Unveiling the nucleation and coarsening mechanisms of solution-derived self-assembled epitaxial Ce 0.9 Gd 0.1 O 2-y nanostructures

Self-assembling approaches based on chemical solution deposition (CSD) are ideal methods for the cost-effective production of epitaxial nanostructures with high-throughput. Therefore, an in-depth investigation of the nucleation and coarsening processes involved in the self-assembling of nanostructures is mandatory to achieve a good control over nanostructure shape, dimensions and orientation. Heteroepitaxial Ce 0.9 Gd 0.1 O 2-y (CGO) is an ideal model system to unveil the underlying nanostructure development mechanisms in addition to their promising properties for catalysis, gas sensors and ionic


INTRODUCTION
Nanoscale materials have been investigated for many years due to the reduced dimensionality and new phenomena appearing at their interfaces.Thus, the smart engineering of nanomaterials is fundamental to design novel functional devices with good properties.Bottom-up approaches based on self-assembling and self-organization processes are extremely adequate for the fabrication of nanomaterials, either as thinfilms or nanostructures.They are also envisaged as alternative methods to substitute top-down techniques (e.g.lithography) achieving higher throughput, smaller dimensions and coverage of larger areas homogeneously [1][2][3][4][5][6][7][8] .Self-assembling has been a notably prolific topic for metals and semiconductors with significant contributions aiming to produce interfacial nanostructures through strain engineering 4,8,9 .
The study of the fundamental principles involved in self-assembling and self-organization of oxides, i.e. nucleation and coarsening, is essential for the fabrication of nanostructured devices with compelling functional properties that usually depend on the particular crystalline orientation [34][35][36][37][38] .These principles are of thermodynamic and kinetic origin, and act as driving forces enabling the growth of nanostructures.
However, a large controversy has been generated around the role played by these competing thermodynamic and kinetic mechanisms (e.g.strain relaxation, surface energies, nucleation barriers and atomic diffusion) in the self-assembling and self-organization of epitaxial nanostructures [39][40][41][42] .Essentially, the formation of thermodynamically stable nuclei occurs when they grow beyond a critical size overcoming a nucleation barrier.Atomic diffusion mechanisms also contribute to nucleation since they enable the transference of atoms between nuclei, but also the merge of nuclei as whole entities 43,44 .Once nuclei are stable, they grow through coarsening and coalescence which are thermally activated processes 45 .Eventually, nanostructures will reach a stable maximum size which is determined by thermodynamic factors such as the surface energy of island facets, the energy of the interface and strain relaxation mechanisms, but also when the diffusion of atoms to and from nanostructures reaches an equilibrium 46- 48 .It is also important to notice that the growth of nanostructures will also depend on the type of system investigated, i.e. an open system where deposition and kinetic evolution of the system occur simultaneously or a closed system where material deposition is finished before the kinetic evolution is enabled.
Chemical solution deposition (CSD) has appeared among the different processing techniques as an adaptable, low-cost and very powerful ex-situ approach for the bottom-up fabrication of functional oxide architectures with good stoichiometric control.For instance, it has been employed to successfully fabricate a broad range of oxides such as Pb(Zr,Ti)O3, (Ba,Sr)TiO3, LaNiO3, Ce1-xGdxO2-y (CGO), (La,Sr)MnO3, BiFeO3 and YBa2Cu3O7-x (YBCO) either as polycrystalline and epitaxial films, or in the form of interfacial nanostructures 10,35,38,[48][49][50][51][52][53][54][55] .Some attempts have been made to evaluate nucleation and coarsening in closed systems.It is worth mentioning the work done on YBCO and CGO films to determine nucleation barriers and the morphological evolution with the annealing temperature and time 38,56 , and on the study of CGO nanostructures to control the final island morphology, orientation and dimensions with the annealing temperature, time and ambient atmosphere 48,[57][58][59] .Nevertheless, these studies face some limitations since CSD usually uses tubular furnaces which are slow processes.The heating ramps below 0.5 ºC s -1 prevent the independent observation and analysis of nucleation mechanisms which usually depend on temperature and are also strongly linked with the kinetic evolution of the system (i.e.coarsening).Rapid Thermal Annealing (RTA) furnaces are more suitable tools to achieve a precise control over nucleation and growth since they can achieve very fast heating ramps of 20 ºC s -1 and above.Despite the promising advantages of RTA for their use in CSD processing, only few cases have been reported for purposes not related to nucleation and coarsening phenomena.For example, it has been employed in the production of multilayered functional oxide thin-film capacitors and electrodes for electronic devices 60,61 or to avoid the formation of non-desired phases during film growth 62 .Recently, we have shown that RTA can provide an accurate control over epitaxial film growth, and also we used it to investigate the mechanism of orientation-symmetry breaking of CGO nanowires 48,54,63 .
