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The nature of the structural phase transition from the hexagonal (4H) phase to the cubic (3C) phase of silver
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 115405 (http://iopscience.iop.org/0953-8984/26/11/115405) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 15/06/2017 at 17:03 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115405 (8pp)
doi:10.1088/0953-8984/26/11/115405
The nature of the structural phase transition from the hexagonal (4H) phase to the cubic (3C) phase of silver Indrani Chakraborty1 , Sharmila N Shirodkar2 , Smita Gohil1 , Umesh V Waghmare2 and Pushan Ayyub1 1
Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai 400005, India 2 Theoretical Science Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560064, India E-mail: [email protected] Received 31 July 2013, revised 19 January 2014 Accepted for publication 21 January 2014 Published 3 March 2014
Abstract
The phase transition from the hexagonal 4H polytype of silver to the commonly known 3C (fcc) phase was studied in detail using x-ray diffraction, electron microscopy, differential scanning calorimetry and Raman spectroscopy. The phase transition is irreversible and accompanied by extensive microstructural changes and grain growth. Detailed scanning and isothermal calorimetric analysis suggests that it is an autocatalytic transformation. Though the calorimetric data suggest an exothermic first-order phase transition with an onset at 155.6 ◦ C (for a heating rate of 2 K min−1 ) and a latent heat of 312.9 J g−1 , the microstructure and the electrical resistance appear to change gradually from much lower temperatures. The 4H phase shows a Raman active mode at 64.3 cm−1 (at 4 K) that undergoes mode softening as the 4H → 3C transformation temperature is approached. A first-principles density functional theory calculation shows that the stacking fault energy of 4H-Ag increases monotonically with temperature. That 4H-Ag has a higher density of stacking faults than 3C-Ag, implies the metastability of the former at higher temperatures. Energetically, the 4H phase is intermediate between the hexagonal 2H phase and the 3C ground state, as indicated by the spontaneous transformation of the 2H to the 4H phase at −4 ◦ C. Our data appear to indicate that the 4H-Ag phase is stabilized at reduced dimensions and thermally induced grain growth is probably responsible for triggering the irreversible transformation to cubic Ag. Keywords: structural phase transition, transformation kinetics, polytypes of silver, density functional theory, stacking fault energy, size-driven transition (Some figures may appear in colour only in the online journal)
1. Introduction
polytypes can coexist in the same thermodynamic regime. Even though extensive accounts on polytypism and polytypic phase transitions are available in the literature [1–3], the precise nature of such transitions and the origin of such large periodicities in polytypes remain debatable. Polytypic phase transitions may be reversible (e.g. the 2H → 12R transition in PbI2 [4]), or irreversible (e.g. the 2H → 3C transition in SiC [5]). The one generic feature that appears to link all
Polytypes are crystallographic modifications of the same substance, in which two dimensions of the unit cell are similar but the third is a variable integral multiple of a common unit [1]. Polytypes differ from polymorphs (or allotropes) in that they are usually not represented by definite thermodynamic phases with specific domains of temperature and pressure: multiple 0953-8984/14/115405+08$33.00
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polytypes is the presence of stacking faults [3]: it is empirically evident [1] that polytypes of a given system are formed by the creation and periodic arrangement of stacking faults in the corresponding basis structure. The periodicity and symmetry of a polytype are determined by the nature and frequency of the stacking faults. The present study is motivated by the fact that much of the previous work on polytypic phase transitions refers to compounds and alloys, and there is little understanding of polytypism and inter-polytype transitions in elemental metals. Silver is commonly known to have a face-centered cubic (fcc) structure, which can be visualized as a periodic stacking of hexagonally close-packed (hcp) atomic layers along the cubic [111] axis. The various polytypes differ from each other merely in the periodicity of the stacking. The stacking sequence of the hcp layers in 3C-Ag (fcc) is ABCABC. . . along the [111] axis, while it is ABCBABCB. . . for 4H-Ag and ABAB. . . for 2HAg. We earlier observed traces of the hexagonal 4H polytype in sputter-deposited nanocrystalline silver films [6], and later reported the growth of phase-pure 4H and 2H polytypes as thick films as well as nanorods [7]. We have recently reported the optical, electronic, vibrational and mechanical properties of the anisotropic, hexagonal (4H) form of silver [8]. Here, we present a detailed study of the irreversible phase transformation from the hexagonal 4H form to the cubic 3C form of silver and suggest that the 4H polytype appears to be a size-stabilized phase.
Figure 1. X-ray diffraction patterns from as-deposited 4H-Ag and
from samples heated to progressively higher temperatures and cooled down to room temperature, showing a gradual phase transformation from the 4H to 3C over a relatively broad temperature range.
No. 411402] preferentially orientated along the [0004/1010] direction, with an interplanar spacing of 0.250 nm in this direction. The coherently diffracting crystallite size calculated using the Scherrer formula is 26 nm. Thermal evolution of the microstructure was investigated by heating parts of the same 4H-Ag film to progressively higher temperatures in a nitrogen atmosphere and holding for 10 min at each temperature. With increasing temperature, the (0004/1010) line of 4H gradually loses intensity in comparison to the (111) line of 3C, and is surpassed at 160 ◦ C. The Scherrer size corresponding to the completely transformed 3C-Ag film annealed at 400 ◦ C was >100 nm. The transformation is obviously irreversible, as the XRD patterns were recorded only after cooling down the samples to room temperature. The complex microstructure of the 4H phase consists of long, cylindrical entities that are arranged roughly parallel to each other in aggregated structures spanning a few µm (figure 2(a)). These cylindrical particles are themselves made up of 20–30 nm diameter crystallites stacked end to end (see inset). This size matches quite well with the Scherrer size calculated for the as-deposited 4H films. Figures 2(b)–(d) depict the evolution of this microstructure as it is heated to progressively higher temperatures, using the protocol described earlier. In the sample heated to 120 ◦ C (figure 2(b)), almost all the 30 nm particles have fused together to form larger cylindrical structures of about 40 nm diameter and 130–180 nm length (inset). These cylindrical particles are still arranged in roughly parallel aggregates. At 160 ◦ C (figure 2(c)), none of the cylindrical particles are visible, having fused into roughly equi-axed particles with an average diameter of 150 nm, which are loosely aggregated into 0.5–1.0 µm clusters. Finally, in films annealed at 400 ◦ C (figure 2(d)), we observe a relatively uniform, compact microstructure consisting of 1–2 µm grains of the pure 3C phase. Thus, both XRD and SEM results indicate that the 4H → 3C transformation clearly occurs through a process of aggregation and grain growth.
