Multiterminal Geometry Research Articles

Phase-engineering the Andreev band structure of a three-terminal Josephson junction

In hybrid Josephson junctions with three or more superconducting terminals coupled to a semiconducting region, Andreev bound states may form unconventional energy band structures, or Andreev matter, which are engineered by controlling superconducting phase differences. Here we report tunnelling spectroscopy measurements of three-terminal Josephson junctions realised in an InAs/Al heterostructure. The three terminals are connected to form two loops, enabling independent control over two phase differences and access to a synthetic Andreev band structure in the two-dimensional phase space. Our results demonstrate a phase-controlled Andreev molecule, originating from two discrete Andreev levels that spatially overlap and hybridise. Signatures of hybridisation are observed in the form of avoided crossings in the spectrum and band structure anisotropies in the phase space, all explained by a numerical model. Future extensions of this work could focus on addressing spin-resolved energy levels, ground state fermion parity transitions and Weyl bands in multiterminal geometries.

Effectuating tunable valley selection via multiterminal monolayer graphene devices

Valleytronics using two-dimensional materials opens unprecedented opportunities for information processing using a valley polarizer as a basic building block. Various methodologies, such as strain engineering, the inclusion of line defects, and the application of static magnetic fields, have been widely explored for creating valley polarization. However, these methods suffer from low transmission or lack of polarization directionality. To overcome the above limitations, we propose an all-electrical valley polarizer using zigzag edge graphene nanoribbons in a multiterminal device geometry. The proposed device can be gate-tuned to operate along two independent regimes: (i) a terminal-specific valley filter that utilizes band-structure engineering, and (ii) a parity-specific valley filter that exploits the parity selection rule in zigzag edge graphene. We show that the device exhibits intriguing physics in the multimode regime of operation affecting the valley polarization; we investigate various factors influencing polarization in wide-device geometries, such as optical analogs of graphene Dirac fermions, angle-selective transmission via $p\ensuremath{-}n$ junctions, and the localization of edge states. Furthermore, we evaluate the performance of the proposed device structures in the presence of Anderson short-range disorder at the edges and the bulk, and we find it to be resilient to edge disorder even for a higher disorder strength. The device geometry is optimized to achieve maximum valley polarization, thereby paving the way for a physics-based tunable valleytronic device.

Multistage Kondo effect in a multiterminal geometry: A modular quantum interferometer

Quantum systems characterized by an interplay between several resonance scattering channels demonstrate very rich physics. To illustrate it we consider a multistage Kondo effect in nanodevices as a paradigmatic model for a multimode resonance scattering. We show that the channel crosstalk results in a destructive interference between the modes. This interplay can be controlled by manipulating the tunneling junctions in the multilevel and multiterminal geometry. We present a full-fledged theory of the multistage Kondo effect at the strong-coupling Fermi-liquid fixed point and discuss the influence of quantum interference effects to the quantum transport observables.

Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures.

We consider the problem of heat transport by vibrational modes between Langevin thermostats connected by a central device. The latter is anharmonic and can be subject to large temperature difference and thus be out of equilibrium. We develop a classical formalism based on the equation of motion method, the fluctuation–dissipation theorem and the Novikov theorem to describe heat flow in a multi-terminal geometry. We show that it is imperative to include a quartic term in the potential energy to insure stability and to properly describe thermal expansion. The latter also contributes to leading order in the thermal resistance, while the usually adopted cubic term appears in the second order. This formalism paves the way for accurate modeling of thermal transport across interfaces in highly non-equilibrium situations beyond perturbation theory.

Quantum anomalous Hall edge channels survive up to the Curie temperature

Achieving metrological precision of quantum anomalous Hall resistance quantization at zero magnetic field so far remains limited to temperatures of the order of 20 mK, while the Curie temperature in the involved material is as high as 20 K. The reason for this discrepancy remains one of the biggest open questions surrounding the effect, and is the focus of this article. Here we show, through a careful analysis of the non-local voltages on a multi-terminal Corbino geometry, that the chiral edge channels continue to exist without applied magnetic field up to the Curie temperature of bulk ferromagnetism of the magnetic topological insulator, and that thermally activated bulk conductance is responsible for this quantization breakdown. Our results offer important insights on the nature of the topological protection of these edge channels, provide an encouraging sign for potential applications, and establish the multi-terminal Corbino geometry as a powerful tool for the study of edge channel transport in topological materials.

