfacial energies As discussed earlier, the equation of the biphase boundary is obtained by equating ⌬g’s of the two relevant biphases. In the absence of coherency strains, the relevant ⌬g’s are represented by Eq. (1) and the location of the biphase boundary is then given by 1 ⫺⌬GZr ⫽ (1⫺fNb) l 2(gbcc/bcc⫺gbcc/hcp)
(7)
with the slope of the biphase boundary being ⌬GZr 2(gbcc/bcc⫺gbcc/hcp)
(8)
Using the experimental results obtained in this study, the slope of the biphase boundary has been determined from Fig. 2. The bulk free energy change per unit volume when Zr transforms from hcp to bcc, as computed using solution thermodynamics based models such as CALPHAD, is ⌬GZr = 3.6 × 108 J / m3 [13]. Substituting this value into Eq. (8) yields the result, ⌬gexp ⫽ gbcc/bcc⫺gbcc/hcp ⫽ ⫺250 mJ / m2. Fig. 3. (a) A Nb-rich section of the biphase diagram with points corresponding to hcp Zr/bcc Nb (#1), semi-coherent bcc Nb/bcc Zr (#2) and coherent bcc Nb/bcc Zr (#3) structures. (b) Electron diffraction patterns exhibiting the structures for the three multilayers marked #1, #2, and #3 with corresponding intensity profiles.
reducing l at a constant fNb, the structural transformation from hcp to bcc Zr occurs prior to the formation of a coherent Nb/Zr interface in the multilayers. Thus, it appears that in addition to the biphase boundary between bcc Nb/hcp Zr and bcc Nb/bcc Zr, there exists a second boundary between semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr. The slopes of these biphase boundaries may be used to deduce information concerning the energetics associated with the phase stability of hcp and bcc Zr.
A number of different factors contribute towards ⌬g including a chemical and a structural contribution. The chemical component arises from the change in bonding across the interface while the structural component arises from the change in misfit at the interface accompanying the change in phase. The structural component of ⌬g, ⌬gstruc, can be evaluated based on the difference in the line energy per unit interfacial area between the misfit dislocation networks for a semi-coherent bcc Nb/bcc Zr and a bcc Nb/hcp Zr interface, i.e. ⌬gbcc/bcc ⫽ gⲚbcc/bcc⫺gⲚbcc/hcp where gⲚ is the line energy per unit area of misfit dislocations at the interface. To calculate a first approximation to the structural component of the interfacial energies, the Bollman’s O-lattice [14] approach to modeling misfit dislocation networks has been adopted. Such networks have been constructed for the bcc/hcp and bcc/bcc interfaces. Furuhara and Aaronson
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[15] have shown studies in which reasonable agreement can be obtained from the calculated and experimental values of the O-lattice network to that observed in α/β interfaces. Specifically for the Zr– Nb system, Perovic and Weatherly [16–18] have extensively studied the α/β interfaces in binary Zrrich Zr–Nb alloys. A complex dislocation network consisting of 具a典 and 具a + c典 dislocations have been experimentally observed at the α/β interfaces when these two phases exhibit the Potter orientation relationship: {0 1 1}β / / {1¯ 0 1 1}α 具1¯ 1 1典β / / 具1¯ 21¯ 0典α In case of the Nb/Zr multilayers being investigated in the present study, the Burgers orientation relationship, rather than the Potter relationship, was observed for the bcc Nb/hcp Zr combination. Therefore, a two-dimensional centered rectangular lattice was constructed for the {0 0 0 1} basal plane of Zr [19] in contact with the {0 1 1} plane of Nb in the Burgers or near-Burgers orientation relationship. To maintain a simple two-dimensional interface, only 具a典 type dislocations were used to construct the network. The Burgers vector is of the type b = 1 / 3 具21¯ 1¯ 0典 (with reference to the hcp Zr layer). Fig. 4(a) is a schematic representation of the assumed dislocation network based on the O-lattice approach. As seen, the Burger’s vectors are neither parallel nor perpendicular to the dislocation lines but are inclined at a certain angle. Using the equation by Matthew [20], the energy per unit length of a misfit dislocation line segment between two isotropic solids is El ⫽
2
2
mNbmZrb 1⫺ncos a (lnh / b ⫹ 1) 2π(mNb ⫹ mZr) (1⫺n)
(9)
where m and n denote shear modulus and Poisson’s ratio, b is the length of the Burger’s vector of the misfit dislocations, a is the angle between the Burgers vector and the dislocation line, and h, for this case, being the layer thickness of the Zr layer. n was taken to be the average of Nb’s 0. 397 and Zr’s 0.38 Poisson’s ratio [21]. The shear moduli of bcc Nb and hcp Zr used in the calculation are 37.5 and 35.5 GPa, respectively [21]. With the aid of Eq. (9), the energy per unit length for the two sets
Fig. 4. (a) O-lattice construction of a proposed bcc Nb/hcp Zr two-dimensional dislocation network with corresponding Burgers vectors. (b) O-lattice construction of a proposed bcc Nb/bcc Zr two-dimensional dislocation network with corresponding Burgers vectors.