2. Experimental details
Single-phase 4H-Ag films were grown by a simple twoelectrode electrodeposition procedure. The substrates were commercially available porous anodic alumina (PAA) templates, on which was sputter-deposited a ≈200 nm thick film of cubic silver to act as the cathode. The porous sides of the templates were coated with an insulating layer to prevent infiltration within the pores. The electrolyte was a solution of 4 g AgNO3 , 4 g anhydrous citric acid and 1.5 g boric acid in 100 ml water; and the anode was a 99.9% pure silver rod of 1 cm diameter held 5 cm away from the cathode. Initially a 1 V pulse was applied for 4 s, followed by recurrent potential pulses of 200 mV for 5 s at 10 s intervals. The total deposition time for a 30 µm thick film was 4 h. The samples were characterized using a Philips X’Pert Pro Powder x-ray diffractometer and a Zeiss Ultra 55 Plus field emission scanning electron microscope (SEM). Differential scanning calorimetric (DSC) measurements were made with a Perkin-Elmer DSC 7. Raman spectra were recorded in the backscattering geometry using a Jobin–Yvon T-64000 triple grating spectrometer with the 647.1 nm Kr-laser line and a CCD detector. The effective spectral resolution obtained was 0.7 cm−1 . Low temperature measurements were made in a continuous flow liquid helium cryostat with quartz windows. 3. Results 3.1. Microstructural evolution
The x-ray diffraction (XRD) pattern for the as-deposited 4H-Ag sample (figure 1) shows single-phase 4H-Ag [PCPDF 2
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Figure 2. Plan view SEM images of 4H-Ag films heated to different temperatures and cooled down: (a) As-deposited 4H-Ag film shows needle-shaped crystals arranged in roughly parallel, micron-sized aggregates. Inset shows that each needle is actually a linear stack of 20–30 nm nanoparticles. (b) On heating the above sample to 120 ◦ C, the primary particles fuse together into longer (≈150 nm) cylindrical structures. (c) At 160 ◦ C, the acicular microstructure is totally replaced by roughly equi-axed, ≈150 nm particles of 3C-Ag. (d) Finally at 400 ◦ C, we observe only micron-sized particles of 3C-Ag.
Seen in conjunction with our earlier observation of 4H-Ag in sputter-deposited nanocrystalline Ag films with average crystallite size ≈30 nm [6], it is reasonable to presume that the 4H polytype of Ag is a size-stabilized phase. 3.2. Evolution of electrical resistance
In a companion study that has just been reported separately [8], we have shown that the 4H phase is much less metallic than the 3C phase, with an in-plane resistivity that is over two orders of magnitude higher. Thus, we can use resistivity as a probe for the 4H → 3C transformation. Figure 3 shows the temperature dependence of the four-probe electrical resistance of a 4H-Ag film on a PAA substrate heated at the rate of 3 K min−1 across the phase transition. The initial rise in the resistance up to 60 ◦ C reflects the normal metallic behavior of 4H-Ag. This is followed initially by a gradual fall up to ≈140◦ and then a sharp fall, ultimately flattening off at about 190 ◦ C. The resistance is nearly constant at higher temperatures and is about 24 times lower than the resistance of the as-deposited 4H film. The resistance curve does not necessarily represent the 4H:3C phase fraction, as the sharp drop probably has a percolative origin.
Figure 3. Variation of the resistance of a 4H-Ag film on a porous
alumina substrate as it is heated across the 4H → 3C transformation. 3.3. Thermal analysis
Differential scanning calorimetry (DSC) measurements were made on 8 µm thick 4H-Ag films grown on PAA. Since the phase transition parameters depend appreciably on the heating rate, we report data (see figure 4) for a series of 3
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Flynn–Wall–Ozawa equation [11, 12] simply states that for each value of α, −0.4567E a + A0 , RT i AE a − 2.315. where A = Log g(α)R Logβ =
Ti is the isoconversion temperature for a given value of α, and g(α) is a function that describes the nature of the conversion. Logβ versus 1/Ti was plotted for nine values of α, and E a calculated from the slope of each. Figure 5(c) shows that the activation energy falls linearly with the degree of conversion and clearly indicates an autocatalytic transformation. The activation energy obtained from the two methods is quite similar. An isothermal DSC scan of the 4H film recorded at 175 ◦ C, showed that the transformation is quite rapid and is basically completed within a couple of minutes. For the isothermal run, the α versus t plot was sigmoidal in nature. A plot of the time derivative of the fraction transformed, dα/dt, against α (figure 5(d)) shows that the transformation is at its fastest at α ≈ 40%, further indicating an autocatalyzed process. For an autocatalytic process, the Prout–Tompkins rate equation [13, 14] (also known as the two-variable Sest´ak– Berggren equation) is given by:
Figure 4. DSC scans for the 4H-Ag film on anodic alumina substrates for different rates of heating. The onset temperature of the 4H → 3C transition for each heating rate is indicated.
different rates ranging from 2 to 30 K min−1 . The welldefined exothermic peak in the DSC data indicates that the higher energy hexagonal 4H phase undergoes a strongly first-order phase transition in transforming irreversibly to the lower energy cubic 3C phase. The transition (onset) temperature varies from 155.6 ◦ C (at 2 K min−1 ) to 192.6 ◦ C (at 30 K min−1 ). The enthalpy of transition was obtained by dividing the area under the exothermic peak in the DSC data by the estimated mass of 4H-Ag in the film and using the instrumental calibration factor. The value of the enthalpy ranged from 312.9 J g−1 (at 2 K min−1 ) to 252.9 J g−1 (at 30 K min−1 ). As a consistency check, we also scanned a PAA substrate sputter-coated with bulk cubic Ag, which expectedly yielded a flat baseline in the same temperature range. We now proceed to a more detailed investigation of the transition kinetics. Firstly, we obtain the phase fraction, α (fraction of 4H transformed to 3C at a particular temperature T ) by computing the integral curve obtained from the DSC scan, normalized by the total area under the transition peak. Figure 5(a) shows the temperature dependence of α for different heating rates. The preferred techniques for performing kinetic analysis from DSC scans are the ‘isoconversion’ or ‘model-free’ methods, wherein the unknown form of the reaction model is temporarily eliminated by comparing measurements made at common values of α corresponding to different heating rates [9]. The two most useful isoconversionbased approaches for extracting the activation energy of a phase transition from DSC scans are due to Kissinger and Flynn–Wall–Ozawa. The Kissinger equation [10] is given by: Ln(Tp2 /β) = −Ln(A R/E a ) + E a /RTp .
