Atomically Thin 1D Loop in Nanoscale

The Aharonov-Bohm oscillations observed in HfO2 and Al2O3 films have a period B0 ∼ 13—17 T for pressure-metallized samples and ∼ 15—20 T for voltage-metallized samples. These large periods correspond to nanosized loops, which proved to be the smallest one ever found. To see this, one can directly estimate the loop radiusrfrom the interference relation,

B0πr2 =h/2e= 2.07×10−15 T·m2. For B0 = 15 T, it is r = 6.6 nm. In the literature of Aharonov-Bohm oscillation of the WL/WAL origin, B0 in the hollow cylinder experiment of Sharvin and Sharvin is much smaller, of the order of 10−3 _{T, which corresponds to}

r = 0.8 µm consistent with the radii of the cylinders used in the experiments [36, 40, 45]. Nanoscale oscillation has been observed, in a multi-walled carbon nanotube [46], which features a period of 9 T, corresponding to a radiusr ∼8.6 nm. (Recently, h/e oscillations were also reported in nano wires made of Bi2Se3, a topological insulator, with a feature size of several hundreds of nm [112].) In fact, since the strongest field of DC magnets in the world is currently limited to 45 T, being able to provide a period of 90 T for the Aharonov- Bohm oscillation, the smallest radius that can be measured is capped at 2.7 nm. This is

only a little smaller than the smallest loops (3.6 nm) suggested by our data.

One remarkable feature of the oscillations in this study is how little attenuation they suffer. When the magnetic field is normal to the film surface, the amplitude of the Aharonov-Bohm oscillation in HfO2 decreases by only 5% and 1% after one period in pressure-metallized (Fig. 4.4b) and voltage-metallized films (Fig. 4.3b), respectively. This is the cleanest and least attenuated Aharonov-Bohm oscillation ever reported. In comparison, the amplitude decay is over 20% in cylindrical Mg and Li [36, 40] and over 50% in multi-walled carbon nanotubes. Such small attenuation is possible if and only if the loop is made of an atomically thin wire, i.e.,W ∼0 in Eq. (1.33), and if there is no misorientation between the loop normal and the field, i.e.,θ= 0◦. It was not possible to perfectly align the loop with the field in our experiment at the SCM1 station at the National High Magnetic Field Laboratory, because there is no 3D goniometer there. Yet, despite some misorientation that must have existed, very little attenuation was seen. This implies W ∼ 0 in our loop. Since the same fitting for different sets of W and θ is obtained if the same W∗ is used in Eq. (1.34), it is W∗

that sets the upper limit of W given the fact that the exact θis not known in our analysis. The value ofW∗ calculated from above is 0.9 nm for the 5% attenuated oscillation and 0.6 nm for the 1% attenuated oscillation, seen respectively in pressure- and voltage-metallized HfO2. Therefore, there are indeed very small and about the atomic scale.

4.4.2. 3D Mesh vs. 1D Filament in HfO2 and Al2O3

The picture of Fig. 4.9 is proposed to account for (a) the positive MR cusp at low field (b) the Aharonov-Bohm oscillation at high fields, and (c) 1D-like QCC at low temperature including the presence of Tmax and Tsat, in voltage-metallized HfO2 and Al2O3 films. It contains a 3D mesh network with at lease some reduced metal ions in series with some 1D filamament(s) and a single dominant loop. (Smaller loops may also be present, but it is the largest loop that is responsible for the Aharonov-Bohm oscillation observed. The loop is also likely to be the most resistive part of the conducting path, thus more prominently featured in the resistance measurement.) The development of these nanostructures follows

the sequence of decreasing resistance: With decreasing resistance, the loop becomes more prominent (i.e., larger in radius), and the mesh more electrically reduced (i.e., the WAL cusp more pronounced). This is in concert with more total-current passage during the voltage-metallization process. Meanwhile, there is no metal ion reduction and presumably no 3D mesh in pressure-metallized samples, since in these samples only the Aharonov-Bohm oscillations were observed.

