Many paths to interference: a journey between quantum dots and single molecule junctions

Quantum interference effects are at the core of modern physics. Especially fascinating is the fact that interference of electron waves is directly observable through the conductance of a phase-coherent conductor. In general, constructive interference leads to enhanced conductance values, whereas destructive interference results in suppression. We have studied these interference effects in a series of molecules contacted to metallic electrodes: (i) the effect of quantum interference through a single para- and meta coupled benzene ring was shown using a mechanically controlled break-junction [1], (ii) in a anthraquinone transistor where charging the molecule led to a tenfold increase of the off-resonant conductance indicating electrical control over interference effects [2], (iii) a donor-acceptor pair coupled to a OPE3 backbone was used to switch the conducting pathway from linearly to cross-conjugated with a corresponding ten-fold conductance decrease [3] and (iv)finally, pi-stacked dimers were employed to mechanically control quantum interference effects [4]. In the latter experiment, the conductance through the molecule pair changes by two orders of magnitude while changing the electrode distance by 50 picometer. Research supported by the national funding agencies NWO/OCW/FOM and by an ERC Advanced grant (Mols@Mols). References: [1] C.R. Arroyo et al., Angewandte Chemie Int. Ed. 52 (2013) 3152–3155C.R. Arroyo et al., Nanoscale Research Letters 8 (2013) 234. [2] M. Koole et al., Nano Letters 15 (2015) 5569-5573. [3] H. Lissau et al., Nature Communications 6 (2015) 10233. [4] R. Frisenda et al., Nature Chemistry 8 (2016) 1099–1104.

We present a method based on graph theory for evaluation of the inelastic propensity rules for molecules exhibiting complete destructive quantum interference in their elastic transmission. The method uses an extended adjacency matrix corresponding to the structural graph of the molecule for calculating the Green function between the sites with attached electrodes and consequently states the corresponding conditions the electron-vibration coupling matrix must meet for the observation of an inelastic signal between the terminals. The method can be fully automated and we provide a functional website running a code using Wolfram Mathematica, which returns a graphical depiction of destructive quantum interference configurations together with the associated inelastic propensity rules for a wide class of molecules.

Molecular junctions, i.e. single molecules bound to electrodes, are interesting systems to study nonequilibrium quantum transport at the nanoscale. An important transport mechanism in molecular junctions is electronic-vibrational coupling. In this talk, various mechanisms and phenomena of vibrationally coupled electron transport are analyzed, including current-induced vibrational excitation [1], nonadiabatic effects [2], and fluctuations [3]. Furthermore, quantum interference effects and their quenching due to vibrationally induced decoherence are discussed [1,4]. The studies employ a combination of generic as well as first-principles based models and different transport methods, including nonequilibrium Green's functions [1,2] and the hierarchical quantum master equation approach [5]. [1] R. Härtle, U. Peskin, M. Thoss, Phys. Status Solidi B 250, 2365 (2013). [2] A. Erpenbeck, R. Härtle, M. Thoss, Phys. Rev. B 91, 195418 (2015). [3] C. Schinabeck, R. Härtle, H. B. Weber, M. Thoss, Phys. Rev. B 90, 075409 (2014). [4] R. Härtle, M. Butzin, O. Rubio-Pons, M. Thoss, Phys. Rev. Lett. 107, 046802 (2011); Phys. Rev. B 87, 085422 (2013); S. Ballmann, R. Härtle, P.B. Coto, M. Mayor, M. Elbing, M. Bryce, M. Thoss, H.B. Weber, Phys. Rev. Lett. 109 , 056801 (2012), [5] C. Schinabeck, A. Erpenbeck, R. Härtle, M. Thoss, Phys. Rev. B 94, 201407(R) (2016).

Landau-Zener-Stückelberg (LZS) interferometry has been studied in many systems including coupled quantum dots containing one or two electrons. In this talk I will present two recent results from our group where experiments were performed under modified conditions yielding new observations. Firstly LZS experiments will be presented in GaAs/AlGaAs quantum dots containing holes. For holes, in contrast to electrons, it will be shown that due to the spin-orbit interaction spin is not conserved during tunneling. This leads to a more complicated LZS behavior as the magnetic field is applied. This is due to the presence of two channels for interferometry, one related to spin non-flip tunneling and one related to spin-flip tunneling. Secondly, experiments on the singlet triplet qubit (S,T+ ) in GaAs/AlGaAs quantum dots will be presented. In previous measurements it had been assumed that the coupling between the states was directly related to the hyperfine interaction. However, it will be shown that the role of the spin-orbit interaction cannot be ignored if experiments are performed in the presence of phonons with a suitable energy. Amazingly the latter can be controlled experimentally. For both sets of experiments the results will be compared to theory.

