Investigation of Statistical Metal-Insulator Transition Properties of Electronic Domains in Spatially Conﬁned VO 2 Nanostructure

: Functional oxides with strongly correlated electron systems, such as vanadium dioxide, manganite, and so on, show a metal-insulator transition and an insulator-metal transition (MIT and IMT) with a change in conductivity of several orders of magnitude. Since the discovery of phase separation during transition processes, many researchers have been trying to capture a nanoscale electronic domain and investigate its exotic properties. To understand the exotic properties of the nanoscale electronic domain, we studied the MIT and IMT properties for the VO 2 electronic domains conﬁned into a 20 nm length scale. The conﬁned domains in VO 2 exhibited an intrinsic ﬁrst-order MIT and IMT with an unusually steep single-step change in the temperature dependent resistivity ( R - T ) curve. The investigation of the temperature-sweep-rate dependent MIT and IMT properties revealed the statistical transition behavior among the domains. These results are the ﬁrst demonstration approaching the transition dynamics: the competition between the phase-transition kinetics and experimental temperature-sweep-rate in a nano scale. We proposed a statistical transition model to describe the correlation between the domain behavior and the observable R - T curve, which connect the progression of the MIT and IMT from the macroscopic to microscopic viewpoints. in an with with a and °C °C °C °C the and with a width of were using a was by etching (RIE: Kyoto, Kyoto, in


Introduction
A correlated electron material (CEM) exhibits rich and fascinating properties originating from the interplay between the spin, lattice, charge, and orbital degrees of freedom of the material. Vanadium dioxide (VO 2 ) is one of the representative and most extensively studied CEMs [1]. VO 2 displays an interesting metal-insulator transition (MIT) and insulator-metal transition (IMT) at a convenient temperature, which is accompanied by the structural phase transition. In the MIT process the high-temperature, tetragonal and metallic phase transits to the low-temperature, monoclinic and insulating phase, and in the IMT process vice versa. Figure 1 shows the temperature dependent resistivity (R-T) curve for a VO 2 film sample. A gradual change in resistivity, as much as three orders of magnitude, was observed in the heating and cooling processes. The transition temperature from the insulator to the metal (T I→M ) was~77 • C and that from the metal to the insulator (T M→I ) was~65 • C. This transition has been attracting considerable interest for fundamental reasons [1,2] and possible applications [3][4][5][6][7][8][9][10][11][12]  Multiple phase coexistence has been observed not only in VO 2 , but also in many different CEMs, such as manganite [29][30][31] and nickelate [32][33][34]. The coexistence of metal and insulator domains around T I→M and T M→I , indicates that the properties of the CEMs are dominated by competing nanoscale and/or microscale electronic phases. The evolution of the ratio of the metal (or insulator) phase, i.e., the microscopic and macroscopic dynamics, determines the electronic transport property in the system. Therefore, a quantitative evaluation of the domain dynamics, which provides a seamless picture from the nanoscale to the macroscale, is indispensable.
Many researchers have been trying to capture a nanoscale electronic domain and investigate its exotic properties. A particularly effective approach to identify the phase transition for phase-separated CEMs is to spatially confine the target materials in ultrasmall scales equal to the average size of a single domain [18,26,35,36]. In the phase separation in the macrosize samples, such as VO 2 thin films, metallic and insulating domains with a huge difference in conductivity are randomly mixed; the domains have a fractal nature near the transition temperatures. The percolation model can describe the transition behavior, namely, the relationship between the conductivity and the fraction of metal domains: P Met [19,[26][27][28]. The counterpart of P Met , the occupation probability of the insulator phase: P Ins is defined as P Ins = 1 − P Met . The isotropic two-dimensional (2D) square-lattice percolation model can be applied for the thin film sample. The abrupt decrease and increase of the resistivity at P Met~0 .6 and P Met~0 .4 (=P Ins~0 .6) was reported in heating and cooling process, respectively, which correspond to the formation of the connection of metallic and insulating path between electrodes [19,[26][27][28]. Thus, for a macrosize sample, the R-T curve shows huge but gradual change. In this manner, T I→M and T M→I are defined as temperatures at P Met~0 .6 and P Ins~0 .6, respectively.
