Nano-mechanics driven by Andreev tunneling
A. V. Parafilo, L. Y. Gorelik, M. Fistul, H. C. Park, R. I. Shekhter
NNano-mechanics driven by Andreev tunneling
A. V. Parafilo, L. Y. Gorelik, M. Fistul,
1, 3, 4
H. C. Park, and R. I. Shekhter Center for Theoretical Physics of Complex Systems, Institute for Basic Science,Expo-ro, 55, Yuseong-gu, Daejeon 34126, Republic of Korea Department of Physics, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden Theoretische Physik III, Ruhr-University Bochum, Bochum 44801 Germany National University of Science and Technology ”MISIS”,Russian Quantum Center, Moscow 119049, Russia Department of Physics, University of Gothenburg, SE-412 96 G¨oteborg, Sweden (Dated: September 8, 2020)We predict and analyze mechanical instability and corresponding self-sustained mechanical os-cillations occurring in a nanoelectromechanical system composed of a metallic carbon nanotube(CNT) suspended between two superconducting leads and coupled to a scanning tunneling micro-scope (STM) tip. We show that such phenomena are realized in the presence of both the coherentAndreev tunneling between the CNT and superconducting leads, and an incoherent single electrontunneling between the voltage biased STM tip and CNT. Treating the CNT as a single-level quantumdot, we demonstrate that the mechanical instability is controlled by the Josephson phase difference,relative position of the electron energy level, and the direction of the charge flow. It is found nu-merically that the emergence of the self-sustained oscillations leads to a substantial suppression ofDC electric current.
Introduction.
Modern nanomechanical resonators [1]characterized by low damping and fine-tuning of theresonant frequency are widely used nowadays as super-sensitive quantum detectors [2]-[6] and as the mechan-ical component for various nanoelectromechanical sys-tems (NEMS) [7],[8]. The latter represent a promis-ing platform for studying the fundamental phenom-ena generated by the quantum-mechanical interplay be-tween nanomechanical resonator and electronic subsys-tem [9],[10].Large amount of fascinating physical phenomena havebeen predicted and observed in various NEMS, e.g. en-ergy level quantization of a nanomechanical oscillator[11], a strong resonant coupling of nanomechanical oscil-lator to superconducting qubits [12], mechanical cooling[13–15], a single-atom lasing effect [12, 16], mechanicaltransportation of Cooper pairs [17] and the generationof self-driven mechanical oscillations by a DC chargeflow [18–23], just to name a few.Significant part of these effects are based on the res-onant excitation of low damped mechanical modes by coherent quantum dynamics occurring in the electronicsubsystem. A straightforward method to establish co-herent quantum dynamics in mesoscopic devices, e.g.,the quantum beats, the microwave induced Rabi os-cillations etc., is to use the macroscopic phase coher-ence of superconducting (SC) elements incorporatedinto NEMS, see, for example, the review [24]. In partic-ular, in superconducting hybrid junctions [25]-[31] thecoherent electronic transport is determined by the pres-ence of Andreev bound states [32],[33]. The applied DCor AC currents induce the transitions between Andreevbound states, and the coherent high-frequency oscilla-tions in an electronic subsystem occur [14]. These coher-ent charge oscillations can excite the mechanical modes in the resonant limit only, when the frequency of me-chanical mode matches Andreev energy level difference.On other hand, an incoherent quantum dynamics oc-curring in the electronic subsystem can induce the me-chanical instability and subsequent formation of the self-driven mechanical oscillations in hybrid junctions. In-coherent quantum fluctuations of electric charge can beeasily mediated by tunneling of a single electron. Theself-driven oscillations generated by a DC electronic flowhave been predicted in [18, 19], later observed in a car-bon nanotube (CNT) based transistor [20], and studiedin detail [21],[22], see, e.g., [23] for recent experiment.A nontrivial interplay between coherent and incoher-ent electric charge variation and its influence on the per-formance of NEMS can be achieved in a nanomechan-ical Andreev device , where normal and SC metals arebridged by a mechanically active mediator.In this Letter, we present a particular NEMS setupwhere the mechanical oscillations are strongly affectedby a weak coupling to the electronic part of a system.We demonstrate that in the adiabatic limit as the fre-quency of mechanical oscillations is much smaller thanthe typical frequencies of electron dynamics, simultane-ous presence of coherent Andreev tunneling and inco-herent single electron tunneling can induce mechanicalinstability of the resonator and result in the appearanceof the self-sustained mechanical oscillations.
