Tunable RKKY interaction in a double quantum dot nanoelectromechanical device
TTunable RKKY interaction in a double quantum dot nanoelectromechanical device
A. V. Parafilo and M. N. Kiselev
The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy (Dated: September 25, 2018)We propose a realization of mechanically tunable Ruderman-Kittel-Kasuya-Yosida interaction ina double quantum dot nanoelectromechanical device. The coupling between spins of two quantumdots suspended above a metallic plate is mediated by conduction electrons. We show that the spin-mechanical interaction can be driven by a slow modulation of charge density in the metallic plate.We propose to use St¨uckelberg oscillations as a sensitive tool for detection of the spin and chargestates of the coupled quantum dots. Theory of mechanical back action induced by a dynamicalspin-spin interaction is discussed.
I. INTRODUCTION
Recent progress in theory and experiment onnanometer-sized devices sheds light on a significant roleof the spin degree of freedom (see Refs [1]-[6]) in opto-and electromechanical systems. The nanomechanical quantum - classical hybrid systems [7]-[9] are impor-tant for both fundamental research and applications[10]. The range of problems addressed by the nanome-chanics varies from designing new tools for a quantuminformation processing to a development of highly sen-sitive methods of, for example, mass, force and cur-rent detection in metrology [7],[10]. While the nano-optomechanics is dealing with coupling of light withmechanical degrees of freedom [9], the nanoelectrome-chanics (NEM) works with mechanically nanomachinedelectrons [10]. Several new directions of NEM, in partic-ular those which are focused on an investigation of me-chanical systems coupled to the spins (spintromechanics[11] and optomagnonics [5]) emerged recently thanks toa progress in both experiment [4], theory [5] and ma-terial science [6]. The ability to manipulate nanoelec-tromechanical systems via electron’s spins leads to a va-riety of new phenomena [12]-[16]. Since typical mechan-ical displacements of NEM devices are in a range fromangstroms to nanometers, its detection requires utiliza-tion of very sensitive methods. Quantum interferom-etry [17] provides one of such sensitive tools [18],[19].In most cases measurements of a back action inducedby the quantum electron spin [20] and charge system[21],[22] onto a mechanical resonator gives yet anothersensitive tool for measuring the out-of-equilibrium prop-erties of the quantum system operating in many cases ina regime of strong electron-electron interaction and/orresonance scattering, see examples in Ref.[11].One of the most intriguing examples of spin-relatedphysics in NEMS is the Kondo effect in shuttling devices[23],[24]. Kondo physics in quantum dots (QD) [25]manifested itself as a many-body effect associated withthe creation of a cooperative singlet state composed ofconduction electrons in the leads and a localized QDspin S =1 /
2. Complete screening of a spin impurity inthe QD occurs at temperatures well below the Kondotemperature T K , the typical energy scale of the interac-tion. Formation of a Kondo singlet is accompanied by the saturation of the nano-device’s electric conductanceat the unitary limit 2 e /h . Mechanical motion of theQD results in the appearance of a time dependency of T K , which allows us to employ this effect as a dynamicalprobe of the Kondo cloud [24]. Quantum engineering ofNEM-QD devices opens a possibility for investigatingcompeting interactions and emergent symmetries in thepresence of resonance electron scattering, for example:two channel Kondo effect in a side-coupled QD [26], twoimpurity Kondo effect in parallel- and serially-coupleddouble quantum dot (DQD) [27],[28], or SU(4) Kondoeffect in a single-wall carbon nanotube based QD [29],[30].To demonstrate back action based on spin exchangein a mechanical resonator we concentrate on studying aparallel DQD system with spin-spin coupling controlledby its nanomechanical motion. In the regime of resonantscattering of mobile electrons on localized spins, thetwo-impurity Kondo model arises [31]. It is well known(see, e.g., Refs. [31],[32]) that at temperatures T