On the other hand, the discrete T I→M and T M→I were observed on the microsize sample whose size were close to the electric domains [13,17,18,20,26,37,38]. The size of the formed domains varies from the order of 100 nm on Al 2 O 3 single-crystal substrates [17,18] to the order of 10 µm on TiO 2 (001) substrates [13,19,20]. The generation and extinction of the electronic domains can be attributed to nucleation, which initiates first-order phase transitions. Although there have been significant advances in understanding the general nature of the transition, the mechanism responsible for the domain generation, i.e., nucleation, is not yet well established. The nucleation rate and corresponding resistivity Crystals 2020, 10, 631 3 of 14 behavior strongly depend on the temperature (T) sweep conditions [39,40] and sample size [39][40][41]. However, the examination of the possible role of the temperature and sample size has been limited. A more complete understanding of these features associated with the phase-separation phenomena is indispensable to the advancement of our knowledge of the detailed physics of the phase transition in VO 2 . In this article, we investigate the domain dynamics in spatially confined VO 2 samples. The advantage of nano-confined sample is the quantification of the domain generation (nucleation) process. An analysis based on the statistical transition model revealed the correlation between the domain size and the T I→M and T M→I distribution, which can explain the macroscopic progression of the IMT and MIT.

Materials and Methods
A VO 2 thin film with a thickness of 50 nm was deposited on a commercial Al 2 O 3 (0001) substrate by pulsed laser deposition (PLD) (ArF excimer: wavelength of 193 nm) using a V 2 O 5 sintered ceramic target, as reported elsewhere [18]. The base pressure in the PLD chamber was less than 2 × 10 −6 Pa. The deposition was performed under an oxygen pressure of 1 Pa at 350 • C with an energy density of 2 J cm −2 and a pulse repetition rate of 2 Hz. To observe the structural change of the VO 2 film with the temperature, temperature-dependent crystal structure of the VO 2 film was confirmed by reflection high-energy electron diffraction (RHEED: R-DEC RDA-003G, Tsukuba, Ibaraki, Japan) and X-ray diffraction (XRD: Rigaku SmartLab, Akishima, Tokyo, Japan). In-situ RHEED observation in the PLD chamber was performed at 2 × 10 −6 Pa in the temperature range from 30 • C to 100 • C. Before the RHEED observation, the VO 2 film was cooled to 30 • C. RHEED patterns were observed using an electron beam of 15 keV energy and ∼1 mm diameter. These patterns were filtered using an image software program to emphasize spot features. θ-2θ scans were performed in an ex-situ XRD system with Cu Kα1 radiation with a Ge(220) monochromator and a sample heating stage. The temperature sequences was 120 After the RHEED and XRD measurements, VO 2 wire structures with a width (W) of 0.6-2.0 µm were produced using a lithographic technique. The VO 2 film was etched by reactive ion etching (RIE: Samco RIE-10NR, Kyoto, Kyoto, Japan) in SF 6 gas, employing a photoresist mask. The optimized RIE conditions were 10 sccm SF 6 , 1 Pa, and 60 W. In order to measure the temperature-dependent dc transport property of the VO 2 samples, a pair of Au electrodes with a gap length (L) of 20 nm or 2 µm were attached to the VO 2 wires. The electrodes with L = 20 nm were produced through the two-times tilted Au/Ti deposition technique in accordance with previous reports [40,42]. The sample structure was confirmed by field-emission scanning electron microscopy (SEM: JEOL JSM7610F, Akishima, Tokyo, Japan). The temperature dependent resistivity (R-T) measurements were carried out for the VO 2 wire samples with heating and cooling in 0.1 • C steps with temperature-sweep-rates of 0.5, 2, and 3 • C/min. As a reference, R-T measurements of a VO 2 film (8 mm × 10 mm × 50 nm) were also performed. T I→M and T M→I for the film sample are calculated from the peak of d(log R)/dT vs. T plot of the heating and cooling curves, respectively.