of misfit dislocations, E1l and E2l , corresponding to the angles a1 (49.6°) and a2 (10.4°), has been calculated. The energy per unit interfacial area was acquired using the following equation, gⲚbcc/hcp ⫽
E1l l1 ⫹ E2l l2 l1l2sinq
(10)
where l1 and l2 are the length of dislocation line segments (⬇7.7 nm each) in a unit cell of the network and q (20.8°) is the angle between l1 and l2. This calculation yields a value of gⲚbcc/hcp⬇500 mJ / m2 A similar two-dimensional O-lattice lattice with in-plane dislocations of the type b = a具1 0 0典 and a / 2具1 1 1典 was constructed for modeling the misfit at the bcc Nb/bcc Zr interface. Fig. 4(b) is a schematic representation of the resulting hexagonal net-
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work of pure edge dislocations for the bcc/bcc interface. The shear modulus of bcc Zr used for the calculation was 17 GPa [22] and the length of the dislocation segments, l1 and l2, predicted by the O-lattice construction, was 1.8 and 3.0 nm, respectively. The estimated dislocation line energy per unit area from Eq. (8) was found to be gⲚbcc/bcc⬇140 mJ / m2 Using these values, a calculated ⌬g for the transformation is ⌬gbcc/bcc ⫽ gⲚbcc/bcc⫺gⲚbcc/hcp ⫽ ⫺360 mJ / m2. This difference in dislocation line energies is larger than the interfacial energy difference experimentally obtained from the slope, e.g. ⫺250 mJ/m2. Subsequently, the driving force for transformation could be completely driven by the reduction in the structural misfit dislocation line energies between the two systems. The chemical component of the interfacial free energy was estimated using a simple nearestneighbor bond model. In this approach, the interfacial energy for a perfectly coherent bcc/bcc or hcp/hcp interface can be estimated in terms of the binary solution thermodynamics interaction energy, w, hcp hcp hcp whcp Nb–Zr ⫽ fNb–Zr⫺1 / 2(fNb–Nb ⫹ fZr–Zr)
(11)
hcp A⫺B
where f is the bond energy for the A–B bond in the hcp structure. An equivalent definition holds for wbcc Nb–Zr. w is related to the regular solution parameter, ⍀, by the relationship ⍀ = Zw where Z is the coordination number. For a (0 1 1) bcc Nb/bcc Zr interface, the excess energy per atom (with respect to the bulk) on the Nb side equals ⌬ENb atom bcc = 2fbcc Nb–Zr⫺2fNb–Nb while the excess energy per bcc atom on the Zr side equals ⌬EZr atom = 2fNb–Zr⫺ bcc 2fZr–Zr. Therefore, the chemical component of the chem ) can be approxiinterfacial energy per atom (gatom mated as follows: Nb Zr bcc gchem atom ⫽ (⌬Eatom ⫹ ⌬Eatom) / 2 ⫽ 2[fNb–Zr bcc bcc bcc ⫺0.5(fbcc Nb–Nb ⫹ fZr–Zr)] ⫽ 2wNb–Zr ⫽ 0.25⍀Nb–Zr.
Based on a recent assessment of the Nb–Zr ⫺20 binary phase diagram [13], ⍀bcc Nb–Zr = 2.8 × 10 chem ⫺21 J / atom. Therefore, gatom = 7×10 J / atom. Multi-
plying with the number of atoms per unit area on 2 the (0 1 1) bcc plane, gchem bcc / bcc⬇91 mJ / m . Using an equivalent nearest-neighbor approximation and a ⫺20 J / atom [13], value of ⍀hcp Nb–Zr = 4.1 × 10 chem 2 ghcp / hcp⬇44 mJ / m . The difference in the number of out-of-plane bonds between the (0 1 1) bcc plane (2 bonds) and the (0 0 0 1) hcp plane (3 bonds) makes it not possible to estimate the chemical component of the bcc Nb/hcp Zr interfacial energy using this simple nearest-neighbor bond approach. However, the above calculations for coherent bcc/bcc and hcp/hcp interfaces serve as a guide in assessing the order of magnitude of the chemical component of the interfacial energy for this system. 4.2. Coherency biphase boundary in Nb/Zr multilayers In case of Nb/Zr multilayers, the experimental results indicated that the bcc coherency at the interface is established at a smaller layer thickness as compared with the initial bcc phase stability thickness. When a semi-coherent (assuming no strain in the multilayer) bcc Nb/bcc Zr multilayer transforms to a coherent bcc Nb/bcc Zr structure, there is no change in crystal structure. Thus, ⌬G = 0, and equating ⌬g’s for coexisting phases (semicoherent and coherent) yields, 2⌬gsemi ⫽ 2⌬gcoh ⫹ Ym2ol
(12)
Therefore, Eq. (12) reduces to the equation of the coherent/semi-coherent biphase boundary given as 1 lcoh/semi
⫽⫺
Ybcc/bccm2bcc/bcc . semi 2(gcoh bcc/bcc⫺gbcc/bcc)
(13)
Note that Ybcc/bcc depends non-linearly on fNb, as expressed in Eq. (4). In order to estimate the value of Ybcc/bcc, the elastic modulus of bcc Zr was taken to be 90 GPa [22]. The difference between the structural interfacial energies of a semi-coherent bcc Nb/bcc Zr interface and a coherent bcc Nb/bcc Zr interface can again be estimated as the line energy (per unit area) of misfit dislocations at the semi-coherent {0 1 1} bcc Nb//{0 1 1} bcc Zr interface. This assumes that
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the coherency strains are completely relieved by misfit dislocations. Thus,
5. Summary
inc Ⲛ 2 gcoh bcc/bcc⫺gbcc/bcc ⫽ ⫺gbcc/bcc ⫽ ⫺140 mJ / m .