Superpositions of indirectly coupled states are possible in quantum mechanics even when the intermediate states are far apart in energy. This is achieved via higher-order transitions in which the energetically forbidden intermediate states are only virtually occupied. Interest in such long-range transitions has increased recently within the context of quantum information processing with the possibility of low dissipation transfer of quantum states or coherent manipulation of two distant qubits .The recently achieved control and tunability of triple quantum dots allow to investigate phenomena relying on quantum superpositions of distant states mediated by tunneling. Recent experiments in these devices show clear evidence of charge and spin electron exchange between the outermost dots[1-3]. We investigate long-range transport and quantum interferences in ac driven triple dots where transitions between distant and detuned dots are mediated by the exchange of photons[4]. We propose the phase difference between the two ac voltages as an external parameter, which can be easily tuned to manipulate the current characteristics. For gate voltages in phase opposition we find quantum destructive interferences among long-range and direct photon-assisted transitions, analogous to the interferences in closed-loop undriven triple dots. As the voltages oscillate in phase, interferences between multiple virtual paths give rise to dark states. Those totally cancel the current, and could be experimentally resolved. 1. M. Busl et al., Nature Nanotech. 8, 261 (2013). 2. F. Braakman et al., Nat. Nanotech.,8,432 (2013). 3. R. Sánchez, G. Granger, L. Gaudreau, A. Kam, M. Pioro-Ladrière, S. A. Studenikin, P. Zawadzki, A. S. Sachrajda and G. Platero, PRL, 112, 176803 (2014). 4. F. Gallego-Marcos, R. Sánchez and G. Platero, PRB, 93, 075424 (2016)

We explain how the electrical current flow in a molecular junction can modify the vibrational spectrum of the molecule by renormalizing its normal modes of oscillations. This is demonstrated with first-principles self-consistent transport theory, where the current-induced forces are evaluated from the expectation value of the ionic momentum operator. We explore here the case of H2 sandwiched between two Au electrodes and show that the current produces stiffening of the transverse translational and rotational modes and softening of the stretching modes along the current direction. Such behavior is understood in terms of charge redistribution, potential drop, and elasticity changes as a function of the current.

Although the dream of manipulating quantum interference in single molecules has been discussed for many years, experimental evidence of the effect of quantum interference on the room-temperature electrical conductance of single-molecules was reported only recently1. In this talk, I will present a brief outline of current understanding of quantum interference in single-molecules2 and then discuss recent results3 demonstrating how room-temperature quantum interference and room-temperature phonon interference can be exploited to increase the thermoelectric performance of single molecules and assemblies of molecules connected to nano-gap electrodes. 1. J. Am. Chem. Soc., 2011, 133, 11426, Nature Nano. 2012, 7, 305; Nat. Nano., 2012, 7, 663, Phys. Rev. Lett. 2012, 109, 056801; Nano. Lett. 2012, 6, 1643-1647; JACS 2012, 134, 5262; Beilstein J. Nanotech. 2011, 2, 699 and refs. therein 2. Lambert, Chem. Soc. Rev. 44, 875-888 (2015); Geng, et al., J. Am. Chem. Soc. 137, 4469 (2015); Sangtarash et al., J. Am. Chem. Soc. 137 11425 (2015); D. Manrique, et al., Nature Comm. 6 6389 (2015); Berritta et al, Nanoscale 7 1096 (2015) 3. Evangeli et al., Nano Letters 13, 2141-2145 (2013); Garcia-Suarez et al, Nanotechnology 25, 205402 (2014); Sadeghi et al, Nano Lett. 15, 7467-7472 (2015); Manrique et al., Nano. Lett. 16, 1308−1316 (2016); Ismael et al, Nanoscale 7 17338 (2015); Rincón-García at al, Nature Materials, 15, 289–293 (2016); Han et al., Nature Comm. 7 11281 (2016)

To be done.

