I. MOTIVATION

Spintronics takes advantage of the electronic in designing a variety of applications, in- cluding for giant sensing, quantum , and quantum-information processing [1, 2]. The spins of mobile can be manipulated by the spin-orbit interac- tion (SOI), which causes the spin of an moving through a spin-orbit active material to rotate. Transport properties of spin-active electric weak links has been the subject of our ASG’s at PCS in 2018 and 2019. The main results obtained so far during one and a half year’s work, where the emphasis has been on new functionalities of spin-active devices, can be grouped depending on the area of research as follows:

A. Electric transport in spin-orbit-interaction (SOI) active electric weak links

a) The theoretical concept of “Rashba splitting” of electrons transferred through an SOI- active electric weak link was formulated and applied to a supercurrent flowing through a superconducting weak link [3, 4]. We have shown that a transfer of Cooper pairs through an SOI-active weak links results in their spin-polarization, which makes it possible to generate a supercurrent that carries a finite spin.

b) Extra spin and charge can be accumulated in a small quantum dot by letting a super- current of spin-polarized Cooper pairs flow through it. New types of superconducting proximity states carrying both charge and spin were theoretically considered [5].

c) Electric- and magnetic gating effects on electron- and spin transport through a spin- orbit-active nanowire were suggested as tools for detecting and measuring spin-orbit coupling in nanomaterials [6].

B. Spin generation in mechanically driven SOI devices

We have predicted theoretically that a non-zero spin current can be generated in me- chanically driven SOI-active electric weak links [7]. As a result a nanometer size point-like spin current-source is suggested for possible nano-spintronic applications.

1 C. Heat transport in magnetic shuttle structures

We predicted theoretically that heat transport through an electric weak link made of magnetic material can be controlled via the spin of the electrons that carry the heat [8]. This is due to the strong coupling of the electronic spin to the mechanical vibrations of the electric weak link, a coupling which causes a mechanical instability which can be seen as a thermal breakdown, similar to the electric breakdown occurring in electric transport. This prediction has important consequences for applications including those addressing the important issue of heat removal from electronic nanodevices.

The electric charge of electrons becomes an important player in the of mesoscopic transport if discrete electron tunneling events are important elements of the process. If a weak link is connected to electrodes by high-resistance tunneling barriers, the picture of a continuous charge flow valid for bulk conductors is replaced by the image of discrete tun- neling events between spatially localized electronic states. In such a picture the discrete nature of the charge carried by single electrons qualitatively change the physics of small size conductors. The charging energy U needed to accumulate an extra electric charge e in the small volume of a mesoscopic conductor determines the role of the Coulomb correlations dominating the physics of mesoscopic nanodevices. At low temperatures, T  U, thermal fluctuations of the number of electrons accumulated in a conductor is suppressed and new physical phenomena come into play. In this case the parity of the number of electrons be- comes a good quantum number, allowing for qualitatively new physical phenomena. Two examples of such phenomena relevant for the present application are: 1) the parity effect in superconducting quantum dots [9, 10] and 2) parity controlled spin-accumulation in a dot [11]. The first example amounts to the quantization of charge in a superconducting quantum dot in units of 2e, which allows for the formation of a superconducting “Cooper pair box” a key ingredient for the implementation of a qubit for using supercon- ducting devices. The second example allows for electrostatic control of the spin-dependent magnetic exchange force determining mechanically assisted charge- and heat transport in magnetic nanodevices. Our previous theoretical predictions and recent results obtained within the ongoing ASG program demonstrate qualitatively new performances occurring from the electrostatically controlled [13] electron number parity, such as the phenomenon

2 of Coulomb promotion of mechanically assisted transport and mechanically assisted Cooper pair transfer through superconducting Josephson junctions [12, 13]. The interplay between the two fundamental electronic degrees of freedom, charge and spin, and the quantum orbital motion of electrons in mesoscopic nanodevices will be in fo- cus of the proposed research program for the new ASG. Possible nanodevice applications employing mechanical degrees of freedom of normal metals, semiconductors, magnets and superconductors will be explored in in close cooperation with experimentalists in two labo- ratories that we collaborate with. The very choice of the directions of our planned research is significantly affected by the desire to establish such a close cooperation as well as looser cooperation with the members of PCS.

