Atomic resolution imaging of HfO 2 nanorods

High aspect ratio, monoclinic hafnia nanorods were grown via a non-hydrolytic sol–gel synthesis, as described previously31. Unit cells for the three polymorphs of HfO 2 are shown in Fig. 1, illustrating the slight displacive distortions necessary to undergo a diffusionless martensitic transition from monoclinic to tetragonal to cubic on heating13,14. Figure 2 shows high angle annular dark field (HAADF) ‘Z-contrast’ imaging of the nanorod before heating, with accompanying simulated image (calculated with QSTEM software)32. The rods are monoclinic with multiple twin boundaries occurring along the (100) planes. These twins are believed to form to accommodate shear strain during the tetragonal to monoclinic phase transformation upon cooling during synthesis31. Upon increasing the length of the nanocrystals while keeping the diameter constant, the number of twins are monotonically increased31. The stabilization of such ferroelastic domains and coherent twin boundaries has previously been examined by dark-field TEM studies31. The nanorods represent excellent model systems since defects tend to migrate to surfaces and are annealed out during hot colloidal synthesis33,34. Indeed, no cation vacancies can be observed through atomic resolution imaging. The coherent twin boundaries visible in Fig. 2 are the primary imperfections discernible in these crystals. Measurements of bulk samples or even ensemble measurements of nanostructures furthermore cannot capture specific nucleation events given the large nucleation volumes and the facile propagation of transformations across the 3D material akin to twinning dislocations during transformation twinning. The dimensionally confined nanorods thus represent ideal model systems for examining discrete nucleation events and the propagation of the nucleated phase.

Figure 1: Crystal structures of HfO 2 polymorphs. The unit cell of three phases of HfO 2 , showing the small atomic displacements, and preservation of the coordination environments, during transformation between the phases. The monoclinic to tetragonal (a,b) and tetragonal to cubic (b,c) transitions occur at ∼1700 and ∼2600 °C, respectively, at atmospheric pressure. Arrows in (a) show displacements necessary to transform to the tetragonal phase (b). Hafnium atoms are shown as olive spheres, whereas oxygen atoms are shown as red spheres. Full size image

Figure 2: Z-contrast image of twinned monoclinic nanorod. (a) False-coloured HAADF image of a hafnia nanorod with twin domains imaged in the [110] zone coloured yellow and blue. Scale bar, 2 nm. (b) FFT of nanorod with twin domain spots masked in yellow and blue. (c,d) Magnified image and simulation, respectively, of a single twin boundary. The plane is mirrored about the (100) plane. The simulation provides a qualitative view of the expected contrast and is not intended to demonstrate quantitative contrast matching. Scale bar, 5 Å. Full size image

Monoclinic to tetragonal transition of an HfO 2 nanorod

In Fig. 3 we show an individual nanorod as it undergoes a structural change during heating in situ in the STEM (Supplementary Movie 1). The nanorod in this figure was stepped from room temperature to 600 °C at a reported rate of 106 °C s−1 and held at this temperature for 1 h; this temperature was chosen in order to target the midpoint of the mixed phase region observed with in situ heating XRD31. The sample holder allows for rapid and homogeneous temperature equilibration and allows for the target temperature to be reached within a few seconds35. Such a setup thus allows for isothermal kinetics to be mapped without the substantial thermal gradients that can result in cracking of bulk samples. Other wires heated under similar conditions and showing similar structural changes are described in Supplementary Figs 1 and 2, though in several of these cases the crystallites within each wire are too small to clearly identify their crystal structures during transformation.