The R-T curves for the samples were simulated using the random resistor network model [18,26,35]. The net resistivity of the system, R sample , was evaluated by the cycle current equation, where H is a cycle basis matrix, R is a resistance matrix, I is a current vector, and V is a voltage vector, on the basis of the Kirchhoff's laws [43]. For three-dimensional (3D) cubic lattice with a resistor on each edge, R is a diagonal matrix where each element is the edge (q) resistance. Elements of I and V correspond to a loop current and applied voltage in possible closed loop (cycle p) circuits. H is the matrix of the cycle number by the edge number; the element is defined as + 1 (flow directions of cycle p and edge q are the same), −1 (opposite), or 0 (cycle p does not contain edge q at resistor q). The model of the 3D resistor network with two side electrodes with applied voltage V 0 and current I 0 leads to R sample = V 0 /I 0 . The number of the domains, leading to the edge resistance, in 3D sample is defined as N D = N L × N W × N H . The numbers of domains in the length, width, and height directions are defined as N L = L/D ave , N W = W/D ave , and N H = H/D ave , respectively. D ave is the mean domain size. Since the deposited VO 2 thickness of 50 nm is enough smaller than the reported D ave [17], the calculations were performed using a 2D resistor network with N H = 1. The population of metallic domains at each temperature: P Met (T) was given by a Gaussian function. The metal and insulator domain arrangement in N L × N H domain geometry, i.e., either a metal or insulator at the (i, j) site (i = 1 · · · N L , j = 1 · · · N W ) is randomly selected according to P Met (T). No interaction among the domains was considered.

Phase-Separated VO 2 Structure
XRD and RHEED results showed that an epitaxial VO 2 film was grown on the Al 2 O 3 (0001) substrate. Figure 2a shows the XRD θ-2θ scan for the VO 2 film at 20 • C. The strong reflections from the VO 2 (020) and VO 2 (040) planes at 2θ = 39.9 • and 86.1 • , respectively, indicate VO 2 growth on the Al 2 O 3 (0001) substrate with its (010) plane parallel to the substrate surface [44]. Figure 2b shows the VO 2 (020) peak at various temperatures. With decreasing temperature from 120 • C to 20 • C, the peak position shifted toward a higher angle. The estimated lattice constant b is plotted against temperature in Figure 2c. The lattice constant b rapidly changed from 4.518 Å at 80 • C to 4.512 Å at 50 • C, that is, the thermal expansion coefficient was ∼ 4 × 10 −4 • C −1 , while the thermal expansion coefficient was only ∼ 7 × 10 −6 • C −1 (∼ 1 × 10 −5 • C −1 ) below 50 • C (above 80 • C). This irregular thermal expansion has been established as a characteristic of the phase transition of VO 2 [45]. The thermal expansion of the Al 2 O 3 substrate in the c direction, was ∼6 × 10 −6 • C −1 , as shown in the inset of Figure 2c, which is consistent with the value of 5 × 10 −6 • C −1 (c direction) for the Al 2 O 3 single crystal [46] within the experimental error. Thus, Figure 2c clearly indicates the structure transition from rutile VO 2 (P4 2 /mmm) at a high temperature to monoclinic VO 2 (P2 1 /c) at a low temperature [16]. The temperature dependence of the lattice constant indicates that during the phase transition in the cooling process, the initial state is rutile at T ≥ 90 • C, which is followed by a coexisting rutile/monoclinic state at 50 • C < T < 80 • C and a monoclinic state at T ≤ 50 • C.
RHEED patterns directly reveal the coexisting states in VO 2 through the phase transition. Figure 3a shows a typical 2110 -incident RHEED pattern from VO 2 at 70 • C. The diffraction pattern consisted of two phases with different periodicities in the 0110 direction. The most prominent diffraction peaks of one phase are noted by (020) r , (040) r , (031) r , (051) r , (022) r , and (042) r which are also indicated by pink arrows; these peaks have a period of 2.2 Å −1 in the in-plane direction, which corresponds to the reciprocal lattice unit, c * r = 2π/c r = 2.20 Å −1 , along the c axis of the rutile VO 2 phase. At a different incident azimuth angle, 0110 The key point of the structural transformation between rutile and monoclinic VO 2 observed in XRD and RHEED is not gradual with homogeneous changes, but comprises partial generations of drastic change from rutile to monoclinic with the spatial coexistence of the two phases, that is, phase-separated structure. This observation indicates that the gradual resistance change observed in an R-T curve ( Figure 1) is associated with the propagation of the local crystallographic transformation, which is consistent with previous studies [37,47]. The key point of the structural transformation between rutile and monoclinic VO2 observed in XRD and RHEED is not gradual with homogeneous changes, but comprises partial generations of drastic change from rutile to monoclinic with the spatial coexistence of the two phases, that is, phase-separated structure. This observation indicates that the gradual resistance change observed in an R-T curve (Figure 1) is associated with the propagation of the local crystallographic transformation, which is consistent with previous studies [37,47].