In this work, we will make use of strain-engineering by changing the single crystal substrate and its orientation to study the nucleation and coarsening mechanisms of different CSD-derived gadoliniumdoped ceria (Ce0.9Gd0.1O2-yor CGO) nanostructures as a model system.[72] These CSD nanostructures are archetypical examples of closed system growth.Hence, nucleation and coarsening are initiated upon ulterior thermal annealing which leads the system to evolve towards thermodynamic equilibrium.Although some works made some attempts to investigate the nucleation and coarsening of CGO nanostructures by means of tubular furnaces 57,58 , to our knowledge, this is the first time that an in-depth study of the coarsening of nanostructures in a closed system is conducted.Instead, all previous kinetic studies of nanostructures published until now have been performed on open systems 40, 73 .Particularly, we will grow isotropically-strained heteroepitaxial CGO nanodots (NDs) on (001)oriented Y2O3:ZrO2 (YSZ) and LaAlO3 (LAO) substrates, and CGO nanowires (NWs) on (011)LAO substrates.The nucleation and coarsening phenomena exhibited along in-plane and out-of-plane directions will depend on the particular orientation of nanostructures, as well as the presence of low energy facets that can sustain large amounts of oxygen vacancies such as the {111} orientation which has been widely investigate for applications in catalysis, gas sensors and ionic conductivity 36,[74][75][76][77][78] .Thus, we will perform a detailed study of the formation process of nanoislands from a metal-organic precursor film, and we will evaluate the influence of growth conditions such as the heating ramp, dwell temperature and annealing time on nucleation and coarsening.

EXPERIMENTAL SECTION
Heteroepitaxial Ce0.9Gd0.1O2-y(CGO) nanostructures are grown by Chemical Solution Deposition (CSD) methodology.Solutions with a 0.25 M concentration are prepared by mixing Cerium(III) and Gadolinium(III) acetylacetonates (Sigma-Adrich) in propionic acid.The amount of Ce and Gd are, respectively, 1.64 wt.% and 0.18 wt.%.The solutions are stirred and heated at 50 ºC for 30 min in order to dissolve the metalorganic salts and form cerium -gadolinium propionates.Ultra-diluted solutions with a metal-ion concentration of 0.008 M are prepared by adding proportional amounts of propionic acid.Any remaining impurities are removed by seeping the solution through a 0.2 µm filter.Solution deposition is done by spin coating at 6000 rpm, 3000 rpm•s -1 for 2 min on 5x5 mm 2 (001)Y2O3:ZrO2 (YSZ), (001)LaAlO3 (LAO) and (011)LAO single crystal substrates thoroughly cleaned by sonication in acetone and methanol.A thermal treatment at 900 ºC during 5 h under O2 is performed on the substrates before deposition to obtain atomically flat terraces.Afterwards, we perform a pyrolysis at 300 ºC during 10 min in O2 using a tubular furnace in order to completely eliminate the organic compounds and carbon impurities and obtain flat homogeneous nanocrystalline films 79 .Nucleation and kinetic evolution of CGO nanostructures is studied using a Rapid Thermal Annealing (RTA) furnace (AS-Micro from Annealsys) under controlled and stagnated oxygen atmosphere (filled at 5 l min -1 for 1 min), at temperatures ranging from 800 to 1100 ºC for variable annealing times (0 to 30 min), and heating ramps from 0.5 to 20 ºC s -1 .
The morphology of nanostructures is investigated by atomic force microscopy (AFM) with a 5100 system from Agilent Technologies.The characterization is done in intermittent contact mode using silicon tips of around 10 nm in diameter, and cantilevers with force constants around 2.8 Nm -1 .The topography measurements are analyzed with the MountainsMap 7.0 software from Digital Surf.
High resolution and scanning transmission electron microscopy (HRTEM and STEM) observations are carried out to investigate the crystalline structure, orientation, epitaxial relationship with the substrates, morphology and faceting of the nanostructures.Sample preparation is done by mechanical polishing and Ar-ion milling.TEM images are obtained by means of FEI Tecnai F20 and Jeol J2010F systems equipped with field emissions gun operated at 300 kV.Lateral resolutions achieved are of 0.14 nm.
Reflection high energy electron diffraction (RHEED) patterns are obtained with a Staib Instruments system using an electron beam of 30 kV, a current of 50 μA and beam incidence angles with respect to the sample surface below 2º.The investigated incident directions correspond to the <100> and <110> substrate orientations.The diffraction pattern has been simulated with a software based on the kinematic theory of electron scattering.