Based on a classical thermodynamic model of phase stability in multilayered materials, the phase stability of bcc Zr in Nb/Zr multilayers has been investigated in detail. Using a previously reported experimental result for the phase stability of bcc Zr in these multilayers, coupled with a thermodynamics model, a biphase diagram has been proposed. This diagram consists of a single bcc Nb/hcp Zr–bcc Nb/bcc Zr biphase boundary. A series of new multilayers with Nb-rich volume fractions and varying bilayer thicknesses on either side of the biphase boundary have been sputter deposited. Plan-view electron diffraction patterns are in agreement with the predictions afforded by the proposed biphase diagram. From the experimentally determined slope of the bcc Nb/hcp Zr– bcc Nb/bcc Zr biphase boundary, the reduction in the interfacial energy accompanying the change in phase stability from hcp to bcc Zr has been estimated, ⌬gbcc / bcc = ⫺250 mJ / m2. First approximation calculations were used to assess the structural and chemical contributions to the interfacial energy. The O-lattice generated dislocation network suggests that the structural component of the interfacial free energy is large enough to drive the change in phase stability. Furthermore, at a fixed volume fraction of Nb, the phase stability path of hcp to bcc Zr has been found to be semi-coherent bcc then coherent bcc to the Nb interface with decreasing bilayer spacing l. Coupling the simple thermodynamics of the proposed model with linear elasticity theory, a coherency boundary between semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr has been calculated. Reasonable agreement was found between calculated and experimentally determined values for the interfacial energies associated with these phase stability regions.
With the aid of Eq. (13), it was possible to calculate an approximate coherent/semi-coherent biphase boundary on the Nb/Zr diagram. This boundary is indicated by the dotted curve in Fig. 5. The points marked #2 (semi-coherent bcc Nb/bcc Zr) and #3 (coherent bcc Nb/bcc Zr) lie on either side of the dotted curve suggesting good agreement with experiment. But this curve falls above points marked #5 (semi-coherent bcc Nb/bcc Zr) and #6 (coherent bcc Nb/bcc Zr), indicating poor agreement at this different volume fraction of Nb. The dotted curve was drawn from Eq. (13) assuming the calculated ⌬g of ⫺140 mJ/m2. By curve fitting this line such that it intersects between both sets of points labeled #2/#3 and #5/#6, an experimentally calculated interfacial energy of the bcc Nb/bcc Zr can be obtained. This is seen as the solid curve in Fig. 5. The value of ⌬g of the solid curve was ⫺160 mJ/m2; this is in reasonable agreement with the O-lattice calculation used above.
Fig. 5. A Nb-rich section of the Nb/Zr biphase diagram with the points corresponding to all the multilayers deposited and characterized in the present study. The calculated coherency boundary separating semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr biphase fields is also plotted on this diagram. Note that the data points #2 and #3 (same as in Ref. Fig. 3) and #5 and #6, exhibit the experimentally determined semicoherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr structures. The dotted curve corresponds to the calculated ⌬g of ⫺140 mJ/m2 where the solid curve had a ⌬g of ⫺160 mJ/m2.
Acknowledgements The authors would like to acknowledge the support of the Center for the Accelerated Maturation of Materials (CAMM) at the Ohio State University.
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References [1] Prinz GA. Phys Rev Lett 1985;54:10511. [2] Wormester H, Huger E, Bauer E. Phys Rev Lett 1996;77(8):1540. [3] Flyan L, Cunningham JE. Phys Rev B 1986;34(3):2022. [4] Lupis CHP. Chemical thermodynamics of materials. New Jersey: Prentice Hall PTR, 1983. [5] Lowe WP, Geballe TH. Phys Rev B 1984;29(9):4961. [6] Dregia SA, Banerjee R, Fraser HL. Scripta Mater 1998;29(2):217. [7] Banerjee R, Zhang XD, Dregia SA, Fraser HL. Acta Mater 1999;47(4):1153. [8] Banerjee R, Dregia SA, Fraser HL. Acta Mater 1999;47(15):4225. [9] Dregia SA, Hirth JP. J Appl Phys 1991;69(4):2169. [10] National Instruments, LabVIEW Software version 2.5, www.ni.com. 2003. [11] Chang LL, Giessen BC, editors. Synthetic modulated structures. New York: Academic Press, Inc; 1985.
[12] Touloukian YS, editor. Thermophysical properties of matter (the TPRC data series; a comprehensive compilation of data). New York: IFI/Plenum; 1979. p. 1970. [13] Guillermet AI. Zeitshrift fur Metallkunde 1991;82(6):478. [14] Smith DA, Pond RC. Int Metals Rev 1976;205(June):191. [15] Furuhara T, Aaronson HI. Acta Metall Mater 1991;39(11):2857. [16] Perovic V, Weatherly GC. Acta Metall Mater 1989;37(3):813. [17] Zhang WZ, Perovic V, Perovic A, Weatherly GC, Purdy GR. Acta Metall Mater 1998;46(10):3443. [18] Perovic V, Weatherly GC, Simpson CJ. Acta Metall Mater 1983;31(9):1381. [19] Kelly A, Groves GW. Crystallography and crystal defects. Virginia: Tech Books, 1970. [20] Matthews JW. In: Nabarro FR, editor. Dislocations in solids, vol. 2. Amsterdam: North-Holland; 1979. p. 502. [21] Brandes EA, editor. Smithells metals reference book. 6th ed. London: Butterworths and Co Ltd; 1983. [22] Chen Q, Sundman B. Acta Mater 2001;49:947.