Transport Properties in the Phase-Separated VO2
For phase-separated materials, by utilizing the spatial confinement effect, it is possible to isolate individual contributions from specific interactions through their inherent length scale [35,36]. To realize the nano-confinement effect, a pair of Au/Ti electrodes with a gap length of 20 nm was attached along the length of a VO2 wire as shown in Figure 4a,b shows a typical R-T curve from the VO2 wire sample with W of 600 nm and L of 20 nm. Interestingly, one-step rapid resistivity changes were observed in both the heating and cooling processes, in contrast to the continuous changes in a macroscopic VO2 film sample (Figure 1). These one-step resistivity changes indicate the occurrence

Transport Properties in the Phase-Separated VO 2
For phase-separated materials, by utilizing the spatial confinement effect, it is possible to isolate individual contributions from specific interactions through their inherent length scale [35,36]. To realize the nano-confinement effect, a pair of Au/Ti electrodes with a gap length of 20 nm was attached along the length of a VO 2 wire as shown in Figure 4a,b shows a typical R-T curve from the VO 2 wire sample with W of 600 nm and L of 20 nm. Interestingly, one-step rapid resistivity changes were observed in both the heating and cooling processes, in contrast to the continuous changes in a macroscopic VO 2 film sample (Figure 1). These one-step resistivity changes indicate the occurrence of the nano-confinement effect; the isolation of a single domain into a nano-space of 600 nm × 20 nm was realized. Although the VO 2 wire sample with the nano-gap electrodes showed one-step resistance changes, their change ratio through IMT and MIT was smaller than those in the film sample, which is mainly because of the unintended damages during dry etching process. However, the one-step change which is absent in the macrosize sample clearly showed that this sample exhibit a prototype of first order transition.  To consider the phase-separated condition quantitatively, we have proposed the statistical transition model [18,35], where the IMT and MIT property of a single domain, the number of domains ND, in the sample, and their statistical behavior, that is, the TI→M and TM→I distribution, are taken into account. The step resistivity change reflects the IMT and MIT from a single domain, showing the ideal first-order transition. Figure 5b shows the ideal R-T curve for a VO2 single domain through the IMT and MIT. For example, in the heating (cooling) process, the domain has two transport properties defined as RIns(T) and RMet(T); RIns(T) switches with RMet(T) at TI→M (RMet(T) switches with RIns(T) at TM→I). Therefore, assuming the simplest first-order IMT and MIT, the resistivity of a single domain Rs(T) is described as where ( ) ≫ ( ). A, B, and C are coefficients, R0 is the residual resistivity in the metallic phase, Eg is the activation energy in the insulating (semiconducting-like) phase, and kB is the Boltzmann constant. The parameters in Equation (2) were obtained by fitting for an R-T curve for a The gap distance of the electrodes was 20 nm, which corresponds to the VO 2 region for the electronic resistivity measurement. (b) Normalized R-T curve for the VO 2 wire with 20 nm-gap electrodes. One-step changes in resistivity were observed in both the heating (red) and cooling (blue) processes. The resistivity was normalized by the value at 60 • C. The temperature-sweep-rate was 3.0 • C/min. VO 2 undergoes a MIT and IMT accompanied by a structural transition: from a high-symmetry, metallic rutile phase at to a low-symmetry insulating monoclinic phase with V-V dimerization in cooling (MIT) process and vice versa in heating (IMT) process [2]. The IMT in VO 2 is understood by the collapse of an insulating state (electron crystal) to a metallic state (electron liquid). This is a simple picture for a nano-scale single domain. The observation of the step changes in resistivity due to the spatial confinement effect shows that the single domain has an ideal first-order IMT and MIT with one-step resistivity change. Therefore, the ensemble of the step IMTs and MITs in a macrosize sample, where the IMT and MIT in numerous domains contributes to the observable IMT and MIT as an R-T curve, should show a gradual resistivity change without step changes as shown in Figure 1. Such a spatial distribution of the metal and insulator domains is illustrated in Figure 5a.  