Model.
We consider a metallic single-wall carbonnanotube suspended between two grounded SC elec-trodes and coupled to a scanning tunneling microscope(STM) tip via electron tunneling. The two SC elec-trodes are characterized by the same modulus ∆ anddifferent phases φ L,R of SC order parameter, and cor-responding Josephson phase difference, φ = φ R − φ L .We study the case where the CNT mean-level spacing a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p x F Gate h ! " ! S S V N V g FIG. 1. Scheme of the superconducting (SC) nanoelectrome-chanical device. A single-wall carbon nanotube (CNT) issuspended between two SC leads which are characterized bythe phases of SC order parameter, φ L,R . A normal metalelectrode (STM tip) placed near the CNT-QD allows to in-ject electrons in CNT. The nanoelectromechanical force F between the CNT and gate electrode, which is located onthe distance h from the CNT, is controlled by a gate voltage V g . is greater than temperature k B T and the bias-voltage eV applied between STM tip and CNT. It allows us totreat the CNT as a movable single-level quantum dot(QD). The capacitive coupling between the CNT and agate is controlled by a gate voltage V g . We aslo assumethe dynamics of the CNT bending is reduced to the dy-namics of the fundamental flexural mode. The schemeof the described model is presented in Fig.1.The Hamiltonian of the model reads as follows H = H N + H S + H CNT + H tun . (1)The first two terms in Eq.(1) are the Hamiltonians of anSTM tip (normal lead) and two SC leads, accordingly: H N = (cid:88) kσ ( ε k − eV ) c † kσ c kσ , (2) H S = (cid:88) kjσ (cid:110) ξ kj a † kjσ a kjσ − ∆ e iφ j ( a † kj ↑ a †− kj ↓ + H.c. ) (cid:111) . (3)Here, c kσ ( c † kσ ) and a kjσ ( a † kjσ ) are annihilation (cre-ation) operators of electrons in the normal and j -th SCleads ( j = L, R ) with energies ε k and ξ kj , correspond-ingly. The index σ = ↑ , ↓ indicates the spin of electronsin the leads.The Hamiltonian of the single-level vibrating CNT-QD reads as follows H CNT = (cid:88) σ ε d † σ d σ + (cid:126) ω p + ˆ x ) − F ˆ x (cid:88) σ n σ . (4)The quantum dynamics of the electronic degree of free-dom is described by the first term in Eq. (4), where ε is the QD electron energy level, and d σ , d † σ are annihila-tion and creation operators of the electrons in the QD, n σ = d † σ d σ [34].The second term in Eq. (4) characterizes the CNTvibrations with the frequency ω , and the dimension-less operators ˆ x = ˆ X/x , ˆ p = x ˆ P / (cid:126) are canonically conjugated displacement and momentum of the CNT-QD. Here, x = (cid:112) (cid:126) /mω is the amplitude of the zero-point oscillations of the CNT, and m is the mass of theCNT. Electromechanical interaction determined by thethird term in Eq. (4), is achieved through the electro-static interaction of the charged CNT-QD with the gateelectrode. The interaction strength is F ∝ ( ex /h ) V g β [19],[35], where h is the distance between the CNT andgate electrode, and β ∼ . H tun = (cid:88) kσ e − ˆ x/λ (cid:16) t nk c † kσ d σ + ( t nk ) ∗ d † σ c kσ (cid:17) + (cid:88) kjσ (cid:16) t sk a † kjσ d σ + ( t sk ) ∗ d † σ a kjσ (cid:17) , (5)describes the tunneling processes between the CNT andi) the STM tip with deflection dependent hopping am-plitude, i.e. t nk exp( − ˆ x/λ ), where λ = l/x and l is thetunneling length of the barrier; ii) SC leads with thehopping amplitude t sk . Mechanical instability.
