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
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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 . 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 . for the case of zero relaxation and ea = eb = t' and t 2.6 g • (kBT/T) versus electrostatic energy ec for various 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 . 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) 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. Black is centered at the resonance peak, red to the left of the resonance peak, and green to the right of the resonance peak. (Bot Right) Fast Fourier transforms of the AB oscillations shown in to the left 43 2.12 Current versus bias voltage, for no relaxation and strong relaxation, with all parameters as set earlier 45 2.13 Current versus dot energy ec and magnetic flux <&/&o for Vbias = 500/ieV(Left) and Vbias = -bOOfieV (Right) 46 2.14 Current versus dot energy ec for V^as = 500fieV for various values of $ at 50 mK. (Top Leit) 3.1 Schematic of a double dot system connected via two tunnel couplings representing two spatially separated paths enclosing a magnetic flux We use a quantum master equation approach to study the transport properties of a triple quantum dot ring. Unlike double quantum dots and triple quantum dot chains, this geometry gives two transport paths with a relative phase sensitive to magnetic flux via the Aharonov-Bohm effect. This gives rise to a coherent population trapping effect and what is known as a "dark state". Unlike other master equation techniques valid only in the high bias voltage limit, our treatment reproduces such results as well as giving an analytic zero-bias conductance formula. As well as providing a more robust signature of this "dark state" physics, our model further predicts a negative differential resistance in connection with high bias rectification already predicted. vn Resume Nous utilisons unc approche dequation quantique maitrcssc pour etudier les proprictcs de transport des points quantiques triples en forme d'anneau. Contrairement aux points quantiques doubles et triples en forme de chaines, cette geometrie offr e deux chemins pour le transport avec une phase quantique relative qui est sensible au flux magnetique en raison de l'effet Aharonov-Bohm. Ceci mene a un effet de piegeage de population coherent et cela est connu sous le nom d:un "etat sombre". Contrairement a d'autres techniques d'equation maitresse qui sont seulement valides dans la limitc d'un potentiel electrique eleve, notre methode reproduit les resultats de ces derniers en plus de donner une expression analytique pour la conductance differentielle de zero potentiel electrique. En plus de donner une optique plus robuste de la physique "detats sombres". notre modele predit une resistance differentielle negative qui est reliee au phenomene deja predit de rectification a potentiel eleve. vm Acknowledgments Adam would like to acknowledge his supervisor Aashish Clerk for the countless hours of guidance and insight, as wTell as Louis Gaudreau and Andy Sachradja for the useful conversations and data. Adam would also like to thank all of his friends and especially his family for all of the support and encouragement needed to carry on. IX x Acknowledgments 1 Introduction As semiconductor fabrication has reached the nanoscale, quantum effects have become increasingly significant. When a device confines electrons to a small enough region, a so called quantum dot is formed, having a discrete atom-like energy spectrum. This makes semiconductor quantum dots not only a promising medium for next generation quantum computers, but also a novel means to study analogs of fundamental atomic and molecular properties. This earned them the name "artificial atoms" as coined by the title of a paper by Kasterfl]. Easily tunable to contain only a few electrons at a time, quantum dots provide an ideal grounds to study coherent transport, as covered extensively in the review by Hackenbroich et al. [2]. A series of experiments by Yacoby, Schuster et al.[3, 4, 5] carefully study the phase coherent electron transport through a single quantum dot. With double dot fabrication underway for several years already[6, 7, 8, 9], many of their electronic transport properties are also now very well understood, as outlined in the review by van der Wiel et al. [10]. Most recently, triple dot systems have begun to be realized experimentally, showing novel coherent transport properties. After a brief overview of the physics and experimental characterization of single and double dot systems, we will briefly review three recent triple dot experiments. From these experiments, we will see how a triple dot can have a geometry funda mentally distinct from a double dot system, as well as the novel effects that result. With this recent experimental progress in controlled triple dot fabrication, there have been several recent theoretical efforts towards understanding coherent transport in triple dot systems, all of which leave some questions unanswered. Before outlining 1 2 1 Introduction a few interesting new triple dot models already proposed, we will first give a brief overview of some basic transport properties of single and double dot systems. After highlighting the success and shortcomings of a few recent triple dot theory papers, we will then see the regimes to which they are limited, motivating the presentation of our novel triple dot model. 1.1 Experimental Background 1.1.1 Single Quantum Dots: The Coulomb Blockade A quantum dot (QD) is formed when a potential well confines electrons to a small enough region, that a discrete atom-like energy spectrum is formed. The fabrication of such a device generally begins with with the formation of a heterostructure of two different semiconductor materials (GaAs/AlGaAs). The free electrons of the system are thus confined to the interface of the GaAs and the AlGaAs, restricting their motion to a two-dimensional plane creating a two-dimensional electron gas (2DEG). The free electrons in the 2DEG can be further confined to a point by either etching techniques, or more commonly via metal gate electrodes patterned on the surface of the 2DEG stack. With clever patterning of these gate electrodes, these dot structures can be coupled to electrical contacts via tunnel barriers allowing transport properties of such devices to be studied[10]. The voltages of the gate electrodes can then be tuned experimentally, controlling the electrostatic energies of the electrons trapped in the dot's potential well. As show in figure 1.1, a single quantum dot coupled to two lead contacts can be treated as three small chunks of metal, modeled by a capacitor and voltage node network. The dot in the center is both capacitively coupled and tunnel coupled to the leads such that a voltage difference between the two leads can pass a current, while the dot's gate voltage is only capacitively coupled to control the dot's potential. The systems electrostatic energy is derived in terms of the voltages and capacitive couplings between each element, as shown in the review by van der Wiel et al.[10] to 1.1 Experimental Background 3 Figure 1.1: Network of capacitors and voltage nodes for a single quantum dot with one gate voltage and two lead contacts having additional tunnel couplings to pass a current. give [-(n - n0)le| + CLVL + CgVg + CRVRf U{n) (1.1) 2d where C\ = CL + Cg + CR is a sum of all capacitances connected to the dot, n is the number of electrons in the dot and n0 is the number of electrons on the dot with no applied voltage. The above equation can be reduced to the more compact form. 2 E{n) = Ec{n - N) ;i.2) 2 where J\f = CgVg/\e\ is the now dimensionless gate voltage. EQ = e /Ci is the charging energy of the dot, or energy required to add another electron to the dot. This equation tells us how many electrons will be in the dot for a specific value of gate voltage, this being the n for which E(n) is minimized. Although classically we do not think of the amount of charge on a capacitor a quantized, in the weak coupling regime such that the rate at which electrons tunnel on and off the dot is much less that the charging energy, we will have a well defined number of electrons on the dot, quantizing the amount of charge. Up until now our description of a quantum dot has actually been purely classical, only considering the electrostatic potential energy of an electron. In general even a free electron can have any additional amount of kinetic energy, what makes a quantum dot quantum is that this confinement of a single electron results in a discrete particle-in-a-box-like energy spectrum of possible excited states. The electrostatic energy equation above gives the ground state energy for different 4 1 Introduction electron numbers in the dot, not this full spectrum of each state. This intrinsic level spacing of the dot's excited states is inversely proportional to the dot's diameter which when reaching the nanometer scale makes single states easy to isolate. In the central model of this thesis we will assume that this level spacing is actually large enough to be completely ignored. In this case, although a quantum dot will only have a ground state, by tuning its gate voltage its potential well is altered possibly resulting in a change in the ground state configuration. The amount that the gate voltage must change in order for the ground state to change in number by one electron, is proportional to the charging energy of the dot. This change in ground state number occurs whenever E{n) = E(n + 1), a degeneracy which corresponds to gate voltages of TV = (n+l)/2 in dimensionless units (as defined after equation 1.2). By slightly offsetting the chemical potentials of the left and right lead contacts a bias voltage is created and a current will attempt to pass through the dot from one lead to the other. For a bias voltage and lead temperature both much less than the charging energy of the dot, current will pass most easily at the degeneracy point just mentioned as an electron can be added and then removed with no cost in energy. Away from this degeneracy point, the leads will not be able to provide the required charging energy to add an electron to the dot. This is known as the Coulomb blockade regime as discussed by Averin and Grabertfll, 12]. In this regime, measuring the current through the dot as a function of its gate voltage, a series of sharp peaks known as Coulomb Blockade peaks are found[3, 4]. These peaks are spaced apart by a voltage proportional to the charging energy of the dot, occurring at the gate voltage values specified earlier. These peaks do of course have some width, which is a function of the temperature of the the leads as well as the strength of the lead-dot coupling. Although two charge states need not be completely degenerate, far from the degeneracy points only higher order non-energy conserving processes are possible (but vanishingly rare). We will see that in double and triple dot systems current resonances like these can be much sharper than temperature. Although the current through a single quantum dot only is only a function of the 1.1 Experimental Background 5 magnitude of the transmission through it, the transmitted wave function is in fact a complex quantity with a possibly well defined phase relative to the incident wave. From clever experiments by Yacoby et al.[3, 4, 5], this phase factor was in fact probed by placing a single quantum dot in one arm of a two-path interferometer for electrons. This is what as known as an Aharanov-Bohm (AB) ring, as the relative phase of two transport paths is proportional to any magnetic flux enclosed within them[13]. Therefore, the current produced by the two interfering paths oscillates with respect to the magnetic field strength. Tuning the dots gate voltage to a Coulomb peak to get a current, one can then vary the magnetic field to see oscillations. What was found was that the oscillations on either side of a Coulomb peak were always out of phase by IT. This abrupt phase flip is now understood in terms of an Onsagger symmetry present in two terminal devices, as discussed extensively in the review on coherent transport in mesoscopic devices by Hackenbroich et al.[2]. Later we will see how these oscillations also become possible in triple dot ring systems, as well as consider the effects of decoherence on these oscillations. 1.1.2 Double Quantum Dots: Stability Diagrams Figure 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. A double quantum dot system is created when some extra gate electrodes are added to create a double potential well, with each dot coupled to one lead. This results in two dots both capacitively and tunnel coupled to each other and one lead each, while only coupled capacitively to two gate electrodes V9l and Vg2 as shown in figure 1.2. Again, the electrostatic potential for the system modeled as a capacitor 6 1 Introduction network was done by van der Wiel et al.[10], resulting in the now slightly more cumbersome equation 2 2 U{na, nb) = -n aECa + -n bECb + nanbECm + f{Vgi,Vg2) f(Vgi.V92) = ~[CgiVgMaECa + nbECm) + Cg2Vg2(naECm + nbECh)\ (1.3) E 2 + \£flK C + \c g2VlECb + Cg2Vg2Cg2Vg2EcJ., 1 th where, Eci = |r(l — ~(fc~)~ is the charging energy for the i dot, and d is the sum th _ i -1 of all capacitances coupling to the i dot. Ecm = ^ (%f ~~ I) is the energy of the capacitive coupling of the two dots. Fortunately this can again be simplified greatly to . E 2 E{na, nb) = J2 d {m ~ M) + ECm (n„ - Na)(nb - Mb). (1.4) i=a,b The dimensionless gate voltages of each dot are defined as before in equation 1.2. and na and nb now represent the number of electrons in each dot, a and b. We can now have states with different electron number or configuration, (na,nb), the pair which minimizes E{na.nb) being the ground state configuration. We will now see how a map of the ground state configuration for different values of the two gate voltages is both measured and understood. A charge stability diagram is a map specifying the ground state charge config uration of the system with respect to its two gate voltages as shown in figure 1.3 from the review by van der Wiel et al.