Figure 6 shows R-T curves for the VO2 sample with different temperature-sweep-rate with W of 0.6 μm and L of 20 nm. We can see the variation of the transition behavior with the temperaturesweep-rate. At the high temperature-sweep-rate of 3 °C/min, the R-T curve showed one-step resistivity changes in both the cooling and heating processes, indicating that ND = 1 during the IMT and MIT. Interestingly, at the temperature-sweep-rate of 2 °C/min (Figure 6b), multi-step resistivity changes, that is, multiple points in the transient stage in the cooling process, were observed while the one-step resistivity change was maintained in the heating process. The R-T curves at the temperaturesweep-rate of 0.5 °C/min consisted of the many small step changes, as shown in the insets (Figure 6c). These results clearly showed that reducing of the temperature-sweep-rate induced the multi-step resistivity changes. The change from one-step to multi-step resistivity changes indicates an increase in the ND of the sample, i.e., a decrease in domain size. Our observed tendency is consistent with previous reports. Lopez et al. reported the effect of sample size on TI→M and TM→I for VO2 nanoparticles [41], where an increase (decrease) in TI→M (TM→I) with sample miniaturization was To consider the phase-separated condition quantitatively, we have proposed the statistical transition model [18,35], where the IMT and MIT property of a single domain, the number of domains N D , in the sample, and their statistical behavior, that is, the T I→M and T M→I distribution, are taken into account. The step resistivity change reflects the IMT and MIT from a single domain, showing the ideal first-order transition. Figure 5b shows the ideal R-T curve for a VO 2 single domain through the IMT and MIT. For example, in the heating (cooling) process, the domain has two transport properties defined as R Ins (T) and R Met (T); R Ins (T) switches with R Met (T) at T I→M (R Met (T) switches with R Ins (T) at T M→I ). Therefore, assuming the simplest first-order IMT and MIT, the resistivity of a single domain R s (T) is described as

Domain Dynamics of VO2 Domains: The Effect of the Temperature-Sweep-Rate
where R Ins (T) R Met (T). A, B, and C are coefficients, R 0 is the residual resistivity in the metallic phase, E g is the activation energy in the insulating (semiconducting-like) phase, and k B is the Boltzmann constant. The parameters in Equation (2) were obtained by fitting for an R-T curve for a film sample.
In the temperature-induced IMT of VO 2 , metal domains initially nucleate at seed sites in a sample, then grow and spread with increasing temperature. The domain dynamics, i.e., the temperaturedependent population of the metal domains, strongly affects the observable transport property of the sample. To express the phase-separated condition as domain dynamics, the T I→M and T M→I distribution is defined in this model. The direct observation of the coexistence of metal and insulator domains in the vicinity of the observable T I→M and T M→I [13][14][15][16][17][18][19][20][21][22][23][24][25] indicates that the T I→M and T M→I values of the single domains are distributed. The distribution of T I→M and T M→I for each domain can be described using the Gaussian function as where σ is the half width of the Gaussian distribution and T center is the temperature where the population of metal domains, P Met is equal to that of insulator domains, P Ins , i.e., P Met (T center ) = P Ins (T center ) = 50%. f(T) gives the probability of the IMT and MIT at a certain temperature. Here, the values of σ and T center in the heating process are treated to be different from those in the cooling process [48]. Thus, the metal population is obtained as P Met (T) = N D ·f(T) = 1 − P Ins (T).