(2)
dα = K (T )α m (1 − α)n , dt
(3)
where dα/dt is the rate of transformation at temperature T , K (T ) is the rate constant and (m + n) is the order of the reaction. It is clear from figure 5(d) that the dα/dt versus α data fits the Prout–Tompkins equation (3) perfectly in the range 0.1 ≤ α ≤ 0.7 with the rate constant, K = 20.38 min−1 , m = 1.02 and n = 1.93 being the fit parameters. The total reaction order is thus close to 3. However, the curve deviates markedly from the autocatalysis equation for α > 0.7. That the 4H → 3C transformation is autocatalytic, implies that the initial formation of 3C nuclei in the 4H matrix promotes further reaction due to the appearance of dislocations or cracks at the reaction interface (branching). Such imperfections act as nucleating centers for the growth of the 3C phase, thereby showing autocatalytic behavior. Termination occurs when the reaction begins to spread into material that has already transformed [15]. Once the reaction has spread sufficiently through the material, the autocatalytic behavior is replaced by normal nucleation and growth processes till the conversion is complete. 3.4. Vibrational spectroscopy
In its cubic form, silver is not Raman active, but the hexagonal (4H) modification exhibits at least one Raman active mode at ≈ 64.3 cm−1 at 4 K. In our earlier paper [7], we had presented a simulation of the phonon dispersion curves and Raman active modes for 4H-Ag from a first-principles density functional theory calculation. The simulation indicated that the observed mode is a doubly degenerate, transverse optic mode. Here, we report the evolution of the Raman mode as the temperature
(1)
Tp is the temperature at which maximum conversion occurs, β is the heating rate and E a is the activation energy, which can be obtained from a plot of Ln(Tp2 /β) versus 1/Tp . The activation energy for the 4H → 3C transition obtained from the Kissinger plot (figure 5(b)) is 103.2 kJ mol−1 . The 4
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Figure 5. (a) Plots of the fraction (α) of 4H transformed to 3C versus temperature, showing the change in transformation kinetics with
heating rate. (b) Kissinger plots for determining the activation energy of the 4H → 3C transformation. (c) Plot of the activation energy against the fraction transformed, following the Flynn–Wall–Ozawa method. (d) A plot of dα/dt against α obtained from the isothermal DSC scan at 175 ◦ C fits the autocatalytic rate equation perfectly in the range 0.006 ≤ α ≤ 0.7.
of the phonon dispersion revealed that it is only marginally stable, with an energy surface that is rather flat or weakly varying with respect to several modes [7]. A 2H sample kept at −4 ◦ C showed a gradual spontaneous transition from the 2H to the 4H phase over a time period of 1 month (figure 7), and on heating a transition from 4H to 3C. This implies that the 4H state is energetically intermediate between 2H and 3C, in agreement with our earlier DFT-based predictions.
is increased across the 4H → 3C transformation temperature. Figure 6(a) shows the background-subtracted, high-resolution Raman data recorded at three different temperatures. We observe that the Raman mode of 4H-Ag becomes broader and less intense as the transition temperature is approached. The mode frequency—determined from Lorentzian fits of the spectra—shows a small but distinct tendency to soften close to 350 K (figure 6(b)). However, it was difficult to track the peak position at temperatures closer to the transition. This type of mode softening is known to be associated with several classes of temperature- or pressure-driven structural phase transitions, such as the displacive-type ferroelectric–paraelectric phase transition, the transition from α-quartz to β-quartz, and so on [3].
4. Discussions
We now attempt to elucidate the mechanism controlling the 4H → 3C phase transition. Liang et al [16] have shown that intense electron beam irradiation converts 4H-Ag nanowires to the 3C form, and ascribed it to knock-on displacement of Ag atoms by electrons generating lattice defects. Ab initio calculations of stacking fault energies of various noble and transition metals and the energy differences between their hcp and fcc phases [17] have shown that Ag has the lowest stacking fault energy among the noble metals. As an example of the significance of stacking faults in polytypic phase transitions, we point out that the free energy difference between the pristine and faulted structures of 4H-SiC leads to the growth of stacking faults at temperatures above 60 K and the consequent
3.5. The 2H polytype
Other than the 4H polytype of Ag (which is stable at room temperature), we have reported the growth of a 2H polytype (metastable at room temperature) in the form of single crystalline films and nanorods by optimizing the electrochemical growth kinetics [7]. 2H-Ag spontaneously transforms to the stable 3C phase at room temperature over a period of days. Local stability analysis of the 2H structure through a simulation 5
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Figure 7. XRD patterns of: (top) a 2H-Ag film (with a minor
fraction of 3C-Ag and (bottom) the same film after spontaneous transformation to 4H-Ag after a period of one month.
ABAC ACAC ABAC. In Hagg notation, the 4H polytype is [+ − −+]n and the two stacking fault sequences are [+ − − + + − + − + − − +] corresponding to ABAC ABAB ABAC and [+ − − + − + − + + − − +] corresponding to ABAC ACAC ABAC, which implies that the two are equivalent. Henceforth, we therefore consider only the stacking fault with the ABAC ABAB ABAC sequence. The stacking fault energy, γs (T = 0), associated with this planar defect at 0 K is ≈3.3 mJ m−2 . All the Brillouin zone (BZ) integrations were carried out on a Monkhorst Pack [22] mesh of 9 × 9 × 3 and 9 × 9 × 1 of k-points for pristine and faulted structures respectively. Such a choice was made after verifying that a larger mesh of k-points, e.g. 15 × 15 × 5, has no significant impact on the results obtained. For example, the energies of the 4H polytype with respect to the 3C polytype obtained with 9 × 9 × 3 and 15 × 15 × 5 k-meshes were 1.3 meV/atom and 1.5 meV/atom, respectively. Not only do the energy differences converge within ≈0.2 meV/atom with respect to k-mesh sampling, we also find no significant change in the electronic Fermi surface and band structure of the 4H polytype calculated with a 15 × 15 × 5 mesh. This justifies our use of a 9 × 9 × 3 mesh of k-points sampling the BZ integrations for the pristine structures. In the case of the faulted structures, we use a 9 × 9 × 1 mesh because the corresponding Brillouin zone is shorter by a factor of three along the c-direction. While the 4H polytype is stable against formation of stacking faults at 0 K, the low stacking fault energy suggests that wide faults should be present in the 4H polytype. We now explore a possibility of stabilization of the stacking fault in the 4H polytype due to vibrational free energy (Fvib ) at finite temperatures. The vibrational free energy is calculated as [18]: }ωqi kB T X X Fvib (T ) = log 2 sinh , (4) Nq q 2kB T
Figure 6. (a) Raman spectra of 4H-Ag films on porous alumina substrates recorded at three temperatures, showing the doubly degenerate transverse optic mode of 4H-Ag that appears at 64.3 cm−1 at 4 K. (b) Mode softening as the 4H → 3C transition temperature is approached.