[10]. Regions of parameter space with different ground state configurations are separated by lines. With Cm = 0 we see that each independent gate voltage can be tuned to add an electron to its respective dot, gen erating a set of horizontal and a set of vertical lines, one for each dot. The spacing of these parallel lines is proportional to the charging energy of the particular dot they represent and are thus called charging lines. Although the point of intersection of two perpendicular charging lines is a quadruple point (QP) where four charge states are degenerate, any capacitive coupling between the two dots lifts this degeneracy splitting the the QP into two triple points (see dashed box in right of figure 1.3). In 1.1 Experimental Background 7 general, only triple points are available in double dot systems, and one must look to systems of more dots for degeneracies involving more states. When Cm 7^ 0, changing one dots energy effects the other dot, resulting in nega tively sloped set of parallel charging lines for each dot (as seen in the right of figure 1.3). With strong inter-dot coupling it is therefor possible to add an electron to one dot, by varying the gate voltage of the other dot. With non-zero inter-dot coupling, one also notices positively sloped lines connecting the two triple points created by the split QP (see dashed box in right of figure 1.3). These positively sloped lines generally separate two regions of different charge configuration, but equal number. At the triple point found at the left end of these lines for example, the ground state is threefold degenerate between (na, Ub). (na + 1, rib). (na, n_, + 1). It is at these point that sequential tunneling across the dots is maximized. Sequential tunneling is the simplest form of essentially classical transport through a double dot system and will be discussed in detail in section 1.2.1. This occurs when the classical electrostatic energies of the three states just mentioned are equal, allowing the system to easily switch from one to the other shuffling electrons across the system one by one. After measuring the stability diagram, the system can then be tuned to any degeneracy point of interest. (a) n_n (b) , , , (0.2) 'j (1.2)', (2,2) \ 50,2) (1,2) (2.2) 1 _ _•____ <___ CM (0.1 >• ',0,1)1 (2,1)', (0.1) (1,1) (2 1) — __ A- i-__ J-__ 2s? " I (0,0) Wl.0)'l(2 0)'. (0,0) * 0,0) (2.0) ' \ \ 1. V.. > Figure 1.3: (a) Stability diagram for a double dot system with no inter-dot capacitive coupling, (b) Stability diagram for double dots with non-zero inter-dot coupling. These figures were taken from the the review Electron Transport Through Double Quantum Dots by van der Wiel et al.[10] 8 1 Introduction Measuring the stability diagram is achieved by placing a quantum point contact (QPC) near the double dot system[14, 15, 16, 17]. A QPC is a separate current channel that is highly sensitive to the charge configuration of the nearby dot system. When the number or configuration of charges in the quantum dot system changes, the potential in the nearby QPC will change giving a sudden change in its conductance, or a spike in the derivative of its conductance versus the gate voltage being changed. It is d/dVgi of gqpc, that is actually plotted in a stability diagram, which gives lines separating regions with different ground state charge configurations. Although the classical electrostatic energy of equation 1.4 will tell us where these lines should be, it cannot explain the variable thickness found in experimentally measured lines[26]. This thickness is due to quantum tunneling between the two states being separated. This tunneling allows superpositions of the two states to exist when their energy difference is small enough. The charge transfer lines are thus much broader and generally less well defined than the charging" lines separating states with a different electron number. Furthermore, it is found experimentally that at the point where two charging lines of different dots cross, the lines develop a curvature, and an anti-crossing is formed by the splitting of the quadruple point into two triple points. This curvature as well as the width of these charge addition lines are quantum mechanical effects not captured by the electrostatic Hamiltonian. A fully quantum mechanical Hubbard Hamiltonian (see line one of equation 2.2 for example) combining the inter-dot tunnel couplings with the electrostatic energies must then be used to fully reproduce an experimentally measured stability diagram. In this way the dot charging energies, gate voltage coupling constants, and inter-dot tunneling strengths can all be extracted from experiment. In this full quantum Hamiltonian for a double dot system, quantum tunneling allows the system to exist in a superposition of electronic charge states. There have in turn been many proposals for the use of semiconductor quantum dots as qubits[18, 19, 20, 21. 22, 23, 24] in next generation quantum computers. Although many exploit the intrinsic quantum spin of the electrons (which we neglect in this entire thesis). 1.1 Experimental Background 9 charge qubits have also been proposed. This is a system where the two possible charge states correspond to the 0 and the 1 of the bit, were now superposition states with long lifetimes are desired. In our triple dot model, we will not only consider novel coherent transport properties potentially useful for future quantum computation algorithms. 1.1.3 Triple Quantum Dots: Chains and Rings A triple quantum dot can be formed in much the same way as a double dot, as for instance when one dot in a double dot split in two to make the first ever triple dot ring of three potential wells arranged in a triangle[25, 26] (see device image and schematic in figure 1.4). In this case two dots arc coupled to one lead on the left, while the third dot couples to the lead on the right. Another star-like triple dot ring was recently intentionally created (see figure 1.6), where each dot has its own lead so that current could be measured through two independent paths simultaneously. A third sort of triple dot was also created using electrons already confined to one dimension in a carbon nanotube. In this case a triple dot chain is formed in which only one transport path is possible as depicted in figure 1.5. Although the triple dot chain system is not fundamentally distinct from a double dot, in a triple dot ring two possible transport paths enclose a magnetic flux $ when a perpendicular field is applied, giving rise to Aharonov-Bohm oscillations not possible in double dot systems. The first ever triple dot system was recently created and characterized by Gaudreau et al.[25, 26], when one dot in a double dot system was split in two forming a triple dot ring. They were able to identify the system as a triple dot from the stability diagram (see figure 1.4), even though it had been designed as a double dot. For a double dot system where each dot is coupled to a gate voltage, the stability diagram should have a set of parallel charging lines for each dot. Since one dot is mainly coupled to one gate voltage, its charging lines should be nearly perpendicular to that axis, while the other dots charging lines are perpendicular to the other gate voltage axis (as previously discussed in section 1.1.2). What was found was a single set of charging lines for the right dot c, perpendicular to the x-axis gate voltage (see vertical lines in figure 1.4 labeled with a green c). Instead of a single set of horizontal charging 10 1 Introduction V„(V) Figure 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 ct al.[26] lines for the left dot however, two sets of nearly parallel lines were found, as can been seen labeled by alternating red and blue dots in figure 1.4. The two new lines are hard to distinguish as they are nearly parallel, but in fact one set of lines is brighter than the other as it couples to the QPC more than the other. These two lines do however form an anti-crossing where they should intersect as is marked by the large dashed oval in figure 1.4. The fact that the anti-crossing formed by dot a and dot b lines is much more pronounced than those between dots a and c or b and c (marked in smaller dashed circles in figure 1.4) is because the tunnel coupling between dots a and b is much greater than between either dot and dot c. This limit will in fact turn out to be very convenient in for our model, as will be explained in section 2.2. Although this triple dot only has two gate voltages, in general with three dots one should have three gate voltages, and the stability diagram would become a three dimensional object. Viewing a planar projection of the stability diagram we do still recover the same features as before with a set of parallel charge addition lines for each dot, their anti-crossings measuring the respective inter-dot tunnel couplings. Even though the triple dot in figure 1.4 only has two main gate voltages labeled as IB and 5D, there are still additional gate voltages necessary to define the full dot potential wells, labeled 2B, 3B, 4B, 3T. In fact, while the main two gate voltages are used J.I Experimental Background 11 to make the stability diagram, these additional gates can be tuned to modify it. In this way, Gaudreau et al.[25] were able to bring two triple points (one from the large dashed oval and one from the small dashed oval) together, forming quadruple point between four states, such as (na, nb, nc) G {(0, 0, 0), (1, 0. 0), (0,1, 0), (0, 0,1)}. As discussed is section 1.1.2, the QP just mentioned is the point at which sequential tunneling across the dot is maximized. Sweeping one of the gate voltages to move the system through this QP, resulted in generally ordinary resonances with a width set by temperature. However, since the three dots could be in a ring enclosing a magnetic flux, Gaudreau et al. then measured the current versus magnetic field for fixed gate voltages on either side of this resonance. In order to find Aharonov-Bohm oscillations[26], it was necessary to be close enough to the resonance to get enough current flowing, but too close to the resonance peak and the oscillations disappeared. Measuring these oscillations in current, at fixed gate voltages on either side of this resonance, it was found that the oscillations had flipped by TT: most of the time. Although these oscillations are of a similar nature to those for the Coulomb peaks in a single dot (as discussed earlier), we now have more than one dimension through phase space with which to cross a resonance, not all of which lead to this flip in the AB phase. We aim to shed some light on the geometric mechanism behind this flipping in the presentation of our model in section 2.2, and our analytic calculations in section 3.2. Figure 1.5: Schematic of a triple dot chain system, in which only dots a and c couple to external lead contacts, while the middle dot a is tunnel coupled to both dots a and c. More and more triple dot systems are now being built and characterized, such as the carbon nanotube (CNT) triple dot chain of Grove-Rasmussen et al.[27] and the triple dot ring created by Rogge et al.[28, 29]. As CNTs can have remarkably high coherence lengths[27], they are a perfect medium with which to create coherently coupled quantum dot systems. With electrons already confined to one dimension, top 12 1 Introduction gates can be added which pinch the tube off into a chain of dots (as depicted in figure 1.5). In this system, they were able to clearly resolve that dot b was strongly coupled to dot a on the left, but weakly coupled dot c on the right. Although the CNT prevents the dots from forming a ring to enclose any magnetic flux, the high coherence time makes them ideal for observing higher order effects due to the two strongly coupled dots. In these experiments each dot had its own gate voltage, but those of the left and right dots both cross-couple to the middle dot. This was discovered in stability diagrams of both Vgi vs V92 as well as Vg2 vs V93. Here it was found that two of the dots were more strongly coupled, resulting in a much bigger avoided crossing of the two dots charging lines, and in turn a much bigger shift in the triple point formed there. JlAL Drain 1 *" " *" G5 Figure 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]. In Rogge's triple dot star, each dot has its own lead, but the three dots are still not all tunnel coupled to complete a closed ring[28]. Opposite to the work of Gaudreau et al., rather than run the current from two dots to one dot, the current runs in two independent paths from one dot to the other two (see figure 1.6). In this case the two paths do not interfere, and no AB oscillations can be observed as the dots are not tunnel coupled into a complete ring. However, Rogge et al. were able to map the current for each path while blocking the second path, and then overlay the two with the charge stability map. Although the two paths are independent, one was 1.2 Theoretical Overview 13 found to have two weakly coupled dots displaying atom-like electronic states, while the other contained two strongly coupled dots displaying molecule-like states[29]. They were then able to measure, signatures of these states in higher order transport measurements, rather than just stability diagrams. In this thesis, we will further investigate the effects of strong dot coupling and molecular states in a geometry creating two interfering transport paths. First of all, we shall present some basic transport theory applied to single and double quantum dots, before reviewing a few newer models of triple dots and their shortcomings. 1.2 Theoretical Overview We begin this section with an overview of the basic electronic transport properties of single and double dots, as have been well summarized in the review papers by Van der Wiel et al.[10] and Hackenbroich et al.[2]. These well established foundations are forcing theorists to investigate more detailed models in search of more exotic phenomena[30, 31, 32] devices continue to become more sophisticated. For instance, Hettler et al. predict a novel negative differential conductance (NDC) effect[38], due to strong inter-dot tunneling in a double dot system. With the recent experimental realization of triple dot systems, theory papers are beginning to analyze the new ring geometry attained. A simple non-interacting tight binding approach by Delgado et al.[33] gives some results on AB oscillation periodicity. More sophisticated master equation approaches do treat interactions and make novel predictions of rectifying "dark states" [40, 41], but are only valid in the high bias voltage regime. After re viewing the success' and shortcomings of these three papers, we will present a novel master equation technique, wdiich treats electron-electron interactions while being valid for both high and low bias voltages. Not only will this model fill in the nec essary gaps left by these other theories, but it will bring new understanding to the geometric mechanism behind the "dark state" rectification previously overlooked. 14 1 Introduction 1.2.1 Transport Basics Reviewing elementary transport, we can begin with the simplest one dimensional problem of solving the Schrodinger equation for a square potential barrier. In this problem, solved in every undergraduate quantum mechanics course one finds an in coming wave, a reflected wave, and a transmitted wave. Of course it is required that T + R = 1, where T is the probability of the wave being transmitted through the barrier and R is the probability of it being reflected back. We can now consider a single quantum dot (or any system) coupled to two leads, as a coherent scattering region. Once the transmission probability T is known for the system, the correspond ing current due to an applied voltage can easily be calculated with the Landauer formula. This treatment rests on the assumptions that the current is composed of a flow of independent non-interacting electrons, which scatter elastically through/off some form of potential barrier. The resulting formula is ( / = -J dE[f(E - fiL) - f(E - nR)]T(E), (1.5) where E is the energy of the incoming electron, while fiL = eVbias and /jR = 0 specify the voltage applied across the system. The zero-bias conductance of the system is defined as g = dl/dV\}\as\vbias=Q, which is then given by ' = M-^r)n£) (L6) Of course the transmitted wave function will have a phase, but the current through a system is a function of T — \t\2, where the phase of t cannot be directly measured. The problem now is to calculate the transmission probability of the system. This can be done via scattering theory as done by Delgado et al.