Phase stability of bcc Zr in Nb/Zr thin film multilayers G.B. Thompson, R. Banerjee ∗, S.A. Dregia, H.L. Fraser Department of Materials Science and Engineering, The Ohio State University, 477 Watts Hall, 2041 College Road, Columbus, OH 43210, USA Received 5 February 2003; received in revised form 1 July 2003; accepted 3 July 2003
Abstract A change in phase stability from hcp Zr to bcc Zr occurs in Nb/Zr multilayers when the bilayer thickness is reduced to the nanometer scale. This phase stability in these multilayers has been described using a model based on classical thermodynamics. Using a previously reported experimental observation of hcp to bcc transformation in these multilayers, a phase stability diagram (referred to as the biphase diagram) has been proposed. Subsequently, a range of multilayers with varying volume fractions and bilayer thicknesses have been sputter deposited. The crystal structures in these multilayers have been determined using X-ray and electron diffraction. The hcp and bcc Zr phases within the Nb/Zr multilayers are in agreement with the predictions afforded by the proposed biphase diagram. The sequence of the Zr bcc phase stability was accomplished by forming its β-Zr (high temperature bcc phase) prior to forming a bcc coherent interface with Nb. First approximations of the structural and chemical contributions to the interfacial energy accompanying the changes in hcp to bcc phase stability for Zr have been evaluated. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Pseudomorphic; bcc Zr; Multilayers; Nb/Zr; Biphase stability diagram
1. Introduction When the layer thickness in a multilayered thin film is reduced to the nanoscale regime, it often exhibits structural transitions resulting in one or more of the layers adopting a crystal structure different from its bulk equilibrium form [1–3]. Based on the competing influences of the bulk chemical free energy, the interfacial free energy, and the strain energy, it is possible to model and conse-
Corresponding author. Tel.: +1-614-292-6263; fax: +1614-292-7523. E-mail address: [email protected] (R. Banerjee). ∗
quently predict the critical transition thickness in these systems. This paper focuses on the use of a classical thermodynamics model to predict hcp and bcc phase stability in the Zr layer in Nb/Zr multilayers. 1.1. Nb/Zr multilayers The bulk equilibrium structures of Nb and Zr, in the standard condition (room temperature and 1 atm), are bcc and hcp, respectively. Pure elemental Zr exhibits an allotropic transformation from the hcp α-Zr phase to the bcc β-Zr phase at a temperature of 840 °C [4]. Stabilization of a bcc phase in the Zr layers of sputter deposited Nb/Zr multilayers
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00380-X
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at room temperature has been reported by Lowe and Geballe [5]. A series of multilayers of varying equal Nb and Zr layer thicknesses were characterized by the authors [5] using X-ray diffraction (XRD). Their results reveal that when the bilayer thickness, l is reduced below ~3.1 nm (individual layer thickness ~1.55 nm), the bulk bcc Nb/hcp Zr structure transforms to a metastable bcc Nb/bcc Zr structure. Lowe and Geballe [5] did not vary the individual layer thickness within the bilayer pair in their multilayers. In the present study, a detailed investigation of the formation of bcc Zr in Nb/Zr multilayers as a function of varying individual thicknesses has been undertaken. The experimental results have been coupled with modeling efforts. 1.2. The biphase diagram In a recent paper, Dregia et al. [6] proposed a classical thermodynamics model to explain phase stability in multilayered systems. The details of this model have been described elsewhere [6]. Based on this model, a Nb/Zr multilayer can be represented by a reference bilayer consisting of a single layer of Nb, a single layer of Zr, and two Nb/Zr interfaces. The individual layer thickness of this reference bilayer is described in terms of the volume fraction of one of the constituent layers, e.g. fNb, while the bilayer spacing (l), the sum of the individual thickness of a Zr and Nb layer pair, represents the length scale. Thus, fNb and l can be considered to be the two degrees of freedom in this multilayered system. Varying either or both of these degrees of freedom could potentially result in structural transitions in the multilayer. For a Nb/Zr bilayer, the free energy change per unit interfacial area of the multilayer accompanying a structural transition, ⌬g, is given as ⌬g ⫽ 2⌬g ⫹ ⌬GNbfNbl ⫹ ⌬GZr(1⫺fNb)l
phase stability in Nb/Zr multilayers. Thus, in addition to the allotropic free energy differences between the metastable and stable (with reference to the bulk) forms of Nb and Zr, other contributions to the free energy, such as the strain energy, can play a significant role in determining the phase stability. In order to maintain simplicity in the approach, it is assumed that all energetic contributions that scale with the volume are contained within the ⌬Gi terms, and all energetic contributions that scale with the interfacial area are contained in the ⌬g term. A useful way to depict phase stability in Nb/Zr multilayers is to plot the volume fraction fNb vs. 1/l. Such a plot of phase stability has been previously referred to as a biphase diagram [6–8]. An initial approximation to a biphase diagram for Nb/Zr multilayers may be deduced from Lowe and Geballe’s [5] report of a structural transition from hcp Zr to bcc Zr for l ⬍ 3.1 nm at a fNb = 50%. Based on this data point, a single biphase field boundary has been constructed. This is seen in Fig. 1. Thermodynamically, the hcp to bcc phase stability in Zr would be most probable in the Nb-rich regions of the biphase diagram. It has been seen elsewhere [6] that the other constituent phase can exhibit a structural change in regions where it has low volume fraction. The focus of this paper will be on the possible hcp to bcc phase stability of Zr
(1)
where ⌬g is the interfacial energy difference between a Nb/Zr interface in the transformed state and a Nb/Zr interface in the bulk equilibrium state, ⌬Gi is the allotropic volume free energy change for component i, fNb is the volume fraction of Nb, and l is the bilayer spacing. Eq. (1) provides a basic thermodynamic framework for representing
Fig. 1. A proposed, hypothetical biphase stability diagram based on a previously reported [5] structural transition of Zr from hcp to bcc in Nb/Zr multilayers. This point is designated as a hexagonal mark.
