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Self-sustained oscillations in nanoelectromechanical systems induced by Kondo resonance

We investigate instability and dynamical properties of nanoelectromechanical systems represented by a single-electron device containing movable quantum dot attached to a vibrating cantilever via asymmetric tunnel contact. The Kondo resonance in electron tunneling between source and shuttle facilitates self-sustained oscillations originated from strong coupling of mechanical and electronic/spin degrees of freedom. We analyze stability diagram for two-channel Kondo shuttling regime due to limitations given by the electromotive force acting on a moving shuttle and find that the saturation amplitude of oscillation is associated with the retardation effect of Kondo-cloud. The results shed light on possible ways of experimental realization of dynamical probe for the Kondo-cloud by using high tunability of mechanical dissipation as well as supersensitive detection of mechanical displacement.

dynamical probe for the Kondo-cloud by using high tunability of mechanical dissipation as well as supersensitive detection of mechanical displacement.
The recent progress in nano-technology made it possible to fabricate nano-devices in which mechanical degrees of freedom are strongly coupled not only to electronic charge [nanoelecromechanics (NEM)] but also to spin degrees of freedom [nanospintromechanics (NSM)]. 1,2 While manipulating the charge degrees of freedom requires the energies/external voltages determined by the Coulomb interaction in the nano-device (charging energy of quantum dot), the spin manipulation needs much smaller scales of the energy determined by the exchange interaction. Therefore, on the one hand, the spin manipulation is free of a heating problem, and, on the other hand, it allows one to achieve very high efficiency of devices. 3 A special case where the spin degrees of freedom are dominant in quantum transport is the Kondo-effect which manifests itself as a resonance scattering of electrons on the impurity spin. 4,5 The retardation effects in NEM devices result in two-channel Kondo tunneling which enhances both spin and charge transport due to maximal overlap of the wave functions of the electrons in the leads. 2,3 These processes are mediated by the spin flip of the localized state in the dot. One more facet of the Kondo effect is formation of a screening cloud of conduction electrons which "dresses" the spin of quantum impurity. The typical length scale of screening cloud is ∼ 1µm. 6 There has been several proposals to detect the size of the Kondo cloud, 3,7-13 however, the unambiguous results are still unavailable and there has been no conclusive measurement due to the quantum fluctuations with zero averaged spin. 14 We are interested in new effects where a moving quantum impurity is nano-machined by attaching it to a nano-mechanical device. Such devices are realized as quantum dots incorporated in a mechanical system which oscillates between two metallic leads. These mechanical systems include long cantilever nanorelay, 15 atomic force microscope with a tip 16,17 and nanoisland attached either to the cantilever 18 or to one of the leads. 16,17 The basic understanding of the NEM/NSM has been achieved in both theoretical [19][20][21][22][23][24] and experimental 25,26 studies of single electron shuttling.
Alternatively, the mechanical system can play itself a role of one of the contacts when the quan-tum dot (impurity) is deposited on a top of metallic cantilever. 18 In these cases either one or two tunnel barriers change its shape in the process of mechanical motion thus providing a coupling of mechanical and electronic/spin degrees of freedom. The temporal dynamics of the Kondo cloud is governed by two main effects. First, the cloud adiabatically follows a position of quantum impurity. Second, the size of the cloud changes in time due to change of tunnel matrix elements. Both these effect are accompanied by the retardation processes similar to those which determine the polaronic effects due to strong electron-phonon interaction. How does the dynamics of the Kondo cloud affects the mechanical system? How can one probe this dynamics? Is it possible to control the cloud's size? Some of these questions have been addressed in a recent publication. 3 This study has shown that the mechanical dissipation is controlled by the kinetics of Kondo screening if electric dc current is transmitted through the system in the presence of external magnetic field. Besides, the characteristic time determining the kinetics of Kondo screening may be measured through the mechanical quality factor. Thus, the strong coupling of spin with mechanical subsystem allows a superhigh tunability of mechanical dissipation as well as supersensitive detection of mechanical displacement.
