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Observation of Cooper Pair Splitting and Andreev Bound States in Carbon Nanotubes

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Observation of Cooper Pair Splitting and Andreev Bound States in Carbon Nanotubes Observation of Cooper Pair Splitting and Andreev Bound States in Carbon Nanotubes Inauguraldissertation zur Erlangung der W¨urdeeines Doktors der Philosophie vorgelegt der Philosophisch-Naturwissenschaftlichen Fakult¨at der Universit¨atBasel von Jens Oliver Schindele aus Deutschland Basel, 2014 Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakult¨at auf Antrag von Prof. Dr. C. Sch¨onenberger Prof. Dr. A. Levy Yeyati Dr. T. Kontos Basel, 18.02.14 Prof. Dr. J¨orgSchibler Dekan Contents 1. Introduction1 2. Carbon nanotube quantum dots7 2.1. Electronic structure of carbon nanotubes............8 2.1.1. Different forms of carbon................8 2.1.2. From graphene to CNTs.................9 2.2. Metallic contacts to Carbon nanotubes............. 13 2.2.1. Schottky barriers..................... 14 2.2.2. Character of the chemical bonding........... 18 2.2.3. Contact to metallic CNTs................ 20 2.3. Carbon nanotube quantum dots................. 20 2.3.1. Discrete energy spectrum................ 20 2.3.2. Coulomb blockade and single electron tunnelling... 21 2.3.3. Coulomb diamonds.................... 24 2.3.4. Co-tunnelling and excited states............ 25 3. Quantum dots coupled to superconductors 29 3.1. Superconductivity........................ 30 3.1.1. Basic phenomena..................... 30 3.1.2. Cooper pairs....................... 30 3.1.3. The energy gap...................... 32 3.1.4. Proximity effect and Andreev reflections........ 36 iii Contents 3.1.5. Crossed Andreev reflection and elastic co-tunnelling. 38 3.2. Cooper pair splitters....................... 40 3.2.1. Basic idea and working conditions........... 40 3.2.2. Quantitative description of the Cooper pair splitter.. 44 3.3. Andreev bound states...................... 47 3.3.1. The superconducting atomic limit........... 47 3.3.2. Finite superconducting gap............... 51 4. Device fabrication and measurement set-up 55 4.1. Standard fabrication process................... 55 4.2. Residues from lithography resist................ 58 4.2.1. Minimizing resist residues................ 60 4.3. Measurement set-up....................... 61 5. Novel fabrication approaches 65 5.1. Aluminium oxide protection layer against resist contamination 66 5.1.1. Conclusions........................ 67 5.2. Cleaning and selective CNT etching with hydrogen radicals. 68 5.2.1. Generation of hydrogen radicals from a hydrogen plasma 68 5.2.2. Selective etching of carbon nanotubes......... 69 5.2.3. Etching of PMMA.................... 71 5.2.4. Transport measurements on hydrogen radical inert CNTs........................... 71 5.2.5. Conclusions........................ 75 5.3. Bottom-gate defined quantum dots............... 76 5.3.1. Sample fabrication.................... 76 5.3.2. Transport characteristics................. 76 5.3.3. Conclusions........................ 78 6. Near-unity Cooper pair splitting efficiency 81 6.1. Introduction............................ 81 6.2. Device and measurement set-up................. 82 6.3. Experimental signature of Cooper pair splitting........ 83 6.4. Evaluation of the splitting efficiency.............. 87 6.5. Semi-classical master equation model.............. 89 6.6. Discussion of the model..................... 91 6.7. Outlook: niobium based Cooper pair splitters......... 92 6.8. Conclusions............................ 94 7. Non-local spectroscopy of Andreev bound states 95 7.1. Introduction............................ 95 7.1.1. Local spectroscopy of ABS............... 95 iv Contents 7.1.2. Non-local spectroscopy of ABS............. 97 7.2. Experiment............................ 98 7.2.1. Device and measurement set-up............. 98 7.2.2. Local transport measurements............. 100 7.2.3. Non-local conductance correlations........... 100 7.3. Model............................... 104 7.3.1. Rate equation....................... 105 7.3.2. Model results and comparison with experiment.... 108 7.4. Discussion and Conclusion.................... 109 8. Summary and outlook 111 A. Detailed fabrication recipes 127 A.1. Wafer characteristics....................... 127 A.2. Wafer cleaning.......................... 127 A.3. CVD catalyst........................... 127 A.4. CVD growth........................... 128 A.5. E-beam lithography....................... 129 B. Solution for the CPS master equation model 131 C. Additional data to Chapter 7 135 Curriculum Vitae 137 Publications 139 Acknowledgements 141 v 1 Introduction Okay, I still get nervous with it. And therefore, some of the younger students... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. So that's why I like to investigate things. Richard P. Feynman, [1] Decision making can be hard work, mainly because of the anticipation of possible consequences. Quantum mechanics allows the tiny objects which build up our world to be be indecisive: particles may be delocalized, oc- cupy several energy states simultaneously or posses a unidirectional mag- netic moment that points in two directions at the same time. However, this schizophrenic indecisiveness, known as quantum superposition, has conse- quences, too; predictable consequences that are observable in experiments (e.g. as interference patterns that appear on a screen). Despite the pre- dictive success of quantum mechanics, the counterintuitive behaviour of its objects is a hard pill to swallow and there is a native temptation to view quantum superpositions merely as mathematical constructs, instead of 1 1. Introduction something that occurs for real in nature. In other words, quantum mechan- ics correctly predicts measurement outcomes, but does it also provide an adequate description of the physical reality? The answer to this question is well hidden by another feature of quantum mechanics, named the wavefunc- tion collapse. Any attempt to observe the superposition itself, instead of its consequences, will force the quantum object into a decision and destroy the signature of the indecisiveness. In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) set up a famous gedankenexperiment, aiming to lift the epistemic curtain that hides the true nature of quantum superposition states [2]. Instead of just one particle, they considered two particles in a common superposition state. According to the rules of quantum mechanics a measurement on one of the particles instantaneously forces both particles into a decision, thus deter- mining the state of the second particle. One says, the two particles are entangled. But what if the entangled particles are spatially separated from each other? EPR felt that the laws of special relativity would be broken. Special relativity seems to demand that any action on, say, particle A, can affect particle B only after a finite time has passed. Thus, if the predictions of quantum mechanics are correct { and in this respect EPR seemed to have no doubts {, it follows that the state of particle B was unambiguously de- termined already before the measurement on particle A was performed. In this case, the description of the particle by a superposition state reflects our lack of knowledge about the particle, rather than an inherent uncertainty of the particle's properties. Accordingly, EPR expressed their hope that quantum mechanics may emerge from some deeper theory. A theory that contains no such elements as the schizophrenic indecisiveness of quantum objects; a theory that allows a straightforward connection to the elements of an objective reality. However, the EPR paper could not convince the community of quantum physicists. Many researchers took a viewpoint that is nicely summarized in a wording by Pascual Jourdan [3]: "Observations not only disturb what has to be measured, they produce it (...)." So why couldn't the EPR argument persuade this group of people? EPR assumed that a measurement at point A can not "in any way" disturb instantaneously a physical system at point B[2]. But this criterion is too harsh when applied to quantum mechan- ics. Quantum mechanics allows a physical quantity at B to instantaneously acquire a definite value by virtue of a measurement at A. After a famous Einstein quote, this non-local element of quantum mechanics is often nick- named the spooky action at a distance ("spukhafte Fernwirkungen"). As an advocate of Jourdan's viewpoint one can turn the table: because the spooky actions at a distance do merely act on the elements of quantum theory, and not on the elements of an objective reality, they do not violate the laws of 2 special relativity. Shortly after the publication of EPR's work Niels Bohr wrote a reply where he stated that EPR's assumption of no action at a distance can not be applied to quantum mechanics [4]. However, Bohr's point was not to disproof EPR's belief in the existence of a measurement independent reality. His point was to reveal it as a belief that can not be backed up empirically. The opposite view, as expressed by Jourdan, has to be treated as a belief in the same manner. Bohr clarifies [4]: "(...) there can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules which allow to predict the results (...) [of a given experiment]". Thus, we are back at our starting point. The fact that we can only observe the consequences
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