Enhanced flexoelectricity at reduced dimensions revealed by mechanically tunable quantum tunnelling

Flexoelectricity is a universal electromechanical coupling effect whereby all dielectric materials polarise in response to strain gradients. In particular, nanoscale flexoelectricity promises exotic phenomena and functions, but reliable characterisation methods are required to unlock its potential. Here, we report anomalous mechanical control of quantum tunnelling that allows for characterising nanoscale flexoelectricity. By applying strain gradients with an atomic force microscope tip, we systematically polarise an ultrathin film of otherwise nonpolar SrTiO3, and simultaneously measure tunnel current across it. The measured tunnel current exhibits critical behaviour as a function of strain gradients, which manifests large modification of tunnel barrier profiles via flexoelectricity. Further analysis of this critical behaviour reveals significantly enhanced flexocoupling strength in ultrathin SrTiO3, compared to that in bulk, rendering flexoelectricity more potent at the nanoscale. Our study not only suggests possible applications exploiting dynamic mechanical control of quantum effect, but also paves the way to characterise nanoscale flexoelectricity.


Supplementary Figure 1. Growth and electrical characterisation. a,
In situ growth monitoring employing reflection high-energy diffraction (RHEED). The growth mode of the metallic SrRuO3 (SRO) layer exhibited the typical transition from layer-by-layer to step-flow, as evident by the evolution of RHEED intensity (blue curve). The SrTiO3 (STO) layer, however, grew in a layer-by-layer manner. Here, we show representative RHEED intensity profiles collected during the growth of 30 unit cell-thick SRO and 11 unit cell-thick STO layers. The RHEED patterns obtained after growth (shown underneath the intensity profiles) feature sharp Bragg reflexes, indicative of atomically smooth surfaces. b, The piezoresponse force microscopy (PFM) phase image of an eleven unit cell-thick STO film taken after electrical poling with ±4V of applied bias. No phase contrast is discernible across the poled area, suggesting that our STO film was paraelectric. We obtained similar PFM phase images for the nine-and five-unit cell-thick films. Source data are provided as a Source Data file.

Supplementary Note 1
To extract barrier heights from the tunnelling I-V curves, we used an analytical equation describing direct tunnelling through trapezoidal tunnel barriers 1,2 : where c is a constant and α(V) ≡ [4d(2me . Also, b, me, d, and φ1,2 are the baseline, the electron mass, barrier width, and barrier height, respectively. Supplementary Figure 2. Tunnelling currents across a nine-unit cell-thick SrTiO3 film with increasing strain gradients. Tunnelling currents were measured as the applied strain gradient increased from 1.14 × 10 7 m -1 (a) to 1.7 × 10 7 m -1 (l). Figure 2 of the main text presents spectra a, f, and l. The solid red lines in Figures a-e indicate the fits to Supplementary Eq. 1.
Note that in a-c, we fitted the entire spectra (i.e., -1 V to +1 V), but we used smaller bias windows to fit the tunnelling currents of d and f. Source data are provided as a Source Data file.

Supplementary Note 2
We considered an STO thin film of thickness hf, with the top surface in contact with an AFM tip and the bottom interface coherently constrained by the substrate. At the top surface, the normal stress distribution (as a function of the distance from the contact center) is described by the Hertz contact mechanics of the spherical indenter, as follows: = 0 where stress is related to strain via This approach allows us to extend reliably the Hertz contact mechanics to the flexoelectric materials for obtaining stress/strain distribution under the force imparted by the tip.   slightly under compressive strain (decreasing the crystal volume) 3 . Also, according to our strain analysis ( Supplementary Fig. 4), pressing by the AFM tip decreased the physical thickness of STO by a few %. We thus incorporated strain-induced systematic changes into the tunnel barrier profiles (Supplementary Figs. 7a,b). However, even after these changes, the |I+1 V/I-1 V| values increased only negligibly (Supplementary Fig. 7c). Therefore, any effect of strain per se does not explain our experimental observations.   Using the one-dimensional WKB approximation, we can simply describe the tunnelling current density for a low T and small V, as follows: where T(E), f(E), and U(x) represent the transmission probability, the Fermi-Dirac distribution, and the tunnel barrier profile, respectively.
Using Supplementary Eq. 4, we obtain the tunnelling current density for a trapezoidal barrier profile ( Supplementary Fig. 8), as follows:  Using Supplementary Eq. 4, we obtain the tunnelling current density for a triangular barrier profile ( Supplementary Fig. 9), as follows: Finally, using Supplementary Eqs. (5)(6)(7)(8), we obtain tunnelling I-V curves for systematically modified tunnel barrier profiles ( Supplementary Fig. 10). In these calculations, we assume d = 3.5 nm.  To understand how the strain affects the band structure of SrRuO3 (SRO) and

Supplementary
subsequently the tunnelling transport, we additionally performed first-principles DFT calculations. We fixed the in-plane lattice parameter of SRO to that of STO substrate, and imposed compressive strain u33 (ranging from 0 to -8%) in the out-of-plane direction. This assumption closely accounts for the strain distribution, obtained from the phase-field simulations ( Supplementary Fig. 4). As shown in the Supplementary Figs. 11a Fig. 11c). Thus, we conclude that the effect of strain on SRO is not significant. A nonlinear flexoelectric response could arise under large strain gradients, as demonstrated in several material systems 7,8 . In the case of a centrosymmetric material like STO, the quadratic flexoelectric term should be zero, so we additionally considered the cubic flexoelectric term, i.e., P/ε = f•(∂ut/∂x3) + g•(∂ut/∂x3) 3 , where f and g are the first-order and third-order flexocoupling coefficients. For simplicity, by assuming f = 2.6 V (i.e., bulk flexocoupling coefficient) 9 , we fitted our data and found that g is minuscule, as small as 3.8 × 10 -14 V m 2 ( Supplementary Fig. 13). However, when ∂ut/∂x3 is huge, e.g., much larger than 10 5 m -1 , the effective flexocoupling coefficient, i.e., feff = (P/ε)/(∂ut/∂x3) = f + g•(∂ut/∂x3) 2 , might become significantly enhanced.

Supplementary
Supplementary Figure 13. Analysis under the assumption of the third-order flexoelectricity. Filled symbols are calculated from Fig. 4a in the main text. The error bars represent the standard deviations of the total electric field, (φ2 -φ1)/ed, calculated by fitting the tunnelling spectra in the Supplementary Figs. 2 a-e to the Supplementary Eq. 1. The gray line shows a fit to f•(∂ut/∂x3) + g•(∂ut/∂x3) 3 with assuming f = 2.6 V. Bragg peak from the STO film. Therefore, this data suggests that our ultrathin STO barrier layer is strain-free. Source data are provided as a Source Data file