The critical size limit of electric polarization remains a fundamental question in nanoscale ferroelectric research 1 . As such, the viability of ultrathin ferroelectricity greatly impacts emerging low-power logic and nonvolatile memories 2 . Size effects in ferroelectrics have been thoroughly investigated for perovskite oxides -the archetypal ferroelectric system 3 . Perovskites, however, have so far proved unsuitable for thickness-scaling and integration with modern semiconductor processes 4 . Here, we report ultrathin ferroelectricity in doped-HfO2, a fluorite-structure oxide grown by atomic layer deposition on silicon. We demonstrate the persistence of inversion symmetry breaking and spontaneous, switchable polarization down to 1 nm. Our results indicate not only the absence of a ferroelectric critical thickness, but also enhanced polar distortions as film thickness is reduced, contradictory to perovskite ferroelectrics. This work shifts the focus on the fundamental limits of ferroelectricity to simpler transition metal oxide systems -from perovskite-derived complex oxides to fluoritestructure binary oxides -in which 'reverse' size effects counter-intuitively stabilize polar symmetry in the ultrathin regime.Ferroelectric materials exhibit stable states of collectively ordered electrical dipoles whose polarization can be reversed under an applied electric field 5 . Consequently, ultrathin ferroelectrics are of great technological interest for high-density electronics, particularly field-effect transistors and nonvolatile memories 2 . However, ferroelectricity is typically suppressed at the few nanometer scale in the ubiquitous perovskite oxides 6 . First-principles calculations predict six unit cells as the critical thickness in perovskite ferroelectrics 1 due to incomplete screening of depolarization fields 3 . Atomic-scale ferroelectricity in perovskites often fail to demonstrate polarization switching 7,8 , a crucial ingredient for application. Furthermore, attempts to synthesize ferroelectric perovskite films on silicon 9,10 are plagued by chemical incompatibility 4,11 and high temperatures required for epitaxial growth. Since the discovery of ferroelectricity in HfO2-based thin films in 2011 12 , fluorite-structure binary oxides (fluorites) have attracted considerable interest 13 as they enable lowtemperature synthesis and conformal growth in three-dimensional (3D) structures on silicon 14,15 , thereby overcoming many of the issues that restrict its perovskite counterparts in terms of complementary metal-oxide-semiconductor (CMOS) compatibility and thickness scaling 16 .

Ferroic-order parameters are useful as state variables in non-volatile information storage media because they show a hysteretic dependence on their electric or magnetic field. Coupling ferroics with quantum-mechanical tunnelling allows a simple and fast readout of the stored information through the influence of ferroic orders on the tunnel current. For example, data in magnetic random-access memories are stored in the relative alignment of two ferromagnetic electrodes separated by a non-magnetic tunnel barrier, and data readout is accomplished by a tunnel current measurement. However, such devices based on tunnel magnetoresistance typically exhibit OFF/ON ratios of less than 4, and require high powers for write operations (>1 × 10(6) A cm(-2)). Here, we report non-volatile memories with OFF/ON ratios as high as 100 and write powers as low as ∼1 × 10(4) A cm(-2) at room temperature by storing data in the electric polarization direction of a ferroelectric tunnel barrier. The junctions show large, stable, reproducible and reliable tunnel electroresistance, with resistance switching occurring at the coercive voltage of ferroelectric switching. These ferroelectric devices emerge as an alternative to other resistive memories, and have the advantage of not being based on voltage-induced migration of matter at the nanoscale, but on a purely electronic mechanism.

Two methods commonly used to measure the normal spring constants of atomic force microscope cantilevers are the added mass method of Cleveland et al. ͓J. P. Cleveland et al., Rev. Sci. Instrum. 64, 403 ͑1993͔͒, and the unloaded resonance technique of Sader et al. ͓J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 ͑1999͔͒. The added mass method involves measuring the change in resonant frequency of the fundamental mode of vibration upon the addition of known masses to the free end of the cantilever. In contrast, the unloaded resonance technique requires measurement of the unloaded resonant frequency and quality factor of the fundamental mode of vibration, as well as knowledge of the plan view dimensions of the cantilever and properties of the fluid. In many applications, such as frictional force microscopy, the torsional spring constant is often required. Consequently, in this article, we extend both of these techniques to allow simultaneous calibration of both the normal and torsional spring constants. We also investigate the validity and applicability of the unloaded resonance method when a mass is attached to the free end of the cantilever due to its importance in practice.

Mapping energy transformation pathways and dissipation on the nanoscale and understanding the role of local structure on dissipative behavior is a grand challenge for imaging in areas ranging from electronics and information technologies to efficient energy production. Here we develop a novel Scanning Probe Microscopy (SPM) technique in which the cantilever is excited and the response is recorded over a band of frequencies simultaneously rather than at a single frequency as in conventional SPMs. This band excitation (BE) SPM allows very rapid acquisition of the full frequency response at each point * Corresponding author, sergei2@ornl.gov 2 (i.e. transfer function) in an image and in particular enables the direct measurement of energy dissipation through the determination of the Q-factor of the cantilever-sample system. The BE method is demonstrated for force-distance and voltage spectroscopies and for magnetic dissipation imaging with sensitivity close to the thermomechanical limit. The applicability of BE for various SPMs is analyzed, and the method is expected to be universally applicable to all ambient and liquid SPMs.