[33], or with Green's functions. For details of the green's function approach see the book by Bruus and Flensberg[34]. For a single quantum dot with a single level coupled to a left and right lead via Ti and TR, the transmission probability is found to be where td is the energy of the single dot level, and F = TL + FR is the sum of the tunneling rates into and out of the dot via the left and right leads. Substituting 1.2 Theoretical Overview 15 this transmission probability back in to the Landauer equation for conductance, and letting E = x • kBT, results in = 1 / , 1 oT/kBT f/kBT 9 X 2 l 4 J cosV{x/2) (1 + a) x2 + ff/kBT\ ' • a where I have let Tr, = T, TR = aT, and f = ^ 'T. This makes the last term aa lorentzian of width T. so in the limit that T/kBT —> 0, we get a delta function which kills the integral, giving n 2a T 8(1 + a) kBl We have thus seen that the conductance through a single resonant level produces a Lorentzian with respect to the energy difference of the dot level and the lead electrons, which in the limit of weak lead-dot coupling (as will be used in the central works of this thesis) has a peak that scales as Y/kBT. Thus if we choose a = 1 for equal lead couplings, we find that T/kBT • g = n/8 ~ 0.39. Another way to calculate the current through a single quantum dot level is via the use of a master equation. In this approach, we consider two probabilities: P0 is the probability that the dot will be empty, and Pd is the probability that the dot will be full. If we then assume a low bias voltage regime in which an electron can tunnel on and off the- dot via both the left and right leads, we can write the classical master equation ^o = Toff-Pd ~ ronP0 (1.10) where ron = TL0 + r^0 and T0ff = T0L + T0R. Transitions on and off the dot can occur via the left or right lead, resulting in the rates Ti0 = Tif(e(i — Hi/e) and Toi = Ti[l — f(cd — fJi/e)], where i € {L, R} and e^ is the electrostatic energy of the electron on the dot (often rescaled to be zero). The Fermi functions are present to either require and electron in the lead with the energy of the dot level, or a hole in the lead with that energy. For a derivation of these rates via Fermi's golden rule, see section 2.2. Solving these equations for the steady state probability distribution, 16 1 Introduction with the additional normalization constraint that PQ + Pd = 1 results in the solutions J- ~T r1 x oft 1 off Letting TL = TR = T, we can then solve for the current between the left lead and the dot (which must equal that trough the right), giving / = e(rL0P0 - T0LPd) = f (h - (1 - WljT^)) (L12) -= —(fL - fa), where we have abbreviated /, = f(cd — //j/e), and the denominator D = 1 + (JL + /R)/(2 — fL — fR). If we now set the bias voltage such that HL = Vbias and //# = [), then the conductance can be calculated via the formula g = 9//dVbias|vbias=o> giving g{) = ^^^] (1 13) {iA6) = r f(E)[i-f(E)} kBT D Again, we see that the conductance scales as T/ksT, but now we see that the reso nance width is proportional to kBT due to the derivative of the Fermi function. For simplicity, we have used (—df(E)/dE) = f(E)[l — f(E)]/kBT, where the two fermi functions now tell us that we need an electron available in the lead from f(E), as well as an empty spot in the a lead from 1 — f(E). In this classical master equation approach, all of the quantum mechanics is taken into account by the T's which can be calculated via Fermi's golden rule, as is done in section 3.1.3. Figure 1.7: Double quantum dot schematic in which the tunnel rate from the left lead to dot a is F/,, and from dot b to the right, lead is T^, while the two dots have tunnel coupling tab- In order to generalize this approach to a double dot system (depicted in figure 1.7), we must use a quantum master equation for the density matrix of the now three 1.2 Theoretical Overview 17 possible states, (na.nb) G {(0, 0). (1, 0), (0, 1)}. The density matrix and quantum master equation will be described in section 3.1.2, or can be seen in detail in the quantum optics text by Scully[35]. In this case, we will only three diagonal entries giving the probability of these three possible states of the system. Although the full master equation involves the off diagonal elements as well which will not be shown here (see section 3.1.2), this can be reduced to an equivalent classical master equation of probabilities and classical transition rates: Po = —(FLO + TR0)P0 + TOLPU + ^mPb P0 = rLOp0 - r0Lpa - f(pa - pb) (1-14V ); Pb = r*0Po - T0RPb + T(Pa -Pb) I* 12 r p \tab\ ± off v C-ff + Xh where T is the effective classical rate at which electrons tunnel back and forth between dots a and b. For an explicit derivation of this classical master equation for a double dot see section 3.3. In the limit that T0L = TR0 = 0 and lead tunneling can only occur from left to right, these equations can be solved in the steady state limit to give the high bias voltage sequential tunneling current formula as first derived by Nazarov et al.[36] and Stoof[37], r |2 I(AE) = e ^ f|2 —. (1.15) where in this equation, TL is the rate at which electrons tunnel from the left lead onto dot a, TR is the tunnel rate from dot b to the right lead, tab is the tunnel coupling between the two dots (as shown in figure 1.7), and AE = £(1,0) — E(0,1). This is sequential tunneling, the lowest order most classical form of transport in which an electron hops across the dots, one dot at a time. In this equation the resonance width is set by TR not by the temperature of the electrons in the leads. Via the uncertainty principle, these decay rates into and out of the dot via the leads (or the states lifetimes), are proportional to an energy uncertainty or intrinsic level linewidth also denoted V, as shown in figure 1.8. This decay rate or tunnel rate is proportional 18 1 Introduction to the quantum tunneling strength between the lead and dot. Thus in the weak lead- dot coupling regime with high bias voltages, as two dot levels are swept past each other, the current resonance will be proportional to the Ps of the two dot levels. For F Energy TLU ML •tr R kBT\ VR Figure 1.8: Schematic of the two dot energies, and two lead chemical potentials, in the high bias voltage regime. 1.2.2 Beyond Sequential Tunneling As we have just seen, sequential tunneling is the lowest order form of tunneling and can be described by an effectively classical master equation (with some work). If one thinks of the tunneling hops as Fermi's golden rule transitions, higher order transition rates exist which involve two or more hops through virtual states. The difference between a regular state and a virtual state is that the electron is there for such a short time that the uncertainty principle allowTs transitions to a virtual state that do not conserve energy. These are known as co-tunneling events, as they involve multiple simultaneous transitions which can even involve more than one electron. In fact, a single electron appears more delocalized over the entire system during these processes, rather than being highly localized on one dot after another. These highly delocalized states are known as molecular states, caused by the strong tunnel 1.2 Theoretical Overview 19 coupling between the dots. These molecular states are in fact energy eigenstates of the Hamiltonian including both the electrostatic charge state energies as well as the inter-dot tunneling. In a theory paper, Hettler et al.?? use a novel technique in which lead electrons tunnel directly into the molecular states of a strongly coupled double dot system. Rather than a sequential tunneling current resonance when the two dots electrostatic energies are degenerate, high current is found when the molecular state energies are on resonance with the leads [38]. The molecular states of the system correspond to those in which the matrix representation of the system Hamiltonian is diagonal (as derived for a double dot by van der Wiel et al.[10]). These two molecular eigenstate energies can of course not be degenerate, and have a difference proportional to the inter-dot tunneling. However, only one eigenstate is needed for transport as it can be composed of a superposition of both dots. Including Coulomb interactions, Hettler's model allows for a system with two electrons, where both molecular eigenstates are filled. With this model, Hettler et al. were able to investigate a novel mechanism for an interesting effect known as negative differential conductance (NDC), in which the current through the system actually decreases as a function of bias voltage. The first and most trivial case of NDC found is due to the added capacitive coupling of the lead contacts bias voltage to the charge state electrostatic energies. Thus, as the bias voltage is increased, the eigenstate energy can actually be shifted out of the bias voltage window causing the current to suddenly shut off. This effect is of course also possible in the sequential tunneling regime, as the lead voltages actually couple directly to the charge state electrostatic potentials, only modifying the molecular states indirectly. The second and more interesting mechanism of NDC found is a direct consequence of this co-tunneling via the molecular eigenstates. Since the molecular eigenstates have an energy difference, it is not until a sufficiently high bias voltage that the higher energy state will become available. In the case of strong inter-dot tunneling £ob > ^TLTR/2 it was found that at high biases, the second eigenstate filling doubled the current like an ordinary extra transport channel. However, in 20 1 Introduction the case of tab < ^/YiTR/2 it was found that the current actually dropped as the resonance was broadened[38]. Actually, Hettler didn't provide any explanation for this effect, as it was first discovered by Djuric using a more conventional master equation technique[39]. We will present another NDC mechanism only possible in triple dot rings, by using a similar technique of tunneling the lead electrons directly into the systems molecular eigenstates. However, before moving on to our model of a triple dot ring, let first consider some other recent models and their shortcomings. 1.2.3 Triple Dot Rings and Aharonov-Bohm Oscillations Due to the recent success in triple dot fabrication, theory papers have begun to deal with the AB oscillations found in triple dot rings. One such paper comes from Delgado et al.[33], in which a non-interacting tight-binding approach is used. Discretizing the leads as a one dimensional chain, a transfer matrix T can be written to transform the wavefunction amplitude at one lead to that at the other lead. The transmission through the system is then a function of the matrix elements of T, giving the simplest expression if first diagonalized. Diagonalizing this matrix requires the molecular eigenstates and energies of the triple dot system, which Delgado et al. have found for the specific case in which each dot has equal energy, and all three have equal coupling to one another. Although sequential tunneling would be maximal here, for strongly coupled dots we expect the current peak to be shifted off this naive triple point. Furthermore, we find solving for the eigenstates in the limit that taf, ^> tac to be simple and illustrative. Delgado et al. did find Aharanov-Bohm oscillations with a periodicity of $0, as well as anomalous sharp dips in the conductance at $o/2, but the cause of which remained a mystery. We will not only explain these dips, but also see how they are smoothed out by decoherence effects. Another effort to analyze the triple dot ring was made my Michaelis et al.[40] who found an all electronic analogue of coherent population trapping in quantum optics known as a "dark state" [42, 43]. Here it was found that this "dark state" provides a novel mechanism for rectification in the high bias voltage regime. In this work Michaelis et al. used a quantum master equation for the density matrix of the 1.2 Theoretical Overview 21 system, coupling the left lead to two dots, which can tunnel to a third dot which couples to a right lead (as in figure 2.1, but with t' = 0). Although this does not allow magnetic flux to be enclosed by the dots giving AB oscillations, the effect found is related. Looking at the steady state current limit, it was found that an anti- bonding superposition state of the first two a,b dots is an eigenstate of the system which cannot tunnel into the third dot due to deconstructive interference of the two possible paths. The rectification behavior found was thus explained by the fact that this anti-bonding eigenstate can still decay back into the left lead if the bias voltage is reversed[40]. Although we too find this high bias rectification behavior, we will show- that the mechanism behind it is slightly more subtle than this, involving a second molecular state to complete the rectification. Michaelis et al. also introduce dephasing terms to the master equation representing charge noise in the system. Decoherence of the "dark state" in turn prevents it from fully blocking the current. In this case it is found that in the limit of no dephasing the current is blocked by the "dark state", but in the limit of strong dephasing the current is again blocked due the quantum Zeno effect of the environment repeatedly measuring the occupancy of each dot. For intermediate dephasing rates, the current that leaks through provides a measure of the decoherence rate or coherence time of the system. Although the "dark state" responsible for the current blocking discussed above is found for a system in which all three dots have the same energy, and there is no tunneling between the left two dots, in further works continued by Emary et al.[41] the robustness of this "dark state" is explored for less arbitrary choices of parameters. It is found that while the right dot c can have any energy, if the left two dots a and b have energies ea = A and e?, = — A then a "dark state" can always be found by offsetting the tunneling from n to c and b to c such that A = A0 = tab(tlc — tlc)/2tuctbc. The current is then compared to this energy detuning A — A0 where it is found that the current blocking is lifted as the "dark state" is lost, as expected. AB oscillations were then studied where it was found that the "dark state" caused the current to drop to zero with a periodicity of $0 when tab ^ tac, but with a periodicity of "&0/2 22 1 Introduction when tai, = tac. Although the mechanism behind this difference in periodicity was not well understood, many references to systems with <&0 periodicity[3, 4, 44] as well as systems with 1.2.4 What's Next? In this thesis we will present a theoretical model for coherent transport through a triple dot ring, resulting in a novel negative differential resistance effect which clarifies the mechanism behind the "dark state" rectification predicted by Michaelis et al.[40]. We use a quantum master equation like Michaelis and Emary, but rather than having the leads tunneling to the dots themselves, we couple the leads directly to the system's molecular eigenstates, as done by Hettler[38]. The lead tunnel rates then follow from Fermi's golden rule (FGR). This approach is justified in the limit that inter-dot tunneling is much stronger than coupling to the leads, and should not be neglected in the lead coupling as done by Emary (See discussion in section 3.1.2). This treatment is valid for both zero bias conductance, as well as the high bias voltages in which it reproduces the results of Michaelis and Emary. Furthermore, we couple our system to a bosonic bath as done by Brandes et al.