appearance of new soft phonon modes [18]. Particle size also appears to play an important role in the 4H → 3C phase transition, which proceeds through extensive microstructural changes and grain growth. We now discuss the role of stacking faults and particle size in the 4H → 3C phase transition. 4.1. Calculation of stacking fault energy
Our calculations are based on density functional theory (DFT) as implemented in the Quantum Espresso code [19] with a plane wave basis set, and ultrasoft pseudopotentials [20] to represent the interaction between ionic cores and valence electrons. Exchange–correlation energy of the electrons is treated with a generalized gradient approximated (GGA) functional of the Perdew–Burke–Ernzerhof parametrized form [21]. An energy cutoff of 40 Ryd on the plane wave basis was used in representation of wavefunctions and a grid with a cutoff of 320 Ryd in representation of the density. Stacking faults were introduced in a supercell of 1 × 1 × 3 dimensions with a stacking sequence of ABAC ABAB ABAC. In the 4H polytype with ABAC stacking sequence, the only possible stacking faults have sequences ABAC ABAB ABAC and
i
where kB , h¯ and T are the Boltzmann constant, Planck’s constant and temperature respectively. Nq denotes the number 6
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3C to 4H polytype. With the available data at T = 0 and P = 0, our conjecture for a P–T phase diagram for the 3C and 4H polytypes of silver is shown in figure 8(b). 4.2. Role of particle size
Our crystallographic and microstructural data are consistent with the conjecture that the 4H → 3C transformation in Ag is essentially size-driven. So far, the 4H-Ag phase has mainly been observed in confined geometries, such as nanocrystals [6], nanorods [7, 23] or nanocrystalline films (this work), lending credence to the belief that 4H-Ag is a size-stabilized phase [24] such as γ -Fe2 O3 , γ -Al2 O3 , etc. In fact, Liu et al [25] have shown both theoretically and experimentally that in nanowires, the 4H-Ag phase is energetically preferred in the diameter range of 10–50 nm, with 25.5 nm being energetically the most favorable. We have earlier emphasized the role of slow growth kinetics in suitably controlling the particle size and stabilizing the higher polytypes of electrochemically grown silver [7]. We further note that though DSC data indicate that the 4H → 3C transformation is a sharp, first-order phase transition with an onset at or above 155 ◦ C, all other data (crystallographic, microstructural, vibrational and transport) suggest a broad transition with a much lower onset. A possible explanation for this is to assume the existence of a critical grain size below (above) which the 4H (3C) phase is more stable. Thermally induced grain growth is essentially a heterogeneous process and there is a probability that a few grains grow beyond the critical size at temperatures much below the ‘sharp’ transition, and these 3C nuclei eventually catalyze a more rapid transition at a higher temperature.
Figure 8. (a) Temperature dependence of the stacking fault energy of the 4H-Ag polytype. Red (dashed) and black (solid) curves denote γs (T ) and 1Fvib (T ), respectively. Note that the vibrational contribution to the free energy at 0 K is higher than that of the stacking fault energy. (b) Pressure–temperature phase diagram for 3C and 4H phases of silver. The 4H structure transforms to 3C under 3.4 GPa pressure at 0 K.
5. Conclusions
of wavevectors in the Brillouin zone (BZ) at which the frequencies are calculated, q is the wavevector, i denotes the mode of vibration (i = 1 to 3Na , Na = number of atoms), ωqi corresponds to the phonon frequency at wavevector q and the ith mode of vibration. In the calculation of free energy, we have omitted the zero frequency acoustic modes at the 0-point. We estimate the variation of stacking fault energy as a function of temperature, using γs (T ) = γs (0) + 1Fvib (T ), where 1Fvib (T ) is the difference between the vibrational free energy of faulted and pristine structures obtained using equation (4). As T → 0, we observe that 1Fvib (T ) (≈5 mJ m−2 ) is larger than γs (0), i.e. the vibrational free energy also does not stabilize the faulted structure at 0 K. At finite temperatures, the energy of the stacking fault continues to increase with temperature (figure 8(a)). In the temperature range of the phase transformation (433–443 K), the stacking fault energy is ≈0.039 J m−2 , implying that the faults in the 4H polytype become narrower with temperature. Since 4H is a much more faulted structure than 3C, this implies that the 3C structure is favored over the 4H structure at higher temperature. We have also studied the possibility of a structural phase transition occurring due to application of external pressure. We estimate a critical pressure of 3.4 GPa (at 0 K) for the system to undergo a phase transition from the
The irreversible phase transition from the metastable, hexagonal 4H polytype of silver to the face-centered cubic 3C polytype was studied using x-ray diffraction, electron microscopy, electrical transport, Raman spectroscopy and differential scanning calorimetry. Calorimetric data suggest an exothermic, first-order transition, with an onset at ≈156 ◦ C for the slowest heating rate. However, a more gradual but extensive microstructural change and grain growth is initiated at much lower temperatures. In fact, the resistance starts to drop from as low as 60 ◦ C. The enthalpy of transformation (313 J g−1 ) and the activation energy (103 kJ mol−1 ) were obtained from a kinetic analysis of the calorimetric data. Analysis of isothermal DSC data suggests that the phase transition is mainly of an autocatalytic type. Raman spectroscopic measurements at different temperatures tell us that the TO mode of 4H-Ag (which appears at 64 cm−1 at 4 K) exhibits a definite mode softening as the transition temperature is approached. A first-principles density functional theory calculation shows us that the stacking fault energy for the 4H structure becomes increasingly positive with rising temperature. Since the 4H has a much more faulted structure than 3C, this suggests that the stability of 4H decreases at higher temperatures. Energetically, the 4H phase lies in between the highly anisotropic 2H form of Ag (unstable at ambient temperature) and the stable, cubic 3C form. 7
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The nature of the structural phase transition from the hexagonal (4H) phase to the cubic (3C) phase of silver
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 115405 (http://iopscience.iop.org/0953-8984/26/11/115405) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 15/06/2017 at 17:03 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115405 (8pp)
doi:10.1088/0953-8984/26/11/115405
The nature of the structural phase transition from the hexagonal (4H) phase to the cubic (3C) phase of silver Indrani Chakraborty1 , Sharmila N Shirodkar2 , Smita Gohil1 , Umesh V Waghmare2 and Pushan Ayyub1 1
Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai 400005, India 2 Theoretical Science Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560064, India E-mail: [email protected] Received 31 July 2013, revised 19 January 2014 Accepted for publication 21 January 2014 Published 3 March 2014
Abstract
The phase transition from the hexagonal 4H polytype of silver to the commonly known 3C (fcc) phase was studied in detail using x-ray diffraction, electron microscopy, differential scanning calorimetry and Raman spectroscopy. The phase transition is irreversible and accompanied by extensive microstructural changes and grain growth. Detailed scanning and isothermal calorimetric analysis suggests that it is an autocatalytic transformation. Though the calorimetric data suggest an exothermic first-order phase transition with an onset at 155.6 ◦ C (for a heating rate of 2 K min−1 ) and a latent heat of 312.9 J g−1 , the microstructure and the electrical resistance appear to change gradually from much lower temperatures. The 4H phase shows a Raman active mode at 64.3 cm−1 (at 4 K) that undergoes mode softening as the 4H → 3C transformation temperature is approached. A first-principles density functional theory calculation shows that the stacking fault energy of 4H-Ag increases monotonically with temperature. That 4H-Ag has a higher density of stacking faults than 3C-Ag, implies the metastability of the former at higher temperatures. Energetically, the 4H phase is intermediate between the hexagonal 2H phase and the 3C ground state, as indicated by the spontaneous transformation of the 2H to the 4H phase at −4 ◦ C. Our data appear to indicate that the 4H-Ag phase is stabilized at reduced dimensions and thermally induced grain growth is probably responsible for triggering the irreversible transformation to cubic Ag. Keywords: structural phase transition, transformation kinetics, polytypes of silver, density functional theory, stacking fault energy, size-driven transition (Some figures may appear in colour only in the online journal)
1. Introduction
polytypes can coexist in the same thermodynamic regime. Even though extensive accounts on polytypism and polytypic phase transitions are available in the literature [1–3], the precise nature of such transitions and the origin of such large periodicities in polytypes remain debatable. Polytypic phase transitions may be reversible (e.g. the 2H → 12R transition in PbI2 [4]), or irreversible (e.g. the 2H → 3C transition in SiC [5]). The one generic feature that appears to link all
Polytypes are crystallographic modifications of the same substance, in which two dimensions of the unit cell are similar but the third is a variable integral multiple of a common unit [1]. Polytypes differ from polymorphs (or allotropes) in that they are usually not represented by definite thermodynamic phases with specific domains of temperature and pressure: multiple 0953-8984/14/115405+08$33.00
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polytypes is the presence of stacking faults [3]: it is empirically evident [1] that polytypes of a given system are formed by the creation and periodic arrangement of stacking faults in the corresponding basis structure. The periodicity and symmetry of a polytype are determined by the nature and frequency of the stacking faults. The present study is motivated by the fact that much of the previous work on polytypic phase transitions refers to compounds and alloys, and there is little understanding of polytypism and inter-polytype transitions in elemental metals. Silver is commonly known to have a face-centered cubic (fcc) structure, which can be visualized as a periodic stacking of hexagonally close-packed (hcp) atomic layers along the cubic [111] axis. The various polytypes differ from each other merely in the periodicity of the stacking. The stacking sequence of the hcp layers in 3C-Ag (fcc) is ABCABC. . . along the [111] axis, while it is ABCBABCB. . . for 4H-Ag and ABAB. . . for 2HAg. We earlier observed traces of the hexagonal 4H polytype in sputter-deposited nanocrystalline silver films [6], and later reported the growth of phase-pure 4H and 2H polytypes as thick films as well as nanorods [7]. We have recently reported the optical, electronic, vibrational and mechanical properties of the anisotropic, hexagonal (4H) form of silver [8]. Here, we present a detailed study of the irreversible phase transformation from the hexagonal 4H form to the cubic 3C form of silver and suggest that the 4H polytype appears to be a size-stabilized phase.
Figure 1. X-ray diffraction patterns from as-deposited 4H-Ag and
from samples heated to progressively higher temperatures and cooled down to room temperature, showing a gradual phase transformation from the 4H to 3C over a relatively broad temperature range.
No. 411402] preferentially orientated along the [0004/1010] direction, with an interplanar spacing of 0.250 nm in this direction. The coherently diffracting crystallite size calculated using the Scherrer formula is 26 nm. Thermal evolution of the microstructure was investigated by heating parts of the same 4H-Ag film to progressively higher temperatures in a nitrogen atmosphere and holding for 10 min at each temperature. With increasing temperature, the (0004/1010) line of 4H gradually loses intensity in comparison to the (111) line of 3C, and is surpassed at 160 ◦ C. The Scherrer size corresponding to the completely transformed 3C-Ag film annealed at 400 ◦ C was >100 nm. The transformation is obviously irreversible, as the XRD patterns were recorded only after cooling down the samples to room temperature. The complex microstructure of the 4H phase consists of long, cylindrical entities that are arranged roughly parallel to each other in aggregated structures spanning a few µm (figure 2(a)). These cylindrical particles are themselves made up of 20–30 nm diameter crystallites stacked end to end (see inset). This size matches quite well with the Scherrer size calculated for the as-deposited 4H films. Figures 2(b)–(d) depict the evolution of this microstructure as it is heated to progressively higher temperatures, using the protocol described earlier. In the sample heated to 120 ◦ C (figure 2(b)), almost all the 30 nm particles have fused together to form larger cylindrical structures of about 40 nm diameter and 130–180 nm length (inset). These cylindrical particles are still arranged in roughly parallel aggregates. At 160 ◦ C (figure 2(c)), none of the cylindrical particles are visible, having fused into roughly equi-axed particles with an average diameter of 150 nm, which are loosely aggregated into 0.5–1.0 µm clusters. Finally, in films annealed at 400 ◦ C (figure 2(d)), we observe a relatively uniform, compact microstructure consisting of 1–2 µm grains of the pure 3C phase. Thus, both XRD and SEM results indicate that the 4H → 3C transformation clearly occurs through a process of aggregation and grain growth.