0, the in-teraction between spins is antiferromagnetic and thecorresponding ground state of the system is a singlet(the total spin S =0). The ferromagnetic RKKY interac-tion I RKKY ( R ( t )) < S =1 ground state. As a result, the potential energy as-sociated with the RKKY exchange interaction gives riseto an additional displacement-dependent force. Thisforce acts on the mechanical resonator being sensitiveto the spin configuration of the DQD NEM device.The paper is organized as follows: Section II is de-voted to formulation of the model describing a drivendouble-quantum dot nanoelectromechanical device. Wepresent a short derivation of the mechanically nanoma-chined effective two-impurity Kondo model. The RKKYinduced back action in mechanical subsystem and thedynamics of the QD oscillations are discussed in Sec.III. The analysis of the St¨uckelberg interference patternis presented in Sec. IV. The summary and discussionsare given in Sec. V. II. MODEL
The Hamiltonian of the mobile DQD system reads: H = H DQD + H leads + H tun + H vib , (1)where the first term characterizes the DQD H DQD = (cid:88) i =1 , ,σ,σ (cid:48) [ ε i ˆ n i,σ + U ˆ n i, ↑ ˆ n i, ↓ ] + U ˆ n ,σ ˆ n ,σ (cid:48) . (2)Here ε i is an electron energy level on the i th QD,ˆ n i,σ = ˆ d † i,σ ˆ d i,σ is a density operator of electron with spinprojection σ = ↑ , ↓ and ˆ d i,σ ( ˆ d † i,σ ) is its annihilation (cre-ation) operator, U is the Coulomb interaction betweenelectrons in each QD, and U is a capacitive couplingbetween QDs. We assume that each QD is singly oc-cupied and the system is in the particle-hole symmetric a) b) x y y r x r l R drain source source drain x R eff V S S
T T x (cid:31) RKKY I (cid:33) RKKY I FIG. 1. (Color online) (a) Mobile double quantum dot NEMdevice: Each quantum dot (light blue circle) is sandwichedbetween its own source-drain electrodes (pink). The quan-tum dots are suspended above a metallic plate (gray) formedby 2DEG. The dots are attached to the mechanical oscilla-tors (not shown) and move in the x direction parallel tothe metallic plate (motion is excited by external gates notshown on the picture). The time-dependent distance be-tween QD is denoted by R ( t ); r y and r x are Cartesian projec-tions of the DQDs equilibrium (rest) position, l is a lengthof mechanical oscillator. (b) Sketch of RKKY interactionbetween two local spins attached to mechanical resonator V eff = I RKKY ( R ) (cid:104) S S (cid:105) , where sign of interaction is deter-mined by R = | R | . The ground state of the two-localized-electron system is singlet (S) if I RKKY ( R ) > regime, ε i = − U/ H leads in Eq. (1) describes the elec-trons in the leads and metallic plate, H leads = (cid:88) α (cid:88) k,i,σ ξ ikα ˆ c i † kσα ˆ c ikσα + (cid:88) k σ (cid:15) ( t )ˆ a † k σ ˆ a k σ , (3)where operators ˆ c ikσα (ˆ c i † kσα ) and ˆ a k σ (ˆ a † k σ ) denote elec-tron annihilation (creation) in ( i, α )th electrode and2DEG (here α stands for source/drain). Their ex-citation energies ξ ikα = ε ikα − µ iα and (cid:15) ( t )= (cid:15) k − µ ( t ) arecounted from the chemical potentials µ iα , µ ( t ). Notethat we consider the case of a time-dependent chemi-cal potential and density of the electrons in the 2DEG, µ ( t )= (cid:15) F + eV sin(Ω t ) (periodic modulations with a fre-quency Ω), while the amplitude of its modulation is lim-ited by a condition | eV |