Fano-Kondo resonance versus Kondo plateau in an Aharonov-Bohm ring with an embedded quantum dot

We theoretically examine the transport through an Aharonov-Bohm ring with an embedded quantum dot (QD), the so-called QD interferometer, to address two controversial issues regarding the shape of the Coulomb peaks and measurement of the transmission phase shift through a QD. We extend a previous model [B. R. Bulka and P. Stefanski, Phys. Rev. Lett. 86, 5128 (2001); W. Hofstetter, J. Konig, and H. Schoeller, ibid. 87, 156803 (2001)] to consider multiple conduction channels in two external leads, L and R. We introduce a parameter p_{\alpha} (|p_{\alpha}| \le 1) to characterize a connection between the two arms of the ring through lead \alpha (=L, R), which is the overlap integral between the conduction modes coupled to the two arms. First, we study the shape of a conductance peak as a function of energy level in the QD, in the absence of electron-electron interaction U. We show an asymmetric Fano resonance for |p_{L,R}| = 1 in the case of single conduction channel in the leads and an almost symmetric Breit-Wigner resonance for |p_{L,R}| < 0.5 in the case of multiple channels. Second, the Kondo effect is taken into account by the Bethe ansatz exact solution in the presence of U. We precisely evaluate the conductance at temperature T=0 and show a crossover from an asymmetric Fano-Kondo resonance to the Kondo plateau with changing p_{L,R}. Our model is also applicable to the multi-terminal geometry of the QD interferometer. We discuss the measurement of the transmission phase shift through the QD in a three-terminal geometry by a "double-slit experiment." We derive an analytical expression for the relation between the measured value and the intrinsic value of the phase shift.

Role of Quasiparticles in an Electric Circuit with Josephson Junctions.

Although Josephson junctions can be viewed as highly nonlinear impedances for superconducting quantum technologies, they also possess internal dynamics that may strongly affect their behavior. Here, we construct a computational framework that includes a microscopic description of the junction (full fledged treatment of both the superconducting condensate and the quasiparticles) in the presence of a surrounding electrical circuit. Our approach generalizes the standard resistor capacitor Josephson model to arbitrary junctions (including, e.g., multiterminal geometries and/or junctions that embed topological or magnetic elements) and arbitrary electric circuits treated at the classical level. By treating the superconducting condensate and quasiparticles on equal footings, we capture nonequilibrium phenomena such as multiple Andreev reflection. We show that the interplay between the quasiparticle dynamics and the electrical environment leads to the emergence of new phenomena. In a RC circuit connected to single channel Josephson junction, we find out-of-equilibrium current-phase relations that are strongly distorted with respect to the (almost sinusoidal) equilibrium one, revealing the presence of the high harmonic ac Josephson effect. In an RLC circuit connected to a junction, we find that the shape of the resonance is strongly modified by the quasiparticle dynamics: close to resonance, the current can be smaller than without the resonator. Our approach provides a route for the quantitative modeling of superconducting-based circuits.

On the Accuracy of Conductance Quantization in Spin-Hall Insulators

In contrast to the case of ordinary quantum Hall effect, the resistance of ballistic helical edge channels in typical quantum spin-Hall experiments is non-vanishing, additive and poorly quantized. Here we present a simple argument connecting this qualitative difference with a spin relaxation in the current/voltage leads in an experimentally relevant multi-terminal bar geometry. Both the finite lead resistance and the spin relaxation contribute to a non-vanishing four-terminal edge resistance, explaining poor quantization quality. We show that corrections to the four-terminal and two-terminal resistances in the limit of strong spin relaxation are opposite in sign, making a measurement of the spin relaxation resistance feasible, and estimate the magnitude of the effect in HgTe-based quantum wells.

