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Strain-gradient induced topological transition in bent nanoribbons of the Dirac semimetal Cd3As2

Dirac semimetal is an ideal parent state to realize various exotic states of matters, such as quantum spin Hall state, Weyl semimetal phase and Majorana zero modes. Topological phase transition allows for the switching between these different topological states. Here, in this paper, we exhibit experimentally an effective approach of inducing topological phase transition in Cd3As2 nanoribbons, by applying a bending strain profile onto the sample. The local strain varies linearly from compression to tension through the cross-section of a bent nanoribbon. The strain gradient causes obvious lattice deformation and breaks the C4 rotational symmetry, thus opening an energy gap at the Dirac points and making the bulk gapful. When further increasing the strain strength, the local strain effect dominates over the symmetry-breaking effect, where spatially-varying band shift becomes prominent across the nanoribbon. Our results demonstrate the effect of strain gradient on the evolution of energy band structures, which should be valuable for further study of strain-mediated topological phase transition.


INTRODUCTION
The combination of topology and physics has ushered a variety of quantum materials, such as topological insulators [1,2], Dirac semimetals [3][4][5][6], Weyl semimetals [3][4][5][6] and topological superconductors [1]. Topological Dirac semimetals (TDSs), featured with gapless Dirac cones in the bulk band structure, are of vital importance as a parent state of various other topological phases. For instance, reducing the material thickness would possibly trigger a phase transition from TDSs to topological insulators [7,8]. The quantum confinement effect in an ultrathin TDS film would lead to a crossover to a two-dimensional topological insulator, with the emergence of quantum spin Hall state [8,9]. Upon breaking the time-reversal symmetry via external magnetic field or magnetic exchange interaction, the original Dirac cone would split into Weyl cones, accompanied by the presence of magnetic Weyl semimetal phase [3][4][5][6]. Besides, when proximitized with s-wave superconductors, TDS is proposed to support topological superconducting phase and Majorana zero modes [10][11][12][13], which are significant for implementation of topological qubit and fault-tolerant quantum computation. TDS provides an ideal platform to study the phase transition between different quantum states, which is valuable for fundamental and application-based studies and has attracted current research interest.
As a prototypical TDS, Cd3As2 has two gapless Dirac cones along the direction protected by the C4 rotational symmetry [8]. Pristine Cd3As2 has demonstrated abundant novel features, including chiral anomaly [14], π Aharonov-Bohm oscillations [15,16], quantum Hall effect [9,17,18] and spin-polarized Fermi arc surface states [19][20][21]. In previous literatures on topological phase transitions in Cd3As2, widelyacknowledged useful methods include introducing interplay with external electromagnetic fields [22][23][24], or reducing sample dimensions to introduce quantum confinement effect [8,9]. Experimental works have demonstrated feasibility of magnetic field-driven topological phase transition in Cd3As2 into Weyl semimetals, quantum Hall insulators, or trivial insulators [9,14,17,18,25,26], and predictions that Dirac semimetal Cd3As2 can turn into a 2D topological insulator in an ultrathin film are also proposed [8]. Recently, an emergent type of mechanism that induces topological phase transitions in topological semimetals has been proposed, that is, introducing certain strain profiles onto the whole lattice [27][28][29][30]. Rather than introducing external electromagnetic fields or reducing the sample thickness, the strain-induced band modulation can be easily achieved during sample fabrication, which is more static and intrinsic as a sample property, and more intuitive in the symmetry analysis. The correlation between strain profile and gauge field theories further enriches the physical implication of relevant topological phase transitions [31][32][33][34].
In this work, we demonstrate the variation of electronic properties modulated by static bending strain applied on Cd3As2 nanoribbon devices through transport experiments. Compared to the case of straight nanoribbons, the bent nanoribbon displays some distinct features, i.e., a second turning point in the − curve and flat region in the − curve. This can be well understood by considering that the introduced strain gradient breaks the pristine lattice symmetry and opens a global bulk gap near the neutrality point. When further increasing the bending strength applied onto the nanoribbon, the local strain induced energy band shift dominates and renders the bulk gap closing.

METHODS
Cd3As2 nanoribbons were grown by chemical vapor deposition method as reported in our previous work [35]. Cd3As2 nanostructures with a typical scale of several tens of microns in length and several hundreds of nanometers in diameter were obtained and examined using scanning electron microscopy (SEM), as shown in Fig. 1 In the following, we mainly elaborate on the three typical devices, marked as device A, B and C (see Appendix A).  From the above perspective, the lattice deformation is varying spatially in a 3 nm 10 μm continuous manner, making the system a complex "parallel" configuration of many local lattice structures. Thus, a theoretical analysis of band evolution is necessary here to shed light on the transition of transport properties in a bent nanoribbon, serving as a guidance for experimental exploration. The topological semimetal nature of unbent Cd3As2 arises from the inverted bands due to strong spin-orbit coupling [8], as depicted in Fig. 2(c). The strength of coupling, as well as all the hopping amplitudes in the system, should be functions of lattice parameters. Therefore, the strain profiles applied on Cd3As2 nanoribbons in our experiment generate corrections to local lattice parameters, and directly changes the low energy band structures. By unidirectionally increasing or decreasing certain lattice parameters, the conduction and valence band shall accordingly move, as indicated by relevant DFT calculations [36,37], towards a lower or higher energy. In the case of a weak bending profile, a considerable energy gap induced by the breaking of C4 rotational symmetry shall emerge [8,25,37,38], while the overall energy shifts of bands remain sufficiently small. There may exist a global energy gap in the whole sample, and the system would exhibit insulating behavior when the Fermi level is tuned into the global gap [ Fig. 2(d)]. In the strongly bending case, however, the shift of energy band becomes more significant, and no universal gap for the whole sample exists [ Fig. 2(e)]. Thus, the Fermi level always intersects with energy bands, and the corresponding transport measurements manifest a p-type to n-type crossover in one single device. We believe that the bending gradient , which determines the size of gap induced by symmetry reduction, and the sample width , which relates to the maximal band shift, together decide the final energy band structure of the system.
structure. On the other hand, local strain modifies lattice parameters, which further induces slight energy shift in conduction and valence bands. (e) Case of strong bending strain. With the band shifts further, the system will enter an "indirect gapless" regime, and the Fermi level will always intersect with conduction or valence bands.