Electron counting techniques are used to distinguish a spin-conserving fast tunneling process and a slower process involving spin flips in AlGaAs/GaAs-based double quantum dots. By studying the dependence of the rates on the interdot tunnel coupling of the two dots, we find that as many as 4% of the tunneling events occur with a spin flip related to spin-orbit coupling in GaAs. Our measurement has a fidelity of 99% in terms of resolving whether a tunneling event occurred with a spin flip or not. We control the double dot tunnelling direction with respect to the crystal lattice and with respect to external magnetic field and map out a landscape in which spin-orbit scattering is maximised or almost completely suppressed.

Rafael Sánchez (1), Björn Sothmann (2), Andrew N. Jordan (3,4) 1 Universidad Carlos III de Madrid, Leganés (Spain) 2 Universität Duisburg-Essen (Germany) 3 University of Rochester, Rochester (U.S.A.) 4 Chapman University, Orange (U.S.A.) Chiral properties of charge and heat transport in the quantum Hall regime introduce peculiar three-terminal thermoelectric [1,2] and thermal [3] effects. Electron propagation along edge channels allows one to define and control quantum coherent trajectories. Scattering of different channels occurs in constrictions such as quantum point contacts. The thermoelectric response usually relies on the breaking of electron-hole symmetry due to energy-dependent scattering at the constriction. In some cases, the occurrence of close-loop trajectories gives rise to Fabry-Pérot-like resonances which induce a finite response even for electron-hole symmetric barriers. This is however not a sufficient condition [3]. References: [1] R. Sánchez, B. Sothmann, A. N. Jordan. Phys. Rev. Lett. 114, 146801 (2015) [2] B. Sothmann, R. Sánchez, A. N. Jordan, Europhys. Lett. 107, 47003 (2014) [3] R. Sánchez, B. Sothmann, A. N. Jordan. New J. Phys. 17, 075006 (2015)

Current fluctuations in out of equilibrium nanoscale systems can yield information about relevant transport mechanisms not accessible from the knowledge of the average current only. In my presentation I will discuss the Fano stability diagram of a symmetric triangular triple quantum dot, as a function of its filling N. A nontrivial electron bunching mechanism arising from the presence of many-body states combined with orbital interference and Coulomb interactions is unravaled for filling N>1. An expression for the Fano factor and the associated dark states is provided for generic N.

Single electron spins in quantum dots is one of the strongest candidates to encode information in it towards building a quantum computer. As the length scale of the system becomes larger, following the current increase in number of coupled quantum dots, we need a technique to couple spins between long range to continuously scale-up the quantum computational architecture. In this talk I will focus on the demonstrations in Vandersypen lab. where we coherently transfer spins through a quantum dot array. We find that quantum dots can both be used as a mediator for coupling distant spins[1], and as a bus to shuttle spins across the array[2]. References: [1] T.A. Baart et al., Nat. Nano. 12, 26-30 (2017). [2] T. Fujita et al., arXiv:1701.00815 (2017).

Using recently developed single-electron approach for electron transport in mesoscopic (molecular) systems we derived the Landauer-type formula for time-dependent potentials, which can be applied for randomly fluctuating potentials, as well. Using this formula we investigated conditions for appearance of dc current under zero bias voltage. We demonstrate that the effect of quantum coherence plays a crucial role in this phenomenon.

In recent years, there is substantial progress in creation of specially separated spin entangled electrons in solid state using the splitting of Cooper pairs, which is necessary ingredient of quantum communication and computing. It is also possible to generate a Josephson supercurrent form by split nonlocal Cooper pairs [1]. This new Josephson current required further studies especially its interference properties. While the behavior of single electrons under the influence of Aharonov-Bohm (AB) and Aharonov-Casher (AC) effects is well understood, it raises the question of the impact of these effects on nonlocal superconducting Cooper pairs that for s-wave superconductors are in spin singlet state. We analyze a normal ring, where a single electron interference is possible and two parallel nanowires connected to two superconducting electrodes, where a single-electron interference can be absent but a cross Andreeev reflection is possible. At low transmission, we can link the AB effect only to local Cooper pairs and the AC effect to nonlocal Cooper pair transport. We demonstrate that by using two quantum dots we can obtain different AC phases for the nonspin-flip and spin-flip transport processes that leads to a beating in the AC effect. 1. R. S. Deacon, A. Oiwa, J. Sailer, S. Baba, Y. Kanai, K. Shibata, K. Hirakawa, and S. Tarucha, Nature Communications 6, 7446 (2015).