In order to rigorously demon-strate the phenomenon of mechanical instability in theSC hybrid junction, we analyze the dynamics of theCNT’s flexural mode by using the reduced density ma-trix technique. By treating the tunneling Hamiltonian(5) as a perturbation and tracing out the electronic de-grees of freedom in the normal and SC leads, one canget a quantum master equation for the reduced densitymatrix operator (in (cid:126) = 1 units):˙ ρ = − i [ H CNT , ρ ] + i Γ S ( φ )[ d †↑ d †↓ + d ↓ d ↑ , ρ ] − (cid:88) σ L [ ρ ] . (6)Here, Γ S ( φ ) = 2 πν | t sk | cos( φ/
2) is the Josephson phasedependent strength of the intra-QD electron pairing in-duced by the coherent Andreev tunneling, ν is the elec-tron density of states in the leads, and L [ ρ ] is a Lind-bladian operator in the high-voltage regime eV (cid:29) ε , ω [36],[37]: L [ ρ ] = Γ2 (cid:40) { e − xλ d σ d † σ , ρ } − e − ˆ xλ d † σ ρd σ e − ˆ xλ , V > , { e − xλ d † σ d σ , ρ } − e − ˆ xλ d σ ρd † σ e − ˆ xλ , V < , (7)where Γ = 2 πν | t nk | is the QD energy level width pro-duced by a single electron tunneling. The quantum mas-ter equation (6) is justified in the deep sub-gap regimeunder the following assumptions: all relevant energiesare smaller than the SC gap, eV, k B T, ε (cid:28) ∆.Density matrix ρ acts in the finite Fock space ofthe two-fold degenerate single-electron level in the QD.The four possible electronic states are | (cid:105) , | σ (cid:105) = d † σ | (cid:105) ( σ = ↑ , ↓ ), and | (cid:105) = d †↑ d †↓ | (cid:105) . In this representationthe reduced density matrix ρ contains five nonzero ele-ments: ρ , ρ ↑↑ = ρ ↓↓ ≡ ρ , ρ , ρ , and ρ . Using thenormalization condition ρ + 2 ρ + ρ = 1 one caneliminate the ρ component of the density matrix from a) b)c) d) -2.7 x -2 x -3 x -3 x -3 FIG. 2. Phase diagrams of the mechanical instability show-ing pumping coefficient η (0) as a function of the Josephsonphase difference φ , the QD level width Γ / Γ S (0), and theQD energy level ε (0) /ω for: a) α = 0 . λ − = 0 .
05, and ε (0) = 0; b) α = 0 .
2, Γ / Γ S (0) = 0 .
3, and λ − = 0; and forgeneral case Γ / Γ S (0) = 0 . λ − = 0 .
05 when c) α = 0 . α = − .
2. The red and blue color schemes indicatethe mechanical instability ( η >
0) and the damping ( η < Q − = 0 and κ = 1. further consideration. Therefore, the joint dynamics ofthe electronic and mechanical subsystems is determinedby the matrixˆ (cid:37) = 12 (cid:18) ρ − ρ ρ ρ ρ − ρ (cid:19) . (8)If the amplitude of the CNT displacement is largerthan the amplitude of zero-point oscillation, one cantreat the dynamics of the CNT bending as a classi-cal with time-evolution governed by Newton’s equation.Introducing the dimensionless time units as ω t → t we obtain a closed system of the relevant equations forthe CNT displacement x and matrix ˆ (cid:37) Eq. (8) in thefollowing form:¨ x + Q − ˙ x + x = α + α Tr { σ ˆ (cid:37) } , (9) ω ˙ˆ (cid:37) = − i [ ε ( x ) σ − Γ S ( φ ) σ , ˆ (cid:37) ] − Γ( x ) (cid:16) ˆ (cid:37) − κ σ (cid:17) , (10)where dimensionless parameter α = F/ω , σ i ( i =1 , ,
3) are the Pauli matrices, ε ( x ) = ε − αx , Γ( x ) =Γ exp( − x/λ ), and κ = sgn( V ). An environment in-duced damping of the mechanical subsystem is deter-mined by the term ∝ Q − , where Q ∼ [20] is thequality factor. In the adiabatic limit, ω / Γ (cid:28)
1, weobtain ˆ (cid:37) ( t ) from Eq. (10), and the non-linear part ofEq. (9) is presented in the following form:Tr { σ ˆ (cid:37) ( t ) } = κ (cid:18) − S ( φ ) D ( x ( t ) , φ ) (cid:19) + ˙ x ( t ) η ( x ( t )) , (11) where D ( x, φ )= ξ ( x ) + Γ ( x ), ξ ( x ) = 2 (cid:112) ε ( x ) + Γ S ( φ )is the energy difference between two Andreev levels ofthe QD-SC subsystem, and a mechanical friction coef-ficient η ( x ), induced by interaction with the electronicdegree of freedom, reads as η ( x ) = α I ( x ) (cid:18) λ − C ( x ) + α ε ( x )Γ ( x ) C ( x ) (cid:19) . (12)Here, I ( x ) = κ x )Γ S ( φ ) /D ( x, φ ) is the DC flow ofelectrons between the STM tip and SC leads, and C ( x ) = 6Γ ( x ) − ξ ( x ) D ( x, φ ) , C ( x ) = 20Γ ( x ) + 4 ξ ( x ) D ( x, φ ) . (13)The frequency of a typical CNT-based resonator is ω ∼ x ≈ V g ∼