II. OBJECTIVES:

(a) To build collaborative scientific research with the experimental group of Dr. Junho Suh (through a theoretical study SOI-induced nanomechanics; microwave induced spin- current generations, etc.).

(b) To build collaborative scientific research with the experiment group of Dr. Chulki Kim (KIST, Seoul) (through the formulation of a theory of superconducting nanomechan- ics).

(c) To develop theoretical studies of the interplay of spin and charge in many-body struc- tures with SOI.

(d) To extend further the cooperation and collaborative research with PCS members (pre- liminary discussions show a significant potential in involving such PCS members as Drs. M. Fistul, A. Parafilo, Sang-Jun Choi, Kunwoo Kim, and Sungjong Woo in the planned research on the nanoelectromechanics of magnetic and superconducting nanodevices, and Nojoon Myoung (Chosun University) and Sejoong Kim (UST) for spin-active 2D materials).

(d) To organize an International workshop on Mesoscopic Nanoelectromechanics, which we consider as a special objective of the planned ASG activity for the year 2020. Such an international event, which will be held at PCS IBS, is specially motivated by the

3 obvious advantage for the ASG in particular and for the wider community of Korean scientists in general, especially the younger ones in Daejeon, in getting some of the leading world experts in our field to Daejeon for an intensive period of interaction. Further collaboration with theoreticians and experimentalists at the cutting edge of our field would be a both desirable and probable outcome. Moreover, the format of an ASG-related Workshops at PCS would provide an ideal opportunity to advertise to the relevant international community the quite considerable progress we have been fortunate to make since the launch of our ASG one year ago. This progress has been achieved in the broad area of functional spin-driven mesoscopics, which is the basis for the operation of a number of possible future nanodevices. It involves, e.g., spin-orbit- interaction driven transport in semiconductors, metals, magnets and superconductors, where the interplay between quantum coherence and electron-electron correlations are crucial, and many-body effects in Kondo transport.

III. RESEARCH DIRECTIONS

A. High frequency properties of SOI-active weak links:

a) Spin generation and charge accumulation in AC-driven Rashba weak links; - ASG members responsible for the research: M. Jonson, O. Entin-Wohlman, A. Aharony, J. Suh

b) DC charge transport controlled by AC driven SOI; - ASG members responsible for the research: M. Jonson, O. Entin-Wohlman, A. Aharony, H.C. Park, D. Radi´c,J. Suh

c) Nano-mechanics of suspended nano-wires, driven by high frequency SOI; - ASG members responsible for the research: I. Krive, H.C. Park, M. Jonson, O. Entin-Wohlman, J. Suh.

B. Properties of SOI-active NEM devices with coupled spintronic, electric and mechanical degrees of freedom (Spintro-electro-mechanics of SOI-active NEM devices):

a) Mechanically generated spin-currents in SOI-active weak links; - ASG members responsible for the research: O. Entin-Wohlman, A. Aharony, M. Jonson, J. Suh

4 b) SOI-induced mechanical force and Rashba pumping of nanovibrations; - ASG members responsible for the research: O. Entin-Wohlman, A. Aharony, H.C. Park, D. M. Jonson, J. Suh.

C. Nano-electro-mechanics of superconducting weak links:

a) Electro-mechanics of a superconducting Cooper-pair box (CPB) (formulation of approach); - ASG members responsible for the research: L. Gorelik, D. Radi´c,H.C. Park, C. Kim

b) Cohesive and Coulomb forces driving mechanical CPB vibrations; - ASG members responsible for the research: L. Gorelik, D. Radi´c,H.C. Park, C. Kim

c) Resonant generation of pronounced CPB vibrations in a DC voltage biased device; - ASG members responsible for the research: L. Gorelik, D. Radi´c,H.C. Park, C. Kim

d) Shuttle instability due to mechanical transportation of Cooper pairs; - ASG members responsible for the research: L. Gorelik, H.C. Park, C. Kim.

D. Coulomb promotion of spintro-mechanics in magnetic shuttle devices - ASG members responsible for the research: I. Krive, D. Radi´c,H.C. Park, M. Jonson.