Theoretical formalism for the study of nucleation
The classical nucleation theory (CNT) is the simplest formalism that describes nucleation events thermodynamically.It states that small clusters of a new phase stabilize after they become large enough to overcome a thermodynamic barrier G*.The energy barrier is calculated by assuming a change in the Gibbs free energy ΔG of the system.In the case of CSD-derived nanostructures, the driving force governing nucleation is the reduction in free energy between the amorphous initial state and the final crystalline one 10 .The most common form of CGO nuclei, as we will see later, is a trapezoidal-prismatic shape (Figure 1) with a height h, width a and length b, and the lateral facets associated with the width and length are tilted at angles δ and θ, respectively.Then, the variation of the Gibbs free energy can be written as a function of the effective diameter S (= √ • ) 44,58,63,80 : where  S1) 59,[80][81][82] .Then, the nucleation barrier as a function of the supersaturation can be written as 48,58,63,80 : At high temperatures, i.e. small undercooling (ΔT=Tm-T), the supersaturation Δμ is usually described in terms of the latent heat of melting per unit volume Lm, the melting temperature Tm and  the unit cell volume 44 : The nucleation rate dN/dt is a kinetic parameter quantifying the number of nuclei that stabilize after overcoming the nucleation barrier.Typically, this process is thermally activated and also depends on atomic diffusion: The first term considers the effect of atomic mobility which is thermally activated.Qn is the activation energy for atomic diffusion, and kB is the Boltzmann constant.The second term quantifies the number of nuclei that are able to surpass the nucleation barrier.Typically, Qn is assumed to be constant, while G* has a strong dependence with temperature.At the high temperatures investigated the "energy barrier" term dominates the nucleation rate and, hence, it will be the only term considered in our description of nucleation.A more detailed explanation of nucleation and the different regimes can be found elsewhere 80 .

Figure 1.
General island shape considered for the theoretical analysis of nucleation.

Shape of nanostructures
The shape of nanostructures provides vital information for the study of nucleation and coarsening (i.e.island facets, lattice mismatch, etc).We have evaluated two different systems: CGO isomorphic islands (i.e.nanodots or NDs) grown on (001)YSZ and (001)LAO (Figure 2a,b), and CGO anisotropic parallel nanowires (NWs) grown on (011)LAO (Figure 2c).These AFM images show a narrow size distribution with average sizes of approximately 22.6±2.7 nm and 21.3±4.1 nm in diameter for CGO NDs on (001)YSZ and (001)LAO substrates, respectively, and 21.6±2.9nm in width and 166.7±41.9nm in length for CGO NWs on (011)LAO substrates.Figure 2d shows the HRTEM characterization of CGO NDs on YSZ.The NDs' shape consists of a truncated pyramid as it has been reported previously for different nanoislands including CGO 48,53,58,83,84 .The top face of NDs corresponds to the (001)CGO orientation and they grow cube-on-cube on top of the substrate with a relation (001)CGO [100]||(001)YSZ [100]   (Figure 2d).The orientation of the lateral facets in Figure 2d seems to coincide with (011) planes.However, CGO NDs grow on (001)YSZ with a square-shaped base rotated 45º from the <100> substrate axes, as opposed to (001)LAO where they grow parallel (Figure 2a and 2b).The TEM specimen was cut parallel to substrate axes, therefore, the lateral facets are the expected (111)CGO planes, i.e. the configuration with lowest surface energy (Table S1), as it is depicted in the solid-sphere models from Figure 2e and 2f 48,58,83 .Figure 2d also indicates that NDs are fully relaxed on top of (001)YSZ with a compressive lattice mismatch ε=(aYSZ-aCGO)/aCGO ~ -5% (aYSZ=5.145Å and aCGO=5.413Å). Figure S1 shows that CGO NDs on (001)LAO have the same island shape, i.e. a truncated pyramid with (111) lateral facets and a (001) top facet.However, for the present case, the CGO lattice is 45º rotated with respect to the LAO, and thus, the epitaxial relationship is (001)CGO[110]||(001)LAO [100], as it has been reported before 58 .The lattice mismatch is then ε=(aLAO-d( 110

Heating ramp and temperature dependence of nucleation
The fast heating ramps obtained with rapid thermal annealing (RTA) furnaces allow tuning of the growth conditions for the separate evaluation of nucleation and coarsening events.Figure S3 shows that our pyrolyzed CGO ultradiluted solutions at 300 ºC for 30 min on (001)YSZ, (001)LAO and (011)LAO substrates derive in very smooth films with RMS roughness below 0.5 nm. Figure S4 presents the RHEED measurements of pyrolyzed CGO films, CGO nanostructures and LAO substrates.These analyses reveal a diffuse halo characteristic of amorphous content (Figure S4a and d).We also observe some faint bright spots arranged on a half circle that can be associated to the substrates (Figure S4b and e) since the depth information is about 5 nm, and the precursor film is 2 nm-thick (Figure S5).Additionally, we see that the halo has some features resembling the typical diffraction patterns of epitaxial CGO nanostructures (Figure S4c and f) which may indicate the presence of nanocrystalline grains.A more detailed study of the local structure has been performed by HRTEM for precursor CGO films pyrolyzed on (001)YSZ, (001)LAO and (011)LAO substrates (Figure S5).We confirm the nucleation of either epitaxial/uniaxial or polycrystalline CGO grains embedded into a 2 nm amorphous matrix at temperatures as low as 300 ºC.