100 mV, h ∼ − m, andthe tunneling length l (cid:39) − m we estimate dimension-less coupling constants to be α ∼ . λ − ∼ − .After substituting Eq.(11) in Eq.(9), we found non-linear equation for the CNT deformation local in time.In the limit α, λ − (cid:28) x c = α + κα ε (0) + Γ D (0 , φ ) + O ( α , αλ − ) . (14)The stability of the static solution is studied by lineariz-ing Eq. (11). In the limit Γ (cid:29) ω , the time evolution ofthe small CNT deviation from its equilibrium position δx ( t ) = x ( t ) − x c is given by [38] δ ¨ x + (cid:0) Q − − η (0) (cid:1) δ ˙ x + δx = 0 . (15)The static solution x c of the system at η (0) > Q − be-comes unstable with respect to the generation of me-chanical oscillation with amplitude exponentially in-creasing in time. Development of instability results inthe appearance of self-sustained mechanical oscillations,governed by the nonlinearity of r.h.s. Eq. (9).Next, we analyze the influence of various parameterson the coefficient η (0) which we call a pumping coeffi-cient in what follows. First, we note that η (0) linearlyincreases with the electromechanical coupling α and theDC flow ∝ I (0). Moreover, the pumping coefficient η (0)changes a sign depending on the direction of the elec-tronic flow, i.e. the sign of eV . At | eV | (cid:29) ε , bias volt-age affects the phenomenon under consideration solelyby this means. Below we analyze the case of eV > η (0) on the parameters φ , Γ / Γ S (0) and ε (0) obtainedfrom Eqs. (12) and (13) are shown in Fig. 2 (redcolor scheme indicates η (0) >
0, while blue scheme – η (0) < ε (0)= ε =0, the pumping coeffi-cient η (0) ∝ κα/λ is determined by the ratio betweenΓ and Γ S ( φ ), since only the first term in Eq. (12) con-tributes. The pumping coefficient changes its sign whenΓ = (cid:112) / S ( φ ), see Fig. 2(a). If the dependence of theelectron hopping on the amplitude of the CNT oscilla-tions is negligible, i.e. λ − = 0, the pumping coefficient η (0) ∝ κα ε (0) is determined by the sign of ε (0). Suchbehavior is illustrated in Fig. 2(b). General case, whenboth terms in Eq. (13) contribute into the pumping coef-ficient Eq. (12), is shown in Fig. 2(c) and (d) for positive( α >
0) and negative ( α <
0) electrostatic interaction,respectively.The origin of the pumping processes, and correspond-ing mechanical instability can be qualitatively explainedas follows: since two electronic states | (cid:105) and | (cid:105) in theQD are not the eigenstates of the QD-SC subsystem,the quantum Rabi oscillations emerge with a frequencyproportional to the energy difference between Andreevlevels ξ ( x, φ ). These Rabi oscillations occur in the formof periodic in time single-Cooper pair transfer betweenSC leads and the QD. However, an incoherent singleelectron tunneling from the STM tip to the QD can in-terrupt the coherent oscillations as well as resume them.As this takes place, the averaged charge in the QDis governed by the interplay between two processes: i)a coherent Rabi oscillations and ii) an incoherent singleelectron tunneling. Both processes and their main char-acteristics, Γ( x ) and ξ ( x ), are controlled by the CNTdisplacement and vary in time if δ ˙ x ( t ) (cid:54) = 0. Such vari-ations give rise to a correction of the average charge inthe QD, that is proportional to the velocity of the QD,thereby generating effective friction force. We note thatthe amplitude of the effective friction force is determinedby two terms (see Eq. (12)), where the first term is in-duced by the time variation of the hopping amplitudeof single electron tunneling ˙Γ( x ( t )) ∝ λ − ˙ x , while thesecond term is generated by the time variation of theRabi frequency ˙ ξ ( x ( t )) ∝ αε (0) ˙ x . DC electric current.