In this paper we address a question of whether such a strong coupling between mechanics and spintronics can drive system from nearly adiabatic regime of small amplitude mechanical vibrations to a steady state regime with large amplitude self-sustained oscillations. As an example of such regime we consider an instability associated with appearance of self-sustained oscillations in the system induced by "Kondo friction". It will be shown that this regime can be controlled by electric (source-drain voltage, gate voltage) and magnetic fields. We analyse the sensitivity of solutions to the initial conditions and construct the complete phase diagram of the model. We show that the system possesses reach non-linear dynamics (Hopf-pitchfork bifurcation). We demonstrate that by controlling the displacements (velocities) of mechanical system with a high precision one can manipulate both spin and charge tunnel currents.
A sketch of the system under consideration is presented in Fig.1. A nano-island is mounted on the metallic cantilever attached to the drain electrode. The distance between the source electrode and the island, and thus the tunnel coupling between them, depends on the cantilever motion. An external magnetic field is applied perpendicular to the cantilever far from island. In our consideration the flexural vibration of the nanowire is restricted to the dynamics of the fundamental mode only. It is treated as a damped harmonic oscillator with frequency ω 0 , and quality factor Q 0 .
The equation of mechanical motion for Kondo-NEM coupling device is given by: where u describes the cantilever's displacement of free end (see Fig.1) and m is effective mass. The right hand side of Eq. (1) includes the Lorentz force acting on a metallic cantilever in the presence of the effective magnetic field B and the "Kondo-force" associated with the coordinate dependence of the ground state Kondo energy E gs ∼ T K (u). 27,28 Here L is the cantilever length and the current Here G 0 =e 2 /h is the unitary conductance per spin projection, V bias is bias voltage, λ is the tunneling length for the source-island tunnel barrier. The Kondo temperature for moving island is where E c is the charging energy of the dot, Γ d = Γ 0 and Γ s (u) = Γ 0 exp {2(u − u 0 )/λ } are the island-drain (d) and island-source (s) tunnel rates, D 0 is effective bandwidth for the electrons in the leads. Thus, I dc describes the Ohmic regime where the time dependence is associated with Breit-Wigner factor given by the time-dependent tunnel widths. In contrast to it, the major time dependence of I ac is connected with time modulations of the Kondo tempera- The last contribution to the current I is I em f = −G 0u LB. This term is related to the voltage difference E = −uLB between the electrodes induced by motion of the metallic cantilever in the effective magnetic field B. As a result, the velocity dependent current term I ac is modified by the Thus, the electromotive force ∼ B 2 is immaterial in the regime of weak magnetic fields.
First we analyse the amplitude dynamics and stability of the system without emf term, and then consider a regime where the emf term plays an important role. It is convenient to introduce the dimensionless equation of motions using Eq. (1) scaled by tunnel length λ , (x ≡ u(t)/λ ) and dimensionless time scaled with ω −1 0 : In these notations α = 2G 0 V bias BL 8Γ 0 mω 2 0 λ 2 are the dimensionless Lorentz and Kondo forces respectively, β = π 4 E c Γ 0 is the coupling strength of electronic states, γ = 1 Q 0 is the mechanical friction coefficient and the x 0 is dimensionless parameter x 0 = u 0 /λ characterizing the asymmetry of the system at the equilibrium position such that Γ l (x 0 )/Γ 0 = 1. The retardation time associated with dynamics of the Kondo cloud τ β =¯h ω 0 k B T min K β 2 is parametrically large compared to the time of Kondo cloud formation. 3 The correction to the quality factor Q 0 of mechanical system due to retardation effects is determined by functional form of f (x) = tanh(x) cosh 2 (x) e − β 2 (1+tanh(x)) . The time dependent Kondo temperature in the strong coupling limit at T ≪ T min K is given by The k B T min K plays the role of the cutoff energy for Kondo problem. As is mentioned above, we consider adiabatically slow motion of the QD,hω 0 ≪ k B T min K provided the condition {k B T, gµ B B, eV bias } ≪ k B T min K is fulfilled.