Schwartz and Schneller identified an amorphous metal-oxygen-metal structure in lead/titanium oxides, while the crystallization of other oxides such as barium/strontium titanates may present intermediate crystalline phases before reaching the final phase 10,49 .The structure observed for the CGO system is the final oxide phase which indicates that CGO has a large supersaturation and atomic mobility at low temperatures, and thus, the driving force is considerably large to allow crystallization from an amorphous phase.Roura et al. have also reported the nucleation of ceria thin films at very low temperatures right after decomposition, i.e. at 200 ºC when annealed for 1 h 79 .The study of nucleation of CGO nanostructures at high temperatures requires to minimize the appearance of CGO nuclei at low temperatures, thus, pyrolysis times of 10 min have been used.
High temperature nucleation studies of CGO NDs on (001)YSZ and (001)LAO, and CGO NWs on (011)LAO substrates have been conducted at 1000 ºC in O2, heating rates of 0.5-20 ºC s -1 and no dwell time (i.e.quench).The AFM images in Figures 3a-c reveal an increase of island dimensions as the heating rate decreases.Analyzing the data extracted from AFM images, we observe that the density of nanoislands (Figure 3d) decreases from approximately 2000 to 1000-500 islands nm -2 (NDs) and from 1500 to 500 islands nm -2 (NWs) by reducing the heating rate (from 20 to 0.5 ºC s -1 ), whereas the mean volume (Figure 3e) grows from 200-300 to 1200-1400 nm 3 (NDs) and from 800 to 4000 nm 3 (NWs).These phenomena suggest that coarsening has a large contribution on growth at slow heating rates.In addition, these results evidence that all studies about nucleation which have been mostly conducted with conventional thermal treatments (i.e.heating rates <0.5 ºC s -1 ) are strongly affected by coarsening.Therefore, heating rates of 20 ºC s -1 and above must be employed in order to perform an independent study of nucleation and coarsening.NWs also present a larger volume increase as compared to NDs which indicates larger coarsening.Figure S6 shows the evolution of the different contributions to the mean volume.Figures S6a   and S6b show that the equivalent diameter S increases from 10 to 20 nm when the heating ramp is slowed down from 20 to 0.5 ºC s -1 , whereas the height h grows only from 2.5 to 3.5 nm.This indicates a larger contribution of S to the volume increase.For the case of NWs (Figure S6c and d), we see that the height h and the width a have a similar evolution with the heating ramp to that of NDs.Instead, the length b experiences a fast increase in size from 40 to 90 nm which suggests a stronger contribution of the length to the rise in volume, also confirmed by calculating the lateral aspect ratio c (Figure S6d).We have demonstrated before that the growth of NWs is thermodynamically promoted along their length 48,57,63,80 .The non-linear growth of the volume can be related to a variation of the nucleation rate with temperature as we will see later.Despite that, it would be possible that some coarsening is still present.4a-c illustrate the growth of CGO nanoislands with temperature.Quenched samples are annealed on (001)YSZ, (001)LAO and (011)LAO substrates at 20 ºC s -1 in O2 and temperatures between 900 and 1100 ºC.These results confirm that temperature indeed has an influence over nucleation rates.This is illustrated more clearly in Figure 4d where the density of nanostructures experiences a notable decrease as the annealing temperature rises from 900 to 1100 ºC.We have fitted the density of nanoislands considering that the nucleation barrier G* is described by Equation 2 and 3, and that the nucleation rate is defined by the second term in Equation 4. The parameters used for data fitting are h=1.1 nm, v=0.158 nm 3 , Tm=2400 ºC and the surface energies reported in Table S1.Then, we can obtain the initial density of nuclei N0 and the latent heat of melting per unit volume Lm for each system.These values show that (001)CGO NDs grown on (001)LAO and (001)YSZ have similar latent heats, 4.8×10 9 and 5.0×10 9 J m -3 respectively, whereas N0 is around two orders of magnitude larger for CGO NDs grown on (001)LAO (4.2×10 6 islands μm -2 ) than for (001)YSZ (2.7×10 4 islands μm -2 ).For the case of (011)CGO NWs on (011)LAO, Lm is around three times lower than that of NDs (1.6×10 9 J m -3 ), whereas N0 is between three and five orders of magnitude larger (5×10 10 islands μm -2 ).It is worth noticing that the values of Lm are in agreement with those of other oxide materials such as La2O3, SiO2, Al2O3 85- 87 .The larger N0 for NDs grown on (001)LAO as compared to those on (001)YSZ could point out towards an enhanced driving force for nucleation, and an even bigger driving force for NWs grown on (011)LAO substrates.It could seem that a small variation of the experimental values leads to a large change in the results obtained.However, the derived parameters must have a physical meaning; the nucleation barrier must be positive and increase with temperature which is highly influenced by the value of Lm.For CGO NWs, we have assumed that the initial nuclei have a lateral aspect ratio c of 1.8-2.5;otherwise, no fitting is possible or G* has negative values.