Figure 3: Structural phase transformation of a single hafnia nanorod heated in the STEM. Top: False-coloured HAADF images highlighting the structural phases present in each frame. (a) Before annealing, the nanorod is monoclinic. (b–l) Over a period of about 40 min the nanowire converts completely from single-crystalline twinned monoclinic to polycrystalline tetragonal hafnia with retention of the overall morphology. The transformation is nucleated at a surface defect that extends into a twin boundary. The nucleated tetragonal domain advances one discrete lattice plane at a time across two distinct twin boundaries, eliminating the boundaries in the process. Conservation of a one-to-one lattice correspondence across the phase boundary between advancing tetragonal and receding monoclinic phases suggests the propagation of a transformation dislocation. (m–r) Upon cooling, the nanorod is reduced from tetragonal hafnia to hafnium metal. Scale bar, 5 nm. Full size image

The high resolution of the images collected allows us to observe the structural changes directly. To estimate the volume of each phase contained within the wire in Fig. 3 as a function of time, we used a fast Fourier transform (FFT) approach. For each frame, an FFT of the nanorod was acquired, and the spots in this pattern were identified to belong to one or more specific hafnium oxide phase. So identified, the spots were masked, and an inverse FFT (IFFT) was used to regenerate those parts of the image containing the selected phase. Using this technique, we are able to map out the different phases (and their crystallographic zone axes) present in the nanorod over time, directly observing the phase transformation as it occurred. An unknown phase appears transiently (Fig. 3, blue regions), and is neither monoclinic nor tetragonal, but is most similar to the orthorhombic hafnia phase19, although it cannot be identified unambiguously. We speculate that the orthorhombic hafnia (or a distorted version of it) might provide a transition pathway for the monoclinic to tetragonal phase transformation for some orientation relationships but a direct monoclinic–tetragonal transformation appears to be the predominantly observed pathway as further discussed below.

Figure 4 demonstrates this analytical approach for Fig. 3f, chosen as an example. In this panel, collected shortly after two tetragonal regions have nucleated, the FFT/IFFT approach reveals a single-crystalline region of the unchanged monoclinic [110] zone, and two small regions of tetragonal HfO 2 , both oriented along the tetragonal [111] axis (rotated around [111] with respect to one another). A similar analysis was performed on the FFT of each captured frame, in each case identifying the phases present, and their orientations. Supplementary Fig. 3 shows representative FFT analyses of Fig. 3a,f,i,n,r, and a more detailed description of the phase identification procedure is outlined in Supplementary Methods.

Figure 4: A single frame captured during nanorod heating. (a) HAADF image, Scale bar, 5 nm. (b) False-coloured IFFT of the monoclinic phase of HfO 2 . (c) False-coloured IFFT of the tetragonal phase of HfO 2 . (d) Data from (b–c) overlaid onto (a). (e) FFT of (a) with monoclinic and tetragonal spots circled. Full size image

Figure 3 shows clearly that upon heating to 600 °C, nucleation of the tetragonal phase is visible after 20 min. The phase change appears to be defect-nucleated, beginning at a surface defect that extends into one of the previously identified twin boundaries. Using the FFT analysis (Fig. 4) and atomic resolution imaging, we are able to identify the emerging phase as tetragonal HfO 2 , and over a period of about 40 min the nanowire converts completely from single-crystalline twinned monoclinic to polycrystalline tetragonal hafnia, preserving much of the nanorod’s overall morphology. Notably, no migration of oxygen vacancies is observed in this regime.

The tetragonal phase must first be nucleated to initiate the phase transformation and it is clear that the nucleation event in Fig. 3 is associated with the twin planes, which correspond to a relatively higher free energy starting point as compared to the interior of an untwinned monoclinic domain. Indeed, Supplementary Fig. 2 provides another example of nucleation of the tetragonal phase in close proximity to a coherent twin boundary at the bottom end of a particle. Both of these images permit examination of discrete nucleation events and the identification of the nucleation sites, which would otherwise not be possible in extended solids with a large free volume available for nucleating phase transformations. While nucleation phenomena have not been extensively studied in these systems, examination of the displacive monoclinic to tetragonal phase transition of a related binary oxide, VO 2 , suggests a pronounced hysteresis between the forward and reverse transitions since the forward monoclinic→tetragonal transition can readily be nucleated at M1/M2 (two distinct monoclinic polymorphs) phase boundaries and at twin boundaries, whereas the reverse tetragonal→monoclinic transition has to be nucleated at point defects in the absence of high-energy twin planes36,37. Even within these systems, nucleation phenomena have been surmised from the behaviour of strained and defective materials and direct observation remains elusive. In these materials, dimensional confinement reduces the volume density of point defects and thus results in kinetic stabilization of high-temperature phases38.