In this talk I will first review the Leggett-Garg inequalities and quantum-witness equalities, and then discuss their application as a way of distinguishing quantum from classical transport processes. I will discuss applications in electron transport through quantum dots and interferometers, as well in the quantum walks of cold atoms. I will also discuss a extension of Leggett-Garg to the full counting statistics of charge transfer.

Interference in transport through molecules is a well understood effect. We show that interference also can occur in the spin-flip channel of electron transport through molecules with an unpaired spin. We classify molecules where such a possibility exist. It turns out, that at low temperatures not only is the spin-flip channel shut down, but also the normal channel not involving the spin, will be blocked due to the presence of the Kondo cloud. Hence the term 'Kondo blockade'.

Single molecule circuits are ideal systems for studying a wide range of quantum phenomena. Interference effects are exciting examples of such phenomena, whose influence on molecular transport is the focus of much interest. In my talk I will discuss constructive quantum interference and the conductance superposition law in molecular circuits. I will describe first-principles calculations of electron transport through molecules that have either one or two conjugated backbones bound in parallel. In double-backbone molecules, both branches share the linkers that connect them to the electrodes. Molecules with two branches in parallel can exhibit more than twice the conductance of the single backbone counterparts. This increase in conductance results from the in-phase combination of both backbone orbitals. I will discuss the contribution of these coherence effects in conductance and analyze the effect of molecular structure.

Since the concepts for the implementation of data storage and logic gates used in conventional electronics cannot be simply downscaled to the level of single molecule devices, new architectural paradigms are needed, where quantum interference (QI) effects are likely to provide an useful starting point. In order to be able to use QI for design purposes in single molecule electronics, the relation between their occurrence and molecular structure has to be understood at such a level that simple guidelines for electrical engineering can be established. We made a big step towards this aim by developing a graphical scheme that allows for the prediction of the occurrence or absence of QI induced minima in the transmission function and the derivation of this method and the range of its applicability will form the first part of this presentation [1],[2],[3]. In the second part the scheme will be contrasted with a molecular orbital perspective on understanding and predicting QI effects, which was derived from the Coulson-Rushbrooke pairing theorem in quantum chemistry [4]. [1] R. Stadler, S. Ami, M. Forshaw, and C. Joachim, Nanotechnology 15, S115-S121 (2004). [2] T. Markussen, R. Stadler, K. S. Thygesen, Nano Lett. 10, 4260-4265 (2010). [3] R. Stadler, Nano Lett., 15, 7175–7176 (2015). [4] X. Zhao, V. Geskin and R. Stadler, accepted for publication in a special issue on "Frontiers in Molecular Scale Electronics" of J. Chem. Phys. (2017); http://arxiv.org/abs/1612.02266

Implementing atomic and molecular scale electronic functionalities represents one of the major challenges in current nano-electronic developments. Dangling bond structures created on H-passivated silicon surfaces offer a novel platform for engineering planar nanoscale circuits, compatible with conventional semiconductor technologies. These structures are built by selectively removing hydrogen atoms from an otherwise fully passivated Si(100) or Ge(100) substrate. In the first part of this talk, we show how dangling bond loops can be used to implement different Boolean logic gates [1-3]. Our approach exploits quantum interference effects combined with an appropriate design of the interfacing of the dangling bond system with mesoscopic electrodes. We show how OR, AND, and NOR gates can be realized. In particular, varying the length of the loop or the spatial position of at least one of the electrodes has a drastic impact on the quantum interference pattern; depending on whether constructive or destructive interference within the loop takes place, the conductance of the system can be tuned over several orders of magnitude. We also show that by applying a time periodic voltage, mimicking irradiation with monochromatic light, a dramatic enhancement of the current up to the μA range can be achieved, thus counteracting destructive interference effects. In the second part of the talk [4], charge transport signatures of a carbon-based molecular switch consisting of different tautomers of metal-free porphyrin embedded between graphene nanoribbons is discussed. Different low-energy and low-bias features are revealed, including negative differential resistance (NDR) and antiresonances, both mediated by subtle quantum interference effects. We rationalize the mechanism leading to NDR and antiresonances by providing a detailed analysis of transmission pathways and frontier molecular orbital distribution. References 1. A. Kleshchonok et.al. Nanoscale 7, 13967 (2015) 2. A. Kleshchonok et.al. Scientific Reports 5, Article number: 14136 (2015) 3. A. Kleshchonok et.al. Appl. Phys. Letters 107, 203109 (2015) 4. D. Nozaki et.al. J. Phys. Chem. Letters 6, 3950 (2015)