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as predicted in the Nb-rich section of the biphase stability diagram. In Eq. (1), by putting ⌬g = 0 and reorganizing the terms, the slope of the biphase boundary separating bcc Nb/hcp Zr and bcc Nb/bcc Zr biphase fields can be evaluated to be [⌬G] / [2(gbcc / bcc⫺ gbcc / hcp)], which has a negative value. Thus, the proposed biphase diagram, shown in Fig. 1, consists of a single biphase boundary with a negative slope separating the bcc Nb/hcp Zr and bcc Nb/bcc Zr biphase fields. The linear boundary assumes that ⌬G and ⌬g are independent of l. The effects of coherency strains on the boundary are considered separately below. The thermodynamics dictates that this biphase boundary line drawn through the transition point at fNb = 0.5 must approach 1 / l = 0 as fNb approaches 1. Thus as the volume fraction of Nb is increased, the critical value of l at which the transition occurs increases. However, it should be noted that since the slope of the biphase boundary, given by [⌬Gzr] / [2(gbcc / bcc⫺gbcc / hcp)], is a constant, the thickness of an individual Zr layer ( = l(1⫺fNb)) remains constant along the boundary. 1.3. Effect of coherency strains on the biphase diagram The biphase diagram shown in Fig. 1 has been constructed using a simplifying assumption that the ⌬Gi and ⌬g are independent of fNb and l. This would not be the case for multilayers that experience coherency strains, where the equilibrium coherency strain in a layer is inversely proportional to its thickness [9]. If coherency strain energy were included in the thermodynamics, ⌬g would depend non-linearly on volume fraction and lead to curved boundaries in 1/l vs. fNb space. For a strain-free unit reference bilayer, the free energy of formation of a metastable biphase, per unit area of interface, is given by Eq. (1). For a coherently strained Nb/Zr bilayer, the strain energy per unit area of interface [8] is given by Wel ⫽ [YNbe2NbfNb ⫹ YZre2Zr(1⫺fNb)]l
(2)
where ei is the coherency strain and, for each metal, Y is a biaxial elastic modulus defined as Y⫽
E 1⫺n
(3)
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where E is Young’s modulus and n is Poisson’s ratio. The natural misfit, mo, is partitioned such that 兩eNb兩 + 兩eZr兩 = mo, and for force balance, (Yefl)Nb + (Yefl)Zr = 0. Thus, both eZr and eNb can be expressed in terms of mo, and Eq. (2) becomes Wel ⫽ Ylm2o,
(4)
with Y⫽
YNbYZrfNb(1⫺fNb) YNbfNb ⫹ YZr(1⫺fNb)
(5)
Inserting Wel into Eq. (1), the total free energy of formation of a unit bilayer in a coherent metastable Nb/Zr multilayer is ⌬gc ⫽ 2⌬gc ⫹ [⌬GNbfNb ⫹ ⌬GZr(1⫺fNb)
(6)
⫹ Ym ]l 2 o
At the nanometer size scale, in addition to the coherency stresses, the surface and interface stresses can make a contribution to the total strain energy of the multilayer. However, as discussed by Banerjee et al. [8], it is difficult to quantify the interfacial stress, especially in the context of the energetics of a multilayer. Furthermore, very little is known about the magnitude or sense of the interfacial stresses involved in the Nb/Zr system. Therefore, following along the lines of classical treatments of coherency strains in thin films [20], it has been assumed that the contribution of the interfacial stresses is significantly lower than the contribution from coherency strains.
2. Experimental procedure The Nb/Zr films were grown by DC magnetron sputtering in a custom-built stainless steel UHV chamber. Each target was of commercial grade purity. The sputtering gas, Ar, was flowed though a Ti gettering furnace in order to reach a purity of 99.999% before entering the chamber. Sputtering pressures were 2–4 mTorr. The sputtering chamber has a base pressure in the low 10⫺9 Torr regime and is evacuated by molecular sorption and cryogenic pumps. The sputtering rate was power regulated and the deposition runs are automated through LabVIEW software [10]. Each film was
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grown on a Si wafer that had a 200 nm thick amorphous oxide layer. The amorphous layer provided a smooth, non-crystallographic surface from which the film could grow. All films were deposited at ambient temperatures (~40 °C). Structural characterization of the multilayers was carried out using XRD and transmission electron microscopy (TEM). The XRD scans were performed on a Scintag PAD V diffractometer using Cu Kα radiation at 45 kV and 20 mA. Scan rates used were 0.5–1°/min. TEM samples were prepared in the plan-view geometry for characterization using electron diffraction. The TEM samples were analyzed in a FEI/Philips CM200 microscope at an accelerating voltage of 200 keV. The volume fractions were confirmed by compositional analysis of the Zr and Nb in each of the multilayered thin films using energy dispersive spectroscopy (EDS) in a FEI/Philips XL30FEG scanning electron microscope (SEM). Analysis of the XRD and TEM results allowed for the determination of the film’s growth texture, crystal structures, and lattice parameters of the Nb and Zr layers in each of the multilayers. The bilayer spacing, l, for the multilayers was determined from the separation of satellite peaks in the XRD scans [11].
3. Results Using the proposed Nb/Zr biphase diagram as a guide, a series of multilayered films have been deposited. These data points are plotted in Fig. 2(a). Results of both XRD and plan-view electron diffraction studies indicate that the multilayers exhibit crystal structures commensurate with their location in the phase space described by the proposed biphase diagram. For example, the TEM diffraction patterns in Fig. 2(b),(c), correspond to those multilayers (marked #1 and #2) on either sides of the biphase boundary between bcc Nb/hcp Zr and bcc Nb/bcc Zr. It is clearly evident that hcp Zr’s characteristic rings are no longer present in Fig. 2(c) as compared to Fig. 2(b). Furthermore, all the rings of Fig. 2(c) could be consistently indexed to the bcc structure. Thus, Zr has adopted the bcc phase as predicted by the thermodynamics outlined in the biphase diagram construction.