We analyze the dynamics of the i th QD characterizedby the amplitude of its fundamental vibrational mode x i ( t ), whose time evolution is described by Newton’sequation¨ x i + γ ˙ x i + ω i x i = − M ∂∂x i V eff ( | R − R | , t ) , (7)here γ is a phenomenological damping and V eff is aneffective exchange interaction, which in adiabatic ap-proximation ( (cid:126) Ω (cid:28) (cid:15) F ) can be obtained from the linearresponse theory [44]. To derive an effective potentialfor the RKKY interaction between spins located at dif-ferent wires we integrate out completely all states of2DEG at the conduction plate (see Fig. 1). As a re-sult, the effective RKKY potential is given by the realpart of the density-density correlation function [45] ofthe 2DEG [46]. The imaginary part of this correlationfunction contributes to the damping of the mechanicalsubsystem. Performing a Fourier transform of V eff andtreating the second term in Eq.(5) as a perturbation in | γ k i | / ( U (cid:15) F ) (cid:28) V eff = (cid:90) dωe − iωt (cid:101) V eff ( R , ω ) (cid:32) eV i J (cid:88) κ = ± κδ ( ω + κ Ω) (cid:33) , (cid:101) V eff ( | R | , ω ) = J (cid:88) k , q ; i (cid:54) = j (cid:104) S i S j (cid:105) e − i qR ( f k − f k + q )˜ ω i + (cid:15) k − (cid:15) k + q + i + , (8)where R = R − R , δ ( ω ) is the Dirac delta-function,˜ ω i = ω − q ˙ R i , and f k is the Fermi distribution function.The real part of the effective potential is the RKKY in-teraction between two ”impurities”, while its imaginarypart is the spectral function of electron-hole excitationsin 2DEG. The imaginary part of (cid:101) V eff is in general re-lated to the damping mechanisms associated with, e.g.,the creation of collective plasmon or/and magnon ex-citations or tunneling into the leads. We neglect suchcontributions by assuming the adiabaticity condition.The real part of (cid:101) V eff ( | R | ,
0) at zero temperature (see,e.g., Refs. [41],[47]) is given by: V eff ( R ) = J (cid:104) S S (cid:105) A N k ∗ F · [ J ( k ∗ F R ) N ( k ∗ F R ) + J ( k ∗ F R ) N ( k ∗ F R )] , (9)where A is the confinement area of the 2DEG(back gate), k ∗ F ≈ k F +( eV / (cid:126) v F ) sin(Ω t ) is a time-dependent Fermi wave vector, and J m ( z ) , N m ( z ) arethe Bessel functions of the first and the secondkind. For simplicity, we consider only the long-distance limit of Eq. (9) at large k F R , which corre-sponds to a power-law − sin(2 k ∗ F R ) /R RKKY asymp-tote [48]. Here the distance between ”impurities” is R = (cid:113) ( r x + x − x ) + r y and r x , r y are x - y projec-tions of R at the equilibrium position, see Fig. 1(a).Let us assume that the averaged distance between thevibrating QDs R = (cid:113) r x + r y is close to the distance atwhich the RKKY interaction changes sign, 2 k F R ≈ πn (where n is an integer). Then, by expanding the oscillat-ing RKKY function with respect to the small parame-ters ( eV /(cid:15) F , r x ( x − x ) /R (cid:28)
1) and substituting it intoEq. (7) one can obtain the system of coupled equationsof motion for the displacement of the i th QD¨ x i + γ ˙ x i + ω i x i = ± α (cid:32) r y R x − x r x (cid:33) × (cid:18) eV (cid:18) (cid:15) F + J J (cid:19) sin(Ω t ) (cid:19) , (10)where ” + ” is for i =1, ” − ” for i =2, and α =J (cid:104) S S (cid:105) N ( A k F r x / πM R ). The RKKY in-teraction results in three different forces in the r.h.s.of Eq. (10) which (i) lead to the renormalization ofthe i th QD equilibrium position, (ii) create a time-dependent force proportional to sin(Ω t ), and (iii) re-sult in energy transferring between the two oscillat-ing QDs (beating). In particular, the RKKY inter-action between spins provides the coupling betweenthe QDs and the mechanical subsystems. Such spin-mechanical coupling can be easily extended to the quan-tum limit. Expansion of sin(2 k F R ) around the equilib-rium interdot distance πn/ k F and quantization of theQD displacement field leads to the spin-mechanical in-teraction Hamiltonian: H int ∼ λ S S (ˆ b + ˆ b † ) / √