[32]. However, unlike the induced dephasing terms derived by Michaelis and Emary by neglecting the strong inter-dot tunneling, we find a novel way of including this inter-dot tunneling. The effect of this bath coupling is thus found to induce transitions between molecular eigenstates of the system. These transitions are clearly inelastic (unless molecular states are degenerate) either taking energy from the bath or giving energy to the bath. This results in the molecular eigenstates being non-stationary states, allowing the system to relax to the ground state, or be thermally excited to higher energy molecular states. In the strong bath coupling limit we find the system's molecular states thermally distributed, in equilibrium with the bath temperature. 1.2 Theoretical Overview 23 Furthermore, in the zero bias regime with inter-dot tunneling much stronger than temperature, we find that transport involves only the ground and first excited molec ular eigenstates. This allows the molecular eigenstates to be solved for analytically, leading to a rather simple analytic expression for the conductance in both the strong and no bath coupling limits. Cases of intermediate relaxation and high bias voltage can then be evaluated numerically without loss of any of the qualitative understand ing already gained. From these results we find that the ground state can be tuned to a "dark state" via magnetic field, which results in the conductance being reduced to zero. This is understood by an effective tunnel coupling of dots a and b to dot c, which when reduced to zero passes current in neither direction. In fact this effective coupling sets the width of the conductance resonance, which allows for features much sharper than temperature, even at low bias voltages. The width of this resonance we will see actually provides a novel measure of inter-dot tunneling unaffected by thermal broadening. If the second excited state is tuned to be a "dark state", we then find that the ground state is no longer a "dark state". In this case, it is not until a forward bias is increased until the chemical potential is greater than the excited "dark state's" energy, that the current is suddenly blocked. Although the ground state could still pass a current, in the Coulomb blockade regime once an electron gets stuck in the "dark state" it also prevents a second electron from entering the system. Now if the bias voltage is reversed, an electron first tunnels from the right lead to dot c and then cannot tunnel into the excited a,b "dark state" but will still always be able to take the available ground state. Thus the mechanism for rectification requires two molecular eigenstates, one dark and one bright. Dephasing will of course lift the blockade of the "dark state", but investigating the robustness of the rectification and NDR effects we find that they also provide a novel measure of the intrinsic electron-phonon coupling strength. The remaining chapters of this thesis will proceed as follows: In chapter two we will give a brief overview of our model and techniques used to find solutions. We will then present the results of this model, starting with zero-bias conductance in the 24 1 Introduction limits of strong and no bath, where illustrative analytic solutions can be found. We then consider the effects of large bias voltages, predicting a unique XDR effect, as well as clarifying the rectification mechanism proposed by Michaelis et al. In chapter three we will go through all the details and derivations behind our model and results. Finally, chapter four will give the conclusions of this thesis. 2 Model and Results VR Figure 2.1: Schematic of triple dot setup with inter-dot tunneling t and £', lead tunneling rates T, magnetic flux $, and lead chemical potentials ^L,I-l'-Lh and ^R as shown. 2. J Model Hamiltonian We model a triangular triple dot geometry as shown in figure 2.1, where electrostatic gate voltages are used to define the confining potentials of the three dots arranged in a ring. The electrostatic energy of this triple dot system is easily generalized from that of the double dot system given in equation 1.4, resulting in E 2 E(na, nb, nc) = J^ c,{^ ~ AQ + J] EClJ(nt - AQ(^ - M3). (2.1) i=a.b.c i^j=a.b:c th As we saw earlier, Ec, is the charging energy for the i dot, ECl is the energy of the capacitive coupling of the iih and jth dots, where now i.j G {a. 6, c}. Inspired by the experiments of Gaudreau et al.[25, 26], we use two experimentally tunable gate voltages Vgi and Vg2 cross-coupled to the three dimensionless gate voltages A/i, A/2, A/3. These two experimentally tunable gate voltages, along with the experimental coupling 25 26 2 Model and Results parameters, could then be used to reproduce the stability diagrams of Gaudreau et al., allowing us to tune our system to a quadruple point as studied experimentally. In fact, we use charging energies, capacitive couplings, and tunnel coupling parameters from this experiment, as specified in figure 2.2. Working in the Coulomb blockade regime, as the charging energy of the the dots is the largest energy scale in the system, allows us to truncate our system's Hilbert space to states with only zero electrons or one electron in either dot a,b, or c. Thus we consider only the four possible charge states (0,0, 0), (1,0,0), (0,1, 0), (0,0,1), denoted by the kets |0), |a), \b), \c) respectively. Due to the nanometer size of the dots, their intrinsic level spacing is large enough that each dot is assumed to have only a single energy level. With the empty state's energy rescaled to be zero, the addition energy of dot a for instance is defined as ea = e(l, 0, 0) — e(0. 0, 0). Although the dimensionless gate voltages A/i cross couple to the two gate voltages of the experiment, they can easily be remapped to control individual state energies. Henceforth, we shall thus refer to sweeping electrostatic energies rather than gate voltages. To add quantum tunneling to our model we write a Hubbard-like system Hamil- tonian Hsys, as well as a component describing the two leads //res> and their coupling the dot states Hi and HR. In second quantized notation this takes the form: H = Hsys + HL + HR + HTes, Hsys = Y, tifiji + YtUjfVji: i={a,b.c} i^j Hi = ^2\{wk.aPa + wkibfl)ck + II.c], ,2 2, k HR = Y^b"kMpcck + II.c.]. k J Hres = 2^ £fc C kCk + 2^ t-k^k, k k where the charge state creation operators, defined as fi = |0)(?'| for i G {a.b.c}, are not canonical fermionic operators, as only one dot can be occupied at a time. Therefore, despite its appearance, this Hamiltonian does in fact treat electron-electron interactions in the strongly interacting limit that only one electron is allowed in the 2.1 Model Hamiltonian 27 system at a time. The dk operators are however, canonical fermionic operators for the fermi sea of lead reservoir electron states. For simplicity, we take each dots lead = w couplings to be equal, Wk.a k,b = Wk.c = w, and define the first order lead-dot 2 tunnel rate T = 2irg0\w\ (which is derived in section 3.1.3), where go is the average density of states in the leads near the chemical potential. The inter-dot tunneling 2 parameters are defined as tab = \t'\ and tac = tcf, = |f|e'^ , where © = 27r$/ As discussed earlier, we will use a density matrix and quantum master equation as has been done by several others[40, 41, 47, 48]. Since the Hamiltonian in equation 2.2 describes both the four system states as well as a continuum of lead states, the resulting density matrix can become very large. Since we only care about the state of the system, in section 3.1.2 we derive a quantum master equation for the reducesd density operator of the system states via a partial trace of the general time evolution equation ihps = TrR[H, p], as has been done in similar systems by many others[47, 48]. As we find that the lead coupling is also much less than the temperature of the leads, we can ignore any intrinsic level broadening of the system states. The master equation can then be derived by perturbatively coupling the leads to the system states with an offset chemical potential in the leads to give a bias voltage. This is standard in the treatment of open quantum systems[40, 41] and derived in section 3.1.2 of this thesis. Now, considering the regime in which inter-dot tunneling is much stronger than lead-dot tunneling, it is natural to work in the molecular eigenstate basis of the system. We thus diagonalize Hsys giving E 2 3 Hsys = Y, M> ( - ) j = 1.2.3 28 2 Model and Results where the /$ operators are for the system's molecular eigenstates, defined as \Ei) = a,i\a) + h\b)'+Cj|c), with Ex < E2 < £3. This diagonalization can be done analytically in the illustrative limit that t' 3> t as is derived in detail in section 3.2, but nu merical diagonalization can also easily be used for any other parameter values. This is where our approach differs from others. By perturbatively coupling the leads to the molecular eigenstates via Fermi's golden rule (FGR), rather than to the charge states. In fact this perturbative coupling to the molecular eigenstates turns out be be higher order than simply coupling to the charge states. This approach is distinct from the standard approach of coupling the leads to the localized charge states, in that transport now occurs when the molecular eigenstate energies £1,2.3, rather than the localized charge state energies ta.b.c, are aligned within kBT\e^s of the chemical potential of the leads. This is a well justified approach for behavior beyond sequential tunneling as has already been confirmed by the stability diagrams of strongly coupled double-dot systems [38]. To make our model more realistic we treat inelastic effects due to electron-phonon interactions and electromagnetic noise resulting in fluctuating dot potentials. This is modeled by coupling each dot to a bosonic bath of oscillators, introducing twTo new terms to the Hamiltonian, similar to those used by Brandes et al.[32]: He_p = ^(AjrQna)(aJ!Q + a[a), (2 4) _^ .t " where canonical bosonic operators, j sums the bosonic modes, and a G {a, b, c} gives each dot its own independent bath. The bath coupling factors \^a = A are taken to be equal for simplicity, and the bath mode energies CJJIQ are taken to have a linear density of states, for simple Ohmic dissipation. Interpreting the sum of bosonic operators as coordinates of the bath, Ylji^j + SO = ^• we see that the bath couples to the charge as would a fluctuating electromagnetic potential. Treating the bath- dot coupling similarly to the lead-dot coupling results in additional master equation terms as derived in section 3.1.2. Emary et al.[41] give a standard treatment in the 2.1 Model Hamiltonian 29 charge basis ignoring inter-dot tunneling, which results in ordinary charge dephasing terms. But with tunneling much greater than bath coupling, we work in the molecular eigenstate basis, where these fluctuations are found to induce incoherent transitions between molecular states as is derived in section 3.1.4. Taking the bosonic bath to be in thermal equilibrium with a temperature 7bath = Pleads, these incoherent transitions add energy to or remove energy from the system, bringing it into thermal equilibrium with the bath. In the strong coupling limit that these transition rates are much greater than the lead-dot tunnel rates (but still less than inter-dot tunneling), the system's molecular states are found to be in a thermal distribution with a temperature Tbath- The dephasing treatment by Emary et al.[41] in the charge state basis includes no energy exchange between the system and bath. Inducing molecular eigenstate transitions with no energy cost in the high bias voltage regime where all eigenstates are relevant is therefore only valid in the high bath temperature limit where energy of any amount is always available. After adding additional master equation terms to model the incoherent eigenstate transitions (see section 3.1.2 for complete master equation), we now have 16 poten tially coupled differential equations to solve: one for each element of the density matrix. Taking the steady state limit and projecting this master equation onto the eigenstate basis results in only four coupled algebraic equations necessary to charac terize the conductance. In the zero bias regime these can be solved analytically in the limits of both strong and no bath coupling. Intermediate bath coupling strengths can then be investigated numerically, illustrating the robustness of the "dark state" against decoherence. We will also be able to investigate the role of the "dark state" in the Aharonov-Bohm oscillations. Finally, we consider the effects of finite bias volt age and present the resulting negative differential resistance and rectification effects, explained in terms of this "dark state" physics. 30 2 Model and Results 2.2 Zero-Bias Conductance Considering only the four possible charge states |0), \a), |6), \c) (since T Ec), we study transport near a quadruple point (QP) at which all four states have equal electrostatic energies. Naively, we expect the conductance peak here from sequential tunneling, as all energy levels are aligned for the electron to hop along from one dot to the other. However, with leads tunneling directly into the molecular eigenstates due to strong inter-dot tunneling, it is now a molecular eigenstate energies which must equal zero for transport to occur. This corresponds to the conductance resonance (left of figure 2.2) being shifted off the electrostatic charge state QP, which has coordinates (0,0) marked with an x in the right side of figure 2.2. This plot of IQPC = 0 • Po + 0.15PQ + 0.25Pf, + 0.6PC, simulates the stability diagram, where dark blue corresponds to the ground state being |0), light blue corresponds to \a) or |fe), and red to \c). Here we see that the current peak when ea = e6 = 67.8/ieV = t', showing how strong inter-dot tunneling shifts the resonance away from the electrostatic QP. In addition to being energetically aligned with the lead chemical potentials, transport also requires a molecular eigenstates to be delocalized over dots a and b as well as dot c to connect both leads and pass a current. In the light blue region the charge is localized evenly over dots a and 6, and in the red only dot c. In the broad yellow region however, these states merge and the electron is delocalized over the entire system. From figure 2.2 we see that the resonance peak occurs around the point roughly where ea = e^ = t'. In figure 2.3 we compare the energies and steady state probabilities of both the four molecular eigenstates and four electrostatic charge states of the system at a horizontal gate voltage slice through the resonance shown in figure 2.2. From the charge state energies, we see that both dots a and b have equal and constant energies, while the energy of dot c is swept linearly though zero. Then looking at the eigenstate probabilities, we see that each eigenstate is most often composed of either dot c when its line is straight but sloped, and composed equally of dots a and b when horizontal. The two horizontal eigenstate energy lines correspond to the two possible 2.2 Zero-Bias Conductance 31 0.35 o o 0.3 n U K 0.25 ra ST 0.2 T- ^ 0.05 r*t: =4: 0.15 B Ivu 0.1 3= 0.05 jj -0.05 0 0.05 -0.05 0 0.05 ec (meV) ec {meV) Figure 2.2: (Left) Conductance versus e = (ef, + ea)/2 (where ea = 65) and ec with 3? = 0, at a tem perature of at 50mK. Interdot tunnel couplings are set to \tab\ = 67.8/xeV, |iac| = \tcb\ = 9.7/ieV, dot charging energies are Ec„ = 3.1meV, Ecb = 2.8meV, Ecr = 2.28meV, and the capacitive couplings of the dots are Eab = .86rneV, Eac = .28meV, Ebc — A9meV. The lead-dot tunnel rate T = .