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Quantized Transmission in an Asymmetrically Biased Quantum Point Contact Linköping University | Department of Physics, Chemistry and Biology Master’s thesis, 30 hp | Master’s programme in Physics and Nanoscience Autumn term 2016 | LITH-IFM-A-EX—16/3274--SE Quantized Transmission in an Asymmetrically Biased Quantum Point Contact Erik Johansson Examinator, Magnus Johansson Supervisors, Irina Yakimenko & Karl-Fredrik Berggren Avdelning, institution Datum Division, Department Date Theoretical Physics 2016-11-07 Department of Physics, Chemistry and Biology Linköping University, SE-581 83 Linköping, Sweden Språk Rapporttyp ISBN Language Report category Svenska/Swedish Licentiatavhandling ISRN: LITH-IFM-A-EX--16/3274--SE Engelska/English Examensarbete _________________________________________________________________ C-uppsats D-uppsats Serietitel och serienummer ISSN ________________ Övrig rapport Title of series, numbering ______________________________ _____________ URL för elektronisk version Titel Title Quantized Transmission in an Asymmetrically Biased Quantum Point Contact Författare Author Erik Johansson Sammanfattning Abstract In this project work we have studied how a two-dimensional electron gas (2DEG) in a GaAs/AlGaAs semiconductor heterostructure can be locally confined down to a narrow bottleneck constriction called a quantum point contact (QPC) and form an artificial quantum wire using a split-gate technique by application of negative bias voltages. The electron transport through the QPC and how asymmetric loading of bias voltages affects the nature of quantized conductance were studied. The basis is Thomas-Fermi simulations that within the Büttiker model give results somewhat similar to experimental work in aspects regarding electron density effects. An extension of the model to include exchange and correlation interaction was investigated, as well as compared to density functional theory. Nyckelord Keyword QPC, Conductance, Transport, Thomas-Fermi, Büttiker, Asymmetric, Potentials, AlGaAs, GaAs, Heterostructure Linköping University Quantized Transmission in an Asymmetrically Biased Quantum Point Contact Master’s Thesis LITH-IFM-A-EX—16/3274—SE Author: Supervisors: Erik Johansson Correction Irina Yakimenko Theoretical Physics, IFM Karl-Fredrik Berggren Theoretical Physics, IFM Examiner: Magnus Johansson Theoretical Physics, IFM Ausbildung heißt, das zu lernen, von dem du nicht einmal wußtest, daß du es nicht wußtest. Abstract In this project work we have studied how a two-dimensional electron gas (2DEG) in a GaAs/AlGaAs semiconductor heterostructure can be locally confined down to a narrow bottleneck constriction called a quantum point contact (QPC) and form an artificial quan- tum wire using a split-gate technique by application of negative bias voltages. The electron transport through the QPC and how asymmetric loading of bias voltages affects the nature of quantized conductance were studied. The basis is Thomas-Fermi simulations that within the Büttiker model give results somewhat similar to experimental work in aspects regarding electron density effects. An extension of the model to include exchange and correlation interaction was investigated, as well as compared to density functional theory. Acknowledgements I would like to thank my supervisors Irina Yakimenko and Karl-Fredrik Berggren for all help, advice and discussion, and also for giving me the opportunity to do my project course with you that lead to this diploma work. A huge thanks goes out to my classmates for all support and shared moments throughout my studies, especially Jimmy and Andreas who kept fighting alongside me during the master’s degree. I would also like to thank my family for their everlasting support through thick and thin. Lastly, a special thanks goes out to Yvonne for helping me keep my mind off the physics when needed. Contents 1 Introduction 1 2 GaAs/AlGaAs Heterointerface Ballistic Regime 4 3 Model of split gates QPC 5 3.1 Model GaAs/AlGaAs Heterostructure . .5 3.2 Fundamental Theory . .7 3.2.1 Derivation: Landauer-Büttiker Formula . .7 3.2.2 Quantization of Conductance in Nanostructures . .9 3.3 Numerical Schemes . 12 3.3.1 Calculation of total potentials within Thomas-Fermi model . 12 3.3.2 Finite Difference Method . 15 3.3.3 Calculation Method . 16 4 Results of calculations and discussion 17 4.1 Numerical Büttiker Model . 18 4.2 Typical Electron Density - n0 ............................. 19 4.3 Higher Electron Density - 3/2 n0 ........................... 24 4.4 Lower Electron Density - 1/3 n0 ........................... 27 4.5 Comments on Eigenvalues . 30 4.6 Extension of the model - Exchange and Correlation Interactions . 32 4.6.1 Density profiles . 32 4.6.2 Potentials . 34 5 Density functional theory model comparison 41 6 Conclusions 42 6.1 Thomas-Fermi model review . 42 6.2 Future Work . 42 7 References 43 8 Code 44 Appendix A 45 A.1 Typical Electron Density - n0 ............................. 46 A.2 Higher Electron Density - 3/2 n0 ........................... 58 A.3 Lower Electron Density - 1/3 n0 ........................... 63 A.4 Exchange and Correlation . 