2. Experimental details
Single-phase 4H-Ag films were grown by a simple twoelectrode electrodeposition procedure. The substrates were commercially available porous anodic alumina (PAA) templates, on which was sputter-deposited a ≈200 nm thick film of cubic silver to act as the cathode. The porous sides of the templates were coated with an insulating layer to prevent infiltration within the pores. The electrolyte was a solution of 4 g AgNO3 , 4 g anhydrous citric acid and 1.5 g boric acid in 100 ml water; and the anode was a 99.9% pure silver rod of 1 cm diameter held 5 cm away from the cathode. Initially a 1 V pulse was applied for 4 s, followed by recurrent potential pulses of 200 mV for 5 s at 10 s intervals. The total deposition time for a 30 µm thick film was 4 h. The samples were characterized using a Philips X’Pert Pro Powder x-ray diffractometer and a Zeiss Ultra 55 Plus field emission scanning electron microscope (SEM). Differential scanning calorimetric (DSC) measurements were made with a Perkin-Elmer DSC 7. Raman spectra were recorded in the backscattering geometry using a Jobin–Yvon T-64000 triple grating spectrometer with the 647.1 nm Kr-laser line and a CCD detector. The effective spectral resolution obtained was 0.7 cm−1 . Low temperature measurements were made in a continuous flow liquid helium cryostat with quartz windows. 3. Results 3.1. Microstructural evolution
The x-ray diffraction (XRD) pattern for the as-deposited 4H-Ag sample (figure 1) shows single-phase 4H-Ag [PCPDF 2
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Figure 2. Plan view SEM images of 4H-Ag films heated to different temperatures and cooled down: (a) As-deposited 4H-Ag film shows needle-shaped crystals arranged in roughly parallel, micron-sized aggregates. Inset shows that each needle is actually a linear stack of 20–30 nm nanoparticles. (b) On heating the above sample to 120 ◦ C, the primary particles fuse together into longer (≈150 nm) cylindrical structures. (c) At 160 ◦ C, the acicular microstructure is totally replaced by roughly equi-axed, ≈150 nm particles of 3C-Ag. (d) Finally at 400 ◦ C, we observe only micron-sized particles of 3C-Ag.
Seen in conjunction with our earlier observation of 4H-Ag in sputter-deposited nanocrystalline Ag films with average crystallite size ≈30 nm [6], it is reasonable to presume that the 4H polytype of Ag is a size-stabilized phase. 3.2. Evolution of electrical resistance
In a companion study that has just been reported separately [8], we have shown that the 4H phase is much less metallic than the 3C phase, with an in-plane resistivity that is over two orders of magnitude higher. Thus, we can use resistivity as a probe for the 4H → 3C transformation. Figure 3 shows the temperature dependence of the four-probe electrical resistance of a 4H-Ag film on a PAA substrate heated at the rate of 3 K min−1 across the phase transition. The initial rise in the resistance up to 60 ◦ C reflects the normal metallic behavior of 4H-Ag. This is followed initially by a gradual fall up to ≈140◦ and then a sharp fall, ultimately flattening off at about 190 ◦ C. The resistance is nearly constant at higher temperatures and is about 24 times lower than the resistance of the as-deposited 4H film. The resistance curve does not necessarily represent the 4H:3C phase fraction, as the sharp drop probably has a percolative origin.
Figure 3. Variation of the resistance of a 4H-Ag film on a porous
alumina substrate as it is heated across the 4H → 3C transformation. 3.3. Thermal analysis
Differential scanning calorimetry (DSC) measurements were made on 8 µm thick 4H-Ag films grown on PAA. Since the phase transition parameters depend appreciably on the heating rate, we report data (see figure 4) for a series of 3
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Flynn–Wall–Ozawa equation [11, 12] simply states that for each value of α, −0.4567E a + A0 , RT i AE a − 2.315. where A = Log g(α)R Logβ =
Ti is the isoconversion temperature for a given value of α, and g(α) is a function that describes the nature of the conversion. Logβ versus 1/Ti was plotted for nine values of α, and E a calculated from the slope of each. Figure 5(c) shows that the activation energy falls linearly with the degree of conversion and clearly indicates an autocatalytic transformation. The activation energy obtained from the two methods is quite similar. An isothermal DSC scan of the 4H film recorded at 175 ◦ C, showed that the transformation is quite rapid and is basically completed within a couple of minutes. For the isothermal run, the α versus t plot was sigmoidal in nature. A plot of the time derivative of the fraction transformed, dα/dt, against α (figure 5(d)) shows that the transformation is at its fastest at α ≈ 40%, further indicating an autocatalyzed process. For an autocatalytic process, the Prout–Tompkins rate equation [13, 14] (also known as the two-variable Sest´ak– Berggren equation) is given by:
Figure 4. DSC scans for the 4H-Ag film on anodic alumina substrates for different rates of heating. The onset temperature of the 4H → 3C transition for each heating rate is indicated.
different rates ranging from 2 to 30 K min−1 . The welldefined exothermic peak in the DSC data indicates that the higher energy hexagonal 4H phase undergoes a strongly first-order phase transition in transforming irreversibly to the lower energy cubic 3C phase. The transition (onset) temperature varies from 155.6 ◦ C (at 2 K min−1 ) to 192.6 ◦ C (at 30 K min−1 ). The enthalpy of transition was obtained by dividing the area under the exothermic peak in the DSC data by the estimated mass of 4H-Ag in the film and using the instrumental calibration factor. The value of the enthalpy ranged from 312.9 J g−1 (at 2 K min−1 ) to 252.9 J g−1 (at 30 K min−1 ). As a consistency check, we also scanned a PAA substrate sputter-coated with bulk cubic Ag, which expectedly yielded a flat baseline in the same temperature range. We now proceed to a more detailed investigation of the transition kinetics. Firstly, we obtain the phase fraction, α (fraction of 4H transformed to 3C at a particular temperature T ) by computing the integral curve obtained from the DSC scan, normalized by the total area under the transition peak. Figure 5(a) shows the temperature dependence of α for different heating rates. The preferred techniques for performing kinetic analysis from DSC scans are the ‘isoconversion’ or ‘model-free’ methods, wherein the unknown form of the reaction model is temporarily eliminated by comparing measurements made at common values of α corresponding to different heating rates [9]. The two most useful isoconversionbased approaches for extracting the activation energy of a phase transition from DSC scans are due to Kissinger and Flynn–Wall–Ozawa. The Kissinger equation [10] is given by: Ln(Tp2 /β) = −Ln(A R/E a ) + E a /RTp .