E. Electron number parity-controlled thermoelectricity in spin-active NEM devices - ASG members responsible for the research: I. Krive, D. Radi´c,H.C. Park, M. Jonson.

A. High frequency properties of SOI-active weak links

One aim of spintronics is to build logic devices [14], which produce spin-polarized elec- trons, so that one can use their electronic spinors as qubits. In the simplest device, electrons move between two large electronic reservoirs, via a nano-scale quantum network. For this two-terminal case, time-reversal symmetry and unitarity of the Hamiltonian prevent any spin splitting of the transport between the reservoirs [15]. Since the time-independent SOI obeys time-reversal symmetry, it alone cannot generate such a spin splitting. Time-reversal symmetry can be broken by applying a magnetic field,

5 FIG. 1: Schematic visualizations of devices proposed in the text. (a) A spin-orbit-active weak link connects two contacts, L and R, to form a closed circuit. The time-dependent spin-orbit interaction is generated by two perpendicular gates, whose potentials Vy(t) and Vz(t) oscillate slowly in time with frequency Ω. The arrows within the weak link indicate the directions in which polarized electron-spins are flowing. (b) An open-circuit version of (a) where spin is accumulated in two terminals leading to a magnetization that can be measured. either via a magnetic flux, which penetrates SOI-active loops of Aharonov-Bohm interfer- ometers [16–18], or via a Zeeman magnetic field [19, 20]. Alternatives utilize ferromagnetic terminals [21, 22]. In the research direction outlined below we suggest to explore yet an- other means to break time-reversal symmetry. We plan to explore the possibility to activate SOI-induced spin splitting by breaking time-reversal symmetry with an AC-driven spin-orbit interaction. The later can be achieved through microwave activation of Rashba spin-orbit coupling in SOI-active weak link devices (See Fig. 1). The dependence of both charge- and spin transport will be studied depending on microwave power its frequency and its polar- ization. Our preliminary estimation show that detectable in experiment spin and charge accumulations can be achieved in AC driven SOI-active devices promising important spin- tronic applications. The following problems to solve are identified below.

a) Spin generation and charge accumulation in AC-driven Rashba weak links

The dwell time, i.e. the time spent by an electron in the SOI-active part of the device while propagating through it (Fig. 1), sets a frequency scale separating low- frequency (adiabatic) and high-frequency regimes of microwave induced transport.

6 Our preliminary analysis shows that both charge- and spin injection into the metal environments of an SOIactive device takes place in the low frequency adiabatic limit. In the high frequency limit, however, a significant part of the charge and spin generated is accumulated inside the SOI-active weak link. We will study the above effect of frequency controlled spatial separation of the generated charge and spin. A detailed theory based on the non-equilibrium Green’s function (Keldysh) technique will be developed in order to analyze SOI-induced resonant coupling to microwave radiation.

b) DC charge transport controlled by AC driven SOI

Electronic charge transport should reveal SOI induced Rashba oscillations in the case of an AC driven SOI. The strength of the effect should significantly depend on the frequency of the microwave radiation. We expect a weak SOI effect on DC charge transport in the adiabatic regime of microwave irradiation but a significantly amplified effect in the high frequency limit. Both charge-, heat-, and spin transport will be studied theoretically.

c) Nano-mechanics of suspended nano-wires driven by high frequency SOI

An effective way to study SOI phenomena in electric weak links is to focus on the mechanical properties of the weak link, which typically is a suspended SOI- active nanowire, by measuring the back-action produced by electron transport in the nanowire. SOI affected mechanical frequency shifts as well as microwave induced damping (or amplification) of nano-vibrations of the wire will be one subject of the theoretical study within this research direction.

