High temperature quantum transport in graphene/hexagonal-boron nitride heterostructures A Thesis submitted to Lancaster University for the degree of Doctor of Philosophy in the Faculty of Sciences and Engineering 2017 R. Krishna Kumar Department of Physics ` Contents Abstract ..................................................................................................................................... 4 Copyright Statement ................................................................................................................ 6 Declaration................................................................................................................................ 7 Acknowledgements .................................................................................................................. 8 Chapter 1 - Introduction ....................................................................................................... 11 Chapter 2 - Electron transport in graphene/hexagonal-boron nitride heterostructures 14 2.1 The Electronic properties of Graphene and hexagonal boron-nitride ............................ 15 2.2 Quasi-ballistic, Drude-like transport in Graphene ......................................................... 20 2.3 Ballistic transport in Graphene ....................................................................................... 22 2.4 Electron-phonon scattering in graphene ......................................................................... 27 Chapter 3 – Hydrodynamic electron transport in graphene ............................................. 28 3.1 Hydrodynamics .............................................................................................................. 29 3.2 Hydrodynamics in the electron liquid ............................................................................ 30 3.3 Electron-electron scattering in graphene ........................................................................ 33 3.4 Viscous electron flow in graphene ................................................................................. 35 Chapter 4- Electron transport in a periodic potential and magnetic field ....................... 38 4.1 Nearly free electrons in crystals – Bloch electrons ........................................................ 39 4.2 Electrons in magnetic fields – Landau quantization ...................................................... 42 4.3 Shubnikov de-Haas oscillations ..................................................................................... 44 4.4 Magnetic translation group............................................................................................. 45 4.5 Graphene/hBN superlattice ............................................................................................ 48 Chapter 5 – Experimental techniques .................................................................................. 51 5.1 Device fabrication .......................................................................................................... 52 5.2 Transport Measurements ................................................................................................ 58 Chapter 6 Viscous electron whirlpools ................................................................................ 60 ` Chapter 7 Super ballistic flow of viscous electron fluids.................................................... 90 Chapter 8 Brown-Zak Oscillations..................................................................................... 110 Chapter 9 Summary & Outlook ......................................................................................... 138 9.1 Viscous electron flow in graphene ............................................................................... 138 9.2 Electron transport in Brown-Zak minibands ................................................................ 140 9.3 Closing remarks............................................................................................................ 141 Bibliography ......................................................................................................................... 142 ` Abstract The past decade has seen a new paradigm in solid state physics, where a new class of layered crystals can be thinned down to a monolayer and exhibit drastic changes in their electronic and optical properties in comparison to their bulk counterpart. Graphene was the first, and certainly most outstanding, of this set of so called two-dimensional (2D) materials. Aside from its obvious appeal which earnt its discovery the 2010 Nobel Prize, the electronic properties of graphene are truly unique. Perhaps the most familiar is its linear electron dispersion which hosts quasi-particles that obey the Dirac equation. This has enabled the study of a plethora of transport phenomena, as well as the realisation of novel device architectures that will be used in the next generation electronics. In general, experimental signatures of electron transport are most prominent at liquid helium temperatures when lattice vibrations are weak, for example in quantum hall physics. In this Thesis, we explore the regime of intermediate temperatures where the physics of interest is strongest between 100 and 300 K. Equipped with the state of the art high quality graphene samples, we demonstrate novel electron transport unique to graphene. The experimental work consists of two themes. In the first work, we study hydrodynamic electron flow in graphene encapsulated with hexagonal boron nitride devices. At elevated temperatures, electron-electron collisions become significant, and the electron viscosity starts to influence the steady state current distribution in a variety of surprising ways. In the first work, we perform transport experiments on standard graphene hall bars in a unique measurement geometry which allows the detection of negative non-local voltages intrinsic to viscous flow. In another experiment, we study viscous electron flow through graphene nano-constrictions/classical point contacts. Here, we observed anomalous temperature dependence in the conductance measured across the constriction. Specifically, the conductance increases with increasing temperature and even exceeded the semi-classical limit which is expected for single-particle ballistic transport. The underlying mechanism originates from electron-electron collisions, which, counter-intuitively, act to enhance current flow. In the second work, we slightly change our experimental system by studying magneto transport in a graphene/hexagonal boron nitride superlattice. Owed to the large periodicity of the superlattice unit cell, these devices have allowed experimental observation of the long sought Hofstadter butterfly, which addresses the electronic dispersion of electrons in a periodic potential and magnetic field. ` Here, we again go to elevated temperatures, where all the spectral gaps related to Hofstadter butterflies are completely smeared, and instead find a new type of quantum oscillation. These new oscillations are periodic in 1/B with a frequency corresponding to one flux quantum piercing the superlattice unit cell. Whilst these oscillations are related to Hofstadter physics, they are in fact more primal in origin. The most fascinating feature is their robustness with respect to increasing temperature. The oscillations are easily observable at room temperature in fields as low as 3 T and still remained prominent at 373 K, the boiling point of water ` Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. 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Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s policy on Presentation of Theses. ` Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. ` Acknowledgements First of all, I would like to thank