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ADVERTIMENT. Lʼaccés als continguts dʼaquesta tesi queda condicionat a lʼacceptació de les condicions dʼús establertes per la següent llicència Creative Commons: http://cat.creativecommons.org/?page_id=184 ADVERTENCIA. El acceso a los contenidos de esta tesis queda condicionado a la aceptación de las condiciones de uso establecidas por la siguiente licencia Creative Commons: http://es.creativecommons.org/blog/licencias/ WARNING. The access to the contents of this doctoral thesis it is limited to the acceptance of the use conditions set by the following Creative Commons license: https://creativecommons.org/licenses/?lang=en Quantum transport in Solid state devices for Terahertz frequency applications Devashish Pandey Department of Electronic Engineering Universitat Aut´onomade Barcelona (UAB) Supervisor Xavier Oriols Pladevall and Guillermo Albareda Piquer In partial fulfillment of the requirements for the degree of PhD in Electronic Engineering December, 2020 Acknowledgements I would like to thank all the people who directly or indirectly con- tributed to this PhD thesis. First and foremost I would like to thank Dr Xavier Oriols for giving me this opportunity to work under him and helping me in all accounts to settle comfortably after my arrival to Spain. Undoubtedly, I could not have expected a better guide and teacher as Xavier. Apart from a lot of quantum mechanics and quan- tum electron transport related issues he has taught me some crucial things that cannot be learned from a book like research ethics, rigour, the philosophy of physics and above all humility. He somehow taught me a very important thing that the only limit is you. Then, I would like to thank Dr Guillermo Albareda for accepting to be my supervisor and providing his exceptional efforts in raising the quality of my work. He carries a gift of precision be it in conceptual arguments, in MATLAB coding or in research writing. I really learned a lot of this from him particularly, critical thinking and brainstorming. He has been always very kind and patient with my questions and problems. I also want to express my gratitude to the faculty members of the electronic engineering department for their kindness and support. I specially want to thank Dr David Jim´enezand Dr Jordi Bonache for their concern towards me. I want to thank Dr Ferran Martin for creating a joyful and disciplined environment in the department. I am very thankful to Dr Arantxa for solving numerous issues of mine specially related to my PhD submission. I also thank Javier Hell´ın and Javier Gutierrez for solving many of my technical and bureaucracy related problems respectively. Next I would like to thank my dear friends at UAB. Nikos and Ioanna have been like a family member to me and I thank them for their support and good wishes. I really enjoyed the company of Anibal and Noe and thank them for their warmth and support. A special thanks also to Carlos Couso, Marcos and Pau for going out of the way to help me during my initial time in Spain. I also thank Ertugrul, Laura, Matteo, Laura (Italy), Enrique, Zhen, Pedro, Ferney, Jan, Ana, Marti, Jonatan, Amilie, Marta, Li-Juan, Gerard, Ferran, Roushi for being kind to me. And last but not the least I am very grateful to my wife for her unconditional love and support in everything I do, my parents my kind relatives, and my chuddie buddies in India. Dedicated to all my teachers (gurus) in past and present Abstract The work presented in this thesis is dedicated to the understanding of practical and conceptual challenges in simulating dynamical prop- erties beyond the quasi-static approximation, in solid-state quantum devices in scenarios where a full quantum mechanical treatment is necessary. The results of this thesis are particularly relevant for the computation of the fluctuations of the electric current in the THz regime which aids in determining the correlations, the evaluation of tunnelling times that define the cut-off frequency of high-frequency operated devices, or the assessment of thermodynamic work to realize quantum thermal engines. The above mentioned dynamical properties involve multi-time mea- surements and hence are sensitive to quantum backaction. In the context of Orthodox quantum mechanics, the definition of these dy- namical properties cannot be detached from the specification of the measurement apparatus. That is, defining apparatus-independent or intrinsic dynamical properties of quantum systems is incompatible with the postulates of Orthodox quantum mechanics. All in all, a device engineer like me, working on practical problems related with the present and future solid-state devices, is forced to delve into the foundations of quantum mechanics if I really want to properly understand the high-frequency performance of solid-state de- vices. In this regard, I will show that the difficulties associated to the understanding of dynamical properties can be solved by looking be- yond Orthodox quantum mechanics. In particular, I have explored the modal interpretation of quantum mechanics, which is a mathemati- cally precise