B. Spintro-electro-mechanics of SOI-active NEM devices

SOI-active weak links based on suspended nanowires provide a unique opportunity to con- trol trajectories of the 1-D motion of electrons by mechanical deformations of the nanowire. This opens up a new direction for implementing electro-mechanical coupling in the device through the spin-orbit interaction in the nanowire. As a result we expect to be able to ex- plore new perspectives for nanomechanical functionalities of SOI active electric weak links. The experimental work on the nanoelectromechanics of suspended nanowires in the labo- ratory of Dr. Jonho Such is an additional motivation for us to focus on pursuing the new

7 research direction of “Rashba spintro-electro-mechanics”. Below we outline two theoretical projects in this area.

a) Mechanically generated spin currents in SOI-active weak links

Mechanical deformation of the SOI active part of an electric weak link is one way to implement a time-dependent SOI in the device. By breaking time reversal symmetry in this way one enables effects of Rashba spin splitting in the device, such as spin-flip assisted transfer of electrons through the device. An extra spin is produced by each spin flip and as a result a nonzero spin current is injected into the metal reservoirs. We have demonstrated this phenomenon in the low frequency limit by considering mechanical rotations of a mechanically bent nanowire [7]. Extending this analysis to, e.g. flexural vibrations of a bent wire or mechanical pulse operation is something we want to study within this project. Both the low- and high frequency limits of the mechanical operation will be considered. We expect to achieve DC spin-current generation in addition to the predicted AC spin current in the low frequency limit [7].

b) SOI-induced mechanical force and Rashba pumping of nano-vibrations

A question of fundamental interest, which one can put, is to what extent the spintro- electro-mechanical coupling due to the SOI is sensitive to whether the electronic system obeys time-reversal symmetry. Inelastic processes associated with the emission (or absorption) of nanomechanical vibration quanta violate in general the time reverse symmetry of the SOI induces electronic coupling to mechanical degrees of freedom and may open the possibility to detect SOI mechanically even if electronic system with no mechanical degrees of freedom involved is time reverse symmetric. This put in question the role of SOI induced back actions of the mechanical nanowire vibrations. Such back actions are due to special Rashba forces acting on the vibrating nanowire through displacement-sensitive Rashba spin-splitting effects. We will develop a full theory of the SOI-induced back action and explore the possibility of a Rashba mechanism for a nanomechanical instability of the device.

8 C. Nano-electro-mechanics of superconducting weak links

One approach to implementing quantum communication is to employ the mechanical transportation of quantum information between remote quantum bit devices. Supercon- ducting weak link devices such as superconducting charge- and phase qubits are well known devices for use in quantum computation. The problem of how to coherently match nanome- chanics with superconducting qubit devices is presently a hot area of research. One approach to solving this problem is to use the nano-electro-mechanical (NEM) Cooper-pair box de- vice suggested by us some time ago [12, 13]. At the time we showed that a superconducting single-electron , containing a movable quantum dot, coherently couples of Cooper pairs flowing through the device with mechanical vibrations, thus allowing for coherent transportation of a superconducting Cooper pair box which can be considered as a mov- able qubit for quantum communication. Recently experimental research in superconducting nano-electro-mechanics inspired by our theoretical predictions was started by Dr. Chulki Kim in Seoul (KIST), which in its turn is a great stimulus for us to revive our theoretical research in this area. Projects selected for the beginning of our research program are listed below.

a) Electro-mechanics of a superconducting Cooper-pair box (SCPB) (formu- lation of approach)

Superconducting ordering of electrons results in the formation of a new ground state (a condensate of Cooper pairs) and the formation of a gap for elementary quasiparticle excitations. The gap allows us to consider the low-energy properties of superconductors (low compared with the gap and and temperature, as well as with the frequencies and bias voltages applied to the system) by neglecting the quasiparticle branches of the spectrum and restricting the description to the purely coherent dynamics of the superconducting ground state. The superconducting Cooper pair box (SCPB) (Fig. 2) is a device where this quantum dynamics can be described in terms of the electric charge Q = 2Ne and the phase ϕ of the superconducting condensate, which are related by the Heisenberg uncertainty relation δϕ·δN ≥ 1/2. Coulomb correlations, significant for a small enough dot, restrict quantum charge fluctuations resulting in pronounced fluctuations of the superconducting phase. The latter suppress the flow of Cooper pairs through the device (Coulomb blockade of Cooper-pair tunneling). In the limit of

9 I

Superconducting leads

Φ Φ R L

Superconducting Gate grain

FIG. 2: Schematic of the specific system described in the text. A superconducting grain executes periodic motion between two superconducting bulk leads. The presence of the gate ensures that the Coulomb blockade is lifted during the contacts between the grain and superconductor which allows for the grain to be in the Josephson hybrid state. As the grain moves between the leads Cooper pairs are shuttled between them creating a DC-current through the structure.