RESULTS AND DISCUSSION
The results discussed in the main text all come from the device A. Transport measurements were performed in three different bent regions of device A, i.e., region A1, A2 and A3 (see Appendix B). We firstly investigate on the bent region A1 (denoted by the red arrow in Fig. 2(a)). The local curvature radius and width of the bent region are 4.13 μm and 199 nm, respectively, corresponding to a maximal strain amplitude = 2.41%. The − curve corresponding to A1 is plotted in Fig. 3(a).
The − curve exhibits a clear insulating behavior in the high temperature region, where resistivity increases monotonically with decreasing . Such kind of insulating behavior of − curves in high temperature region has actually been observed in several semimetallic systems with high sample quality, including Cd3As2 (Ref. [39]) and ZrTe5 (Ref. [40]) for instance, which is not the point of novelty here in our results. The more noteworthy feature is the second turning point which emerges at a low temperature of around 10 K (see inset of Fig. 3(a)). In previous reports on − curves of topological semimetals, the resistivity increases to only one single maximum with decreasing temperature, then drops monotonically until the temperature approaches 0 K. In our result, the emergent second turning point at 10 K is a very distinctive phenomenon, and has implication on possible phase transitions occurring in the bent Cd3As2 nanoribbon. Especially, the insulating − behavior below 10 K may indicate the opening of a bulk gap in the energy band structure. The metal-insulator transition has been confirmed from − spectrum in many systems [41][42][43][44][45][46][47]. We also performed measurements on gate voltage dependence of the resistivity in the bent region, as shown in Fig. 3(b). Compared to the standard − curves obtained in the straight Cd3As2 nanowires [48], a conspicuously different feature can be observed here, that resistivity maximum no longer exhibits a single peak lineshape, but rather a plateau-like flat region, with plateau width approximately 20 V. The stable maximal resistivity value over a relatively wide range of gate voltage is consistent with the scenario of an energy gap in the bulk band structure [49][50][51]. Once the Fermi level is tuned into the gap, the bulk conduction is greatly suppressed and the overall conduction becomes dominated by other mechanisms, including disorders and surface states [52][53][54], which are little affected by gate voltage. Therefore, a relatively flat region between n-type and p-type conduction in − curve is experimentally observed here. In contrast, for an ideal Dirac semimetal system, the gapless band structure results in a sharp transition between n-and p-type conduction, and the resistivity near the Dirac point should therefore take the form of a narrow peak. Electron and hole mobility can also be extracted from − curve, which take the values = 1.59 × 10 4 cm 2 / (V ⋅ s) and ℎ = 3.81 × 10 2 cm 2 /(V ⋅ s), obviously reduced from the typical values in high-quality Cd3As2 nanostructures [55]. The flat region in our − curve as well as the reduced mobility thus reveal from a parallel aspect the breakdown of Dirac cone structure in the bent nanoribbon.
To better reveal the topological transition occurring in the bent nanoribbon, we also perform magnetotransport measurements on the bent region under an out-of-plane magnetic field . Gate-voltage dependence of the resistivity in the largely-bent nanoribbon region A2 was measured, as shown in Fig. 4(b). Interestingly, it is observed that the typical transfer curve of a gapless semimetallic system, featured with a single sharp resistivity peak, again emerges in the largely bent nanoribbon. From the transfer curve, we can obtain the carrier mobilities = 2.68 × 10 4 cm 2 /(V ⋅ s) and ℎ = 7.32 × 10 2 cm 2 /(V ⋅ s) , much enhanced compared to the weakly bent case A1. Resistivity oscillations are also observed for the largely-bent region, as shown in Fig. 4(c). The collective oscillation is a combination of multiple SdH oscillation patterns with different frequencies, as indicated by the fast Fourier transformation (FFT) spectrum in Fig. 4(d). The multiple-frequency oscillation pattern is also observed in the bent region A3 of the same nanoribbon, which has the similar bending strength and strain gradient as those of region A2 (see Appendix B).

FIG. 4. Transport measurements on the largely-bent region A2. (a) SEM image of
the region A2 from the same nanoribbon as that in Fig. 2  In device A, the bent region A3 has the similar curvature radius and maximal strain amplitude as those of region A2 described in Fig. 4. As expected, multiple-frequency oscillation pattern [Figs. 6(b) and 6(c)] is observed in the region A3. Figure 6(a) displays the evolution of resistivity as varying the temperatures. In striking contrast to case of bent region A1, the upturn below 10 K is missing in the region A3. Instead, the resistivity exhibits a monotonic decay as decreasing the temperature below 75 K. As