Here, we treat the resistor network with the edge resistor having the single domain resistivity R s (T) in Equation (2) under Equations (3) and (4), and obtain R sample (T) from the cycle current equation in Equation (1). Three-dimensional (3D) cubic-lattice networks are treated for films thicker than a metal and insulator domain size; the domain size in a VO 2 film on an Al 2 O 3 substrate was reported to bẽ 100 nm [17]. Since the thickness of the VO 2 samples in this study was 50 nm, we can treat a 2D square-lattice network with the number of the domains N D = N L × N W × N H (= 1). Figure 5c shows a comparison between the observed and simulated R-T curves for the VO 2 film. Here, the resistivity was calculated from Equations (1)-(4) assuming N L = N W = 100. The dashed curves in Figure 5c showed the good fitting of the results with this model, where the T I→M (T M→I ) distribution as shown in Figure 5d (Figure 5e) was obtained. The good agreement between the experimental and simulation results indicates the validity of this model. This means that the observable R-T property of VO 2 is determined by three elements: the resistivity of a single domain R s (T), the T I→M and T M→I distributions, and the N D of the sample. It was reported that the MIT and IMT characteristics including the width of the hysteresis loop in a R-T curve depend on various structural parameters, such as external and internal stresses, grain size in a VO 2 sample since the phase transition in VO 2 is coupled to the defect concentration and the types of crystal defects [37,38,47]. Therefore, R s (T) and the T I→M and T M→I distributions are characteristics of an individual samples that depend on its growth conditions. On the other hand, we can tune N D by controlling the sample size. The step resistivity changes shown in Figure 4b have been observed due to the nano-confinement effects not only in VO 2 but also in other CEMs [13,18,26,35]. The minimization of the sample size results in the step resistivity change at the stochastic T I→M and T M→I corresponding to the reduction in N D . The observed T I→M (T M→I ) of 52 • C (49 • C) for the VO 2 wire sample with W of 600 nm and L of 20 nm (Figure 4b) was lower than that of 77 • C (65 • C) for the film sample (Figure 1). The statistical model shows that T I→M (T M→I ) were distributed within a range of 50 to 100 • C (40 to 90 • C) as shown in Figure 5d (Figure 5e). The stochastic selection of a single domain as a result of the spatial confinement determined the T I→M and T M→I for the 20 nm-gap wire sample while T I→M (T M→I ) is defined as a temperature at P Met (P Ins )~0.6 for the macrosize sample with a high N D . The isolation of one or a few domains into the nanospace makes it possible to investigate the potentially hidden domain behavior in a macrosize sample with a high N D . Thus, the correlation between the domain size and the T I→M and T M→I distribution in the spatially confined structure was examined.
3.3. Domain Dynamics of VO 2 Domains: The Effect of the Temperature-Sweep-Rate Figure 6 shows R-T curves for the VO 2 sample with different temperature-sweep-rate with W of 0.6 µm and L of 20 nm. We can see the variation of the transition behavior with the temperature-sweep-rate. At the high temperature-sweep-rate of 3 • C/min, the R-T curve showed one-step resistivity changes in both the cooling and heating processes, indicating that N D = 1 during the IMT and MIT. Interestingly, at the temperature-sweep-rate of 2 • C/min (Figure 6b), multi-step resistivity changes, that is, multiple points in the transient stage in the cooling process, were observed while the one-step resistivity change was maintained in the heating process. The R-T curves at the temperature-sweep-rate of 0.5 • C/min consisted of the many small step changes, as shown in the insets (Figure 6c). These results clearly showed that reducing of the temperature-sweep-rate induced the multi-step resistivity changes. The change from one-step to multi-step resistivity changes indicates an increase in the N D of the sample, i.e., a decrease in domain size. Our observed tendency is consistent with previous reports. Lopez et al. reported the effect of sample size on T I→M and T M→I for VO 2 nanoparticles [41], where an increase (decrease) in T I→M (T M→I ) with sample miniaturization was observed. They considered that the size dependence is associated with the nucleation statistics where the density of nucleating defects is a power function of the driving force for the MIT and IMT. Yoon et al. found an effect of size on the phase transition speed [39]; the transition speed was roughly proportional to 1/w, where w is the width of the VO 2 microcrystal. They pointed out the possibility of size-dependent thermal dissipation. Oike et al. reported size-dependent supercooling phenomena in strongly correlated electron systems and suggested a nucleation mechanism [40]. In general, systematic shifts of T I→M (T M→I ) toward high (low) temperatures when increasing temperature-sweep-rate are observed, implying that nucleation events are involved in the first-order transition [49]. Namely, the nucleation rate of the electronic domains is consistent with the generation speed of the operational defects in the sample [38,40,47]. Thus, we can expect the phenomena based on the nucleation mechanism [40]. Most first-order phase transitions are initiated by nucleation, which is a critical rate event. The first-order transition could be kinetically suppressed under sufficient rapid cooling or heating; the nucleation growth rate of the domains leading to the first-order transitions is slow compared with the external temperature-sweep-rate. The most extensively studied realization of this scenario is classical liquids [50]. Extension of this manner will be applicable for the IMT and MIT in VO 2 .