AFM images in Figures
We can also calculate the nucleation barriers for each system by employing Equation 2 and the parameters from Table S1.Assuming that T=1000 ºC, we calculate that the nucleation barriers from the density of nuclei in Figure 4d  =(2.5±0.8)×10 -1 J.These values are very similar and between 3 and 18 times larger than the thermal energy of the system at 1000 ºC (kBT~1.4×10 -2 J).The lower nucleation barrier of CGO NDs on (001)YSZ of around one order of magnitude as compared to NDs on (001)LAO does not explain why there is a shift to higher temperatures (1000 ºC and above) in the nucleation of CGO on (001)YSZ when compared to the other systems (Figures 4a-c).We have mentioned before that nucleation rates decrease with temperature.Hence, we have performed a theoretical analysis of the parameters influencing the nucleation barriers.The calculation of nucleation barriers has been done without considering the contribution of strain and interface energies.The contribution of strain is easily addressed by including a term accounting for the misfit strain energy per unit volume (∆  ) in Equation 2 44,63 : CGO NDs grown on (001)YSZ substrates have a larger strain configuration than those grown on (001)LAO ( (001)CGO||(001)YSZ ~ -5% vs  (001)CGO||(001)LAO ~ -1%) 48 .Large values of strain will contribute increasing the nucleation barrier for a given supersaturation as reported before 44,63 .Therefore, the energy required to overcome the nucleation barrier and for the stabilization of nuclei will be higher which means that higher temperatures are needed.On the other hand, nucleation barriers are often calculated considering that the interface energy is negligible (γi=0 J m -2 in Equation 2).Despite that, the interface energy may also contribute shifting nucleation barriers.We can assume that the interface energy between CGO and YSZ, i.e. both have fluorite structures, will be much lower than CGO on LAO (fluorite/perovskite).Figure 4e presents the results of the calculation of nucleation barriers for CGO on (001)YSZ and (001)LAO substrates with interface energies from 0 J m -2 to 0.5 and 1 J m -2 , respectively.
We observe that the nucleation barrier of CGO on (001)YSZ is shifted to lower values of supersaturation (Δμ~ΔT=Tm-T) as compared to CGO on (001)LAO which supports our experimental results, i.e. the nucleation temperature must be higher.If we consider the contribution of interface energies (   =0.5 J m -2 and    =1 J m -2 ), we see that nucleation barriers are displaced to large values of supersaturation (lower temperatures).Given that the interface energy of CGO on (001)LAO must be much larger than on (001)YSZ, the nucleation barrier will be shifted to even higher values of Δμ.Hence, lower temperatures are required to nucleate CGO NDs on LAO substrates.
These results seem to confirm that we have successfully separate nucleation from coarsening.
Nevertheless, the contribution of coarsening cannot be completely discarded due to the large atomic diffusion at such high temperatures.In fact, Meixner et al. developed a theoretical work which suggests that kinetic mechanisms rule over thermodynamics in nucleation, thus, leading to an increase of nanostructure dimensions with temperature 39 .Figure S7 shows the variation of volume with the heating rate at 1000 ºC plotted together with the volume variation with temperature at 20 ºC s -1 for CGO NDs and NWs.The volume increase with temperature for CGO NDs may indicate that some coarsening is still present.Despite that, these values are smaller than the ones obtained using slower heating ramps (<10 ºC s -1 ).The only case where coarsening may have a large contribution is that of CGO NDs grown on (001)YSZ at temperatures above 1100 ºC (Figure S7a). Figure S7c shows that CGO NWs follow a trend equivalent to that of NDs.However, the influence of coarsening seems very important at temperatures above 1000 ºC, given that the mean volume of NWs is similar to that of heating ramps slower than 5 ºC s -1 .Therefore, we can consider that the influence of coarsening in nucleation is limited to high temperatures and slow heating rates.
3.2 Kinetic evolution of CGO nanostructures

Theoretical formalism for the study of coarsening
Crystal growth is a thermodynamically and kinetically governed phenomenon and it has been a long standing issue in epitaxial growth of nanostructures to unveil the role of thermodynamic versus kinetics 39, 40 .As we have described before, nucleation is a kinetically governed phenomenon with energy barriers of thermodynamic origin.Once nuclei are stabilized, they enlarge spontaneously at expenses of other nuclei leading to the uniform size characterizing self-assembly principles 45 .This is again a thermodynamic driven process governed by kinetic parameters such as atomic diffusion constants.This phenomenon is called Ostwald ripening.It is driven by the decrease in the energy of a system due to the reduction of the overall surface-to-volume ratio 88,89 ; hence, promoting the formation of large crystallites from small ones by atomic diffusion, also known as coarsening.It is however, possible that the nanostructures display anisotropic diffusion constants and so their shape aspect can also be determined by kinetic effects.Coalescence is another relevant growth mechanism which can be static, when two nuclei grow independently by Ostwald ripening and, eventually, merge into an individual entity 90 ; or dynamic when two nuclei merge after they diffuse into each other 91,92 .