2, where λ =( − n J N k F A r x x /R is the spin-phonon cou-pling, ˆ b, ˆ b † are boson operators of vibrational quanta.We rewrite the EOM (10) in terms of a normal (i)in-phase, x + x , and (ii) out-of-phase, x − x , modes.The first term in the r.h.s of Eq. (10) is eliminatedby redefinition of the ”impurities” initial deflections x i ∓ α /ω i → x i . We assume that the QD eigenfrequen-cies ω , = ω ± δω differ by a small value δω (cid:28) ω .In addition, we introduce a dimensionless time τ = ω t and dimensionless normal mode displacements in unitsof the length of the nanowires l : ϕ =( x + x ) /l , φ =( x − x ) /l . We denote∆ = 2 δωω , ω d = Ω ω , ˜ α = 2 α ω , (11)and introduce dimensionless force and frequency shifts F = ˜ α eVl J J , α = 2 ˜ α r x (cid:18) r y R (cid:19) , α = l Fr x (cid:18) r y R (cid:19) , The coupled mechanical equations of motion (10) in di-mensionless notations are given by¨ ϕ + ˙ ϕ/Q + ϕ = − ∆ · φ, (12)¨ φ + ˙ φ/Q + [1 − α − α sin( ω d τ )] φ = − ∆ · ϕ + F sin( ω d τ ) , where Q is a quality factor. As a result, the equationof motion for the out-of-phase mode φ describes a para-metric oscillator subject to an external time-dependentdriving force coupled to a nondriven oscillator, associ-ated with the in-phase mode ϕ . Notice that the fre-quency shift for the in-phase mode is negligible com-pared to the shift for the out-of-phase mode which con-sists of a time-independent part α and a part ∝ α periodic in time.The coupling constant α in (12) can be estimatedconsidering realistic parameters for a typical 2DEG: (cid:15) F ∼
10 meV, k F ∼ cm − . Besides, without loss ofgenerality we assume J ∼
10 K and eV /(cid:15) F ∼ . ω ∼
100 MHz is a fundamental frequency ofcarbon nanotube’s bending modes, x ∼ − cm is theamplitude of zero-point oscillations). Furthermore, tak-ing R ∼ r y ∼ − cm and considering the QDs chargingenergies in the range from 1 K to 10 K we obtain: α ∼ J (cid:15) F J (cid:126) ω k F R (cid:18) x R (cid:19) eVU (cid:18) r y R (cid:19) ∼ − ÷ − . IV. ST ¨UCKELBERG INTERFERENCE INCLASSICAL TWO-LEVEL SYSTEM
Finally, we propose an experimental realization of thespin-mechanical coupling based on the investigation ofthe envelope function of vibrating QD’ displacements[10]. An idea is based on the observation that the slowdynamics of the NEM system associated with the drive ω d mimics the dynamics of a driven quantum two-levelsystem. The slowly varying amplitudes of in- and out-of-phase modes play the same role as the spinor in thetime-dependent Schr¨odinger equation of two-level sys-tem [49],[50]. To demonstrate the similarity betweenclassical and quantum driven systems we start with theansatz [49],[50]: ϕ ( τ ) = C · Re { Φ + ( τ ) e iτ − τ/ (2 Q ) } ,φ ( τ ) = C · Re { Φ − ( τ ) e iτ − τ/ (2 Q ) } . (13)The complex amplitudes Φ ± are equivalent to the spinor”wave functions”. Here the constant C accounts forthe normalization condition | Φ + | + | Φ − | ≈ F in the particular solution ofEq. (12)]. Substituting Eq. (13) into Eqs. (12) and per-forming the unitary transformation with operator (cid:99) W = e iα τ/ exp[ − i ( α / ω d ) cos( ω d τ )] we map Eqs. (12) ontoa Schr¨odinger-like equation i ddτ (cid:18) Ψ + ( τ )Ψ − ( τ ) (cid:19) = H TLS ( τ ) (cid:18) Ψ + ( τ )Ψ − ( τ ) (cid:19) , (14) (cid:68) (cid:68) (cid:68) (cid:68) S S
T T
FIG. 2. (Color online) The time averaged over 30 periodsprobability P av to populate the in-phase mode | ϕ | as a func-tion of the dimensionless energy offset (time-independentfrequency shift) α and the driving amplitude α . Thefanlike diagram is obtained from Eq.(12) for the dimen-sionless parameters: driving frequency ω d =10 − , detuning∆=2 · − , and quality factor Q =10 . The initial conditionfor Eq. (12) assumes population of the out-of-phase modeat τ =0: | φ (0) | =1. Initial velocities of mechanical oscilla-tors are equal to zero. We consider the modulations of thedensity in the back gate corresponding to small deviationsof the potential energy around the second node of V eff (Fig.1). The two-spin configuration is locked in the singlet state(main panel) and in the triplet state (inset), see detailed dis-cussion about two-spin configuration in Sec. IV. We neglectthe spin-relaxation processes and the effects of hyperfine in-teraction. where H TLS = − σ x ∆2 − σ z α + α sin( ω d τ )4 , (15)and Ψ ± are linked to Φ ± as Ψ ± = (cid:99) W Φ ± . Theinstantaneous adiabatic eigenvalues of theHamiltonian (15) depend on time τ as follows E = ± (1 / (cid:112) ∆ + ( α + α sin( ω d τ )) /