Current polarity-dependent manipulation of antiferromagnetic domains.

Antiferromagnets have several favourable properties as active elements in spintronic devices, including ultra-fast dynamics, zero stray fields and insensitivity to external magnetic fields 1 . Tetragonal CuMnAs is a testbed system in which the antiferromagnetic order parameter can be switched reversibly at ambient conditions using electrical currents 2 . In previous experiments, orthogonal in-plane current pulses were used to induce 90° rotations of antiferromagnetic domains and demonstrate the operation of all-electrical memory bits in a multi-terminal geometry 3 . Here, we demonstrate that antiferromagnetic domain walls can be manipulated to realize stable and reproducible domain changes using only two electrical contacts. This is achieved by using the polarity of the current to switch the sign of the current-induced effective field acting on the antiferromagnetic sublattices. The resulting reversible domain and domain wall reconfigurations are imaged using X-ray magnetic linear dichroism microscopy, and can also be detected electrically. Switching by domain-wall motion can occur at much lower current densities than those needed for coherent domain switching.

Thermodynamic Bounds on Precision in Ballistic Multiterminal Transport.

For classical ballistic transport in a multiterminal geometry, we derive a universal trade-off relation between total dissipation and the precision, at which particles are extracted from individual reservoirs. Remarkably, this bound becomes significantly weaker in the presence of a magnetic field breaking time-reversal symmetry. By working out an explicit model for chiral transport enforced by a strong magnetic field, we show that our bounds are tight. Beyond the classical regime, we find that, in quantum systems far from equilibrium, the correlated exchange of particles makes it possible to exponentially reduce the thermodynamic cost of precision.

Spin-dependent transport in multi-terminal Aharonov-Casher ring with quantum dot

The Aharonov-Casher (AC) effect is theoretically examined in a mesoscopic ring with an embedded quantum dot in a multi-terminal geometry. We examine a realistic model for such a system with spin-orbit interaction in the ring. A spin-polarized current can be generated in a three-terminal device even in the absence of magnetic field, whereas it cannot in a two-terminal device. The spin polarization is enhanced by the resonant tunneling around the Coulomb peak and Kondo effect in the Coulomb blockade regime. The efficiency for the spin polarization is discussed in realistic devices.

Spin Hall Effect and Origins of Nonlocal Resistance in Adatom-Decorated Graphene.

Recent experiments reporting an unexpectedly large spin Hall effect (SHE) in graphene decorated with adatoms have raised a fierce controversy. We apply numerically exact Kubo and Landauer-Büttiker formulas to realistic models of gold-decorated disordered graphene (including adatom clustering) to obtain the spin Hall conductivity and spin Hall angle, as well as the nonlocal resistance as a quantity accessible to experiments. Large spin Hall angles of ∼0.1 are obtained at zero temperature, but their dependence on adatom clustering differs from the predictions of semiclassical transport theories. Furthermore, we find multiple background contributions to the nonlocal resistance, some of which are unrelated to the SHE or any other spin-dependent origin, as well as a strong suppression of the SHE at room temperature. This motivates us to design a multiterminal graphene geometry which suppresses these background contributions and could, therefore, quantify the upper limit for spin-current generation in two-dimensional materials.

Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform.

Atomically thin two-dimensional semiconductors such as MoS2 hold great promise for electrical, optical and mechanical devices and display novel physical phenomena. However, the electron mobility of mono- and few-layer MoS2 has so far been substantially below theoretically predicted limits, which has hampered efforts to observe its intrinsic quantum transport behaviours. Potential sources of disorder and scattering include defects such as sulphur vacancies in the MoS2 itself as well as extrinsic sources such as charged impurities and remote optical phonons from oxide dielectrics. To reduce extrinsic scattering, we have developed here a van der Waals heterostructure device platform where MoS2 layers are fully encapsulated within hexagonal boron nitride and electrically contacted in a multi-terminal geometry using gate-tunable graphene electrodes. Magneto-transport measurements show dramatic improvements in performance, including a record-high Hall mobility reaching 34,000 cm(2) V(-1) s(-1) for six-layer MoS2 at low temperature, confirming that low-temperature performance in previous studies was limited by extrinsic interfacial impurities rather than bulk defects in the MoS2. We also observed Shubnikov-de Haas oscillations in high-mobility monolayer and few-layer MoS2. Modelling of potential scattering sources and quantum lifetime analysis indicate that a combination of short-range and long-range interfacial scattering limits the low-temperature mobility of MoS2.