The self-sustained oscillationsaffect the DC current flow between the STM tip andSC leads. This phenomenon allows one to verify themechanical instability through the electric current mea-surement.The expression for the DC current is given by I N ( x ( t )) = e Γ ( x ( t )) ( κ − Tr { σ ˆ (cid:37) ( t ) } ) . (16)If the pumping coefficient η (0) < Q − , the mechanicaloscillations of the CNT are damped, and the DC electriccurrent is expressed as I N (0) = e I (0). This expressioncoinsides with the DC current obtained in the absenceof electromechanical interaction. Such dependence isshown in Fig. 3(a). The DC current strongly dependson the Josephson phase difference φ and the QD energylevel ε (0). The current reaches its maximum at ε (0) = 0and vanishes at φ = π . Besides, I N (0) is proportionalto ∝ ΓΓ S , revealing Andreev tunneling [39] since onlytwo electrons (the Cooper pair) can tunnel from the QDto the SC leads.In the regime of mechanical instability η (0) > Q − ,the static solution becomes unstable and CNT vibra-tions develop into pronounced self-sustained oscillations a) b) c) d) FIG. 3. DC electric current I N /I normalized to the max-imum of static current I = e Γ as a function of the Joseph-son phase difference φ and the QD energy level ε (0) /ω atΓ / Γ S (0) = 0 . α = 0, and b) α = 0 . φ = 2 . ε (0) /ω = 3, respectively.These projections are presented in panels c) and d), wherethe charge current ( I N (0) = e I (0)) at α = 0 is shown byblack dashed lines, and the DC current at α = 0 . Q = 10 , κ = 1, and λ − = 0 . of finite amplitude. As a result, the current exhibits pe-riodic oscillations with the frequency ω . The averagedover the period of mechanical oscillations DC currentis obtained numerically and the result is presented inFig. 3(b). The projections of I N at fixed φ and ε (0)are presented in Fig. 3 (c) and (d). As one can seein Fig. 3, pronounced self-sustained oscillations of theCNT-QD suppress the charge current in the region ofparameters obeyed η (0) > Q − condition. The strengthof this current suppression depends on the amplitudeof the CNT self-oscillations and correspondingly on thepumping strength η (0). Conclusions.
We predict the phenomenon of me-chanical instability and corresponding self-sustained os-cillations in a hybrid nanoelectromechanical device con-sisting of a carbon nanotube suspended between twoSC leads and placed near a voltage-biased normal STMtip. Such effect is based on a peculiar interplay of thecoherent quantum-mechanical Rabi oscillations inducedby the Andreev tunneling between the CNT and SCleads, and an incoherent single electron tunneling be-tween the STM tip and CNT. We obtain that the ob-served mechanical instability and self-sustained oscilla-tions of finite amplitude are determined by two param-eters: the relative position of the single-electron energylevel, and the Josephson phase difference between theSC leads. Numerical analysis demonstrates that the pre-dicted mechanical instability develops into pronouncedself-sustained bending oscillations of the CNT resonatorwhich, in its turn, result in a suppression of the DC elec-tric current flowing between the STM tip and SC leads.This effect allows one to detect the predicted mechan-ical instability through the DC current measurement.A SQUID sensitivity to an external magnetic field canbe achieved by using proposed nanomechanical Andreevdevice through the control of the Josephson phase dif-ference by a magnetic flux.
Acknowledgement.