a small Coulomb charging energy U (compared with the Josephson coupling energy) the charge on the dot for low energy charging changes in pairs (2e) and the Coulomb blockade for Cooper-pair tunneling can be lifted at certain values of an applied gate voltage. Lifting the Coulomb blockade opens the possibility for charge/phase quantum dynamics which is controlled by both the applied gate- and the bias voltages. In the case of a movable quantum dot (Fig. 2) such SCPB dynamics couples to the quantum dynamics of the SCPB spatial motion forming a nano-electro-mechanical SCPB device. Within this project we plan to formulate the Hamiltonian for such a coupled dynamics.

b) Mechanical SCPB vibrations driven by cohesive- and Coulomb forces

10 In this project the superconducting NEM-SET device shown in Fig. 2 will be in the focus of our theoretical study. Two electronic energies, which depend on the me- chanical displacement, can be identified in this set-up. The first one has to do with the tunneling of electron pairs between the dot and the superconducting reservoirs of Cooper pairs. This energy corresponds to a lowering of the kinetic energy of the quantum particle (Cooper pair) due to delocalization of its position between the neigh- boring superconductors. This energy, which is called the Josephson coupling energy, depends on the superconducting phase difference between the superconductors and — due to the transparency of the tunnel barriers — exponentially depends on the mechanical displacement of the dot. The other important energy is the electrostatic charging energy, U, which has to do with the electric charge 2e carried by the Cooper pairs of electrons may lead to a Coulomb blockade (or suppression) of Cooper pair tunneling. In addition, if an electric field is present, caused by gate- or voltage biasing of the device, the mechanical-displacement dependent part of the electrostatic energy governs the quantum superconducting phase/charge dynamics. The two forces acting mechanically on the SCPB originating, which result from the displacement-dependent energies mentioned, we call the Josephson cohesive force and the electrostatic Coulomb force respectively. Expressions for the operators of those two forces will be derived and their role in the nanomechanical performance of the device will be investigated. In particular two regimes of superconducting nanoelectromechanics will be considered, as described in the two projects below. c) Resonant generation of pronounced Cooper pair box vibrations in a DC voltage biased device

The Josephson cohesive force acting on the SCPB, being dependent on the supercon- ducting phase difference, is sensitive to the biasing of the device. In the case of an applied DC bias voltage a force that oscillates with frequency 2eV/h is responsible for possibly resonantly driving the nanomechanics. Resonance occurs if eV = n~Ω, where Ω is the mechanical vibration frequency. A quantum theory of such a superconducting driving of the nanomechanics will be developed. Our preliminary analysis shows that such a resonant pumping of nanovibrations results in the development of a coherent superposition of semiclassical vibrations entangled with the superconducting Cooper-

11 pair flow through the device. It is an exciting perspective to search for the prospect of using such entangling for the storage of quantum information.

d) Shuttle instability due to mechanical transportation of Cooper pairs

In the non-resonant regime of voltage biasing the Coulomb force, which has to do with the electric charge carried by Cooper pairs, is expected to play a crucial role in the superconducting nanoelectromechanics. The shuttle mechanism of self-supported mechanical vibrations predicted for non-superconducting devices should play a role also in the case of charged Cooper pairs transferred through the device. However, the quantum coherence of the Cooper-pair flow in combination with the Josephson dynamics of Cooper pair tunneling may result in a qualitative difference between shuttling of non-coherent single electrons and coherent Cooper pairs. The full theory of the phenomenon will be developed.

D. Coulomb promotion of spintro-mechanics in magnetic shuttle devices

Tunneling injection of electrons into a nano-conductor is an obvious way to control the amount of both charge and spin accumulated in a nanometer scale spatial domain. How- ever, in contrast to the amount of electric charge the amount of electron spin that can be accumulated by this process is limited. This is because while electrons with different spin projections can be injected into the conductor, the net spin accumulated depends (assuming a spin-degenerate electronic spectrum) on the parity of the number of injected electrons. The net accumulated spin, at equilibrium, is at most equal to a single electron spin and this occurs only for an odd number of injected electrons. Quantum fluctuations of the electron number destroy all effects originating from parity, thus prohibiting the tunneling accumula- tion of a finite average amount of spin. By suppressing these parity fluctuations the Coulomb blockade phenomenon enhances the probability for a finite spin to be accumulated. This opens an intriguing possibility to use the interplay between single-electronic and spintronic properties for designing the functionality of nanoconductors. Spintromechanics [23] relies on a coupling between mechanical degrees of freedom and the electron spin in magnetic nanoelectromechanical (NEM) devices [24, 25] (see, e.g., the reviews Refs. 26 and 27). The coupling is due to the magnetic exchange interaction be-