A further analysis of these data points near the biphase boundary at f Nb~0.85 has shown the sequence of the bcc transition. The #1 data point in Fig. 3(a) corresponds to a Nb/Zr multilayer in which the layers exhibit the bulk equilibrium structures. This is confirmed by the plan-view electron diffraction pattern shown in Fig. 3(b) (marked #1). This pattern can be indexed consistently as bcc Nb/hcp Zr. As the bilayer spacing (l) is reduced, just above the biphase stability boundary, the Zr transforms to a bcc structure. Indicative of the transformation to bcc Zr is the disappearance of the hcp Zr {101¯ 0} reflection. Interestingly, the diffraction pattern shown in Fig. 3(b) (#2) shows a smearing of the {1 1 0} bcc intensity. The line scan of this ring indicates an asymmetric intensity profile. Using Gatan Digital Micrograph imaging process software, this type of intensity profile in this diffraction pattern can be interpreted on the basis of two {1 1 0} reflections, one from bcc Nb and the other from bcc Zr. The interplanar spacings for these two rings indicated that Zr’s bcc lattice parameter value was 0.354 nm; this lattice parameter is within 1.2% of the lattice parameter of β-Zr extrapolated from high temperatures to room temperature [12]. Nb retained its bulk lattice parameter value of 0.330 nm. The bcc Zr phase formed a semi-coherent bcc/bcc interface with Nb that is essentially strain-free. Upon further reduction of l, the Nb/Zr interfaces became coherent as indicated by a more symmetric intensity distribution in the {1 1 0} diffracted ring (Fig. 3(b) #3). Both layers at this data point have adopted a bcc lattice parameter of 0.338 nm. A similar result of a semi-coherent to coherent transformation of bcc Zr was seen for a set of multilayers grown at fNb = 0.5. These data points are labeled #4, #5, and #6 in Fig. 2. XRD scans indicated that the bulk equilibrium phases of Nb and Zr grew in a texture orientation associated with their closest pack direction, e.g. {0 1 1} and {0 0 0 2}, respectively. For extremely thick bilayers of 500 nm bcc Nb/500 nm hcp Zr, the Zr showed the presence of a {101¯ 0} peak along with the {0 0 0 2}. For the multilayers studied in this paper (l ⬍ 15 nm), only the {0 1 1} and {0 0 0 2} textures were observed. Coupling the XRD results for the films studied
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Fig. 2. (a) Nb-rich section of the Nb/Zr biphase diagram. The solid markers correspond to multilayers deposited and characterized in the present study. The hexagonal marker corresponds to the result reported by Lowe and Geballe [5]. (b) TEM plan-view of a multilayer marked #1 in (a) indicating a bcc Nb/hcp Zr structure. (c) TEM plan-view diffraction corresponding to the multilayer marked #2 in (a) indicating an bcc Zr/bcc Nb structure. Note the loss of the {101¯ 0} ring.
in this paper along with the electron diffraction results from plan-view TEM specimens, a Burgers or near-Burgers orientation relationship between the bcc Nb and hcp Zr phases is reported. This orientation relationship being defined as {0 1 1}bcc / / {0 0 0 1}hcp 具1 1 1典bcc / / {112¯ 0}hcp Upon transformation to bcc Zr, the orientation relationship with bcc Nb was {0 1 1}bcc / / {0 1 1}bcc 具1 1 1典bcc / / {1 1 1}bcc
4. Discussion Using a previously reported experimental data point for the change in phase stability of hcp Zr to bcc Zr in Nb/Zr multilayers, and a classical thermodynamics model, a biphase diagram has been constructed. Subsequently, a whole series of Nb/Zr multilayers with different values of fNb and l have been deposited on the Nb-rich side of the biphase diagram. The results of structural characterization of these multilayers are in agreement with the predictions afforded by the proposed biphase diagram. The experimental results clearly indicate that on
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4.1. Biphase boundaries and associated interfacial energies As discussed earlier, the equation of the biphase boundary is obtained by equating ⌬g’s of the two relevant biphases. In the absence of coherency strains, the relevant ⌬g’s are represented by Eq. (1) and the location of the biphase boundary is then given by 1 ⫺⌬GZr ⫽ (1⫺fNb) l 2(gbcc/bcc⫺gbcc/hcp)
(7)
with the slope of the biphase boundary being ⌬GZr 2(gbcc/bcc⫺gbcc/hcp)
(8)
Using the experimental results obtained in this study, the slope of the biphase boundary has been determined from Fig. 2. The bulk free energy change per unit volume when Zr transforms from hcp to bcc, as computed using solution thermodynamics based models such as CALPHAD, is ⌬GZr = 3.6 × 108 J / m3 [13]. Substituting this value into Eq. (8) yields the result, ⌬gexp ⫽ gbcc/bcc⫺gbcc/hcp ⫽ ⫺250 mJ / m2. Fig. 3. (a) A Nb-rich section of the biphase diagram with points corresponding to hcp Zr/bcc Nb (#1), semi-coherent bcc Nb/bcc Zr (#2) and coherent bcc Nb/bcc Zr (#3) structures. (b) Electron diffraction patterns exhibiting the structures for the three multilayers marked #1, #2, and #3 with corresponding intensity profiles.
reducing l at a constant fNb, the structural transformation from hcp to bcc Zr occurs prior to the formation of a coherent Nb/Zr interface in the multilayers. Thus, it appears that in addition to the biphase boundary between bcc Nb/hcp Zr and bcc Nb/bcc Zr, there exists a second boundary between semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr. The slopes of these biphase boundaries may be used to deduce information concerning the energetics associated with the phase stability of hcp and bcc Zr.