The kinetic mechanisms of grain growth were described long time ago by Burke and Turnbull for polycrystalline materials in isothermal conditions.Assuming that energy minimization and the reduction of the surface area through atomic diffusion are the mechanisms driving grain boundary reconstruction, grain growth is described by a growth rate dS/dt with a general solution 93 : where S and S0 are the average and initial grain diameter, kn=nMγS n-2 is a material constant, M is the grain boundary mobility, γ the grain boundary energy, and n is the growth exponent with values between 2 and 4.Even though this formulism has been used in many cases, it is inadequate to analyze grain growth in the nanometer range where the dependence of atomic diffusion with the grain size and crystalline defects often limit the movement of grain boundaries 89,94 .The time evolution of isothermal grain coarsening S(t), as well as epitaxial crystallization and nanoislands' growth, can be described by a self-limiting model of the form 56,[95][96][97][98] , where S0 and Smax are respectively the initial and final dimensions, and τ is the characteristic relaxation time (at 0.63Smax).The diffusion coefficient at a fixed temperature DT is then calculated with the expression: The temperature dependence of the diffusion can be obtained by considering a thermally-activated process and fitting the data with an Arrhenius law 56,[95][96][97][98] : where D0 is a pre-exponential factor that depends on parameters such as the lattice constant and other geometrical constants, Qd is the activation energy for diffusion and kB is the Boltzmann constant.D0 and Qd are both assumed constant.As we will see in the next section, the kinetic evolution of all the nanostructures investigated here can be properly described using the self-limiting model described by Equation ( 7).

Thermally-activated coarsening of CGO nanostructures
A great discussion has been created in relation to whether thermodynamic or kinetic mechanisms control the formation of nanostructures.We have reported in previous works that coarsening of nanostructures is thermodynamically promoted due to a reduction in the total energy of the systems 48,57,59,63 .Despite that, diffusion mechanisms also play an important role in the definition of nanoisland dimensions 99 .Systems based on the deposition of chemical solutions have the final amount of material already deposited from the beginning of the growth process.Hence, it should be expected that our nanostructures behave close to a system in thermodynamic equilibrium as opposed to physical methods such as pulsed laser deposition where material is being deposited continuously.
Figures 5a-c show AFM images of the size evolution with time of (001)CGO NDs grown on (001)LAO substrates after heating at 20 ºC s -1 in O2, temperatures between 900 and 1100 ºC, and dwell times from 0 to 15 min.Equivalent measurements have been conducted for (001)CGO NDs on (001)YSZ (Figure S8).
A detailed analysis of the time evolution of the equivalent diameter S and height h can be extracted from these AFM images (Figures 5d, 5e and S9).Clearly, both S and h increase with time describing a saturation behavior which is more pronounced at high annealing temperatures.Particularly for (001)CGO NDs on (001)LAO (Figures 5d and 5e), we observe that S increases from 5-12 nm to 20-25 nm depending on the annealing temperature, whereas h grows from 2-3.5 nm to 5-7 nm. Figure S9 shows that S and h evolve respectively with the annealing time from approximately 10 to 22 nm and from 2.5 to 5-6 nm for (001)CGO NDs on (001)YSZ.These results indicate that CGO NDs always reach an equilibrium maximum size, as we predicted before using thermodynamic calculations 48,59 .However, atomic diffusion increases with temperature as we will see later, and thus, the kinetic growth process is speeded up at high temperatures.The height-to-diameter ratios h/S do not show a clear evolution with the annealing time and temperature with values between 0.25 and 0.35 for CGO NDs grown on LAO and between 0.18 and 0.25 on YSZ substrates (Figure S10).However, these results indicate that the lateral dimensions of NDs are always larger than their height (h/S<1).The formation of nanostructures is usually promoted in highly strained systems since they are less restricted in the in-plane direction as compared to films, and thus, they are more free to adapt their structure leading to reduced strain.From a thermodynamic perspective, a stronger relaxation of the system should be obtained for nanostructures with h/S>1 due to the presence of less strained regions 8 .However, kinetic mechanisms have a large influence over their growth.Tersoff and Tromp suggested that there is a tradeoff between strain relaxation and surface energy which leads to nanostructures with larger in-plane dimensions 46 .Essentially, under certain conditions it is more favorable for atoms to diffuse into the island edges instead of diffusing to the top island facet, hence, enhancing strain relaxation and decreasing the overall system energy despite the extra surface energy.