Close inspection of the FFT in Fig. 4 shows close to overlapping pairs of monoclinic (green) and tetragonal (red) spots, indicating sets of parallel planes with similar d-spacings from the two phases, suggesting likely candidates for the monoclinic–tetragonal phase boundary. One such pair are {011} t /{111} m , which forms the lower bounding interface of the larger of the two tetragonal regions, presumably at the location of a transformation dislocation39. Supplementary Fig. 4 highlights the growth of this tetragonal domain during the annealing process. As the dislocation migrates, the {011} t /{111} m phase boundary (highlighted in blue) grows (from 0 to ∼840 s), until it changes orientation over the period of ∼840 to 1320, s. An atomistic model of the {011} t /{111} m interface (Supplementary Fig. 5) illustrates that this boundary is coherent, consistent with a model of migrating transformation dislocation.

Considering Fig. 3a–d, a clear faceted crystallographic relationship is observable across the phase boundary (monoclinic in green and tetragonal in red) with (111) and (111)* planes of the monoclinic phase interfaced with (011) planes of the tetragonal phase. The incipient tetragonal domain advances simultaneously across two distinct twin boundaries that are separated by three lattice planes, eliminating the boundaries in the process since the higher symmetry of the tetragonal phase does not permit twinning. The energetic costs for propagation of the tetragonal phase and for having a phase boundary are thus clearly offset in part by elimination of the interfacial energy at the twinned interface. Notably, the propagation of the tetragonal phase occurs one discrete lattice plane at a time (contrasting Fig. 3c,d,f) with conservation of a one-to-one lattice correspondence across the phase boundary (illustrated in Supplementary Fig. 5); the conservative motion and lattice correspondence suggest that deformation is induced via a transformation dislocation39,40. By Fig. 3h, the twin planes have been eliminated and conservative motion of this domain can no longer be continued without a change of lattice plane. The strain resulting from partial deformation of the lattice likely contributes to further nucleation events.

An evaluation of the energetics of the phase transformation

Figure 3 indicates that the transition temperature is depressed by over 1000 °C as compared to the bulk and exhibits kinetics that are strongly atypical for a displacive transition, suggesting a pronounced modification of both thermodynamic stabilities and activation barriers. Considering the former, the change in free energy (ΔG) across the M→T transformation can be written as

where and values are the chemical free energies of the monoclinic and tetragonal phases (dependent on temperature), respectively; the and values are the strain energies for the monoclinic and tetragonal phases, and the and values are the surface free energies for the monoclinic and tetragonal phases41. The calculated temperature dependencies for the bulk free energy, enthalpy, entropy and specific heat capacity, as determined from density functional theory (DFT) calculations, are shown in Supplementary Fig. 6. At the transition temperature, thermodynamics requires . At a temperature of 600 °C, the chemical free energy difference, for the M→T transformation is expected to be strongly positive and as per the energetics of the bulk phase diagram (Supplementary Fig. 6), the monoclinic phase ought to remain stable in preference to the tetragonal phase. This implies that the differentials in strain energy and surface free energy underpin the observed strong depression of the transition temperature. Supplementary Fig. 7 identifies the surface planes stabilized for the monoclinic and tetragonal phases of HfO 2 . With the exception of a few planes in immediate proximity of the twin planes, the side surfaces of monoclinic HfO 2 are {110} planes with {001} planes binding the ends. In addition to these planes, transformation to the tetragonal phase further exposes {010}, {101} and {112} planes. DFT calculations of surface energies by Ramprasad and co-workers19 suggest that the surface energies of {110} and {001} planes are substantially higher for the monoclinic phase (1.38 and 1.51 J m−2, respectively) as compared to the values for the tetragonal phase (1.08 and 1.21 J m−2, respectively). In other words, as also observed for ZrO 2 (refs 9, 42), the tetragonal phase of HfO 2 has a substantially lower surface energy and thus . This term is furthermore expected to be strongly size-dependent as per:

where the γ T and γ M terms represent the interfacial surface energies of the tetragonal and monoclinic phases, respectively, D is the diameter of the particle and g s =A M /A T the ratio of the interfacial surface areas with the subscripts denoting the values for the monoclinic and tetragonal polymorphs41. Clearly Equations (1) and (2) together predict that the transition temperature will be directly proportional to the diameter of the particle. Ramprasad and co-workers have predicted extended phase stabilities of the tetragonal phase at finite dimensions as indeed observed here and for ultra-small nanocrystals wherein the tetragonal phase can be stabilized at room temperature9,19 based on surface energy considerations. While such surface energy effects have been noted for other systems as well42, the effect of twinning is no less important. The plastic deformation that induces the twin planes shown in Fig. 2 partially alleviates the strain induced upon deformation of a tetragonal particle (the T→M transition) during synthesis. As a result of the energy stored in these deformations that can in turn be dissipated during the M→T transformation, the differential in strain energy further contributes to driving down the transition temperature. Lange41 has developed a size-dependent expression for the energy of the twin surface per unit volume that equates to where γ twin is the twinning energy per unit area, D is the particle diameter and g twin is a dimensionless quantity that can be expressed as , where A twin is the total area of the twin boundaries. Figure 5 plots the energy landscape and activation energy barrier calculated for the transformation from monoclinic to tetragonal HfO 2 as deduced for bulk HfO 2 using DFT calculations. The activation energy barrier for the transformation is deduced to be ca. 207.63 meV atom−1. Twinned regions correspond to relatively high free energy regions of the energy landscape (raised by the interfacial energy of the twin boundaries). Consequently, the activation energy for transformations initiated at twinned domains will be lower. Indeed, Fig. 3 and Supplementary Fig. 2 both suggest that these regions serve as the nucleation sites for initiating the phase transformation. In other words, the high local density of twins and the small particle size both contribute to the >1000 °C depression of the transition temperature from the bulk values and the strongly modified phase equilibrium observed at the nanoscale.

Figure 5: Energy landscape and activation energy barrier for HfO 2 . (a) Energy landscape of the monoclinic to tetragonal phase transformation of HfO 2 (without twin planes) calculated using two parameters: (i) change in lattice and (ii) change in internal coordinates. Here, (0,0) corresponds to the rutile/tetragonal phase and (1,1) corresponds to the monoclinic phase. Calculations were carried out at grid points corresponding to a 9 × 9 grid, which were then interpolated to construct the landscape. A steepest descent algorithm was employed to find the minimum energy path (MEP) of the transformation. Colour bar in meV per atom. (b) Energy barrier for the monoclinic to tetragonal phase transformation. Full size image

Kinetics of the phase transformation

Interestingly the kinetics of this phase transformation are also not typical of the martensitic and athermal processes known to occur in bulk HfO 2 13,31,43,44. In the bulk, the transformation mechanism is a diffusionless process, in which bond angles and distances rearrange without disrupting atomic connectivity, and this mechanism should therefore produce an abrupt change in lattice parameters that propagates across the entire crystal44. Furthermore, the athermal nature of the bulk process suggests that a change in temperature is required for transformation13. Conversely to bulk observations, we observe a relatively slow transformation under isothermal conditions with the rate determining step apparently being the propagation of the transformation dislocation from one lattice plane to another. To investigate the transformation kinetics further we quantified the fraction of the nanorod that had transformed from the monoclinic to the tetragonal phase and applied the Johnson–Mehl–Avrami–Kolmogoroff (JMAK) model to infer the mechanism of phase transformation kinetics. The JMAK model describes thermally activated nucleation and growth kinetics of phase transformations:

where f is the fraction of transformed volume, t is time, k is a constant and n is an integer or half-integer. The exponent n is the Avrami exponent, and describes the rate and geometry of nucleation and growth45,46,47. Within this model, n can be expressed as

where a is the time-dependent nucleation rate, b is the dimensionality of the growing crystals and c is the growth rate47. The value of a gives a measure of nucleation rate; when a=0, no nucleation is observed, whereas an a value of 1 suggests a constant nucleation rate. When 0<a<1, the nucleation rate is decreasing, and a>1 suggests an increasing nucleation rate. The value of b is typically either 1, 2 or 3, corresponding to the dimensionality of phase growth, and c is either 1 or 0.5, corresponding to volume diffusion controlled growth and interface movement controlled growth, respectively47. When the fraction of the transformed phase is plotted versus time, it produces a sigmoidal curve characteristic of nucleation and growth kinetics described by the JMAK model, as shown in Fig. 6a. The data can be further plotted as (ln(1−x)) versus ln(t), where x is the fraction of phase transformed, which produces a straight line with slope equal to n and a y-intercept of k (Fig. 6b). The plot in Fig. 6b yields n=4.38. Because we directly observe interface movement growth (c=0.5) in Fig. 3 and Supplementary Fig. 4, and we know the nanorods are one-dimensional and can therefore assume one-dimensional phase growth (b=1), a value a>1 can be deduced, indicating an accelerating and autocatalytic nucleation rate. This is consistent with our observation, and the data strongly suggest that these nanorods undergo classical autocatalytic nucleation and growth kinetics. Based on Fig. 3, the observed motion of the transformation dislocation is responsible for the initial sigmoidal state and likely constitutes the rate-limiting step of this transformation. Importantly, this implies that through size-confinement, we are able to alter the phase transformation kinetics from a diffusionless mechanism to a nucleation and growth mechanism, potentially allowing for the high-temperature tetragonal phase to be quenched to room temperature if an activation energy barrier of sufficient magnitude can be induced.

Figure 6: JMAK kinetics plots. (a) Fraction of transformed phase versus time, displaying the sigmoidal curve characteristic of nucleation and growth phase transformation kinetics. (b) Linear plot to determine the value of n (slope). Full size image

Reduction to Hf metal upon cooling

Once the monoclinic to tetragonal transformation was complete, we cooled the nanorod over a period of 160 min (at 0.015 °C s−1) to 456 °C (Supplementary Movie 2), to observe any possible structural rearrangements and their kinetics. Interestingly, rather than forming a metastable tetragonal hafnia, or returning to a monoclinic hafnia phase, the Hf–O bonds in the nanorods are seen to distort, and the nanorod then proceeds to lose oxygen, reducing abruptly to hafnium metal at 530 °C, as evidenced by electron energy loss spectroscopy displayed in Fig. 7.

Figure 7: Electron energy loss spectroscopy from two hafnia nanorods before and after heating. (a,c) HAADF image and EEL spectrum, respectively, of a nanorod before heating. The oxygen K-edge is present. (b,d) HAADF image and EEL spectrum, respectively, of a different wire after heating and cooling. No oxygen K-edge is present, indicating the reduction of the rod to hafnium metal. Both scale bars are 5 nm. Full size image

The distortion of the tetragonal phase on cooling is most likely due to oxygen loss, driven by the low oxygen fugacity in the microscope column, and can be measured in our nanorod using the and lattice spacings (Supplementary Fig. 8). Indeed, temperature variant XRD measurements performed by heating the samples up to 900 °C and then back down to room temperature do not show any indications of oxygen loss or stabilization of metallic Hf (Supplementary Fig. 9). FFTs were acquired from frames of the tetragonal nanorods during cooling, and the distances to the and spots were measured. The d-spacing values change by 5% from the native lattice spacings—expanding in the direction and contracting in the direction—before the crystal undergoes transformation to hafnium metal. This gives a quantitative measure of the extent of hafnium reduction which may be tolerated before the zero valent phase is stabilized.