Coherent Electron Transport in Triple Quantum Dots Adam Schneider Centre for the Physics of Materials Department of Physics McGill University Montreal. Quebec Canada A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science © Adam Schneider. 2008 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de ['edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-53766-4 Our file Notre reference ISBN: 978-0-494-53766-4 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1*1 Canada Contents Acknowledgments ix 1 Introduction 1 1.1 Experimental Background 2 1.1.1 Single Quantum Dots: The Coulomb Blockade 2 1.1.2 Double Quantum Dots: Stability Diagrams 5 1.1.3 Triple Quantum Dots: Chains and Rings 9 1.2 Theoretical Overview 13 1.2.1 Transport Basics 14 1.2.2 Beyond Sequential Tunneling 18 1.2.3 Triple Dot Rings and Aharonov-Bohm Oscillations 20 1.2.4 What's Next? ' 22 2. Model and Results 25 2.1 Model Harniltonian 25 2.2 Zero-Bias Conductance 30 2.2.1 No Bath Coupling Limit 33 2.2.2 Strong Bath Coupling Limit 38 2.2.3 Aharanov-Bohm Oscillations 42 2.3 Negative Differential Resistance and The ''Dark State" 43 3 Details of Calculation 50 3.1 Quantum Master Equations 50 3.1.1 Coherent System Evolution 51 3.1.2 Lead-Dot Coupling: A Markovian Master Equation 52 3.1.3 Lead-Dot Coupling: FGR Transition Rates 57 3.1.4 Bath-Dot Coupling: Dephasing and Eigenstate Transitions . 61 3.1.5 Solving A Quantum Master Equation 63 3.2 Analytic Diagonalization 66 3.2.1 Equivalent 2-path Double Dot ... .' 68 3.3 Other Master Equations 70 3.3.1 The Analogous 2-Path Double Dot Master Equation 70 3.3.2 Tunneling into Charge States 72 4 Conclusions 74 References 77 in List of Figures 1.1 Network of capacitors and voltage nodes for a single quantum dot with one gate voltage and two lead contacts having additional tunnel cou­ plings to pass a current 3 1.2 Network of capacitors and voltage nodes for a double quantum dot with one gate voltage and one lead contact for each dot. The dots are tunnel coupled to each other as well as to one lead each so that a current can be passed through the dots 5 1.3 (a) Stability diagram for a double dot system with no inter-dot ca- pacitive coupling, (b) Stability diagram for double dots with non-zero inter-dot coupling. These figures were taken from the the review Elec­ tron Transport Through Double Quantum Dots by van tier Wiel et al.[10] 7 1.4 Stability diagram measured by Gaudreau et al. for the first ever triple dot systems(image shown bottom left). This image was taken from The Stability Diagram of a Few Electron Artificial Triatom, by Gaudreau et al.[26] 10 1.5 Schematic of a triple dot chain system, in which only dots a and c cou­ ple to external lead contacts, while the middle dot. a is tunnel coupled to both dots a and c 11 1.6 Both and image (left), and schematic (right) of the triple dot star system created by Rogge et al. In this case three dots are in a triangular shape, each dot is tunnel coupled to a lead. The middle dot connected to the source lead, and the other two connected to drains allowing two non-interacting paths. This image was taken from Two Path Transport Measurements on a Triple Quantum Dot[28] 12 1.7 Double quantum dot schematic in which the tunnel rate from the left lead to dot a is TL. and from dot b to the right lead is TR, while the two dots have tunnel coupling tab 16 1.8 Schematic of the two dot energies, and two lead chemical potentials, in the high bias voltage regime 18 2.1 Schematic of triple dot setup with inter-dot tunneling / and t', lead tun­ neling rates T. magnetic flux <i>. and lead chemical potentials fiLa^iLb and HR as shown 25 IV List of Figures v 2.2 (Left) Conductance versus e = (eb + ea)/2 (where ea = e^) and ec with $ = 0. at a temperature of at 50mK. Interdot tunnel couplings are set to |/0(,| = 67.8/vel/, |iac| — \lcb\ = 9.7fi,eV, dot charging energies are Eca = S.lmeV, Ech = 2.8meV, EQC = 2.28meV, and the capacitive couplings of the dots are Eab — .86meV, Eac = .28meV, Ebc — A9meV. The lead-dot tunnel rate F = .l[ieV and the temperature is 50m,K (4.3/ieV).(Right) Stability diagram from simulated QPC, where light blue corresponds to the electron equally in dots a and b, dark blue the dots are empty, and red the electron is in dot c 31 2.3 ett = eb = t', with t' = 67.8fieV > t = 9.7/zeV, and 0 = 0. Top Left: Eigenstates energies, with Ground state (red), 1st excited state (green), 2nd excited state (blue), and empty state (black). Chemical potentials (dashed black) for Vbjas = ljueV. Bottom Left: Eigenstate probabili­ ties, with color code as above. Top Right: Charge state Energies, with dot a (red), dot b (green) dot c (blue), and empty state (black). Bot­ tom Right: Charge state probabilities. All other parameters are kept fixed as previously stated 32 2.4 (Top Left): Eigenstates energies, (Bot Left): Eigenstate probabilities, (Bot Right): Charge state probabilities. Now with (p = n. and all other parameters arc kept fixed as previously stated. Color coding as previously used 34 2.5 Differential conductance versus dot energy cc and enclosed flux <I>. for the case of zero relaxation and ea = eb = t' and t <C t', at a temperature of 50 mK. All other parameters are fixed as before 35 2.6 g • (kBT/T) versus electrostatic energy ec for various <f>: (Top Left) 0 = 0, (Top Right) cp = 1.5TT/2, (Bot Left) 0 = 1.9TT/2, (Bot Right) 4> = 7T. For the case of zero relaxation (ie A = 0) and ea = eb = t' and t <C t', at a temperature of 50 mK. All other parameters are fixed as before. The red curve corresponds to current passing through the ground state, while the green curve in the first excited state 36 2.7 (Left) Eigenstate energies versus dot energy ec for $ = 0, in the case of zero relaxation and ea — eb = t' and t = t', at a temperature of 50 mK. All other parameters are as before. Color coding of eigenstates is as before. (Right) g • (kBT/T) versus dot energy ec and enclosed flux $, for the case of zero relaxation and ta = tb = t' and t — t'. at a temperature of 50 mK. All other parameters are as before 37 2.8 g • (kBT/T) versus dot energy ec and enclosed flux $, for t' 3> t in the case of strong bath coupling. All other parameters are as before. ... 38 2.9 g • (ksT/T) versus dot energy ec for various enclosed flux <I>. in the strong bath coupling limit with ea = eh and all other parameters as before, at a temperature of 50 mK. (Top Left) 0 = 0, (Top Right) q> = 1.5TT/2, (Bot Left) 0 = 1.9TT/2, (Bot Right) <p = TT 40 2.10 g • (kBT/T) versus dot energy ec, for various bath coupling strengths, with enclosed flux $/(I>o = 0.5 and all other parameters as before, at 50 mK 41 VI List of Figures 2.11 (Top Left) g • (fceT/T) versus ^/^o for various values of ec, with no bath coupling (A = 0), and all other parameters left as before. Black is centered at the resonance peak, red to the left of the resonance peak, and green to the right of the resonance peak. (Top Right) Fast Fourier transforms of the AB oscillations shown in to the left. (Bot r var us Left)g • (ksT/T) versus $/$o f° i° values of ec, with strong bath coupling, and all other parameters left as before.