Bound on thermoelectric power in a magnetic field within linear response.

For thermoelectric power generation in a multiterminal geometry, strong numerical evidence for a universal bound as a function of the magnetic-field induced asymmetry of the nondiagonal Onsager coefficients is presented. This bound implies, inter alia, that the power vanishes at least linearly when the maximal efficiency is approached. In particular, this result rules out that Carnot efficiency can be reached at finite power, which an analysis based on the second law only would, in principle, allow.

Recursive Greenʼs function method for multi-terminal nanostructures

We present and review an efficient method to calculate the retarded Greenʼs function in multi-terminal nanostructures; which is needed in order to calculate the conductance through the system and the local particle densities within it. The method uses the recursive Greenʼs function method after the discretized Hamilton matrix has been properly partitioned. We show that this method, the circular slicing scheme, can be modified to accommodate multi-terminal systems as well as the traditional two-terminal systems. Furthermore, we show that the performance and robustness of the circular slicing scheme is on par with other advanced methods and is well suited for large variety of multi-terminal geometries. We end by giving an example of how the method can be used to calculate transport in a non-trivial multi-terminal geometry.

Powerful and efficient energy harvester with resonant-tunneling quantum dots

We propose a nanoscale heat engine that utilizes the physics of resonant tunneling in quantum dots in order to transfer electrons only at specific energies. The nanoengine converts heat into electrical current in a multiterminal geometry which permits one to separate current and heat flows. By putting two quantum dots in series with a hot cavity, electrons that enter one lead are forced to gain a prescribed energy in order to exit the opposite lead, transporting a single electron charge. This condition yields an ideally efficient heat engine. The energy gain is a property of the composite system rather than of the individual dots. It is therefore tunable to optimize the power while keeping a much larger level spacing for the individual quantum dots. Despite the simplicity of the physical model, the optimized rectified current and power is larger than any other candidate nanoengine. The ability to scale the power by putting many such engines into a two-dimensional layered structure gives a paradigmatic system for harvesting thermal energy at the nanoscale. We demonstrate that the high power and efficiency of the layered structure persists even if the quantum dots exhibit some randomness.

Homogeneity of bilayer graphene

We present non-linear transport measurements on suspended, current annealed bilayer graphene devices. Using a multi-terminal geometry we demonstrate that devices tend to be inhomogeneous and host two different electronic phases next to each other. Both of these phases show gap-like features of different magnitude in non-linear transport at low charge carrier densities, as already observed in previous studies. Here, we investigate the magnetic field dependence and find that both features grow with increasing field, the smaller one with 0.6meV/T, the larger one with a 5–10 times higher field dependence. We attribute the larger of the two gaps to an interaction induced broken symmetry state and the smaller one to localization in the more disordered parts of the device.

Quantum coherence and its dephasing in the giant spin Hall effect and nonlocal voltage generated by magnetotransport through multiterminal graphene bars

Motivated by the recent experimental observation [Abanin et al., Science 332, 328 (2011)] of nonlocality in magnetotransport near the Dirac point in six-terminal graphene Hall bars, for a wide range of temperatures and magnetic fields, we develop a nonequilibrium Green's function theory of this phenomenon. In the quantum-coherent regime and strong magnetic field, we find large Zeeman-splitting-driven spin Hall (SH) conductance in four-terminal bars, where the SH current is pure only at the Dirac point (DP). In six-terminal Hall bars, this leads to the nonlocal voltage at a remote location due to direct and inverse SH effect operating at the same time in different parts of the device. The ``momentum-relaxing'' dephasing reduces their values at the DP by two orders of magnitude while concurrently washing out any features away from the DP. Our theory is based on the Meir-Wingreen formula for spin-resolved charge currents with dephasing introduced via phenomenological many-body self-energies, which is then linearized for multiterminal geometries to extract currents and voltages. This provides a generalization of the multiprobe Landauer-B\"uttiker formula without employing traditional B\"uttiker voltage probes to introduce dephasing.