4. In the vicin-ity of avoided crossing points, where the distancebetween two levels is minimal, the linearized modeldescribes the Landau-Zener transitions [51] with ef-fective Hamiltonian H LZ = − (∆ / σ x ± ( vτ / σ z , where v = α ω d / P LZ =exp( − π ∆ / v ).In the case of a multipassage process, the transitionprobability accounts for both diabatic and adiabatictransitions and contains the phase responsible for theinterference between two passes [17]. The interferencepattern is visualized by a fan-type diagram, see Fig. 2. The density plot (see Fig. 2) shows the time-averagedprobability to populate the in-phase mode as a functionof the dimensionless ”energy offset” (time-independentfrequency shift) α and the driving amplitude α . Theminima and maxima of the time-averaged probabilitycorrespond to destructive and constructive interferencebetween consecutive energy levels crossings [52]. Usu-ally, in the absence of any dissipation, the maximumvalue of the averaged probability to populate the statesatisfying nonoccupied initial condition is equal to 0.5.However, the maximum value plotted on Fig. 2 is belowthis limit. To explain the probability deficit we point outthat the effects of dissipation in a driven nanomechan-ical system are twofold: On one hand, these effects in-validate at very large times the correspondence betweenthe full-fledged mechanical equations of motion for thein- and out- of-phase modes and its quantum mechan-ical equivalent for slowly oscillating amplitudes (enve-lope curves as ”wave functions”); on the other hand,the evolution of an ”analogous” two-level system be-comes nonunitary. As a result, the maximum value ofthe averaged probability depends on the number of adi-abatic periods used for computing the average value (seeFig. 2).Specific shape of the fan diagram in Fig. 2 indi-cates the crossover from the regime of slow-passage limit α ω d (cid:46) ∆ (bottom part of the figure on the mainpanel) to the regime of the fast-passage α ω d (cid:38) ∆ (top part of Fig. 2). The interference pattern (mainpanel) demonstrates pronounced arcs similar to Ref.[17]. Decrease of the total probability P av with increas-ing α qualitatively reminds the similar effect in thequantum two-level system associated with the presenceof two typical times scales of the same order of magni-tude responsible for the relaxation and dephasing, see,e.g., [17].The standard St¨uckelberg fan diagram [17] is con-structed assuming mutual independence of parameters α and α . In contrast to it, the RKKY-mediatedlevel crossing imposes certain constraint on α ∼ ˜ α and α ∼ ˜ α , making the line α =0, α (cid:54) =0 inaccessible. TheSt¨uckelberg interference is pronounced inside the cone α <α . This condition is achieved by fine tuning thegate voltage and interdot capacitance controlling U .Finally we comment on relations between two-localized spin configurations (singlet/triplet) deter-mined by initial conditions given by RKKY interac-tion and the resulting St¨uckelberg interference pattern.First, the spin configuration affects only the magnitudeof the coupling constants α , . The expectation valueof (cid:104) S S (cid:105) at zero temperature is equal to − (cid:126) / (cid:126) / { eV, U, U } , the relation α Sing1 , = − α Trip1 , holds. Byconstructing the interference diagram we assumed verylong singlet-triplet relaxation times. Thus, the systembeing prepared in certain (singlet or triplet) two-spininitial configuration is locked in the same state dur-ing the evolution. Furthermore, as one can see fromFig. 2, the initial two-spin configuration uniquely de-fines the St¨uckelberg pattern. Therefore, the classicalSt¨uckelberg interferometry can be used for the identifi-cation of the quantum spin states [53]. V. SUMMARY AND DISCUSSION