In order to analyse the amplitude dynamics in the regime of high-quality resonator k B T K hω 0 ≪ Q 0 we apply the Krylov-Bogoliubov averaging method. 29 The equations for amplitude dynamics can be obtained by means of the ansatz x(t) = A(t) sin(ω 0 t + φ ). In this approximation we ignore the dynamics of the phase φ . The equation for amplitude dynamics for Eq. (4) is written as: The results of numerical analysis of Eq. (4) are shown in Fig. 2 A. At zero bias (α = 0), Eq. (4) describes a damped harmonic oscillator with the friction γ. In this case the system is characterized by single stable attracting fixed point at the origin (black line in Fig.2 A.) When the finite bias is applied to the system in the presence of magnetic field perpendicular to the plane, the increase of the Lorentz force results in the change of behaviour of the oscillator at some critical value α = α c . As a result, the equation for the amplitude (6) In the regime α α c , the system is characterized by two attracting and one repelling fixed points in the space of parameters {α, x 0 } determined by equilibrium position of the shuttle, bias and magnetic field as control parameters. In this regime the system shows bi-stability and flows either to the fixed point at the origin corresponding to damped oscillations or to the regime of self-sustained oscillation depending on the initial conditions. At α ≫ α c , the system falls to the self-sustained oscillations regime.
In Fig.2 B we plot a saturation amplitude of the system as a function of α. The hysteresis of the system is originated from the coexistence of two fixed points characterizing a damped and self-sustained oscillation in the intermediate regime. Moreover, there exists a regime of linearly increasing saturation amplitude. Approximating tanh x = x, for |x| < 1, and tanh x = sign[x] for |x| > 1, we rewrite the condition forȦ = 0 as where ξ ≡ A sin θ − x 0 . As a result, the saturation amplitude is found as A sat = 8 Next we analyse a stability of the system by linearizing the equations in the vicinity of the stable fixed point describing the stationary states. It is convenient to rewrite Eq.(4) in equivalent form of two coupled first order differential equations: While position of the fixed point x * can be found from the condition , the corresponding Jacobian matrix is given by where the g(x) = δ ( β 2 cosh(x) − 2 tanh(x)) exp[ β 2 (1 + tanh(x))] − 2 tanh(x) and the δ is the ratio be- tween α and Kondo force at the minimal Kondo temperature. Interestingly, this condition allows a regime of multiple solution for x * depending on |x 0 | and α. In the Fig.3 we plot the stability diagram of the Jacobian matrix in the parameter space {α, x 0 }. current I dc as a function of α at x 0 = 1.0. It is seen that the transition from unstable focus to stable focus can be realized by changing the direction of magnetic field B → −B at given V bias .
The linearized system can be categorized by a stable focus, unstable focus, and a saddle point.
Analysing conditions for each category we obtain the eigenvalues of the Jacobian matrix λ ± First, we consider a regime of single solution for the fixed point x * . In this case instability arises in the absence of stable focus (negative quality factor characterizing an increment of the oscillations). The negative Q corresponds to pumping regime, where the system is effectively "heated" in contrast with damping regime of Q > 0, which may be interpreted as effective cooling of a shuttling device. The positive values of Re(λ ± ) give rise to the regime of instability when both α > 0 and α > γ τ β | f (x un )| are satisfied. The critical values for the instability are given by a functional shape of f (x) with α = γ/τ β | f (x un )|, (green dotted line in Fig.3 B), where Thus, the critical values of α for the unstable regime depending on the quality factor which is given by, α un Fig.3 A) The saddle point solution leads to a bi-stability of the system under certain condition for Q factor.
The approximate solution determining the boundaries for the instability regime of applied magnetic field is given by: This condition is valid for the range of magnetic fields; The upper limit of this domain of validity is determined by the smallest value of two contributions, namely the emf force and the asymmetry condition. Taking into account all necessary constraints for the stability regimes we construct the phase diagram of our model (see Fig. 4). This phase diagram shows the boundaries for the self-sustained oscillations regime (gray). Two green dotted lines correspond to  SiN. The range of quality factors refers to best known nano-mechanical devices. 30 Summarizing, we analysed a full fledged stability diagram of the Kondo shuttle device subject both to variation of external dc electric and magnetic fields and asymmetry of the tunnel barriers.
Kondo effect with its anomalously long relaxation time of dynamical spin screening is an ideal tool for coupling spin and mechanical degrees of freedom. We have shown that the competition between the mechanical damping of the oscillator at zero field, zero bias and contribution coming from the strong resonance spin scattering (Kondo effect)