This phenomenon occurs in open systems where atomic diffusion is faster on the substrate than on nanostructures.It also seems to be the same case for closed CSD systems where atoms are already deposited on the substrate surface prior to the formation of nanostructures (Figure S3). Figure 6 shows equivalent analyses of the evolution with time of the in-plane and out-of-plane dimensions of (011)CGO NWs grown on (011)LAO substrates.The width of NWs a increases in size from 12-16 nm to 16-22 nm depending on the annealing temperature (Figure 6d), while the height h goes from 1-2 nm to 2-7 nm (Figure 6e).Both dimensions show an equivalent saturation behavior to that described for CGO NDs.Equivalently, the length of NWs b grows from 25 to 100 nm at 900 ºC, showing a similar saturation behavior.However, larger temperatures (1000-1100 ºC) show growth in the length from 50-75 nm to around 250 nm which seems to point out towards either longer saturation times or continuous enlargement of NWs.This is more clearly seen in Figure S11a where the length-to-width of NWs b/a increases with the annealing time and temperature.The height-to-width ratio h/a slightly increases with the annealing time and temperature from approximately 0.1 to 0.3 (Figure S11b), whereas the height-to-length ratio h/b (Figure S11c) seems constant with values around 0.02-0.04.Similarly to NDs, the in-plane dimensions are larger than out-of-plane ones which again could be due to faster diffusion of atoms to the lateral facets as compared to the top facet.It is worth noticing that NWs' elongation is in agreement with previous theoretical studies where we have demonstrated that coarsening of NWs along their length is thermodynamically driven by the continuous reduction of the system's energy 63 .Therefore, NWs' elongation seems to be a kinetically-driven thermodynamic process.A self-limited growth model (section 3.2.1)can be used to describe the saturation coarsening behavior observed.Figure 7a presents the dependence with temperature of the diffusion coefficients for the inplane and out-of-plane directions ( ∥  and  ⊥  ) extracted from Equation 7 and 8.We can see that atomic diffusion increases with temperature, as expected from a thermally-activated kinetic process.In addition, atomic diffusion is around one order of magnitude faster along the in-plane direction as compared to the out-of-plane one.Since atomic diffusion follows an Arrhenius-like thermally-activated process (Equation 9), we can extract precise values of in-plane and out-of-plane activation energies ( ∥  and  ⊥  ) and pre-exponential factors ( 0,∥  and  0,⊥  ) from the corresponding Arrhenius plots (Figure 7b).The in-plane activation energy for island coarsening is slightly larger than the out-of-plane counterpart ( ∥  =0.81±0.13eV and  ⊥  =0.69±0.11eV), indicating that a little more energy is required to activate in-plane diffusion.These values are around 6-7 times larger than the thermal energy kBT (~0.1 eV at 1000 ºC) and smaller than those reported in the literature for polycrystalline CGO films produced by spray pyrolysis and PLD, and YSZ films produced by precipitation 56,96,100,101 .These low energy values seem to indicate that surface or grain boundary diffusion could be the main mechanisms contributing to coarsening since these processes typically yield energy values between 0.1 and 1 eV 98 .
On the other hand, the in-plane pre-exponential factor ( 0,∥  =(6.0±0.1)×10 -1 m 2 s -1 ) is two orders of magnitude larger than the out-of-plane value ( 0,⊥  ==(9.8±0.1)×10 -1 m 2 s -1 ).Since activation energies are rather similar, it seems that the difference in pre-exponential factors illustrate the larger atomic mobility along the in-plane direction.Activation energies are related to barriers for atomic jumps, i.e.
diffusion.Therefore, the pre-exponential factors D0 could be related to the number sites available for diffusion.D values reflect basal diffusion stress barriers that control adatom diffusion and are also essential to keep a narrow size distribution 45,99 .In this case, the results obtained may indicate a faster surface mobility in the direction parallel to the substrate than the atomic diffusion along the lateral facets of nanostructures heightwise, and also a faster mobility of atoms on the substrate than on nanostructures due to the high density of adatoms, i.e. surface atoms 46,102 .Nevertheless, several additional aspects should be considered to influence growth of nanodots.We mentioned and demonstrated before that there exists a thermodynamic driving force promoting growth until a stable size is achieved 48,59 .This driving force (or growth rate) decreases as NDs' dimensions approach the optimum size and it also highlights the role of island facets on promoting self-limiting growth and a narrow distribution of sizes independently of the lattice mismatch 99 .In addition, it is very likely that the proximity of nuclei at early stages of growth contributes to a faster atomic diffusion than at later phases.Finally, it has been reported that the presence of a repulsive biaxial strain field stimulates nucleation of NDs while it hinders their growth as they become bigger 103 .In the case of CGO NWs, the analysis of width, length and height diffusion coefficients ( ∥,  ,  ∥,  and  ⊥  ) in Figure 8a also shows the expected rise with temperature associated with a thermallyactivated diffusion process.We can see that the range of  ∥,  and  ⊥  is equivalent to that of NDs inplane and out-of-plane diffusion coefficients (Figure 7a) as expected since they have similar self-limited behaviors.Thus, the same mechanisms as for NDs should be applicable.On the other hand,  ∥,  is between two and three orders of magnitude larger than previous diffusion coefficients; clearly illustrating the anisotropic growth of NWs.The mechanisms leading to such anisotropy will be discussed next.From the Arrhenius plots in Figure 8b, we can determine that the activation energies for width, length and height are respectively  ∥,  =1.06±0.38 eV,  ∥,  =1.38±0.56eV and  ⊥  =1.63±0.56eV.Although these values are quite similar between them, they indicate that less thermal energy is required to activate atomic diffusion along the a direction.Again, surface or grain boundary diffusion seems to be the main mechanism due to the low activation energies of ~1 eV.In this case, the activation energies are between 9 and 15 times larger than the thermal energy kBT and around two times the activation energies of NDs.
