Probing the Critical Nucleus Size in the Metal-Insulator Phase Transition of ${\mathrm{VO}}_2$

In a first-order phase transition, critical nucleus size governs nucleation kinetics, but the direct experimental test of the theory and determination of the critical nucleation size have been achieved only recently in the case of ice formation in supercooled water. The widely known metal-insulator phase transition (MIT) in strongly correlated ${\mathrm{VO}}_2$ is a first-order electronic phase transition coupled with a solid-solid structural transformation. It is unclear whether classical nucleation theory applies in such a complex case. In this Letter, we directly measure the critical nucleus size of the MIT by introducing size-controlled nanoscale nucleation seeds with focused ion irradiation at the surface of a deeply supercooled metal phase of ${\mathrm{VO}}_2$. The results compare favorably with classical nucleation theory and are further explained by phase-field modeling. This Letter validates the application of classical nucleation theory as a parametrizable model to describe phase transitions of strongly correlated electron materials.

Figure 1

Creating a deep supercooled state in the MIT of ${\mathrm{VO}}_2$. (a) The MIT temperature ($T_{\mathrm{MIT}}$) of ${\mathrm{VO}}_2$ nanobeams as a function of the dose of ${\mathrm{He}}^{+}$ ion irradiation. $T_{\mathrm{MIT}}$ is determined from electrical transport measurements. The inset shows schematically a ${\mathrm{VO}}_2$ nanobeam uniformly and globally (i.e., not locally) irradiated with ${\mathrm{He}}^{+}$ ions. (b) Schematics of ${\mathrm{He}}^{+}$ ion irradiation for nucleation shielding in a single ${\mathrm{VO}}_2$ nanobeam. The middle, pristine ${\mathrm{VO}}_2$ segment (gray) is shielded against influence from the Pt contacts (yellow blocks) by two end segments of ${\mathrm{VO}}_2$ that are locally ${\mathrm{He}}^{+}$ irradiated (green). (c) Four-probe measured electrical conductivity of the same ${\mathrm{VO}}_2$ nanobeams with or without irradiation shielding. Deep supercooling of the $M$ phase down to 300 K is observed in the ${\mathrm{VO}}_2$ end shielded with $5\times {10}^{15}\mathrm{ions}/{\mathrm{cm}}^2$ ${\mathrm{He}}^{+}$ irradiation. (d) Scanning electron microscope (SEM) image of a ${\mathrm{VO}}_2$ nanobeam supported by two suspended micropads. Scale bar is $10\mathrm{\mu }\mathrm{m}$. Inset: SEM image of FIB-deposited Pt bonding of the nanobeam onto the underlying electrode to minimize electrical or thermal contact resistance. Scale bar is 100 nm. (e) SEM image of a ${\mathrm{VO}}_2$ nanobeam showing a rectangular cross section. Scale bar is 100 nm. (f) Optical images of a deeply supercooled, shielded ${\mathrm{VO}}_2$ nanobeam as the temperature drops. A $30\text{−}\mathrm{\mu }\mathrm{m}$-long ${\mathrm{VO}}_2$ segment is shielded by two $4\text{−}\mathrm{\mu }\mathrm{m}$-long ${\mathrm{He}}^{+}$-irradiated segments at the two ends (indicated by the two white dashed boxes). Despite the fact that the pristine segments between the shields and the electrodes transition to $I$ phase at normal $T_{\mathrm{MIT}}$ (333 K), the shielded, pristine ${\mathrm{VO}}_2$ segment stays in the $M$ phase (dark) and does not transition to the $I$ phase (bright) until 308 K, which is close to the natural $T_{\mathrm{MIT}}$ (a) of the irradiated shield segments. Scale bar is $5\mathrm{\mu }\mathrm{m}$.

Figure 2

$I$-phase nucleation in supercooled ${\mathrm{VO}}_2$. (a) Schematic showing a disk-shaped nucleation seed (purple) created by ${\mathrm{Ga}}^{+}$ ion irradiation on the surface of a shielded pristine ${\mathrm{VO}}_2$ segment. The ${\mathrm{Ga}}^{+}$ ion penetration depth is $\sim 15\mathrm{nm}$, very shallow compared to the diameter of the irradiated area; hence the nucleation can be approximated as a disk shape with zero thickness. (b) Optical image of the $I$-phase nucleation process along a long, suspended ${\mathrm{VO}}_2$ nanobeam. The nanobeam is divided into eight pristine segments, each shielded by two ${\mathrm{He}}^{+}$-irradiated segments, then a nucleation seed is introduced onto the surface of each of the pristine segments by ${\mathrm{Ga}}^{+}$ irradiation at a dose of $2.2\times {10}^{16}\mathrm{ions}/{\mathrm{cm}}^2$, but with different diameters. The lowest panel shows schematically the case where the diameter of the disk-shaped nucleation seed ($D$) increases from 10 to 180 nm. Scale bar is $5\mathrm{\mu }\mathrm{m}$. (c) Schematic dependence of $I$-phase nucleation temperature ($T_{\mathrm{nuc}}$) on $D$, as predicted by classical nucleation theory. $T_{\mathrm{nuc}}$ is upper bounded by the natural MIT temperature ($T_{\mathrm{natural}}$) and lower bounded by the supercooled temperature ($T_{\mathrm{sc}}$) of the shielded ${\mathrm{VO}}_2$. With reduced interface energy, stable nuclei with smaller $D$ are able to form at a given temperature. (d) Measured (solid symbols) and calculated (open symbols) $T_{\mathrm{nuc}}$ as a function of $D$ for different ${\mathrm{Ga}}^{+}$ irradiation doses. The thick curved bands are a guide for the eye. The critical sizes of the nucleus can be determined from the onset of the rise above the supercooled limit. (e) Measured dependence of $T_{\mathrm{nuc}}$ on ${\mathrm{Ga}}^{+}$ irradiation dose ($n_{\mathrm{Ga}+}$) in the seed at different $D$ intervals. A heavier dose in the seed promotes $I$-phase nucleation as shown by the increased $T_{\mathrm{nuc}}$. $T_{\mathrm{nuc}}$ for nucleation seeds with large $D$ and high doses is found to reach the thermodynamic limit. $T_{\mathrm{natural}}$ and $T_{\mathrm{sc}}$ are marked by the gray shaded areas in (d) and (e).

Figure 3

Measured $D\mathrm{\Delta }T$ of all nucleation seeds with different diameters as a function of the reciprocal ${\mathrm{Ga}}^{+}$ dose density. The black cross hairs and black open circles are $D\mathrm{\Delta }T$ data measured from the nucleation seeds that drive the ${\mathrm{VO}}_2$ to the thermodynamic or natural and supercooled limit, respectively. Between these two limits (solid and colored symbols), $T_{\mathrm{nuc}}$ depends on $D$ and the dose of the nucleation seeds. A linear fit of all colored symbols (black solid line) indicates that the interface energy ($\propto D\mathrm{\Delta }T$) of the nucleation seeds is inversely proportional to their ${\mathrm{Ga}}^{+}$ irradiation dose density $n_{\mathrm{Ga}+}$.