\\ieV and the temperature is 50mK (A.2>iieV).(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. eigenstates formed by the strongly coupled a and b dots, which will be denoted ip+ and tp_ and derived subsequently. We can also see here that an eigenstate made of dot c will form an anti-crossing when its energy becomes degenerate with either -0+ or )/;_. To better understand the derealization of these molecular eigenstates, we can decompose the system Hamiltonian (line 2 in equation 2.2) into left and right parts, by writing Hsys = Hab + HC1 where Hab = taha + ebhb + tabflfb + tbaflfa and Hc = Hsys — Hab. Taking tunneling between dots a and b much greater than between a and c, or b and c, as found experimentally by Gaudreau et al.[26], these molecular eigenstates can easily be found by first diagonalizing Hab (as described in detail in chapter 3.2). This results in bonding and anti-bonding molecular states of dots a and 32 2 Model and Results -0.05 0 0.05 e,. (meV) 0.8 x\ \ \ \ P o Probabilit y \ 0.2 0 Xl- -0.05 0 0.05 0.15 -0.05 0 0.05 0.15 ec (moV) £=( meV) Figure 2.3: ea = eh = £', with t' = 67.8/zeV » i = 9.7/^eK, and 0 = 0. Top Left: Eigcnstatcs energies, with Ground state (red), 1st excited state (green). 2nd excited state (blue), and empty state (black). Chemical potentials (dashed black) for Vb;as = 1/j.eV. Bottom Left: Eigenstate probabilities, with color code as above. Top Right: Charge state Energies, with dot a (red), dot b (green) dot c (blue), and empty state (black). Bottom Right: Charge state probabilities. All other parameters are kept fixed as previously stated. b denoted |0+) and |i/)_) respectively: |-0+) = cos-1 a) + sin-1 b) W-) — — sm -\a) + cos -jo) (2-5) 2\t'\ tan# = . e& — £a In the zero bias regime of differential conductance, with e_ — e+ = 2\t'\ 3> |£|,/SBT, we find the conductance peak around e+ = ec = 0, and can neglect the higher energy \-ip-) state. Although we will not be able to neglect the higher energy anti-bonding 2.2 Zero-Bias Conductance 33 state in the high bias regime, these approximate analytic expressions allow for simple illustrative analytic equations for the systems conductance. The coupling between |^+) and \c) is then given by the following matrix element, 2 2 |*| = |<^+|/fc|c)| (2.6) 2 = |*| (l + sin0cos27r$/$o), where As shown in figure 2.4. for \X\ = 0 at 3>/o = 1/2, an anti-crossing of diameter zero is formed between the two states. An effective coupling of zero results in a "dark state", in which the electron is trapped in the |V-'+) state for forward bias voltages, or in \c) for reverse biases. This is coherent population trapping, as \tp+) is a coherent superposition state. Figure 2.4 also shows the charge state and molecular eigenstate probabilities, from which we see that when \X\ — 0 then region over ec in which the the probability is delocalized over all of the dots is significantly reduced. In the next sections we will present analytic solutions for the conductance through such molecular eigenstates as a function of magnetic flux for both the cases of no bath coupling and strong bath coupling. 2.2.1 No Bath Coupling Limit Solving the master equation in this limit of no bath coupling (as shown explicitly in section 3.1.5). we find the analytic formula for the conductance, g = d//9Vt>ias|H>ias = 34 2 Model and Results -0.05 0 0.05 tc (meV) \ 0.8 \ §0.6 f 0.6 en \ £ 0.4 \f \ 0.2h 0.2 1 \ 0 -0.1 -0.05 0 0.05 0.1 0.15 -0.1 -0.05 0 0.05 0.1 ec (mcV) ec (rncV) Figure 2.4: (Top Left): Eigcnstates energies, (Bot Left): Eigcnstatc probabilities, (Bot Right): Charge state probabilities. Now with 0 = TT, and all other parameters are kept fixed as previously stated. Color coding as previously used. 0 with / EE Y.i=i,2(rioPoo - FoiPu), to reduce to: 9 r T1LT1R (-df(E) 1 - f(E2) g = 2TTF-—^^- [ ———\El T + T dE l-f{El)f{E2) lL 1R (2.7) T2LT2R -df(E)i 2TTT \E2 T2L + T2R V dE 1 - f(El)f(E2) where f(E) is a fermi function with temperature Tieads, with E\ and E2 being the ground and first excited state energies (see equation 3.41). The Fermi function deriva tive terms are the usual terms for resonant tunneling through single level, ensuring the energetic availability of the state, giving a resonance width proportional to /c^T as shown in section 1.2.1. The second Fermi function terms are new blocking fac tors which say that transport through one eigenstate prevents transport through the other eigenstate. This makes sense as both eigenstates are composed of the same 2.2 Zero-Bias Conductance 35 -0.04 -0.02 0 0.02 ec (meV) Figure 2.5: Differential conductance versus dot energy ec and enclosed flux <£>, for the case of zero relaxation and ea = £b = t' and t C i', at a temperature of 50 mK. All other parameters are fixed as before. physical dots, and only one electron is allowed in the system at a time. Tn, and TiR are matrix elements which specify how well the ith eigenstate couples to the left and right leads respectively (see equation 3.42 for expressions and derivation).Since the composition of these molecular eigenstates is a function of the dot energies and the magnetic flux, the matrix elements are not just constant pre-factors anymore, but satisfy the following expression (derived in section 3.2): \X\ TiLTiR = W 4\X\* + (AE)*' (2-8) for i e {1,2}, where AE — (e+ — ec). We now clearly see that the resonance is mod ulated by a Lorentzian of width |X|, the effective coupling of • t' ;» kBT, "Dark state" current blocking is symmetric with respect to the sign of the bias voltage. 0.14 0.25 0.12 ;r- o.i J- 0.2 o. •i ^0.08 t-c £•0.15 ^ 0.06 \ •* 0.1 • 0.04 * i, 0.05 0.02 -0 1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0 e, (meV) f(. (meV) X10"3 (-, 0.15 dl o.i -0.05 0 0.05 0.1 -0.05 0 0.05 e(. (meV) er (meV) Figure 2.6: g • (ksT/T) versus electrostatic energy ec for various $: (Top Left) In figure 2.5, the zero-bias conductance is plotted versus the enclosed flux $/$o and the charge state energy ec, showing that the resonance width goes to zero when ^•/^o — -5 and |^Y| = 0. The resonance has maximal width at contribute. Here the resonance is narrower and more centered around ec = 0. These effects will become clearer by looking at horizontal line cuts of figure 2.5 In figure 2.6 we see four line cuts for various magnetic flux, with the contributions of each molecular eigenstate in a different color. In the top two plots we see the first excited state begin to contribute as \X\ decreases. In the bottom two plots we notice the two states contribute more equally and sharply near X = 0. Now that at this point exactly, the current peak is actually three orders of magnitude smaller and essentially zero. We also notice that when the current is passed mainly through the ground state, the resonance curve (marked in red) is asymmetric. This is due to the ground state composition changing, the energy of which is constant when the state is composed of dots a and b, while changing linearly when composed of dot c. A single eigenstate will therefor have an asymmetric resonance due to its reconfiguration as it delocalizes across the resonance. It is required that kBT > \X\ the excited state to contribute, but at this point the asymmetric resonance is lost due to deconstructive interference responsible for reducing \X\. 0.2 0.15 0.1 > 0.05 I -°-05 -0.1 -0.15 -0.2 -0.1 -0.05 0 0.05 0.1 0.15 -0.04 -0.02 0 0.02 f. (meV) c ec (rneV) Figure 2.7: (Left) Eigenstate energies versus dot energy ec for $ = 0, in the case of zero relaxation and f„ = f;, = t' and t = t', at a temperature of 50 mK. All other parameters are as before. Color coding of cigenstates is as before. (Right) g • (ksT'/T) versus dot energy ec and enclosed flux $, for the case of zero relaxation and e„ = tb = t' and t = t', at a temperature of 50 mK. All other parameters are as before. In the case that t = t', we cannot necessarily ignore the third molecular eigenstate, unless t 2> ksT. The energy difference of the a, b bonding and anti-bonding states, 38 2 Model and Results e+ — e_, will now be small enough that both |^'+) and \ip_) can couple to \c) simultane ously. This can be seen in the right of figure 2.7, where there are no longer clearly two distinct anti-crossings. Even if all three of the systems molecular eigenstates don't contribute to the conductance, analytic expressions will now not be easily found. The eigenstates and their energies can however be calculated numerically, as was clone in the left of figure 2.7. The conductance in this case can be calculated numerically, and is as is done for the conductance in figure 2.7. There does still remain some asymmetry towards the right for the same reason as before, but to a lesser extent as t' is now smaller. A sharp peak and pinch in the resonance at 2.2.2 Strong Bath Coupling Limit Figure 2.8: g • {ksT/T) versus dot energy ec and enclosed flux $, for t' 3> t in the case of strong bath coupling. All other parameters arc as before. In the limit that the incoherent eigenstate transition rates (equation 3.27) induced by the bath are much greater than the lead-dot tunneling rates the master equation is also analytically solvable (see section 3.1.5). In this case1 we find our molecular eigenstate probabilities to have a thermal distribution in equilibrium with the bath 2.2 Zero-Bias Conductance 39 temperature (which we take equal to the lead temperature for simplicity). In this limit we are able to find an analytic conductance formula independent of the particular relaxation rates that reduces nicely to: 2itdY T1LTm-ttEx){\-f{E{)) E) + T2LT2R • f(E2)(l - f(E2))e^- ^ (2.9) + T2LT1R-f(E2)(l-f(E1)) E + T1LT2R • /(/^(l - J{E2))S ^' .. El E2) El E where the normalization factor D = T1(l+f(E2)e^ - )+T2{f(E2)+e^ - ^),with , 0 — l/(A sTbath) and T, = TiL + Tift. Here we see the same two processes as before in the first two lines of equation 2.7, where and electron enters one state and leaves via the same state. In the last two lines however, we find an additional two processes in which the molecular eigenstates are mixed, entering via one state and leaving via the other. The matrix elements in the third term tell us that an electron enters into the excited state, and then leaves via the ground state by emitting energy to into the bath. In the fourth term we see that the electron is thermally raised from the ground to excited state before leaving the system, and thus requires and extra Boltzmann factor to represent this energy cost. We also see that rather than blocking factors on the first two processes, an extra Boltzmann factor is present instead, so that transport through the excited state is proportionally less likely the greater its energy. Simplifying equation 2.9 we find (i + siiig) r \x\2 ;i - /(E^m) n: 21) \(4\X\2 + (AEy) E + {i-f{E2))f(E2)e^- ^\ (2.10) E E + (1 + ,S 2 l^) 2f(El)f(E2)e* > v 4\X\ + {AE)\ The first two lines in equation 2.9 can still contribute an arbitrarily sharp resonance of width proportional to \X\ as with no bath coupling. But, now these can be over whelmed by the two new processes represented in the last line of equation 2.10. As seen in figure 2.8,.the resonance width no longer goes to zero at ^>/ O.H; 0.12 c^ 0.1 ,-.0.08 t; 0.06 • 0.04 0.02 ec (meV) e,. (meV) 0.4 0.5 0.35 0.45 I •I ^ °-3 3" 0.4 j ^0.35 "-"0.25 „ 0.3 i ^ 0.2 ^-0.25 i 1 i J 0.15 _J 0.2 ^0.15 =v 0.1 I 0.1 i I 0.05 0.05 / 0 er (meV) e(. (meV) Figure 2.9: g • (ksT/T) versus dot energy ec for various enclosed flux $, in the strong bath coupling limit with ea = £(, and all other parameters as before, at a temperature of 50 mK. (Top Left) ct> = 0, (Top Right) d) = 1.57T/2, (Bot Left) <5> = 1.9TT/2, (Bot Right) Plotting this conductance, we see that the resonance in figure 2.8 is still somewhat curved towards the right, for values of $ ^ 0. By looking at a series of horizontal line cuts for various values of flux (as shown in figure 2.9) we see that when \X\ is maximal at $ = 0, the current still passes almost entirely through the ground state. This is because the energy difference between the two states is still to large to be overcome by the charge relaxation. As the effective coupling \X\ is decreased, the energy difference between the first and ground state is decreased, and the eigenstate transitions induced by the bath become strong enough to generate some current. Curves in blue and yellow represent the processes in lines three and four of equation 2.2 Zero-Bias Conductance 41 2.9, where an electron enters the ground state and is raised to the excited state before leaving and visa versa. Finally, when \X\ = 0 we see that the resonance is actually at a maximum, due entirely from processes that scatter between molecular states. In this case we again see the equal contribution from both molecular eigenstates, creating a symmetric resonance unlike that of the ground state alone. From figure 2.8 we also notice that the conductance is also maximal when $/<&o = 0.5 exactly, and the effective coupling \X\ = 0. This is because the eigenstates formed by this effective coupling have an energy difference proportional to \X\. Thus when jXj = 0 scattering between the two states has no energy cost. As \X\ is increased, their energies are shifted apart causing the resonance to bend again as the ground state passes the majority of the current. ec (meV) Figure 2.10: g • (fcjgT/T) versus dot energy ec, for various bath coupling strengths, with enclosed flux 'O'/'&o = 0.5 and all other parameters as before, at 50 mK. Although an analytic solution for arbitrary bath coupling strengths is not easily obtained, the system can still be solved numerically. From figure 2.10, with bath coupling jAj2 = 1, where A = ctj = 3j = Sj, we find a resonance much sharper than temperature still remains at ^/^o = 0.5. With the relevant relaxation rates T12 and T^i (see equation 3.27), we define a dephasing rate T^ = (r^ + T2i)/2. Here we find that when |X|,r0 < ksT, we get a resonance width set my max(|A'|, T^). We will thus find a resonance sharper than temperature (a signature of "dark state" physics) 42 2 Model and Results as long as T^ < \X\ < kBT, but once \X\ < kBT < T^ we will have reached the strong relaxation limit. With the relaxation rates given by equation 3.27, we find the strong relaxation limit (at 50mK) is reached with a bath coupling |A|2 = 100, while the critical value at which "dark state" physics can still be resolved is of order unity. We can now investigate the effect of this dephasing on the Aharanov-Bohm oscillations caused by this "dark state" physics. 2.2.3 Aharanov-Bohm Oscillations Examining vertical linecuts of figure 2.5 and figure 2.8, we see how the Aharonov- Bohm oscillations are affected by strong bath induced dephasing. With no dephasing (top left of figure 2.11), we see that if e:c is aligned perfectly at the resonance peak (line shown in black), the AB oscillation has a periodicity of 0.5 0.7 0.6 ;o.4; | 0.5 fo.4 •g 0.3 o £ 0.2 0.1 0.1 0, 0.2 0.4 0.6 0.! 2 3 1/B[AU] 0.7: 0.6 ro.4 CD / .' \ \ | 0.5 0.3 "a. -= 0.2 •I 0.3 3 O "-0.Z 0.1 0.2 0.4 0.6 0.E 0. 2 3 1/B[AU] Figure 2.11: (Top Left) g • (fcgT/r) 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 Left)g • (ksT/T) versus <&/ 2.3 Negative Differential Resistance and The "Dark State" Rectification in a triple dot ring was first reported in the works of Michaelis et al.