Periodic Reporting for period 3 - ENIGMA (EXPLORING NONLINEAR DYNAMICS IN GRAPHENE NANOMECHANICAL SYSTEMS) Reporting period: 2021-11-01 to 2023-04-30 Summary of the context and overall objectives of the project Nanomechanical systems are being adopted in billions of products that address a wide range of sensor and actuator applications in modern technology. The discovery of graphene and the ability to fabricate atomically thin membranes revolutionised the field of nanotechnology and made it possible to make miniaturised resonators that are extremely sensitive to electrical signals and forces. But, soon after its discovery, it was realised that reaching the holy grail in sensing with graphene is associated with severe constraints over its linear dynamic range, since signatures of nonlinearities already emerge at amplitudes that are only a few nanometers. This nonlinear behavior that is considered as a nuisance in the design of graphene nanomechanical systems brings with itself a new unexplored field in nonlinear dynamics, where dynamics of single atom thick membranes is entangled with complex phenomena such as thermal fluctuations and nanoscale forces that are almost non-existent in other mechanical systems. These unique sources of complexity not only affect dynamics but also make mechanical characterization of these novel materials very challenging.The goal of the ERC starting grant ENIGMA is to provide full understanding of nonlinearities in graphene nanomechancial systems in order to avoid them when unwanted while utilising them efficiently when desirable. To realize this vision, ENIGMA focuses on the development of (1) novel numerical and experimental methods that capture nonlinearities and provide insight on their influence on the mechanics of atomically thin membranes; (2) novel methodologies that use nonlinearities for mechanical characterisation of such membranes.Successful completion of the project will contribute to the development of atomically thin resonators with applications in sensing, frequency referencing, time keeping and electronic filtering that are of paramount importance to micro and nanotechnology. Besides graphene, in a longer run, the methodologies developed within the project will also pave the way to exploring the rich dynamics of other two-dimensional nanomaterials like transition metal dichalcogenides, black phosphorous, and hexagonal boron nitride, that are gaining increased interest for applications in the semiconductor industry. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The development of the nonlinear dynamic methodology for graphene nanomechanical systems is associated with 4 intertwined research themes, namely R1. Geometric and material nonlinearities; R2. Opto-thermal effects and thermal fluctuations; R3. Nonlinear dissipation processes; R4. Nanoscale forces.In the period leading to this report all the research themes were tackled with success.In research theme R1, new nonlinear models were developed and successfully implemented on experimental data for estimating the Young’s modulus of sealed graphene membranes and probing highly anisotropic properties of 2D As2S3 layers. Moreover, molecular dynamic simulations were performed to support experimental findings on the interrelation between phonons and thermal bath in nanostructures and how such relation affect's Young's modulus at the nanoscale. In research theme 2, new methodologies and numerical models were built to study the high frequency stochastic switching of nonlinear graphene resonators as a means to boost weak signals in the audible range. Moreover, reduced-order models were built to capture the complex nonlinear dynamics that atomically thin membranes exhibit when driven opto-thermally.Research theme R3 is closely linked with R2 and mainly focuses on understanding the dissipation processes of atomically thin membranes. Our preliminary works related to this theme was quite successful and led to a numerical model and experimental procedure that shed light on the role of mode coupling on the nonlinear dissipation processes of graphene nanomechancial systems. As a continuation of this topic we are currently working towards tuning mode coupling in graphene not only to better understand the routes of nonlinear dissipation but also to unravel new phenomena arising from modal interactions at the nanoscale. In addition, the low-quality factor of graphene based nanomechancial resonators in vacuum, made us also look into other dissipation pathways. One interesting route was found to be eddy current damping which was successfully probed in diamagnetically levitating graphite plates.Finally related to research theme 4 a new interferometry set-up was built and was successfully utilized for probing nanoscale forces of graphene nanomechanical systems. Here, we focused on measuring the forces that are generated by gas molecules passing through nanopores in graphene, and also developed a protocol for measuring the nanomotion of graphene in real-time through a nonlinear optical field. Our first estimates showed that the later can even be used for measuring tiny forces that are generated by micro-organisms. In this framework, new collaborations were initiated with Prof. Cees Dekker from TU Delft, expert in bio-nanoscience, to validate this hypothesis. The primary results of this study led to an ERC Proof-of-Concept grant. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The activities of this period went beyond the state of the art. This includes (i) Methods and models that were successfully used for capturing nonlinear response of graphene membranes leading to multiple publications in Nano Letters and ACS Nano; (ii) Methods for clarifying the microscopic origins of nonlinear dissipation leading to a publication in Nature Communications; (iii) Molecular dynamic simulations and theoretical models based on thermal dynamics of the lattice in nanostructures to support experimental findings on the interrelation of elasticity and thermal bath that was published in physical review letters; (iv) New experimental procedure for measuring the tiny forces that are exerted on graphene that led to an ERC PoC.Until the end of the project I expect the following breakthrough additions based on the ongoing activities: (i) new theoretical and experimental works to investigate the role of wrinkles in atomically thin membranes (related to theme R1); (ii) clarifying the role of nonlinearities in mechanical sensing and their interplay with noise (related to theme R2) (iii) methods and models for tuning and capturing modal interactions in 2D membranes (related to theme R3); (iv) Finalizing the current studies on probing nanoscale forces on graphene(Related to theme R4). I expect that successful completion of the latter particularly will have great societal impact with applications to life sciences and antibiotics research, also relevant to my ERC PoC.