In the frame of this phase-separated scenario, an increase of the transition (nucleation) sites is described as an increase in N D with decreasing the mean domain size: D ave under the slow temperature-sweep-rate. The nucleation of metal and insulator domains in heating and cooling processes, i.e., insulator-metal and metal-insulator transitions, respectively, occurs preferentially at defect sites, such as grain boundaries, surfaces, impurity atoms, vacancies, and dislocations. Since the probability of IMT and MIT depends on the availability of a suitable nucleating defect in the sample space, a rapid temperature sweep, and/or minimization of the sample volume causes a kinetic avoidance of the first-order transition. Naturally, the defects would have preferentially interact with an embryo of a growing metal/insulator domain in order to overcome an activation barrier toward nucleation. During a temperature sweep, the phase-transition kinetics competes with the experimental temperature-sweep-rate. In VO 2 , the transition is mainly driven by phonons in addition to a contribution from electrons [51]. When the temperature-sweep-rate exceeds the phase-transition kinetics, the thermal dynamic motions of V and O atoms leading multiple nucleation at latent defect cites are suppressed, and eventually a transition occurs at an energetically preferential defect site. On the other hand, when the temperature-sweep-rate is sufficiently low, the phase transition is fully completed even at latent defect sites.
size-dependent thermal dissipation. Oike et al. reported size-dependent supercooling phenomena in strongly correlated electron systems and suggested a nucleation mechanism [40]. In general, systematic shifts of TI→M (TM→I) toward high (low) temperatures when increasing temperature-sweeprate are observed, implying that nucleation events are involved in the first-order transition [49]. Namely, the nucleation rate of the electronic domains is consistent with the generation speed of the operational defects in the sample [38,40,47]. Thus, we can expect the phenomena based on the nucleation mechanism [40]. Most first-order phase transitions are initiated by nucleation, which is a critical rate event. The first-order transition could be kinetically suppressed under sufficient rapid cooling or heating; the nucleation growth rate of the domains leading to the first-order transitions is slow compared with the external temperature-sweep-rate. The most extensively studied realization of this scenario is classical liquids [50]. Extension of this manner will be applicable for the IMT and MIT in VO2. Figure 6. Normalized R-T curves for the VO2 wire with 20 nm-gap-electrodes measured at temperature-sweep-rates of (a) 3 °C/min, (b) 2 °C/min, and (c) 0.5 °C/min. Magnified step changes (indicated by arrows) are shown in the insets of (c).
In the frame of this phase-separated scenario, an increase of the transition (nucleation) sites is described as an increase in ND with decreasing the mean domain size: Dave under the slow temperature-sweep-rate. The nucleation of metal and insulator domains in heating and cooling processes, i.e., insulator-metal and metal-insulator transitions, respectively, occurs preferentially at defect sites, such as grain boundaries, surfaces, impurity atoms, vacancies, and dislocations. Since the probability of IMT and MIT depends on the availability of a suitable nucleating defect in the sample space, a rapid temperature sweep, and/or minimization of the sample volume causes a kinetic Figure 6. Normalized R-T curves for the VO 2 wire with 20 nm-gap-electrodes measured at temperaturesweep-rates of (a) 3 • C/min, (b) 2 • C/min, and (c) 0.5 • C/min. Magnified step changes (indicated by arrows) are shown in the insets of (c).