These results remind us of another set of experiments, namely the fracture tests described in [103] and Appendix B. Although both pressure and voltage can metallize films to endow comparable resistance distributions and performance to metallized devices, the statistics of the conducting paths in these metallized devices are quite different and depending on fracture tests. After severing one metallized HfO2 or Al2O3 cell into two halves, we found conductive paths in only one half of the pressure-metallized sample while the other half has the same resistance as the virgin sample. In contrast, both halves of the voltage- metallized samples have resistances much lower than the virgin one, only one half is orders of magnitude less resistive than the other half. This indicates that in a voltage-metallized sample, some conductive paths do exist in both halves. Combining these statistics and the magnetoresistance data, we can associate the clean Aharonov-Bohm oscillations seen in pressure-metallized samples with the halves that have the low resistance. Moreover, it is clear that in these samples the single loop (and very likely, the filament as well) exists in such low-resistance half, while the other half has no conducting paths at all. In contrast, in voltage-metallized samples, while the single loop (and likely too, the filament) again must exist only in one half, it does contain some partially reduced 3D mesh, and the other more resistive half probably also has some 3D meshes to provide conductivity although they are yet to develop into highly conducting filaments. These assignments are made assuming the development of the mesh, the loop and electrical reduction follows the sequence depicted in Fig. 4.9, which seems to be reasonable since in metallization a lower resistance is undoubtedly associated with a later stage in the evolution.

The above picture may be used to interpret the data in Tables 4.3-4.4 and Figs. 4.5-4.8. In HfO2, the initial filaments formed at the onset of voltage-metallization (Fig. 4.9a) may be rather weak, in that they still see relatively a strong random field and allow relatively little conductivity (hence large resistance corresponding to cells (5) and (6) in Table 3), and any loop they contain is too small to be seen (oscillation having a large period in Fig. 4.7.) They also contain relatively few reduced cations, which is why sample (5) gives a weak cusp in Fig. 4.7e and why its still strong 1D WL can overwhelm 1D WAL resulting in the “+” slopes in Fig. 4.5. As metallization continues (Fig. 4.9b-c), the 3D mesh spreads and the random field along the 1D filaments clears up, so an effectively larger loop forms, which has the overall effect of lowering the device resistance (corresponding to cells (4)-(1) in Table 4.3) and allowing a faster oscillation with more periods captured between−18 T and +18 T. Meanwhile, as more cations become reduced, a stronger 3D WAL cusp develops in Fig. 4.7a and the 1D WAL in the filaments now takes over leading to the “−” slopes in Fig. 4.5; it also imparts an opposite phase to the MR oscillation in Fig. 4.7b. This picture is also applicable to voltage metallized Al2O3, although cation reduction is less severe in Al2O3 (Al+3 is more difficult to reduce according to the Ellingham diagram; in addition, we used current compliance for Al2O3 metallization) and its spin-orbit interaction is weaker (Al’s atomic number is much lower than Hf’s.) This explains its weaker WAL; it may even be absent in the 1D filament as we did not observe any “−” slopes in Table 4.4. Neither did we see any WAL oscillation in Fig. 4.8.

This picture in Fig. 4.9 is not inconsistent with the TEM images seen in TiO2 [59] and ZrO2 [61] that are voltage metallized. In both cases, a cone-shaped conducting path thinning down to a filament near the positive electrode was observed. (In the so-called conducting- bridge ReRAM or CBRAM, in which electrode metal electro-migrates into the film, similar TEM images have also been observed [60].) These filaments are narrower and 1D-like near the positive-voltage electrode. A related picture of cone-shaped filaments has also been presented in several simulation models [113, 114, 115]. However, our picture differs from the past work in one important respect: All past work relied on the ionic transport mecha-

nism for filament formation and resistance switching, but this mechanism was unequivocally refuted by our pressure-metallization experiment, , which forms highly conductive samples with a single loop each. Since voltage-metallization produces a single, relatively large loop only at the end of the process, whereas pressure-metallization always produces the same sized loop regardless of pressure, it is clear that the latter method, which involves no ion diffusion and its efficacy is only dependent on reserving the electron-phonon interaction to clear the trapped charge, is much more effective/efficient. Therefore, we suggest that overcoming the electron-phonon interaction to remove the trapped charge is also the critical step in voltage-metallization, and cation reduction is just the side product that unavoidably accompanies voltage metallization that entails a large (integrated) current. If so, redox re- actions and ion migration are only secondary steps of the metallization transition: Whether achieved by applying a pressure or a voltage, the transition is most critically controlled by the formation of 1D filaments with a single loop free of trapped charges. Because much voltage is used to reduce cations instead of clearing trapped charge, and cation reduction happens to more sites, voltage metallization is less efficient producing more mesh first before forming the critical loop/filament.