quantum theory that reproduces all quantum mechanical phenomena. I will show that intrinsic properties can be easily defined in this new (non-orthodox) context. Importantly, I will prove that intrinsic properties can be identified with weak values and hence that they can be measured! Focused on a particular modal theory, viz., Bohmian mechanics, an electron transport simulator will be discussed and applied to address both methodological and practical issues related to the simulation of quantum electron transport. The ontology of Bohmian mechanics naturally enables describing continuously monitored open quantum systems with a precise description of the conditional states for Marko- vian and non-Markovian regimes. This helps to provide an alternate to density matrix approach in the description of open quantum sys- tems, which scales poorly computationally with the number of degrees of freedom. Thus the Bohmian conditional state strategy, which has led to the development of an electron transport simulator, BITLLES will be shown to compute the dwell times for electrons in a two-terminal graphene barrier. It will be demonstrated that Bohmian trajectories are very appropriate to provide an unambiguous description of transit (tunneling) times and its relation to the cut-off frequencies in prac- tical electron devices. Finally, a protocol incorporating collective-like measurements to evade the current measurement uncertainty in the classical and quantum computing electron devices will be discussed. Contents I INTRODUCTION 1 1 From steady state to time resolved quantum transport 2 II THEORETICAL OUTLOOK 11 2 Are electrons there when nobody looks? 12 2.1 Wavefunction perturbation due to measurement and origin of back- action . 13 2.2 Ensemble expectation values in Orthodox theory . 16 2.2.1 One-time expectation value . 17 2.2.2 Two-time expectation values . 18 2.3 Why intrinsic dynamical properties can not be defined in Orthodox theory? . 24 3 Looking for intrinsic properties of quantum systems in non-Orthodox theory 31 3.1 Defining intrinsic properties from modal theory . 33 3.2 Bohmian mechanics as a special case to modal theory . 36 3.3 Interpretation of AAV weak values . 40 3.4 Computing quantum high frequency noise from intrinsic properties 43 vi CONTENTS III METHODOLOGY DEVELOPMENT FOR DEVICE SIMULATION 48 4 Conditional wavefunction approach to many-body correlations 49 4.1 The Stochastic Schr¨odinger Equation (SSE) . 51 4.2 Markovian and non-Markovian regimes . 53 4.3 Bohmian conditional wavefunctions . 54 4.3.1 Application to electron transport . 56 5 GC-TDSE methodology for geometric correlations 60 5.1 Motivation behind the method development . 60 5.2 Methodology . 61 IV APPLICATION TO DEVICE SIMULATION 65 6 Tunneling time in Graphene devices operated at high-frequencies 66 6.1 Motivation of the work . 66 6.2 Numerical calculation of tunnelling times . 70 7 Evading measurement uncertainty in electron devices 73 7.1 Origin of Quantum measurement uncertainty in electronic devices 73 7.2 Collective-like measurement of current in electron devices . 77 V CONCLUSIONS 80 VI APPENDICES 88 A The von-Neumann measurement protocol 89 vii CONTENTS B Two-time correlation functions for an ideally-weak measurement 93 C The perturbed state for an ideally weak measurement 95 D The weak measurement as a displacement operator 97 E Two-time correlations for special operators 103 E.1 Two-time correlation for position operator . 105 E.2 Two-time correlation for position squared operator . 108 F Derivation of AAV weak value 112 G Bohmian trajectories of a Harmonic Oscillator 114 G.1 Intrinsic Bohmian trajectories . 114 G.2 Measured Bohmian trajectories . 115 H Examples of utility of intrinsic properties 119 H.1 The quantum work distribution . 119 H.2 The quantum dwell time . 124 VII COMPENDIUM OF PUBLICATIONS 128 References 248 viii Publications related to this thesis (Compendium of publications included in this thesis) 1. Devashish Pandey, Xavier Oriols, and Guillermo Albareda. From micro- to macrorealism: addressing experimental clumsiness with semi-weak mea- surements. New Journal of Physics, jun 2020 2. Devashish Pandey, Enrique Colom´es,Guillermo Albareda, and Xavier Oriols. Stochastic Schr¨odingerEquations and Conditional States: A Gen- eral Non-Markovian Quantum Electron Transport Simulator for THz Elec- tronics. Entropy, 21(12), 2019 3. Devashish Pandey, Xavier Oriols, and Guillermo Albareda. Effective 1D Time-Dependent Schr¨odingerEquations for 3D Geometrically Corre- lated Systems. Materials, 13(13), 2020 4. Devashish Pandey, Matteo Villani, Enrique Colom´es,Zhen Zhan, and Xavier Oriols. Implications of the Klein tunneling times on high frequency graphene devices using Bohmian trajectories. Semiconductor Science and Technology, 2018 5. Devashish Pandey, Laura Bellentani, Matteo Villani, Guillermo Al- bareda, Paolo Bordone, Andrea Bertoni, and Xavier Oriols. A Proposal for Evading the Measurement Uncertainty in Classical and Quantum Com- puting: Application to a Resonant Tunneling Diode and a Mach-Zehnder Interferometer.
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