This work was supported by theInstitute for Basic Science in Korea (IBS-R024-D1).LYG and RIS thank the IBS Center for Theoreti-cal Physics of Complex Systems for their hospitality.M.V.F. acknowledges the partial financial support of theMinistry of Education and Science of the Russian Fed-eration in the framework of Increase CompetitivenessProgram of NUST ”MISIS” K − − [1] S. Schmid, L. G. Villanueva, and M. Roukes, Fundamen-tals of Nanomechanical Resonators (Springer, Switzer-land, 2016).[2] R. G. Knobel, and A. N. Cleland, Nature (London) ,291 (2003).[3] M. P. Blencowe, Science , 56 (2004).[4] V. Sazonova, Y. Yaish, H. ¨Ust¨unel, D. Roundy, T. A.Arias, and P. L. McEuen, Nature , 284 (2004).[5] B. Lassagne, D. Garcia-Sanches, A. Aguasca, and A.Bachtold, Nano Lett. , 3735 (2008).[6] A. H¨uttel, G. Steele, B. Witkamp, M. Poot, L. Kouwen-hoven, and H. S. J. van der Zant, Nano Lett. , 2547(2009).[7] X. M. H. Huang, C. A. Zorman, M. Mehregany, and M.L. Roukes, Nature (London) , 496 (2003).[8] M. P. Blencowe, Phys. Rep. , 159 (2004).[9] K. L. Ekinci, and M. L. Roukes, Rev. Sci. Instrum. ,061101 (2005).[10] A. N. Cleland, Foundations of Nanomechanics (Springer, New York, 2002).[11] P. Arrangoiz-Arriola, E. A. Wollack, Zh. Wang, M.Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, R.Van Laer, and A. H. Safavi-Naeini, Nature, , 537(2019).[12] J. Hauss, A. Fedorov, C. Hutter, Al. Shnirman, and G.Sch¨on, Phys. Rev. Lett. , 037003 (2008).[13] C. Urgell, W. Yang, S.L. De Bonis, C. Samanta, M. J.Esplandiu, Q. Dong, Y. Jin, and A. Batchtold, NaturePhysics , 32 (2020).[14] G. Sonne, M.E. Pena-Aza, L.Y. Gorelik, R.I. Shekhter,and M. Jonson, Phys. Rev. Lett. , 226802 (2010).[15] P. Stadler, W. Belzig, and G. Rastelli, Phys. Rev. Lett. , 197202 (2016).[16] G. Rastelli, and M. Governale, Phys. Rev. B ,085435 (2019).[17] L.Y. Gorelik, A. Isacsson, Y.M. Galperin, R.I. Shekhter,and M. Jonson, Nature , 454 (2001); A. Isacsson,L. Y. Gorelik, R. I. Shekhter, Y. M. Galperin, and M.Jonson, Phys. Rev. Lett. , 277002 (2002).[18] L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo,R. I. Shekhter, and M. Jonson, Phys. Rev. Lett. ,4526 (1998).[19] Ya. M. Blanter, O. Usmani, and Yu. V. Nazarov, Phys.Rev. Lett. , 136802 (2004).[20] G. A. Steele, A. H¨uttel, B. Witkamp, M. Poot, H. B.Meerwaldt, L. P. Kouwenhowen, and H. S. J. van derZant, Science , 1103 (2009).[21] D. R. Schmid, P. L. Stiller, C. Strunk, A. H¨uttel, Ap-plied Phys. Lett. , 123110 (2015).[22] D. R. Schmid, P. L. Stiller, C. Strunk, A. H¨uttel, NewJ. Phys. , 083024 (2012).[23] K. Willick, and J. Baugh, Phys. Rev. Research ,033040 (2020).[24] A. V. Parafilo, I. V. Krive, R. I. Shekhter, and M. Jon-son, Low Temp. Phys. , 273 (2012)[Fiz. Nizk. Temp. , 348 (2012)].[25] T. Novotny, A. Rossini, and K. Flensberg, Phys. Rev.B , 224502 (2005).[26] A. Zazunov, D. Feinberg, and T. Martin, Phys. Rev.Lett. , 196801 (2006).[27] J. Sk¨olberg, T. L¨ofwander, V.S. Shumeiko, and M. Fo-gelstr¨om, Phys. Rev. Lett. , 087002 (2008).[28] A. Zazunov and R. Egger, Phys. Rev. B , 104508(2010).[29] A. V. Parafilo, I. V. Krive, R. I. Shekhter, Y. W. Park,and M. Jonson, Low Temp. Phys. , 685 (2013).[30] A. V. Parafilo, I. V. Krive, R. I. Shekhter, Y. W. Park,and M. Jonson, Phys. Rev. B , 115138 (2014).[31] J. Baranski, and T. Domanski, J. Phys.: Condens. Mat-ter , 305302 (2015).[32] A. F. Andreev, Sov. Phys. JETP. , 1228 (1964).[33] I. O. Kulik, Zh. Eksp. Teor. Fiz. , 1745 (1969).[34] We neglect the inter-dot electron interaction Un ↑ n ↓ in case when U < eV , since it will give only quali-tative correction to the phenomenon under considera-tion. More precisely, it will renormalize QD energy level, ε → ε + U/ Quantum trans-port: Introduction to Nanoscience (Cambridge Univer-sity Press, 2009).[36] T. Novotny, A. Donarini, and A.-P. Jauho, Phys. Rev.Lett. , 256801 (2003).[37] D. Fedorets, L. Y. Gorelik, R. I. Shekhter, and M. Jon-son, Phys. Rev. Lett. , 057203 (2005); New J. Phys. , 242 (2005).[38] In Eq. (15) we ignore small renormalization of the fre-quency of mechanical oscillations.[39] J. Gramich, A. Baumgartner, and C. Sch¨onenberger,Phys. Rev. Lett.115