12 tween spins accumulated in the movable part of the NEM device (a metal grain or molecule here called a “dot”) and the magnetization in the leads. This makes spintromechanical phenomena an important tool for probing the spin accumulated in a nanoconductor. One can therefore expect a prominent role for Coulomb correlations in the spintromechanical performance of magnetic NEM devices. In the research which we plan along the given direction we will consider the interplay between spintromechanical and single-electron performances of a magnetic NEM system, taking the magnetic shuttle device (see, e.g., Refs. 28, 29) as an example. Our preliminary results demonstrate that a dramatic change of the mechanical behavior of the shuttle device can be induced by using a gate to increase the electron-number (parity) fluctuations in the dot, corresponding to a lifting of the Coulomb blockade of tunneling. As a consequence of the related increase of the fluctuations of the spin-dependent mechanical force on the dot the shuttle instability of the magnetic NEM device, predicted to occur in the absence of parity fluctuations in Ref. 30, is suppressed. We plan to develop a full theory of magneto transport of electric charge and electronic spin through magnetic shuttle device and analyses complete phase diagram identifying the domain of shuttle instability in terms of temperature, magnetic field and applied voltages and degree of the of electrons in magnetic material of the device.

E. Electron number parity-controlled thermoelectricity in spin-active NEM de- vices

Local heating in mesoscopic electronic devices, which becomes even more significant for 1D and 2D systems, is detrimental to the proper operation of the devices. The problem of heat transport is well studied for traditional Coulomb-blockade based but almost nothing is known about thermoelectricity in NEM systems. Therefore research on the intrinsically nonlinear heat transport and thermoelectric effects in shuttle-like mesoscopic devices is interesting and important. In Ref. 8 we have theoretically shown that the shuttling phenomenon, induced by a temperature difference between leads containing fully polarized electrons, is realized for purely magnetic devices when the driving force is provided by a magnetic (exchange) interaction between electron spins on the dot and the magnetized leads. A “shuttle instability” leading to mechanical oscillations of the quantum dot in the

13 Coulomb blockade regime was shown to occur in a finite interval of external magnetic fields. For developed shuttle oscillations the heat current is strongly (exponentially) enhanced, which can be described as the result of a thermal breakdown. The advantage of a thermally induced spintronic shuttle, compared with electrical shuttling, is the absence of electric fields in the device, which operates now by purely magnetic forces and is controlled by magnetic gates (an external magnetic field). The exchange force depends on the spin accumulation in the dot and it is therefore sensitive to electron-number parity effects. The parity of the electron number (even or odd) is in our tunnel device controlled by gate voltage and temperature. In the Coulomb blockade regime the parity is odd, which provides a maximal magnetic driving force and thus promotes magnetic shuttling. When the Coulomb blockade is lifted, strong fluctuations of the electron number on the dot stimulates the appearance of mechanically inactive doubly occupied electronic states on the dot and impedes magnetic shuttling. In the theoretical research in this direction we plan to focus on the following tasks:

a) To calculate the dependence of the critical magnetic field, which separates the shuttling and the damping regimes of mechanical vibrations, on the strength U of the Coulomb correlations. We expect that for non-interacting electrons (U = 0) temperature- induced shuttling is absent;

b) To analyze how the onset of magnetic shuttling depends on the temperature difference between the leads and on their average temperature. We expect a linear dependence of the temperature difference on the frequency of the dot oscillations. Our aim here is to find a region in model parameter space where nonlinear effects in heat transport through an oscillating dot appears in the linear regime of the performance of a standard SET;

c) To investigate the role of strong asymmetry (both in tunneling and magnetic couplings) of the thermal spintromechanical device.

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