Bachelor’s Thesis
Quantum Transport through Single and Double Quantum Dots
Iason Tsiamis 12 June, 2013
Supervisor: Dr Karsten Flensberg
Niels Bohr Institute University of Copenhagen, Universitetsparken 5, Copenhagen 2100, Denmark
Contents 1 Introduction 1.1 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 2-D Electron Gas
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3 Quantum Dots 3.1 Quantum Dots . . . . . 3.1.1 General Features 3.1.2 Hamiltonian . . . 3.1.3 Double Quantum 3.2 Pauli Blockade . . . . . 3.3 Coulomb Blockade . . .
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4 Master Equations 4.1 Sequential Tunnelling . . . . . . . . 4.2 Single Quantum Dot, with spin . . . 4.3 Double Quantum Dot, without spin 4.4 Double Quantum Dot, with spin . .
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5 Conclusions
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6 Appendices 6.1 Double dot without spin (Matlab program) . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Double dot with spin (Matlab program) . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract We study the properties of single and double quantum dots. We focus on the transport of electrons between the dots and the source/drain, and between the first and second dot, in the case of the double dot. We analyse the systems of single dot, double dot without spin and double dot with spin. We find the possible states of each system and the transition rates between its states. We calculate the Master equations, which lead us to calculate the current, produced by the electron transfer, in each system. Finally, we create two programs, where one simulates the double dot system for electrons with and one for electrons without spin, and we compare qualitatively our results with the experimental data.
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Chapter 1
Introduction 1.1 1.1.1
Mathematical Tools Second Quantization
The main tool, used to describe many particle physical systems, is the second quantization. Main features of which are the creation and annihilation operators, which add or remove a particle (boson or fermion) to many-body wave functions, respectively. Below is presented the basic notation used in this thesis. c†kσ : creation fermion operator, creates an electron in state k with spin σ ckσ : annihilation fermion operator, annihilates an electron in state k with spin σ And they act as following in the many-body wave-functions P †
c |n1 ...ni ..i = (−1) i (1 − i)C+ (nνj )|n1 ...n(i+1) ..i P
c|n1 ...ni ..i = (−1) i (1 − i)C− (nνj )|n1 ...n(i−1) ..i Anticommutation relations of fermion operators {c†νj , c†νk } = 0 {cνj , cνk } = 0 {cνj , c†νk } = δνj νk
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Chapter 2
2-D Electron Gas Electron gases of reduced dimensionality play a very important role in modern experimental physics. Especially two-dimensional electron gases in heterostructures are a fundamental part of semiconductor nanostructures. The most recent type of them is the gallium arsenide/gallium-aluminium-arsenide (GaAs/GaAlAs) heterostructure.
Figure 2.1: Layer sequence in a typical GaAs/AlGaAs heterostructure with remote doping.
Firstly we will examine the electrostatic properties of this structure, which is shown in the image above. We choose the z axis in the growth direction of the crystal, with its origin, z=0. For z 0 the electric field in the sample is zero and the conduction band edge is flat. If we place a cylindrical close surface along z with one end face in the region z 0 and the other in the region -s