(2)
dα = K (T )α m (1 − α)n , dt
(3)
where dα/dt is the rate of transformation at temperature T , K (T ) is the rate constant and (m + n) is the order of the reaction. It is clear from figure 5(d) that the dα/dt versus α data fits the Prout–Tompkins equation (3) perfectly in the range 0.1 ≤ α ≤ 0.7 with the rate constant, K = 20.38 min−1 , m = 1.02 and n = 1.93 being the fit parameters. The total reaction order is thus close to 3. However, the curve deviates markedly from the autocatalysis equation for α > 0.7. That the 4H → 3C transformation is autocatalytic, implies that the initial formation of 3C nuclei in the 4H matrix promotes further reaction due to the appearance of dislocations or cracks at the reaction interface (branching). Such imperfections act as nucleating centers for the growth of the 3C phase, thereby showing autocatalytic behavior. Termination occurs when the reaction begins to spread into material that has already transformed [15]. Once the reaction has spread sufficiently through the material, the autocatalytic behavior is replaced by normal nucleation and growth processes till the conversion is complete. 3.4. Vibrational spectroscopy
In its cubic form, silver is not Raman active, but the hexagonal (4H) modification exhibits at least one Raman active mode at ≈ 64.3 cm−1 at 4 K. In our earlier paper [7], we had presented a simulation of the phonon dispersion curves and Raman active modes for 4H-Ag from a first-principles density functional theory calculation. The simulation indicated that the observed mode is a doubly degenerate, transverse optic mode. Here, we report the evolution of the Raman mode as the temperature
(1)
Tp is the temperature at which maximum conversion occurs, β is the heating rate and E a is the activation energy, which can be obtained from a plot of Ln(Tp2 /β) versus 1/Tp . The activation energy for the 4H → 3C transition obtained from the Kissinger plot (figure 5(b)) is 103.2 kJ mol−1 . The 4
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Figure 5. (a) Plots of the fraction (α) of 4H transformed to 3C versus temperature, showing the change in transformation kinetics with
heating rate. (b) Kissinger plots for determining the activation energy of the 4H → 3C transformation. (c) Plot of the activation energy against the fraction transformed, following the Flynn–Wall–Ozawa method. (d) A plot of dα/dt against α obtained from the isothermal DSC scan at 175 ◦ C fits the autocatalytic rate equation perfectly in the range 0.006 ≤ α ≤ 0.7.
of the phonon dispersion revealed that it is only marginally stable, with an energy surface that is rather flat or weakly varying with respect to several modes [7]. A 2H sample kept at −4 ◦ C showed a gradual spontaneous transition from the 2H to the 4H phase over a time period of 1 month (figure 7), and on heating a transition from 4H to 3C. This implies that the 4H state is energetically intermediate between 2H and 3C, in agreement with our earlier DFT-based predictions.
is increased across the 4H → 3C transformation temperature. Figure 6(a) shows the background-subtracted, high-resolution Raman data recorded at three different temperatures. We observe that the Raman mode of 4H-Ag becomes broader and less intense as the transition temperature is approached. The mode frequency—determined from Lorentzian fits of the spectra—shows a small but distinct tendency to soften close to 350 K (figure 6(b)). However, it was difficult to track the peak position at temperatures closer to the transition. This type of mode softening is known to be associated with several classes of temperature- or pressure-driven structural phase transitions, such as the displacive-type ferroelectric–paraelectric phase transition, the transition from α-quartz to β-quartz, and so on [3].
4. Discussions
We now attempt to elucidate the mechanism controlling the 4H → 3C phase transition. Liang et al [16] have shown that intense electron beam irradiation converts 4H-Ag nanowires to the 3C form, and ascribed it to knock-on displacement of Ag atoms by electrons generating lattice defects. Ab initio calculations of stacking fault energies of various noble and transition metals and the energy differences between their hcp and fcc phases [17] have shown that Ag has the lowest stacking fault energy among the noble metals. As an example of the significance of stacking faults in polytypic phase transitions, we point out that the free energy difference between the pristine and faulted structures of 4H-SiC leads to the growth of stacking faults at temperatures above 60 K and the consequent
3.5. The 2H polytype
Other than the 4H polytype of Ag (which is stable at room temperature), we have reported the growth of a 2H polytype (metastable at room temperature) in the form of single crystalline films and nanorods by optimizing the electrochemical growth kinetics [7]. 2H-Ag spontaneously transforms to the stable 3C phase at room temperature over a period of days. Local stability analysis of the 2H structure through a simulation 5
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Figure 7. XRD patterns of: (top) a 2H-Ag film (with a minor
fraction of 3C-Ag and (bottom) the same film after spontaneous transformation to 4H-Ag after a period of one month.
ABAC ACAC ABAC. In Hagg notation, the 4H polytype is [+ − −+]n and the two stacking fault sequences are [+ − − + + − + − + − − +] corresponding to ABAC ABAB ABAC and [+ − − + − + − + + − − +] corresponding to ABAC ACAC ABAC, which implies that the two are equivalent. Henceforth, we therefore consider only the stacking fault with the ABAC ABAB ABAC sequence. The stacking fault energy, γs (T = 0), associated with this planar defect at 0 K is ≈3.3 mJ m−2 . All the Brillouin zone (BZ) integrations were carried out on a Monkhorst Pack [22] mesh of 9 × 9 × 3 and 9 × 9 × 1 of k-points for pristine and faulted structures respectively. Such a choice was made after verifying that a larger mesh of k-points, e.g. 15 × 15 × 5, has no significant impact on the results obtained. For example, the energies of the 4H polytype with respect to the 3C polytype obtained with 9 × 9 × 3 and 15 × 15 × 5 k-meshes were 1.3 meV/atom and 1.5 meV/atom, respectively. Not only do the energy differences converge within ≈0.2 meV/atom with respect to k-mesh sampling, we also find no significant change in the electronic Fermi surface and band structure of the 4H polytype calculated with a 15 × 15 × 5 mesh. This justifies our use of a 9 × 9 × 3 mesh of k-points sampling the BZ integrations for the pristine structures. In the case of the faulted structures, we use a 9 × 9 × 1 mesh because the corresponding Brillouin zone is shorter by a factor of three along the c-direction. While the 4H polytype is stable against formation of stacking faults at 0 K, the low stacking fault energy suggests that wide faults should be present in the 4H polytype. We now explore a possibility of stabilization of the stacking fault in the 4H polytype due to vibrational free energy (Fvib ) at finite temperatures. The vibrational free energy is calculated as [18]: }ωqi kB T X X Fvib (T ) = log 2 sinh , (4) Nq q 2kB T
Figure 6. (a) Raman spectra of 4H-Ag films on porous alumina substrates recorded at three temperatures, showing the doubly degenerate transverse optic mode of 4H-Ag that appears at 64.3 cm−1 at 4 K. (b) Mode softening as the 4H → 3C transition temperature is approached.