A number of different factors contribute towards ⌬g including a chemical and a structural contribution. The chemical component arises from the change in bonding across the interface while the structural component arises from the change in misfit at the interface accompanying the change in phase. The structural component of ⌬g, ⌬gstruc, can be evaluated based on the difference in the line energy per unit interfacial area between the misfit dislocation networks for a semi-coherent bcc Nb/bcc Zr and a bcc Nb/hcp Zr interface, i.e. ⌬gbcc/bcc ⫽ gⲚbcc/bcc⫺gⲚbcc/hcp where gⲚ is the line energy per unit area of misfit dislocations at the interface. To calculate a first approximation to the structural component of the interfacial energies, the Bollman’s O-lattice [14] approach to modeling misfit dislocation networks has been adopted. Such networks have been constructed for the bcc/hcp and bcc/bcc interfaces. Furuhara and Aaronson
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[15] have shown studies in which reasonable agreement can be obtained from the calculated and experimental values of the O-lattice network to that observed in α/β interfaces. Specifically for the Zr– Nb system, Perovic and Weatherly [16–18] have extensively studied the α/β interfaces in binary Zrrich Zr–Nb alloys. A complex dislocation network consisting of 具a典 and 具a + c典 dislocations have been experimentally observed at the α/β interfaces when these two phases exhibit the Potter orientation relationship: {0 1 1}β / / {1¯ 0 1 1}α 具1¯ 1 1典β / / 具1¯ 21¯ 0典α In case of the Nb/Zr multilayers being investigated in the present study, the Burgers orientation relationship, rather than the Potter relationship, was observed for the bcc Nb/hcp Zr combination. Therefore, a two-dimensional centered rectangular lattice was constructed for the {0 0 0 1} basal plane of Zr [19] in contact with the {0 1 1} plane of Nb in the Burgers or near-Burgers orientation relationship. To maintain a simple two-dimensional interface, only 具a典 type dislocations were used to construct the network. The Burgers vector is of the type b = 1 / 3 具21¯ 1¯ 0典 (with reference to the hcp Zr layer). Fig. 4(a) is a schematic representation of the assumed dislocation network based on the O-lattice approach. As seen, the Burger’s vectors are neither parallel nor perpendicular to the dislocation lines but are inclined at a certain angle. Using the equation by Matthew [20], the energy per unit length of a misfit dislocation line segment between two isotropic solids is El ⫽
2
2
mNbmZrb 1⫺ncos a (lnh / b ⫹ 1) 2π(mNb ⫹ mZr) (1⫺n)
(9)
where m and n denote shear modulus and Poisson’s ratio, b is the length of the Burger’s vector of the misfit dislocations, a is the angle between the Burgers vector and the dislocation line, and h, for this case, being the layer thickness of the Zr layer. n was taken to be the average of Nb’s 0. 397 and Zr’s 0.38 Poisson’s ratio [21]. The shear moduli of bcc Nb and hcp Zr used in the calculation are 37.5 and 35.5 GPa, respectively [21]. With the aid of Eq. (9), the energy per unit length for the two sets
Fig. 4. (a) O-lattice construction of a proposed bcc Nb/hcp Zr two-dimensional dislocation network with corresponding Burgers vectors. (b) O-lattice construction of a proposed bcc Nb/bcc Zr two-dimensional dislocation network with corresponding Burgers vectors.
of misfit dislocations, E1l and E2l , corresponding to the angles a1 (49.6°) and a2 (10.4°), has been calculated. The energy per unit interfacial area was acquired using the following equation, gⲚbcc/hcp ⫽
E1l l1 ⫹ E2l l2 l1l2sinq
(10)
where l1 and l2 are the length of dislocation line segments (⬇7.7 nm each) in a unit cell of the network and q (20.8°) is the angle between l1 and l2. This calculation yields a value of gⲚbcc/hcp⬇500 mJ / m2 A similar two-dimensional O-lattice lattice with in-plane dislocations of the type b = a具1 0 0典 and a / 2具1 1 1典 was constructed for modeling the misfit at the bcc Nb/bcc Zr interface. Fig. 4(b) is a schematic representation of the resulting hexagonal net-
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work of pure edge dislocations for the bcc/bcc interface. The shear modulus of bcc Zr used for the calculation was 17 GPa [22] and the length of the dislocation segments, l1 and l2, predicted by the O-lattice construction, was 1.8 and 3.0 nm, respectively. The estimated dislocation line energy per unit area from Eq. (8) was found to be gⲚbcc/bcc⬇140 mJ / m2 Using these values, a calculated ⌬g for the transformation is ⌬gbcc/bcc ⫽ gⲚbcc/bcc⫺gⲚbcc/hcp ⫽ ⫺360 mJ / m2. This difference in dislocation line energies is larger than the interfacial energy difference experimentally obtained from the slope, e.g. ⫺250 mJ/m2. Subsequently, the driving force for transformation could be completely driven by the reduction in the structural misfit dislocation line energies between the two systems. The chemical component of the interfacial free energy was estimated using a simple nearestneighbor bond model. In this approach, the interfacial energy for a perfectly coherent bcc/bcc or hcp/hcp interface can be estimated in terms of the binary solution thermodynamics interaction energy, w, hcp hcp hcp whcp Nb–Zr ⫽ fNb–Zr⫺1 / 2(fNb–Nb ⫹ fZr–Zr)
(11)
hcp A⫺B
where f is the bond energy for the A–B bond in the hcp structure. An equivalent definition holds for wbcc Nb–Zr. w is related to the regular solution parameter, ⍀, by the relationship ⍀ = Zw where Z is the coordination number. For a (0 1 1) bcc Nb/bcc Zr interface, the excess energy per atom (with respect to the bulk) on the Nb side equals ⌬ENb atom bcc = 2fbcc Nb–Zr⫺2fNb–Nb while the excess energy per bcc atom on the Zr side equals ⌬EZr atom = 2fNb–Zr⫺ bcc 2fZr–Zr. Therefore, the chemical component of the chem ) can be approxiinterfacial energy per atom (gatom mated as follows: Nb Zr bcc gchem atom ⫽ (⌬Eatom ⫹ ⌬Eatom) / 2 ⫽ 2[fNb–Zr bcc bcc bcc ⫺0.5(fbcc Nb–Nb ⫹ fZr–Zr)] ⫽ 2wNb–Zr ⫽ 0.25⍀Nb–Zr.