[40], where it was found that |^_) (with 9 = n/2) is an eigenstate of the system which was 44 2 Model and Results unable to tunnel into dot c (ie a "dark state") for forward bias voltages, but could still decay into the left lead with reversed bias. From our treatment at zero-bias when t' 2> t, we found a conductance peak near e+ = ec = e0, depending on the strength of the effective coupling \X\. Using the magnetic field, the effective coupling of these two states can be tuned to zero, in which case no current can pass with either forward or reverse bias voltages. In the forward bias case the electron is found to be trapped in the |^+) state, whereas for a reversed bias the electron gets stuck in |c). Although we do not find the "dark state" physics to provide rectification for zero-bias (and i! 3> t). in agreement with Michaelis et al.[40] we do find rectification in the limit of high bias voltage. Furthermore, by looking at current versus bias voltage we find a negative differential resistance effect which will complete the explanation of this high bias rectification. In the high bias regime that \E+ — E_\ Starting off in the low bias regime, we tune the magnetic flux to zero, keeping \%jj+) aligned with \c) where \X\ is maximal. Now the anti-bonding state \xp_) has an energy difference of e_ — e+ = It' » kgT, so it is not until fii > It' that the anti-bonding "dark state" can begin to play a role. Using a symmetric bias voltage this occurs when eVbias > 4i', in which case the current shuts off as shown in figure 2.12. Although the bonding state can still pass a current, in the steady state limit, an electron will eventually enter into the anti-bonding state and become stuck provided the bias voltage is high enough that it can't tunnel back out into the left lead and try 2.3 Negative Differential Resistance and The "Dark State 45 x10 7 NoRel | Strong Rel to \ \ 11 \ < r o • c / Figure 2.12: Current versus bias voltage, for no relaxation and strong relaxation, with all parameters as set earlier. again. Thanks to the Coulomb blockade, one trapped electron is enough to block all current. If the bias voltage is now reversed, an electron fist tunnels from the right lead into dot c, where it can then still has the option of tunneling to either the bonding or anti-bonding state. In this case, \X\ and |V| cannot both go to zero simultaneously, and one state will always be available to pass current. Whereas in the case of forward bias, as soon as either \X\ or \Y\ become zero current is blocked as can be seen in figure 2.13. With a qualitative understanding of the effects of relaxation, we can easily investi gate its effect in the high bias regime numerically. Although in the forward bias case an electron will eventually scatter into the anti-bonding state, it will now be able to tunnel into the bonding state instead of getting stuck. Thus only a small amount of relaxation is necessary to cause a "dark state" to leak some current through, but in the strong relaxation limit, having both xj)+ and ^- to contribute actually increases the total current as seen in figure 2.12. Therefore, the ratio of high bias current to low bias current could give a measure of level of relaxation and in turn the bath coupling. We can also see that in the forward bias case having the two levels in the bias window increases the total current, whereas in the negative bias case, the current remains unchanged. There is therefore still some asymmetry between the forward and reverse high bias cases, even in the presence of strong relaxation. 46 2 Model and Results er (meV) ec (meV) Figure 2.13: Current versus dot energy fc and magnetic flux /<&o for Vj,ias = 500fieV(Lett) and VbiBa = -500fieV (Right). Full color maps of the current in the high bias voltage regime for both forward and reverse bias are plotted in figure 2.13. In the forward bias case, we see that the current shuts off when either X or Y = 0, whereas in the reverse bias case one state always passes a current. Although this was expected due to our explanation of the high bias rectification, there is still a striking difference in the two colormaps. For forward bias, a zig-zag pattern is formed, while in the reverse bias case we find vertical stripes of current. The separation of these two stripes is proportional to the inter-dot tunneling t'. which if brought sufficiently low would cause the two resonances to merge. In this case it might be possible to get a phase flip in the AB oscillations, but this has not been investigated. To better understand the features that were found in the colormaps, it is noted that in both cases, the lead-dot tunneling and inter-dot tunnelings do not change. All that has changed in the two cases is the probability distribution of the molecular eigenstates (and in turn the charge states). Thus in figures 2.14 and 2.15 we look at these probability distributions for various magnetic fields in both forwards and reverse bias cases. By looking at eigenstate energies shown in figure 2.3, we can then tell which charge states each eigenstate is composed of. In the forward bias case, we see that for (p = 0 and opens up at the point of ec where that particular "dark state" occurs. As one involves the tp+ state while the other the -0- state, the probability must shift from one to the other, which we see happen at 1 0.8 I" I 0.4 0.2 -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2 er (uicV) ec (meV) 1 ^^ X" " -^._ 0.8 /' \ / \ = 0.6 •^ \ \ £0.1 4 \ \ \ 0.2 \ -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 (,• (uieV) e(. (meV) 1 0.8 J 0.6 I, 0.2- -0.2 -0.1 0 0.1 t(. (meV) Figure 2.14: Current versus dot energy ec for Vbias = 500/xeV for various values of $ at 50 mK. (Top Lcft)0 = 0,(Top Rig\\i) 1 1 0.8 0.8 1-0.6 f 0.6 to nJ -Q -Q I 0.4 £0.4 A 0.2 / ;\ 0.2 0 0 -0. o 0.1 -0. -0.1 0 0.1 (inoV) e,, (mcV) L 1 1 • 0.8 0.8 / • f 0.6 f 0.6 .da3 I 0.4 y />\ I 0.4 0.2 / // \ •• \ 0.2 0 y v -0. -0.1 0 0.1 0.2 2 -0.1 0 0.1 0.2 (r (mcV) tr (meV) -0.2 -0.1 0 0.1 e,. (meV) Figure 2.15: Current versus dot energy ec for Vbias = SOOficV for various values of $ at 50 mK. (Top Lcft)0 = 0,(Top Right)0 = 0.l7r/2,(Mid Left)<£ = 7r/2,(Mid Right) t£ = 1.97r/2,(Bot) = TT. 3 Details of Calculation 3.1 Quantum Master Equations In order to write an equation for the time evolution of a system, it is convenient to use a density matrix which contains all relevant information about the state of the system. For some arbitrary system and complete basis of states {ipa}, the diagonal elements 2 of the density matrix will equal the state probabilities (rpa\p\^a) = |^Q| , but the off diagonal elements are (ipa\p\t/jg) = ip^P- ^ the system is only occupying a single state of a basis set, there will be a single 1 along the diagonal of the matrix, with all other entries zero. For a system to have probability distributed over multiple states, we maintain that the electron be found somewhere and that the matrix have a trace of unity. The off-diagonal elements however, correspond to the expectation value of the coherence between two of the states. In our triple dot model Hamiltonian (equation 2.2), we now have four charge states coupled to a continuum of lead states resulting in a potentially infinite dimensional Hilbert space. Rather than solve differential equations equations tracking the time evolution of every lead state as well as the system states, we trace out the lead degrees of freedom and derive a master equation for the reduced density matrix of the four system states. This results in a perturbative treatment of leads coupling to the system, as will be derived explicitly in subsequent sections. In this section we will first review the coherent time evolution of a density operator, followed by the derivation of the master equation for the time evolution of the reduced density matrix for a single quantum dot perturbatively coupled to a lead in this way. 50 3.1 Quantum Master Equations 51 We will then see how to extend this treatment to that of our full triple dot system where strong inter-dot tunneling cannot be neglected. In this case we will show how this perturbative lead coupling is more adequately treated in the basis of the system's molecular eigenstates. Repeating this treatment for the additional Hamiltonian terms in equation 2.4 for the bosonic bath coupling, we then derive the additional master equation terms as used in the dephasing treatment of Michaelis and Emary[40, 41]. Again we will see how to generalize the resulting master equation terms to treat a triple dot system without neglecting the strong inter-dot tuenneling. Finally we put all these components together to present the full quantum master equation used in this thesis, and explain how analytic and numerical solutions can be found. 3.1.1 Coherent System Evolution In order to see how the time evolution of a density operator is written, let us first imagine a simple system of states for which it is diagonal. Denoting these states \tpa), where each is occupied with a probability pa, the density operator for the system is P = ^2Pa\lpa)(lpal (3-1) a We can then write the time evolution of the density operator from the time evolution of states, ihp = ^2pa(ih\i(;a)(i;a\ + ih\ipa)(i;a\) = ^2Pa(HSys\i'a)(^Q\-\^a)(^a\Hsys) (3.2) In general, we our density matrix will not be diagonal, but the final line of this equation can be used for any Hamiltonian and a complete basis set. Using our triple dot system Hamiltonian Hsys as written in equation 2.2 results in the following master equation: P= ~f- Yl c^fllp ~ phi) ~ h Yl taififiP ~ Pfifi)-- (3-3) i=a,b,c i^j = a.b.c 52 3 Details of Calculation which results in 16 coupled differential equations for each element in the 4x4 density matrix of the four system states {|0), \a) \b), \c)}. We can however use a basis set in which the system Hamiltonian is diagonal (such as equation 2.3), in which case the master equation takes the form P=-lT,ei(f}fiP-pf!fo (3-4) i Although both of these mater equations are completely equivalent, it is now much simpler to project this master equation into the eigenstate basis in which Hsys is already diagonal. In this case the 4 differential equations for the diagonal elements of the density operator decouple from the rest, making the task of finding analytic solutions much simplcr(provided you can diagonalize the Hamiltonian in the first place). We will now see how to write a master equation for the reduced density matrix of a single quantum dot coupled to a lead reservoir, and then two possible ways to generalize the treatment to a triple quantum dot system coupled to two leads. As this treatment is perturbative, we will see that the use of the eigenstate basis will in fact differ non-trivially from that of the charge state basis. 3.1.2 Lead-Dot Coupling: A Markovian Master Equation As an illustrative derivation of the master equation terms that will come from the Hamiltonian in equation 2.2, let us first consider a single dot with a single level coupled to a single continuum of lead states described by the following slightly simpler Hamiltonian: H = H$ + HR + HSR Hs = td fc HSR = ^t{c\d + Sck), k 3.1 Quantum Master Equations 53 where we consider Hs + HR = H0 while HSR = Hml. We now choose to work in the interaction picture, where the relevant operators become (3.6) iHot/h iHot/h HSR = e HSRe- , where psCh is actually an operator composed of wavefunctions in the Schrodinger picture, while HSR is a usual time independent Schrodinger picture Hamiltonian. This transformation accounts for the free evolution of the system and reservoir due to Hs and HR respectively, where the only remaining dynamics come from their interaction resulting in i>=^.[HSR,p]. (3.7) HSR is now time dependent and requires the use of the time dependent operators d(t) = de~Kdtlh and Cfc(i) = Cke~%€kt^h. Substituting these into the last line in equation 3.5 and regrouping all of the time dependence into the reservoir operators results in HSR = &FR(t) + FR(t)S, (3.8) c i{ek ed)t/h where S = d and FR{t) = tJ2k ke^ ' - We are now in a position to analyze equation 3.7, but first must consider two necessary approximations that will be made: 1) The reservoir is large, and 2) The reservoir has a broad bandwidth. The first assumption implies that the reservoir is not significantly effected by the interaction, and can be approximated as stationary state in thermal equilibrium at sum temperature T. The result of the second assumption is that the reservoir has a correlation time rc which is much shorter than time scale T5 at which the system evolves. Together these two assumptions result in what is called the "Born-Markov Approximation", allowing us to average our system over a time scale At which lies between the reservoirs correlation time and the system's time scale of evolution, TC TC/TS, providing equations of motion which are valid for times greater than At. With these approximations at hand, we formally integrate equation 3.7 over the 54 3 Details of Calculation time scale At, resulting in i /-At p{t. + bt)-~p{t) = -j dt'{HSR(t'),p(t')}. (3-9) Similarly expanding p(t') and substituting above (which is valid for t' — t < At), results in the second order expansion 1 pt+At -, pt+At pt' Ap = - J dt'[HSR(t'),p(t')} + jr^j dt' J dt"[HSR(t'), [HSR(n P(t")]}, (3.10) where Ap = p{t + At) — p(t). Now rather than solving for the density matrix for the system and all the reservoir degrees of freedom, since we are only interested in what the system states are doing we solve for the reduced density matrix of the system by tracing out the reservoir states, where p~s — Tr^p. 1 pt+At i pt+At pt' Aps = -^J dt'TrR[HSR(t'),p(t')]+—t J dt' J dt"TrR[HsR(t'):[HSR(t"),p(t")}}- (3.11) To proceed in solving this equation, we must make the further assumption that the system and reservoir are initially factorizable, allowing the density operator to be written as p{t)=ps(t)®pR{t). (3.12) For short times thereafter, the density operator remains approximately factorizable, with a correction term that can be neglected since At 2 Defining Aps = Ap^ + Ap^ \ we will first consider the first order contribution 3.1 Quantum Master Equations 55 which can now be written as rt+At Ap™ = -j dt'TrnlSF^t') + FR(t')S, ps(t') ® pR(t')] t t+At iy dt'[S,ps(t')} ®TrR{FR(t')7pR(t')} + H.c. t+M (3.13) dt'[S,ps(l')]^2(FR(t')) + H.c. ih Jt t+At dt'[2S(FR(t')) + 2SHFR(t')),ps(t')]. in Jt <• v ' H's With a large reservoir that doesn't change much on the relevant time scale, (FR(t/)) is essentially constant. This result can be interpreted as an additional term shifting the coherent system dynamics, and can be neglected. ~(2) The second term, Aps\ results in the lowest order non-trivial effects of the reser voir, the integrand of which we evaluate and simplify as follows: / = TvR[HSR(t'l[HSR(n P(t")]] 2 = \t\ TrR[SiF(t') + Fi(t')S, (S^F(t") + F^t")S)ps ®PR-PS® pR^F[t") + F\i")S)] = (&&ps - 2&psSi + psSiSi)TrR(F(l')F(OpR) + (S&ps - SpsS^ - tfpsS + ps^S)TrR(F\t')F(t")pR) + (&Sps - SPsS^ - StpsS + psSS*)TrR(F(t')Fi(t")pR) + (SSps - 2SPsS + PsSS)TrR(Fi(t')Fi(t")pR). (3.14) Noting that TxR(FR{t')FR(t")pR) = (FR(t')FR(t")) is a time-time correlation function of the reservoir operators, we evaluate it as 1 (FR(t')FR(t")) = It^icic^e-^-^'^^- ^'' % (3-15) 2 8(f f t t = \t\ / ^(e)/(e)e- - "» '- "\ t/-oo where ckc/. = hk is the number operator, resulting in the fermionic thermal average {c\ck') — f{e)Sk.k'- This delta function kills one sum over k. while the