The nucleation rate dependency and corresponding resistivity behavior have been observed as the size effects in the phase transition. In the phase transition of VO 2 , the size effects have been typically reported for nano-particle systems; these effects originate from size-dependent thermal dissipation [39,41,[52][53][54][55]. The decreasing domain size consequently induces the shifts of T I→M and T M→I . An increase in the rate of effective thermal exchange in the nano-scale sample leads to a lower (higher) T I→M (T M→I ). Additionally, the increase in hysteresis width in the R-T curve for a sample consisting of smaller nano-particles [41] implies that a wider T I→M and T M→I distribution corresponds to a larger numbers of available defect sites in the sample. The temperature-sweep-rate dependent behavior of the step changes in resistivity in the nano-confined VO 2 sample ( Figure 6) clearly reveals the increases in N D and T I→M and T M→I variation in addition to the shift of T center . The statistical transition model based on the observed results for a spatially confined sample can adequately explain and seamlessly assign the complex domain dynamics in phase-separated VO 2 from the micro-scale to the macro-scale. When the sample size increases from the micro-scale (~D ave ) to the macro-scale (>>D ave ), the N D of the sample drastically increases. For example, N D was estimated to be a few hundreds in a 2 × 2 µm 2 VO 2 sample (W = 2 µm and L = 2 µm). This sample showed a gradual R-T curve without any step changes, which means that the contribution of each domain is buried and is not detectable in larger sample. Actually, we observed the temperature-sweep-rate dependence as a slight temperature shift. In this sample, T I→M reduction was confirmed, while T M→I was insensitive to the temperature-sweep-rate, which reflects the structural stability of VO 2 phases. The low-temperature monoclinic phase is energetically favored over the high-temperature rutile phase [49], and the MIT requires a larger driving forces than the IMT in order to overcome the nucleation barrier in VO 2 [41]. T I→M was reduced to 2-3 • C in the 2 × 2 µm 2 VO 2 sample when the temperature-sweep-rate was changed from 3 • C/min to 0.5 • C/min. According to the IMT statistical model with a random resistor network, the T I→M increase of 2-3 • C was ascribed to the 20-100-fold increase in N D under a constant T center . The T I→M increase may be cause by the decrease of P C . It was reported the critical fraction value: P C depends on the N D and arrangement of the metallic (insulating) domains in a sample [26,27]. That means the degree of the changing ratio, i.e., the shape of the R-T curve (one-step, multi-steps, and gradual changes) depends on the N D ; the changing ratio and sharpness is enhanced with decreasing N D . Additionally, the decrease of N D enhances the contribution of each single domain to the total change in the R-T curve. Therefore, the 20-100-fold increase in N D caused the corresponding~4-10-fold reduction in domain size associated with the 2-3 • C increase in T I→M , which is consistent with previous studies [41,53].

Conclusions
The domain dynamics in phase-separated VO 2 was systematically studied. To identify the single-domain IMT and MIT behavior, a spatially confined VO 2 sample was prepared. The step changes in the resistivity of a nano-confined sample, which are absent in macrosize VO 2 samples, correspond to the first-order IMT and IMT from a single domain and the distributed T I→M and T M→I . Changing the temperature-sweep-rate led to the variation of the transition behavior corresponding to nucleation events. As a result of the competition between the phase-transition kinetics and the experimental temperature-sweep-rate, a realistic domain size and their T I→M and T M→I were decided. We proposed a statistical transition model, where the macroscopic transition behavior, i.e., an observable R-T, was reproduced using the one-step R-T curve for a single domain, the total number of domains in the sample, and their T I→M and T M→I distribution. In the frame of the statistical transition model, the phase-separation scenario and temperature-sweep-rate-dependent behavior were able to be explained quantitatively. The advantage of studying a nano-confined sample is the quantification of the domain generation (nucleation) process. The number of the step changes in an R-T curve and their temperature distribution indicate N D and T I→M and T M→I distribution. Direct understanding of the statistical IMT and MIT properties of electronic domains in the nano-confined sample can advance the detailed physics of phase transition. Furthermore, the number of domains can be controlled by designing the sample structure, i.e., confinement effect. The arbitrary control of N D and T I→M and T M→I temperature distribution would lead to a preferential response, which can be a good guideline for realizing the functional nanoelectronic devices based on the phase-separated CEMs.