What can we learn about the dynamics of transported spins by measuring shot noise in spin–orbit-coupled nanostructures?

We review recent studies of the shot noise of spin-polarized charge currents and pure spin currents in multiterminal semiconductor nanostructures, while focusing on the effects brought by the intrinsic Rashba spin–orbit (SO) coupling and/or extrinsic SO scattering off impurities in two-dimensional electron gas (2DEG) based devices. By generalizing the scattering theory of quantum shot noise to include the full spin-density matrix of electrons injected from a spin-filtering electrode, we show how decoherence and dephasing in the course of spin precession can lead to the substantial enhancement of the Fano factor (noise-to-current ratio) of spin-polarized charge currents. These processes are suppressed by decreasing the width of the diffusive Rashba wire, so that purely electrical measurement of the shot noise in a ferromagnet|SO-coupled-diffusive-wire|paramagnet setup can quantify the degree of quantum coherence of transported spin through a remarkable one-to-one correspondence between the purity of the spin state and the Fano factor. In four-terminal SO-coupled nanostructures, injection of unpolarized charge current through the longitudinal leads is responsible not only for the pure spin Hall current in the transverse leads, but also for nonequilibrium random time-dependent current fluctuations. The analysis of the shot noise of transverse pure spin Hall current and zero charge current, or transverse spin current and non-zero charge Hall current, driven by unpolarized or spin-polarized injected longitudinal charge current, respectively, reveals a unique experimental tool to differentiate between the intrinsic Rashba and extrinsic SO mechanisms underlying the spin Hall effect in 2DEG devices. When the intrinsic mechanisms responsible for spin precession start to dominate the spin Hall effect, they also enhance the shot noise of transverse spin and charge transport in multiterminal geometries. Finally, we discuss the shot noise of transverse spin and zero charge currents in the quantum-interference-driven spin Hall effect in ballistic four-terminal Aharonov–Casher rings realized using high-mobility 2DEG with the Rashba SO coupling. The modulation of the Rashba coupling by the gate electrode imprints the oscillatory signature of constructive and destructive spin interference around the ring on both the spin and charge shot noise, which differ from the corresponding oscillations of the spin Hall conductance, thereby revealing quantum-interference-driven temporal correlations between spin-resolved charge currents of opposite spins.

Elementary charge-transfer processes in mesoscopic conductors

We determine charge-transfer statistics in a quantum conductor driven by a time-dependent voltage and identify the elementary transport processes. At zero temperature unidirectional and bidirectional single-charge transfers occur. The unidirectional processes involve electrons injected from the source terminal due to excess dc bias voltage. The bidirectional processes involve electron-hole pairs created by time-dependent voltage bias. This interpretation is further supported by the charge-transfer statistics in a multiterminal beam-splitter geometry in which injected electrons and holes can be partitioned into different outgoing terminals. The probabilities of elementary processes can be probed by noise measurements: the unidirectional processes set the dc noise level, while bidirectional ones give rise to the excess noise. For ac voltage drive, the noise oscillates with increasing the driving amplitude. The decomposition of the noise into the contributions of elementary processes reveals the origin of these oscillations: the number of electron-hole pairs generated per cycle increases with increasing the amplitude. The decomposition of the noise into elementary processes is studied for different time-dependent voltages. The method we use is also suitable for systematic calculation of higher-order current correlators at finite temperature. We obtain current noise power and the third cumulant in the presence of time-dependent voltage drive. The charge-transfer statistics at finite temperature can be interpreted in terms of multiple-charge transfers with probabilities which depend on energy and temperature.