Based on a recent assessment of the Nb–Zr ⫺20 binary phase diagram [13], ⍀bcc Nb–Zr = 2.8 × 10 chem ⫺21 J / atom. Therefore, gatom = 7×10 J / atom. Multi-
plying with the number of atoms per unit area on 2 the (0 1 1) bcc plane, gchem bcc / bcc⬇91 mJ / m . Using an equivalent nearest-neighbor approximation and a ⫺20 J / atom [13], value of ⍀hcp Nb–Zr = 4.1 × 10 chem 2 ghcp / hcp⬇44 mJ / m . The difference in the number of out-of-plane bonds between the (0 1 1) bcc plane (2 bonds) and the (0 0 0 1) hcp plane (3 bonds) makes it not possible to estimate the chemical component of the bcc Nb/hcp Zr interfacial energy using this simple nearest-neighbor bond approach. However, the above calculations for coherent bcc/bcc and hcp/hcp interfaces serve as a guide in assessing the order of magnitude of the chemical component of the interfacial energy for this system. 4.2. Coherency biphase boundary in Nb/Zr multilayers In case of Nb/Zr multilayers, the experimental results indicated that the bcc coherency at the interface is established at a smaller layer thickness as compared with the initial bcc phase stability thickness. When a semi-coherent (assuming no strain in the multilayer) bcc Nb/bcc Zr multilayer transforms to a coherent bcc Nb/bcc Zr structure, there is no change in crystal structure. Thus, ⌬G = 0, and equating ⌬g’s for coexisting phases (semicoherent and coherent) yields, 2⌬gsemi ⫽ 2⌬gcoh ⫹ Ym2ol
(12)
Therefore, Eq. (12) reduces to the equation of the coherent/semi-coherent biphase boundary given as 1 lcoh/semi
⫽⫺
Ybcc/bccm2bcc/bcc . semi 2(gcoh bcc/bcc⫺gbcc/bcc)
(13)
Note that Ybcc/bcc depends non-linearly on fNb, as expressed in Eq. (4). In order to estimate the value of Ybcc/bcc, the elastic modulus of bcc Zr was taken to be 90 GPa [22]. The difference between the structural interfacial energies of a semi-coherent bcc Nb/bcc Zr interface and a coherent bcc Nb/bcc Zr interface can again be estimated as the line energy (per unit area) of misfit dislocations at the semi-coherent {0 1 1} bcc Nb//{0 1 1} bcc Zr interface. This assumes that
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the coherency strains are completely relieved by misfit dislocations. Thus,
5. Summary
inc Ⲛ 2 gcoh bcc/bcc⫺gbcc/bcc ⫽ ⫺gbcc/bcc ⫽ ⫺140 mJ / m .
Based on a classical thermodynamic model of phase stability in multilayered materials, the phase stability of bcc Zr in Nb/Zr multilayers has been investigated in detail. Using a previously reported experimental result for the phase stability of bcc Zr in these multilayers, coupled with a thermodynamics model, a biphase diagram has been proposed. This diagram consists of a single bcc Nb/hcp Zr–bcc Nb/bcc Zr biphase boundary. A series of new multilayers with Nb-rich volume fractions and varying bilayer thicknesses on either side of the biphase boundary have been sputter deposited. Plan-view electron diffraction patterns are in agreement with the predictions afforded by the proposed biphase diagram. From the experimentally determined slope of the bcc Nb/hcp Zr– bcc Nb/bcc Zr biphase boundary, the reduction in the interfacial energy accompanying the change in phase stability from hcp to bcc Zr has been estimated, ⌬gbcc / bcc = ⫺250 mJ / m2. First approximation calculations were used to assess the structural and chemical contributions to the interfacial energy. The O-lattice generated dislocation network suggests that the structural component of the interfacial free energy is large enough to drive the change in phase stability. Furthermore, at a fixed volume fraction of Nb, the phase stability path of hcp to bcc Zr has been found to be semi-coherent bcc then coherent bcc to the Nb interface with decreasing bilayer spacing l. Coupling the simple thermodynamics of the proposed model with linear elasticity theory, a coherency boundary between semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr has been calculated. Reasonable agreement was found between calculated and experimentally determined values for the interfacial energies associated with these phase stability regions.
With the aid of Eq. (13), it was possible to calculate an approximate coherent/semi-coherent biphase boundary on the Nb/Zr diagram. This boundary is indicated by the dotted curve in Fig. 5. The points marked #2 (semi-coherent bcc Nb/bcc Zr) and #3 (coherent bcc Nb/bcc Zr) lie on either side of the dotted curve suggesting good agreement with experiment. But this curve falls above points marked #5 (semi-coherent bcc Nb/bcc Zr) and #6 (coherent bcc Nb/bcc Zr), indicating poor agreement at this different volume fraction of Nb. The dotted curve was drawn from Eq. (13) assuming the calculated ⌬g of ⫺140 mJ/m2. By curve fitting this line such that it intersects between both sets of points labeled #2/#3 and #5/#6, an experimentally calculated interfacial energy of the bcc Nb/bcc Zr can be obtained. This is seen as the solid curve in Fig. 5. The value of ⌬g of the solid curve was ⫺160 mJ/m2; this is in reasonable agreement with the O-lattice calculation used above.
Fig. 5. A Nb-rich section of the Nb/Zr biphase diagram with the points corresponding to all the multilayers deposited and characterized in the present study. The calculated coherency boundary separating semi-coherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr biphase fields is also plotted on this diagram. Note that the data points #2 and #3 (same as in Ref. Fig. 3) and #5 and #6, exhibit the experimentally determined semicoherent bcc Nb/bcc Zr and coherent bcc Nb/bcc Zr structures. The dotted curve corresponds to the calculated ⌬g of ⫺140 mJ/m2 where the solid curve had a ⌬g of ⫺160 mJ/m2.
Acknowledgements The authors would like to acknowledge the support of the Center for the Accelerated Maturation of Materials (CAMM) at the Ohio State University.
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