appearance of new soft phonon modes [18]. Particle size also appears to play an important role in the 4H → 3C phase transition, which proceeds through extensive microstructural changes and grain growth. We now discuss the role of stacking faults and particle size in the 4H → 3C phase transition. 4.1. Calculation of stacking fault energy
Our calculations are based on density functional theory (DFT) as implemented in the Quantum Espresso code [19] with a plane wave basis set, and ultrasoft pseudopotentials [20] to represent the interaction between ionic cores and valence electrons. Exchange–correlation energy of the electrons is treated with a generalized gradient approximated (GGA) functional of the Perdew–Burke–Ernzerhof parametrized form [21]. An energy cutoff of 40 Ryd on the plane wave basis was used in representation of wavefunctions and a grid with a cutoff of 320 Ryd in representation of the density. Stacking faults were introduced in a supercell of 1 × 1 × 3 dimensions with a stacking sequence of ABAC ABAB ABAC. In the 4H polytype with ABAC stacking sequence, the only possible stacking faults have sequences ABAC ABAB ABAC and
i
where kB , h¯ and T are the Boltzmann constant, Planck’s constant and temperature respectively. Nq denotes the number 6
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3C to 4H polytype. With the available data at T = 0 and P = 0, our conjecture for a P–T phase diagram for the 3C and 4H polytypes of silver is shown in figure 8(b). 4.2. Role of particle size
Our crystallographic and microstructural data are consistent with the conjecture that the 4H → 3C transformation in Ag is essentially size-driven. So far, the 4H-Ag phase has mainly been observed in confined geometries, such as nanocrystals [6], nanorods [7, 23] or nanocrystalline films (this work), lending credence to the belief that 4H-Ag is a size-stabilized phase [24] such as γ -Fe2 O3 , γ -Al2 O3 , etc. In fact, Liu et al [25] have shown both theoretically and experimentally that in nanowires, the 4H-Ag phase is energetically preferred in the diameter range of 10–50 nm, with 25.5 nm being energetically the most favorable. We have earlier emphasized the role of slow growth kinetics in suitably controlling the particle size and stabilizing the higher polytypes of electrochemically grown silver [7]. We further note that though DSC data indicate that the 4H → 3C transformation is a sharp, first-order phase transition with an onset at or above 155 ◦ C, all other data (crystallographic, microstructural, vibrational and transport) suggest a broad transition with a much lower onset. A possible explanation for this is to assume the existence of a critical grain size below (above) which the 4H (3C) phase is more stable. Thermally induced grain growth is essentially a heterogeneous process and there is a probability that a few grains grow beyond the critical size at temperatures much below the ‘sharp’ transition, and these 3C nuclei eventually catalyze a more rapid transition at a higher temperature.
Figure 8. (a) Temperature dependence of the stacking fault energy of the 4H-Ag polytype. Red (dashed) and black (solid) curves denote γs (T ) and 1Fvib (T ), respectively. Note that the vibrational contribution to the free energy at 0 K is higher than that of the stacking fault energy. (b) Pressure–temperature phase diagram for 3C and 4H phases of silver. The 4H structure transforms to 3C under 3.4 GPa pressure at 0 K.
5. Conclusions
of wavevectors in the Brillouin zone (BZ) at which the frequencies are calculated, q is the wavevector, i denotes the mode of vibration (i = 1 to 3Na , Na = number of atoms), ωqi corresponds to the phonon frequency at wavevector q and the ith mode of vibration. In the calculation of free energy, we have omitted the zero frequency acoustic modes at the 0-point. We estimate the variation of stacking fault energy as a function of temperature, using γs (T ) = γs (0) + 1Fvib (T ), where 1Fvib (T ) is the difference between the vibrational free energy of faulted and pristine structures obtained using equation (4). As T → 0, we observe that 1Fvib (T ) (≈5 mJ m−2 ) is larger than γs (0), i.e. the vibrational free energy also does not stabilize the faulted structure at 0 K. At finite temperatures, the energy of the stacking fault continues to increase with temperature (figure 8(a)). In the temperature range of the phase transformation (433–443 K), the stacking fault energy is ≈0.039 J m−2 , implying that the faults in the 4H polytype become narrower with temperature. Since 4H is a much more faulted structure than 3C, this implies that the 3C structure is favored over the 4H structure at higher temperature. We have also studied the possibility of a structural phase transition occurring due to application of external pressure. We estimate a critical pressure of 3.4 GPa (at 0 K) for the system to undergo a phase transition from the
The irreversible phase transition from the metastable, hexagonal 4H polytype of silver to the face-centered cubic 3C polytype was studied using x-ray diffraction, electron microscopy, electrical transport, Raman spectroscopy and differential scanning calorimetry. Calorimetric data suggest an exothermic, first-order transition, with an onset at ≈156 ◦ C for the slowest heating rate. However, a more gradual but extensive microstructural change and grain growth is initiated at much lower temperatures. In fact, the resistance starts to drop from as low as 60 ◦ C. The enthalpy of transformation (313 J g−1 ) and the activation energy (103 kJ mol−1 ) were obtained from a kinetic analysis of the calorimetric data. Analysis of isothermal DSC data suggests that the phase transition is mainly of an autocatalytic type. Raman spectroscopic measurements at different temperatures tell us that the TO mode of 4H-Ag (which appears at 64 cm−1 at 4 K) exhibits a definite mode softening as the transition temperature is approached. A first-principles density functional theory calculation shows us that the stacking fault energy for the 4H structure becomes increasingly positive with rising temperature. Since the 4H has a much more faulted structure than 3C, this suggests that the stability of 4H decreases at higher temperatures. Energetically, the 4H phase lies in between the highly anisotropic 2H form of Ag (unstable at ambient temperature) and the stable, cubic 3C form. 7
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