In summary, we propose a hybrid system coupling twospin impurities embedded in adjacent NEM beams usingthe RKKY interaction. We showed that a nanodevicebased on two suspended quantum dots nanomachined inthe vicinity of a metallic back gate characterized by aslowly modulated density of charges allows us to controlindependently both local and nonlocal spin correlations.The role of the mechanical system is twofold. On onehand, it provides access to RKKY-mediated dynamics.On the other hand, it provides a very sensitive tool forquantum measurements of nanomechanical back action.The interference between two diabatic states of mechan-ical system can be measured with high accuracy throughSt¨uckelberg oscillations. The mechanical system of twocoupled QD oscillators, while being itself deeply in theclassical regime, mimics the dynamics of a quantum two-level system. The slow varying displacement envelopefunctions play the same role as the two-level system’swave functions [49],[50].We have demonstrated that the interference betweentwo classical modes (in-phase and out-of-phase) of amechanical resonator is sensitive to the quantum spinconfiguration of the double quantum dot. As a result,the St¨uckelberg fan diagram provides very accurate in-formation about the spin-spin correlation function. Inparticular, in the presence of competing interactions,such as, for example, a resonance Kondo scattering,the mechanical back action becomes an important toolfor sensing the Kondo screening. The interference pat- tern, being pronounced for both the singlet and thetriplet two-spin configurations, disappears completelywhen two Kondo clouds are formed in the DQD systemto screen the electron’s spins. Moreover, mechanicalback action can be used to probe the quantum criti-cality associated with an antagonism between magnetic(RKKY) interaction and Kondo scattering.The applications of the mobile DQD NEM systemin addition to sensing the spin-spin correlations func-tion include but are not limited by the following prob-lems, to list a few: Competition between resonanceon-site Kondo scattering and spin-spin correlation out-of-equilibrium, nanomechanically induced singlet-triplettransitions in a double-dot device [54], mechanically in-duced drag, classical vs quantum synchronization, etc.Possible experimental realizations of mechanically tunedRKKY can be engineered with coupled suspended car-bon nanotube/metallic quantum wire resonators or insilicon metal-oxide-semiconductor based junctions, inwhich mechanics is modeled by driving the barrier gateswith an ac voltage [55].
ACKNOWLEDGEMENTS
We are grateful to B. Lorenz and S. Ludwig for fruit-ful discussions on St¨uckelberg interference in systems ofcoupled mechanical resonators, S. Ilani for many valu-able suggestions on possible experimental realizationsof quantum nanodevices, F. Pistolesi for critical com-ments on suspended CNTs, R. Fazio and F. Ludovicofor careful reading of the manuscript and valuable com-ments, Leonid Levitov for drawing our attention to Ref.[49] and R. Shekhter and L. Gorelik for inspiring dis-cussions on RKKY interaction. This work was finalizedat Aspen Center for Physics, which was supported byNational Science Foundation Grant No. PHY-1607611and was partially supported (M.K.) by a grant from theSimons Foundation. [1] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui,Nature (London) , 329 (2004)[2] S. Kolkowitz, A. C. Bleszynski Jayich, Q. P. Unterrei-thmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, M. D.Lukin, Science , 1603 (2012)[3] M. V. Gustafsson1, T. Aref, A. F. Kockum, M. K.Ekstr¨om, G. 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