other is con verted to an integral via ^2k —> J deg(t), where g(e) is the density of states of the 56 3 Details of Calculation reservoir. Similarly (c^q.,) = (1 — f{e))Sk,k', while the remaining averages vanish, c c = c c ( \ k') i k k') = 0. Rewriting the equation for Aps , now in terms of these two time correlation functions results in -i /-t-t-ZAH+At t rt'rt A 7,(2[ ) = rf// ,, , t P s' = 7^2 y y ^ [<4(0^(/- 0>(^^s-5tps5)+ t PC POG 2 dt"(Fkt')FR{t")) = \t\ / dt" / deg(e)f(e)e-^)(t>-t") Jt J-OC oo pi' / deg(e)f{e)Jt diV*-^-'") (3.17) 2 = 7t\t\ g(ed)f(ed), where we have used a Wigner-Weisskopf approximation in bringing the time inte gral past the energy integral and setting t = — oo and /.' = 0, to use the identity 5(e) = -— J^° e.'ut to kill the final energy integral. We thus define the lead-dot 2 coupling constant as T = ~-\t\ g0, and rewrite equation 3.7 in form of a master equa tion, commonly known as a "jump term" since it causes an abrupt change to the wavefunction and reduced density operator of the system: A5{2) 1 1 -g- = Y[(dipd--(dSp + pdS))f(ed) + (dpS--{Sdp + pdJd)){l-i{ed))l (3.18) Defining separate transition rates into and out of the dot as Td0 = Tf(ed) and r0d(l — /(e p = Td0(UpL - \(Wp + pLV)) + TM(LpU -\(ULp + ptfL)), (3.19) where we can now take A^ = p, and write L = |0)(d| as the "jump operator" that incoherently transitions the system from the empty state to occupied state. Here we see that a Fermi function is present for transitions into the dot as an electron in the 3.1 Quantum Master Equations 57 lead must be present with the energy of the charge state it is moving into, and that one minus a Fermi function is present for transitions into the lead in which case the lead Fermi sea must have a vacancy with that energy. At this point we can see a sufficiently simple form to these "jump terms'', that we could easily just write them down by hand for any particular transition, then adding on the rate of this transition. The transition rates can then be calculated via Fermi's golden rule to lowest order. Next we will see the simplest way in which to couple two leads to the triple dot system of interest, and then consider another technique more suitable for this case. However, i working in the interaction picture we had to include a time dependance due to the system Hamiltonian. With the two dots being tunnel coupled together, now this system Hamiltonian must include this tunneling. In this case it is more natural to work in the basis of the system Hamitonian's molecular eigenstates. Next, we will see how to write "jump terms" in which electrons tunnel from the leads directly into these molecular eigenstates. 3.1.3 Lead-Dot Coupling: FGR Transition Rates For the full triple dot system we are interested in, we have the four states {|0), \a), |6), |c)} coupled to left and right leads as shown in figure 2.1. Now we can just write down three sets of these jump terms, resulting in the following master equation terms: P = £ {TioWpfi-\(fiflp + PfiPi)] i={afi,c} (3.20) + roi[fiPfl-\(Pifip + ppifi)]}, 58 3 Details of Calculation th where Ti0 and r0i are transition rates from the |0) state to the i state and visa versa, respectively. FGR then gives, for example r«o = f £K/I//UOI2 i lead 2 = J-E K«K /l £«>*(£<* + cl/Jllead^lO)! ^ - ez) i k 2 F 2 = ^IH/i|0)| kE(( S|4)^|FS}| 5((ea + eFS - ek) - (e0 + eFS)) fc (3.21) k 2TT f°° w 2 = -r\ \ / dekg(ek)f{ek)5{tk- ea) = yFl 3o/(ea), where we define the lead-dot coupling constant T = ^|u;|2(7o as before, now assuming that the lead density of states is a constant over the energy range of the dot states. Here we have used \i) = |0) <8> |leadj.) and |/) = \a) 5$ |lead/) as the initial and final states of the system and lead, with |leadj) = |FS) and |lead/) = Cfc|FS), where |FS) is a filled Fermi sea. The initial and final energies are thus Cj = e0 + £FS arid ej = ea + eFs — £fc- Now to include the chemical potential of the left lead we replace the energy of the empty state e0 = 0 with the chemical potential of the left lead e giving the FRG tunnel rate Fa0 = Ff((.a — fii/ )- The FGR tunnel rate into dot b is similarly T^o = r/(e{, — PL/C), while the the rate into dot c, rc0 = r/(ec — ^ij/e), has the chemical potential of the right lead. An applied bias voltage is the result in an offset in the left and right chemical potentials, defined as eVbias = ML — PR for a symmetric bias in the forward direction. Finally, assuming we solve our master equation for the dot state probabilities, we can write the current most generally as IL = e 2, {FioLPoQ — T0iLpii) = 1R = i={a:b} e(T0cRpcc — rc0fi/9oo) = I- For our triple dot system in Coulomb blockade regime, the high bias voltage limit is reached when |e,-| I = e 2. FioLPoo = er0cpcc, but the zero-bias conductance cannot be calculated. i={a.b} This problem can be overcome by perturbatively tunneling the lead electrons directly into the molecular eigenstates of the strongly coupled dot system, a perturbation that in fact goes to higher order that tunneling into the dot state basis. The problem with writing generalizing the the jump terms derived for one dot to three dots, is that using in the interaction picture the time evolution from Ho must now include tunneling between the dots. To include this inter-dot tunneling, it is natural to work in the eigenstate basis of the system Hamiltonian. In order to write these lead-dot coupling master equation terms for tunneling into the eigenstate basis, we can simply write p= E Fioifipfi-l&ilp+pfifi)] «={i,2,3} (3.22) + ^dpfl-\iflhP+pIUi)W: where the sum is now over the three molecular eigenstates, and the jump operators used now become /— |0)(£j|, and \Et) = a2|a) + bi\b) + Q|C). The lead tunneling rates into the ith eigenstate again follow from Fermi's golden rule giving, for the left 60 3 Details of Calculation lead into the ground state for instance, T 2 ioL = ^Y,{\(WLa\i)\ +\(f\HLhm i 2"7T 2 = -J- ^(K^iKlead/l Y^Mflck + 4/o)|lead,>|0>| i k + KE^leadflJ^Mflck + clhWeadmi2)^! - u) k 2 2 2 = T(l(«l/i|0)| + \{a\rb\0)\ ) £ ^-((FSIcDclFS)! ^!^ + eFS - ek) - fa/e + eF3)) k 2 2 = T(\a1\ + \b1\ )f(E1-fJLL/e). (3.23) Each eigenstate now couples to both left and right leads making the total rate into 2 eigenstate i, Ti0 = Ti0L + Ti0jt, where the rate r10fl. = T\ci\ f(Ei — PRI&) is simi larly calculated with HR. In this novel approach, the Fermi functions in the tran sitions to the molecular eigenstates are functions of the eigenstate energies rather than those of the individual charge states. This causes current resonances when a molecular state energy is degenerate with the leads, not when the three electro static charge energies are degenerate favoring sequential tunneling. This has recently been observed experimentally for strongly coupled dot systems and triple dot systems where two strongly coupled dots are contrasted with two weakly coupled dots[28]. Fi nally, we again calculate the current via IL — e \_. 0^ioLpoo — ^mLPa) = IR = i={l,2,3} e 2. (FoiRPn — Fio^Poo) = I, where now it is clear that each eigenstate can COn- tribute to both IL and IR. In this case the zero-bias conductance can be calculated via the formula g = <9//<9Vbias|Vbias=0- It makes no difference whether a symmetric bias or not as we will be taking the bias voltage will be taked to zero anyways, so for simplicity we use /i£ = Vbias and PR = 0. Now, rather than "jump terms" coupling the system to metalic lead contacts, we will further consider the effects of coupling the dots to a bosonic bath and the dephasing induced. 3.1 Quantum Master Equations 61 3.1.4 Bath-Dot Coupling: Dephasing and Eigenstate Transitions Considering again a single dot now coupled to a bosonic bath, we can now follow the same derivation of the lead-dot "jump terms" using the Hamiltonian in equation 2.4 for a single dot. In this case the interaction Hamiltonian takes the form HSR = ^2(Xnd)(aj + a]) i (3.24) = FfS + &F, where now S — rid, and F = 2_]{dj + a])- Substituting S into equation 3.16 now j results in i t-t+At rt' 2) dt t A^ = pji / 'Jt ^"[ P= Y. A[\J)(J\P\3)U\-^(\J)(J\P + P\J)(J\): (3.26) j€{a,b,c} where A is currently an arbitrary dephasing rate which would be calculated via FGR. These "jump terms" result in paa = p^ = pcc = Poo — 0 and Pij — —^-Pij for i,j G {a. b, c}. This gives an exponential decay in coherence between any of the two dots, localizing the electron in a particular dot rather than a molecular eigenstate. However, now that we have written our coherent evolution and lead-dot tunneling in the eigenstate basis, it will be convenient to consider the effects of this charge dephasing on the molecular eigenstates. As we wish to analyze our triple dot system in the regime that inter-dot tunneling is much stronger than the bath coupling, we do not wish to neglect inter-dot tunneling in 62 3 Details of Calculation the derivation of the bath induced "jump terms". In performing a elementary change of basis from {|a). \b), |c}} to {|£q), l^}, l-E^s)}, we find this dephasing to actually induce incoherent transitions between the molecular eigenstates. Knowing that we were interested in the molecular eigenstates, we could have just started with FGR to see what it does to these eigenstates. From this we find the transition rate from the th jth moiecuiar eigenstate to the k eigenstate, rjy = y£K/|tfe-,IOI2*(£/-^) hf 2 = -r- E |A(fc|(na + hb + nc)|j)| 2 <8> Y^ \(bathf\(a-g + al)phath\bathi)\ 8(Ef - E{) 9 A 2 2 = yl ! J2 K^nlj)] (327) m={a,b,c} ® J2 Kbath/lpbathlbath/)!2^!^ - Et) f A 2 = yl P E \{k\hm\j)\ j dsg(e)nB{e)5{e - hukj) m={a,b.c} 2 2 2 = ^-\\\ g{fkokj)nB{hwkj) ^ \(k\m)\ \{j\m}\ , rn={a,b,c} where huikj = Ek — Ej, the initial and final states are defined as \i) = \j) !g> \bath,j) and |/) = |A;) P = YlriALijpL\j ~ ^{L]jUjP + pL\jLij)), (3.28) th where Ltj = \Ei)(Ej\ is the jump operator that takes an electron from the j to ith eigenstate. We then define an effective dephasing rate from the relaxation rates 3.1 Quantum Master Equations 63 for the fist two eigenstates as F^ = (Ti2 + r2i)/2, where the dephasing rate has a clear temperature dependance and an energy cost associated with scattering between molecular states. The dephasing "jump terms" (equation 3.26) used by Emary et al. [41] overlook this energy cost, as they do not induce transitions in the system state or its energy. Transitions between non-degenerate molecular eigenstates do cause a change in the system energy, which must be made up for by the bath, either giving or receiving the necessary energy. This highlights the connection between charge dephasing and energy relaxation, showing that Emary's dephasing model is in fact valid only in the high bath temperature limit where any amount of energy can be absorbed or emitted with equal probability. We now have all the elements necessary to write the complete master equation of our triple dot system, which we will now attempt to solve. 3.1.5 Solving A Quantum Master Equation We can now gather together the master equation components of equations 3.19,3.22, and 3.28, writing the complete master equation for our triple dot system as i + E {rio[f}pfi-\(fJ}p + pfiFi)] .={1,2,3} (329) + roliMl-l(flLp + pJIfm + E rijiLijpLij-TiiLljLijP + pLljLij)] i#j={ 1,2,3} th where q is the i eigenstate energy, ri0 and TQi are the tunnel rates into and out of the eigenstates via the leads, and F^-'s are the transition transition rates from the jth to ith eigenstate. The creation and annihilation operators for the molecular eigenstates are defined as /* = |0)(i?,|, and the eigenstate transition operators as L,j = \Ei)(Ej\. In order to now solve our master equation, we project our quantum operators onto the molecular eigenstate basis, generating a set of coupled differential equations = Pij {AP\J)I where i,j € {0, Ei, E2, E3}. In the eigenstate basis, the differential 64 3 Details of Calculation equations for the diagonal elements of the density operator decouple from the off- diagonal elements, resulting in a system of four coupled differential equations rather than sixteen necessary to characterize the current. Solving for the steady state current through the system in turn results in solving the following system of equations: poo = — (rio + r2o + r30)poo + ToiPn + r02P22 + r03P33 = o p'n = Tiopoo - (r0i + r2i + r3i)pn + ri2p22 + ri3p33 = o (3.30) P22 = T20POO + r2lPn — (ro2 + Tj2 + r32)p22 + r23£>33 = 0 P33 = TsoPOO + r3iPn + T32P22 — (^3 + Ti3 + r23)/?33 — 0. We can then map these four diagonal elements into a vector p = {poo, Pn, P22, P33), which now results in the usual matrix representation of a system of differential equa tions, p = A4p, where p is now a vector and M. is a matrix. As we are only interested in the steady state current of our device, we need only solve for the nullspace of A4. In the regime of zero-bias conductance, where we have seen that only the first two molecular eigenstates contribute to the current resonance as discussed in section 2.2 (see figure 2.3), we can actually solve the above system analytically in the limits of no relaxation and strong relaxation. With the aid of Maple, one can easily find the solutions Poo = r0ir02 + TQIT^ + r^i^i Pu = rioTo2 + ri0r12 + r2ori2 (3.31) P22 = r0ir2o + r10r2i + r2or2i, which when properly normalized are given by pa = Pu/p for i 6 {1,2,3}, with P = Poo + Pn + P22- We then calculate the current through the left lead-dot junction as i = poo(riOi + r2oL) — pnTou — p22ro2L = [(roirioL - ri0r0iL)(r02 + r12) + (r02r2oL - r20r02L )(r0i + r21) (3.32) + (roir2oL — r2or0iL)ri2 + (r02r10i — rior02L)r2i]/p, where we have used the fact that Ta = V0iL + r0iR and similarly I^o = Fi0L + re setting T12 = r2i = 0. four of the final six terms vanish, and the current can be 3.1 Quantum Master Equations 65 reduced to /NoRei = ^^[/(^i-/XL/e)-/(^i-^/e)]ro2+^^[/(^2-^L/e)-/(£?2-/xWe)]roi) p p (3.33) where we have used the fact that (rOiri0i - ri0T0iL) = TiLTlR[f(Ei - phje) - J(Ei - Pn/e)] for i G {1, 2}, with r0;L = TiL[l ~ f(ex - pL/e)\ and TiDL = TiLj{E1 - pL/e). The zero-bias conductance can then be most easily calculated by setting pi = eVt,jas and HR = 0 and then using g = dl/dV^^v^^-Q. This results in the conductance equation 2.7 presented in section 2.2. Solving equation 3.31 in limit that the eigenstate relaxation rates are much larger than the lead-dot tunneling rates requires a bit more work. First we simplify the current equation to / = T1LT1R[f(Ex - pL/e) - f(E, - pR/e)](rQ2 + T12)/p + T2LT2R[f(E2 - iiLje) - f(E2 - pR/e))(Tw + T21)/p E + T2LT1R[f(E2 - pL/e) - /(£?! - pR/e)e^~ ^ (3.34) + J(E2 - fiL/e)/^ - nn/e)^-^ - l)}T12/p E + T^T^fiE, - pL/e)e^~ ^ - f(E2 - pR/e) E • + f(E1 - pL/e)f(E2 - pR/e)(l - e^' ^)]Tl2/p. Finally, to take the strong relaxation limit, we make the simplification p _ (roirpz + r10r02 + r01r2o) ^PP(EI-E2) 7T1 ~ — ^ ri oi + J- io + 120 + (.1 02 + 120 + t loje 12 1+ 01 2 E ~ 7\[1 + /(EOeW^)] + T2[/(£2) - e^~ % (3.35) where TJ = TiL + Tii? for i G {1,2} and ,<3 = l/kBTbath. Also using the fact that r2i/Ti2 = e^El~E2\ and taking derivatives with respect to voltage, the equation 3.34 can be reduced to equation 2.9. As the bias voltage will be taken to zero, chemical potentials in equation 3.35 have been prematurely omitted. Although it may appear that the derivation of the conductance equations is com plete, the price we pay for working in the molecular eigenstate basis is having to 66 3 Details of Calculation actually find analytic equations for them. This is particularly simple and illustrative in the limit if zero-bias conductance where only two eigenstate solutions are needed. 