Considering this, we should expect a slightly delayed diffusion of NWs as compared to NDs.However, the pre-exponential factors of NWs width, length and height ( 0,∥,  =(3.6±0.2)×10 -1 m 2 s -1 ,  0,∥,  =(5.2±0.6)×10 -1 m 2 s -1 and  0,⊥  =(3.4±0.5)×10 -1 m 2 s -1 , respectively) are significantly larger than those of NDs, compensating for the higher Q values and indicating that the driving force for atomic diffusion is larger for NWs as compared to NDs.In addition, the higher value of  0,∥,  in relation with  0,∥,  and  0,⊥  explains why atomic diffusion coefficients are around three times larger, and the faster coarsening and bigger dimensions along the b direction.Atomic diffusion also has a large influence at early stages of NWs' growth.However, we reported before that they may have no thermodynamic restriction towards unlimited growth 63 .In addition, Li et al. mentioned that nanostructures with an anisotropy in their shape and surface energies should grow along the lower surface energy facet in order to reduce the energy of the system 47 .In the case of CGO, the {111} lateral facets have a lower surface energy than {001} facets as reported in Table S1 and Refs. 59,63 We demonstrated previously that the elongation of NWs occurs along the direction with {111} facets as shown in Figure S2 and Refs.57,58,63 .
Consequently, the origin of such anisotropic in-plane diffusion constants could be the result of the dissimilar island facets and their surface energies.Overall, the kinetic evolution of CGO nanostructures seems to direct the growth process towards thermodynamic equilibrium as expected from a closed system 8 .The internal redistribution of adatoms is driven by the need to minimize the surface energy.Thus, we conclude that shape evolution is driven by thermodynamics but mastered by kinetic effects, i.e. enhanced atomic mobility along specific directions (strain is not the driving force since it is isotropic in the systems considered).We determined that nucleation and even epitaxy of CGO nanograins (i.e. the final oxide phase) is possible at temperatures as low as 300 ºC which is rarely the common observations in CSD oxide growth.We also demonstrated, taking advantage of fast RTA furnaces, that it is possible to study nucleation and coarsening independently.The use of fast heating rates in the range of 20 ºC s -1 allows us to reduce the contribution of coarsening in nucleation, as compared to the slow heating rates achieved using conventional thermal treatments (<0.5 ºC s -1 ).The evolution of nanoislands with temperature has also shown a small increase in sizes, although it is not comparable to the growth experienced by decreasing the heating rate.We observed an experimental shift in the nucleation temperature for CGO NDs grown on (001)LAO and (001)YSZ.Thermodynamic calculations reveal that the strain energy shifts the nucleation barriers to smaller values of supersaturation, i.e. larger temperatures.Thus, the larger lattice mismatch of -5% between CGO and YSZ structures as compared to CGO on LAO (ε~-1%) delays nucleation to higher temperatures.In addition, the lower interface energy of CGO on YSZ contributes to lower supersaturation values, and thus, higher nucleation temperatures.
The coarsening of nanostructures has been described with a self-limited growth model.We demonstrated that it is driven by a thermally-activated diffusion process, i.e. atomic diffusion increases with temperature, that leads to thermodynamic equilibrium.For CGO NDs, we have determined that the activation energies for atomic diffusion are quite similar for in-plane and out-of-plane directions, and indicate that surface diffusion may be the main coarsening mechanism.Atomic diffusion is faster along the in-plane direction because is thermodynamically more favorable for atoms to diffuse into island edges instead of the top facet in order to enhance strain relaxation.Thermodynamics also rules the growth of NDs until a stable island dimension, i.e. narrow size distribution, is achieved.The activation energies for atomic diffusion for CGO NWs are larger than those of NDs.Despite that, the coarsening of CGO NWs shows equivalent diffusion trends along their width and height.Thus, similar self-limited diffusion mechanisms control growth along these directions.The coarsening along NWs' length is orders of magnitude faster than the other directions.The large driving force along NWs length is likely promoted by the dissimilar surface energies, and the enhanced growth along the direction with the lowest energy

Figure 3 .
Figure 3. Atomic force microscopy images showing the evolution of nanoislands with the heating rate,

Figure 6 .
Figure 6.AFM images of the influence of coarsening after heating at 20 ºC s -1 for (011)CGO NWs grown
the Gibbs free energy per unit volume,  is the unit cell volume, Δ(> 0) is the change in chemical potential or supersaturation between the epitaxial and amorphous/nanocrystalline state.= [2(   csc  +    −1 csc ) − ( cot  +  −1 cot )],  ≡   +   −   and  ≡   −   +  , with   ,   and   being the surface energies of the substrate, top facet of the island, the substrateinterface energy, respectively.  and   are the surface energies of the lateral facets, and  = �/ is the lateral aspect ratio.Parameter values for the different systems are reported in the literature and can be also found in the Supporting Information (Table