69 A.4.1 Density profiles . 69 A.4.2 Potentials . 72 1 Introduction The quantum-well-based high-mobility semiconductor heterostructures are now commonplace in nanoscience and technology. The high-quality nanostructures based on GaAs/AlGaAs semi- conductors grown by molecular beam epitaxy have become the genuine quantum laboratories to study a number of fundamental issues in low-dimensional physics [1]. The reason is that the split-gate technique inherent in these systems is free of various imperfections and allows one to change the geometry of the system and the density of the underlying two-dimensional electron gas (2DEG) continuously down to the regime of a very low density electron gas when Wigner crystallization and lattice formation appear. The physics of low-dimensional GaAs/AlGaAs based semiconductor structures such as quan- tum dots (QDs), quantum wires (QWs) and quantum point contacts (QPCs) has developed into an important part of nanotechnology, especially in connection with spintronics and quantum information processing. Systems fabricated in this way are also very versatile: QWs can be simply connected to electron reservoirs that serve as source and drain and the conductance mea- surement can be used as a tool for the identification of different electron configurations. It is well known that when the wires become narrow they show the quantum properties at low temperature, i.e., the conductance tends to be quantized in units of the fundamental conduc- tance quanta 2e2=h under changing gate voltage. This phenomenon known as the conductance quantization has been discovered experimentally by van Wees et al. [2] and Wharam et al. [3]. In theory, the existence of this phenomenon follows from the general Landauer-Büttiker formula [4], derived within the frame of a simple model of non-interacting electrons. According to this for- mula, the total conductance is G = N(2e2=h), where N is the number of open subbands (energy levels below the Fermi energy) in the QPC which is defined by the value of the applied gate volt- age. Thus we have a step-like behavior of the conductance as a function of the gate voltage, each step being 2e2=h. In the presence of an external magnetic field the subbands are split, each level contributes e2=h to the conductance, and thus the conductance step becomes e2=h in magnitude. In mid 2015 the group of Prof. M. Pepper and colleagues at University of Cambridge (UK) and London Centre for Nanotechnology, UCL has studied the conductance behaviour in asym- metrically biased split gates QPC. If the voltage on one split gate was fixed and another swept it has been shown that in the cases of highly asymmetric voltages the behaviour of conductance follows one predicted by Büttiker model while for intermediate voltage settings the plateaus of the conductance become smeared or disappear. The (unpublished) experimental measurements are presented in Figure 1(a). An initial gate-1 voltage of Vsg1 = −0:22 V was fixed while sweep- ing gate-2. For the consecutive measurements gate-1 was raised in 20 mV increments. In this measurement series the gate voltages undergo an asymmetric-symmetric-asymmetric transition with respect to their magnitudes. Once again, somewhere along this transition the conductance plateaus become smeared. This occurs for the curves starting close to Vsg2 = −1:50 V. Some other work within this field includes the published paper of S. Kumar et al. from 2014 [5] where they, in connection to this diploma subject, investigated electron transport in a quasi-one dimensional electron gas as a function of the confinement potential. They found a transition similar to the one in Figure 1(a), represented by the blue curve in Figure 1(b). According to them this effect is owing to an anticrossing of the subbands, leading to the formation of a Wigner lattice. Noteworthy is that their gates have different dimensions than that of M. Pepper and the device structure includes an additional uniform top gate placed between the two gates. 1 (a) Image courtesy of M. Pepper and S. Kumar [6]. Here S stands for source and D for drain. (b) Image courtesy of S. Kumar [5]. Here TG stands for top gate and SG for split gate. Figure 1: Experimentally measured conductance. 2 Our motivation is based upon this peculiar experimental feature and it is of our highest interest to gain an understanding of the origin of this transition in conductance. The model device of choice will have the same structure and dimensions as the group of M. Pepper and colleagues. The aim is to study total electron potentials for different settings of the split-gates voltage and to use the Büttiker model to calculate the conductance. The potentials will be generated by Thomas-Fermi simulations. A key point in our investigation is to see if it is possible to describe and understand the transition of conductance curves within the Thomas-Fermi model. The Büttiker model describes one dimensional (1D) conductance and the smooth quantized steps are only achievable in the ballistic regime, wherein quantum effects arise and the charge carrier transport is unscattered and phase-coherent.
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