3.2 Analytic Diagonalization In order to diagonalize the system Hamiltonian given in line one of equation 2.2, we first write it as a matrix in the basis of the three dot states {|a), \b), |c)}, giving £a t' t JJsys — i! th I (3.36) t* t* e. We can begin by diagonalizing the strongly coupled a and b dots (ie the 2x2 sub- matrix in the top-left corner), resulting in c+ 0 X •Wsys — 0 e_ Y (3.37) X* Y* where the e+ and e_ are the energies of the |^+) and \ib-) molecular eigenstates formed by the a and b dots and defined by equation 2.5. Now, in the limit that t' 3> t, also with t' 3> ksT in the zero-bias conductance regime, we can throw out the higher energy \ip~) state, resulting in another 2x2 matrix to diagonalize: e+ X -^ svs (3.35 X* er where the effective coupling X (as shown earlier in equation 2.6) is given by the matrix element of the system Hamiltonian given by line one of equation 2.2, coupling the states \tp+) and |c), X = (e+\HsyS\c) 9 0 i0 ( (3.39) = (cos -(a\ + sin r(6|)(|/.|e 4 4 + M<4<4 + H.c.)\c) 9 -i6 ? , . • 6 = i\t\ i cos 2- e + \t\i i sin-T, 3.2 Analytic Diagonalization 67 where for simplicity here, the phase difference in the two paths is put all in the a-to-c path, which is allowed by the gauge invariance of the magnetic fields vector potential. The magnitude of this coupling is then given by 2 2 \X\ = X*X = \t\ (l + sin 0 cos 2TT$/$0), (3.40) as shown in section 2.2. From equation 2.5 in section 2.2 we saw that 6 is the mixing angle for the \ij;+) and \ipJ) molecular states, and that the two are most evenly delocalized between dots a and b when 9 — n/2, and in turn e0 = £&. This is required in order for X to reach both its maximum and minimum, which occur when 0 = 0 and IE,) = cos - cos -em/2|a) + sin - cos -era/2\b) + sin - -ia/2\c) ii/ 22 !/ 2 2 ' ' 2 e ' ' Q/2 ia/2 ia/2 \E2) = - cos - sin ~e' |a) - sin - sin -e \b) + cos ~e- \c) (3 41) 2 £1.2 = \(e+ + 6C) T ^+-^) + 4|XP 21 A" I ImX tan o = , tan a = . ec — e+ ReX These are the analytic molecular eigenstate equations for the full system's ground and first excited states, as used in equation 2.8. From these equations we see that these molecular eigenstates are most evenly delocalized across the system when both 6 — 7r/2 and 5 = n/2 which occurs when both e„ = e;, and e+ = ec. This is where the triple point had been shifted too in figure 2.2, plus a further shift of amplitude \X\ which can now be made clear. Tuning the system such that e+ = ec = e0, we still find an energy difference between \E\) and I-E2} resulting in the current maximum when Ei = 0. From the energy different E2 — E\ = 2\X\, and the symmetry of the two states energy anti-crossing, we see this shift of \X\. This results in \X\, the effective 68 3 Details of Calculation coupling of \ip+) and |c), directly controlling the size of the anti-crossing and range of ec over which the ground state will be delocalized. Counterintuitively, as this coupling gets smaller and the energy difference between ground and excited state approaches zero, the current peak amplitude is increased as the ground and excited states both contribute currents over a narrower range. This creates the sharper, taller resonance peak seen just before the current shuts off in figure 2.5. We can now use equation 3.41 to calculate the matrix elements as shown in section 2.2 equation 2.8, where we now have i 2 2 T1LT1R = (\{a\E1)\ + \(b\E1)\ )\(c\E1)\ cos - -)- sin -) cos — sin — 2 T 2 2 (3.42) 1 9 „ = - sin2 0 4 '4|JSf|2 + (A£7)2' where in fact TuTm = TILTIR as stated in equation 2.8. Now by diagonalizing the strongly coupled a and b dots first, we have essentially changed the picture from two spatially separate dots with equal energy levels, and converted them to a single dot with two energy levels separated by 2|i'|. In a regime where t' 3> I\ kBT, we can throw out this second state on the left dot, reducing the model to an equivalent double dot system, where each dot has a single level, which are coupled by two possible transport paths. 3.2.1 Equivalent 2-path Double Dot After diagonalizing the strongly coupled dots a and b to be one large dot A, and keeping only its ground state, our triple dot system is reduced to that of a double dot system with two paths connecting the dots as shown in figure 3.1. This can be represented by the following Hamiltonian Heff = (tt + tb)d\dA + (tt + tb)*d\dCl (3.43) 3.2 Analytic Diagonalization 69 HR Figure 3.1: Schematic of a double dot system connected via two tunnel couplings representing two spatially separated paths enclosing a magnetic flux $ which takes the matrix form of eA X* Heff = (3.44) X er where we use the basis {\A), \c)} and have denoted the effective coupling (tt + tb) = X. This effective coupling can then be calculated to be |X|2 = ||^|eW2 + |//6|e-W2|2 2 2 = \tt\ + \tb\ + 2{tt-tb)cos4> (3.45) = 2|£|2(l + cos0), l 2 1 2 where we have taken tt = \t\e l and tb = \t\e~ ^^ . Aside from the extra pre-factor of 2, we see that if the tunnel coupling of the two paths is equivalent, we recover the effective coupling in the triple dot system with 0 = TT/2. This is because in the double-dot system, the large left dot A has no internal structure, while the dots a and b in the triple dot system do. This clearly gives maximal coupling at zero field and zero coupling at = n, so long as \tt\ = \tb\ = t. But, supposing that \tt\ = t and 1^61 = t + ei we find that \X\2 = t2 + (t + e)2 + 2t(t + e)cos |X|2 = T(l + (l-e2)cos^). (3.48) We have now seen that when the energies of dots a and b are equal (9 = TT/2), our system is essentially reduced to that of a double dot system with two transmission paths of equal transmission. In fact, the detuning of ea and e^ is analogous to a detuning of the two transmission amplitudes in the double dot system. This is a parameter not easily tunable in most double dot fabrications. In the next section on other master equations, we consider an effective double dot model, in which the left dot is actually the molecular eigenstate |^+). This model will contain both the two-path double dot structure, as well as the additional internal double dot structure of the left dot, but be solvable exactly. 3.3 Other Master Equations In this section we will consider two other possible master equations that could describe our system. Although the failure at low biases of the master equation in the charge state basis, as used by Michaelis and Emary[40, 41] in the high bias, could not be fully explained, even an effective double dot master equation which is solvable exactly is shown to fail in the low bias regime. Each of these models however is in agreement with the main results of this thesis at high bias voltages. 3.3.1 The Analogous 2-Path Double Dot Master Equation In writing an effective 2-path double dot master equation, we will have a left and right each coupled to a left and right lead, but we will still write these new dot states 3.3 Other Master Equations 71 as \L >= cos||a > + sin||5 > and \R >= \c > in terms of the actual triple dot system. This preserves the original double dot structure in effective left dot, making this model completely analogous to that used in this thesis. This double dot system is now governed by the Hamiltonian HLR= ]T eaclca + Xc[cR + X^RcL, (3.49) a=L,R resulting in the quantum master equation P =-r[tL(nLp ~ pnL) + ej?(»W - pnR) + X(cLcRp - pcLcR) + X*(cRcLp - pcRcL)] i i r C C CaC P + PCaC + T + Yl i «o[ a:P a ~ ^ " «^ 0»{Capc]x - -(clcQp + pc|,CQ)]}. a=L,R (3.50) The steady state conductance requires the solution to the following set of equations: Poo = — (r,L0 + rm)poo + ^OLPLL + F0RpRR = 0 —i PLL = -r(XpRL - X*pLR) + rLOPoo - ^OLPLL = 0 PRR = T(XPRL — X*PLR) + ^ROpOO ~ ^ORpRR = 0 —i 1 e ( PLR. = ~jH( £ ~~ R)PLR + X(pRR - pLL)) - -(T0R + T0L)pLR = 0, where the off diagonal elements of the density matrix no longer decouple as is the basis of the entire systems eigenstates. The last of these equations with its Hermitian conjugate gives the following equation, 3 52 -h{xPRL - x PLR) = --\x\ WQL + Tm? + UtL_tR)M^ - P«*)> ( - ) where (T0L + T0R) = roff. We can then define a new rate f, where f, l*l2 r off (3.53) 2 2 which is a Lorentzian of width rofj, with the additional scaling of \X\ /h . Substitut ing back into the middle two equations above, we now get -T(PLL - PRR) + TLOpm - FOLPLL = pLL = 0 f {PLL - PRR) + FROPOO ~ F0RpRR = pRR = 0 (3.54) 72 3 Details of Calculation where it now appears that T = TLR = FRL is now a classical tunnel rate connecting the left and right dots. These equations are equivalent to the double dot classical master equations that we shown in section 1.2.1. These will in fact give the same current equation but with T instead of tab. These two equations combined with the fact that poo — 1 — PLL — PRR, give the following" solution: PLL = (f ron + TLOrOR)/D PRR = (f ron + TmT0L)/D (3.55) Poo = (froff + roir0fl)/JD D = f (rL + rR + rL0 + rm) + (TLTR - rLOr«0) The current is then given by the following formula, I = e(rioPoo - TOLPLL) f = e—(T0RTL0 - r^oroL) (3.56) f 2 = ?jjl TLTRlf(cL - fiL) - f(eR - PR)}, where the tunnel rates have been set to be: TL0 = jTLf(eL - fiL) TOL = 77L[1 ~ I{(-L ~ PL)] (3.57) r«o = lTRf(eR - pR) r0fl = lTR[l - f(eR - pR)}, where 7 is the lead coupling, and TL,TR are the matrix elements. As this model is equivalent to a double dot model, it is known to be pathalogical at low bias voltages. For instance the current does not go to zero as the bias voltage does. Zero-bias conductance, in turn cannot be calculated in this case. 3.3.2 Tunneling into Charge States Using only the charge state basis, we could write a master equation which now has terms for inter-dot tunneling, as well as the simple lead coupling jump terms which 3.3 Other Master Equations 73 neglect this tunneling. This results in P = — J2 ei(UiP ~ PUi) " ft Yl liMdJP ~ PdUj) i=a,6,c i^j—a,b.c + J2 Ti0[d\pdi - -{did\p + pdi4)] (3.58) i=a,b,c T + J2 oi[dipd\ - -{d\dlP + p4di)], i=a,b.c where the Hamiltonian still only couples the left lead to dots a and b, and the right lead to c. This is essentially the master equation used by Emary et al.[41] with no dephasing. In this case we start off allowing lead tunneling in both directions. In projecting this master equation on the charge state basis in order to begin solving, one no longer finds the differential equations for the diagonal elements to decouple from the off-diagonal elements. This is due to the inter-dot tunneling terms now explicit in our master equaiton. In fact, solving the off-diagonal equations and reducing the system to the four diagonal equations is equivalent to diagonalizing the system Hamiltonian. These equations can of course be solved numerically quite easily, and the current can again be calculated in two ways: II = Poo(Xoa + Tob) — {Paa^ao + Pbb^bo) (3.59) *R Pec*- oc Poo* co- Sensible results however, are only found in the high bias voltage limit of one way tunneling where the leftward direction are reduced to zero, r0a = r0fe = rc0 = 0, while the right moving rates become ru0 — ^bo = Toe = T- In this limit, the results of Emary can in fact be reproduced. In conclusion, we will now summarize the results of this thesis and future directions it could be taken. 4 Conclusions In summary, this thesis has presented a novel modeling technique in the treatment of electron transport through a triple quantum dot ring. The model was inspired by the experiments of Gaudreau et al.[25, 26], from which we have explicitly used experimen tally measured parameters. This produced a regime in which strong electron-electron interactions allowed only single electron transport, putting the system in a Coulomb blockade regime. This can be dealt with easily with a master equation technique, but is entirely neglected in the work of Delgado et al.[33]. Furthermore, experimental data from NRC demanded a model in which inter-dot tunneling was much stronger than lead-dot tunneling. Although Michaelis and Emary do use master equations to model a similar triple dot ring, they completely neglect inter-dot tunneling while treating the lead-dot tunneling. This results in a master equation that fails for low bias voltages. Our model on the other hand uses a novel technique of perturba- tively coupling the leads directly to the molecular eigenstates of the system, which on the other hand does give reasonable results at low bias voltages. We can also reproduce the dephasing jump terms as used by Emary et al.[41], via a perturbative dot-bath coupling which again neglects inter-dot tunneling. But with the desire to include strong inter-dot tunneling, these terms too were adjusted via a treatment in the molecular eigenstate basis. Finally, we were able to find simple analytic solutions for the zero-bias conductance in the regime that t' 3> t. This approach does however require the analytic diagonalization of the system Hamiltonian in order to be used. This can been done analytically in the illustrative limit of /' S> t for low bias voltages, as it is found that only the two lowest energy 74 75 eigenstates contribute. In this limit, the two left a, b dots can be diagonalized first giving a bonding and anti-bonding pair of a, b states. In some cases we can then neglect one of these states and couple the remaining left dot state to dot c and the right. The analytic expressions found in this way give rise to the analogous picture in which two dots are connected by two transport path with a relative phase difference. In this case an effective coupling combining both possible paths can either go to zero if the paths destructively interfere, or be doubled if they constructively interfere. When the current is blocked by destructive interference of the possible tunneling paths, we say the electron is trapped in a "dark state". In the limit of no bath coupling, this effective dot coupling is found to set the width of the resonance, even for low bias. This makes the resonance width a measure of the inter-dot tunneling and a signature of "dark state" physics, unaffected by temperature. We were also able to see the effects of thermal bath coupling on the molecular eigenstates formed by the diagonalization preformed. Strong thermal bath coupling was found to destroy the "dark state" as expected. However, resonances sharper than temperature were found to have a surprising robustness to this dephasing. In the limit that t' = t however, analytic expressions are not as easily available and it is not clear whether a perfect dark state will still be formed. Examining the Aharonov-Bohm oscillations across the sharp resonances found, we did not find any cases of the phase flipping. This is due to the approximation that t' ^> t. In a regime in which both il>_ and IJJ+ states contribute on either side of the resonance, the AB phase would in fact flip. This should be investigated further in future work with this model. The AB oscillations in this limit did still show some interesting behavior. 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B 74, 235309 (2006). \bathf), and g(c) is the bath density of states, which we take to be linear for simple Ohmic dissipation. The bath coupling constant is therefor defined as A = ^\\\2g{JvaJkj)• The sum of matrix elements in the above expression says that for a transition between states k and j, those eigenstates must overlap with the localized dots actually interacting with the bath. Finally, we can write the additional master equation jump terms as