Mechanically Assisted Single-Electronics

Transcription

1 * Mechanically Assisted Single-Electronics Robert Shekhter Göteborg University, Sweden Nanoelectromechanics of CB structures--classical approach: Coulomb blockade of single-electron tunneling (SET) Nanoelectromechanical coupling in SET devices Shuttling of single electrons Shuttling of magnetization and Cooper pairs Quantum nanoelectromechanics: Quantum theory of a shuttle instability Spin-dependent tunneling in CB structures; shuttling of a spin- -polarized electrons Interferrometry of a quantum vibrations

2 The Electronic Shuttle

3 Quantum Nanoelectromechanics of Shuttle Systems δxδp h δx X h Mω R( X + δx ) If >> 1 then quantum fluctuations of the R( X ) grain significantly affect nanoelectromechanics.

4 Conditions for Quantum Shuttling x ~ > 1 λ 1. Fullerene based SET ω λ Tunneling 1 THz x 1 1 λ length Quasiclassical shuttle vibrations..35. Suspended CNT STM ω d 14 1 Hz 1 nm d L X / λ L.5 ω Hz for SWNT with L 1 μm L [μm]

5 Quantum Shuttle Instability e h ω ev Quantum vibrations, generated by tunneling electrons, remain undamped and accumulate in a coherent condensate of phonons, which is classical shuttle oscillations. h ω γ γ < thr ee d = k d Γ λ Shift in oscillator position caused by charging it by a single electron charge Phase space trajectory of shuttling. From Ref. (4) (1)D.Fedorets et al., PRL, 95, 573-1, 5 () D. Fedorets et al. PRL, 9, (4) (3) D. Fedorets, PRB 68, 3316 (3) (4) T. Novotny et al. PRL, (3)

6 Formulation of the Problem H = H + H + H + H + H ( t) Tr R( t) R1( t) () t R1( t) R11( t) H = ε a a, H = T ( x) a c + c a Leads αk αk αk T α αk α α αk α, k α, k h = ( ε ) tσ ( t) = i[ H, σ ( t+ )] H xd cc cc Ucccc cc cc 1 Dot p Leads T Dot bath bath osc + x leadsσ

7 Half-Metallic Conductors Ferromagnetic Metal μh <<E F Half-Metallic Conductors Perovskite Manganese Oxides E F K μh >E F States with minority spin are empty Metal w.r.t. to majority spin electrons; Semiconductor w.r.t. minority spin electrons

8 Spintronics of Nanomechanical Shuttle V/ x +V/ B E Dot A. Pasupathy et al., Science 36,86 (4) B: magnetic field E: electric field D.Fedorets et al., PRL, 95, 573-1, 5

9 Quantum Shuttle Dynamics t σ Time evolution in Schrödinger picture: ( t) = i[ H, σ( t)] Total density operator () leads ( ) t Tr σ t Reduced density operator

10 Density matrix for the spin-polarized shuttle + c + c Four basic vectors for the electronic space = + + c c = Density matrix = 1 = =

11 Spin-vibrational Dynamics Λ Γ + = +Λ Γ = +Λ Γ Γ + Γ Γ + = +Λ Γ + = +Λ Γ Γ + Γ = +Λ Γ Γ + Γ + = γ γ γ γ γ γ () 1 ) ( ], [ () 1 ) ( ], [ () () () () ) ( ], [ } (), { 1 ) ( ], [ () () } (), { 1 ], [ () () } (), { 1 ], [ 1 x ih H i x ih H i x x x x ih H i x ih H i x x x eex H i x x x eex H i v t v t R R L L v t v t L L L v t R R L v t { } [ ],,,, i x p x x γ γ γ Λ

12 Shuttle instability After linearization in x one finds: x& = p d p& = x γ p n x Γ & = + h Γx hλ n n x 1 [ ] xt () x Tr () tx x p() t Tr () t p h nt () = Tr () tn [ ] [ ] Result: an initial deviation from the equilibrium position grows exponentially if the dissipation is small enough: x( t) exp( αt); α = ( γ thr γ)/ ; γ < γ thr Γ h d λ

13 Phase Diagram of Shuttle Vibrations

14 Electromechanical Coupling in Suspended CNT B. J. LeRoy et al., Nature 43, 371 (4) L.M. Jonsson et al. Nano Lett. 5, (5). A.Zazunov, D.Feinberg, T.Martin, Phys.Rev.B 73, (6)

15 How to Detect the Quantum Nanomechanical Vibrations? New principles should be implemented for sensing the quantum displacements

16 Interferometry of quantum vibrations R.Shekhter et al., PRL, 6, (in press)

17 Electronic Transport Through Vibrating CNT

18 Quantum Nano-mechanical Interferometer Classical interferometer Quantum nanomechanical interferometer

19 Classical and Quantum Vibrations

20 Model + σ e m σ Hσ = εαaα, σaα, σ σ= LR, σ= LR, α H = H + H + H + T h 3 + ie + He = d r{ ψ () r A ψ() r + U( y u(,) xz ) ψ () r ψ()} r m r ch Hm = dx x + EI x L 1 ux ( ) () L π

21 Renormalization of Electronic Tunneling H = is is e He = () () + ( ( )) () () y hc x 3 + ψ () r eh + S i d r u xψ r i dxu x ψ r ψ r L ± eh TL TL exp i dxu( x) R R hc

22 Coupling to the Fundamental Bending Mode Only one vibration mode is taken into account h βei ux ( ) = Yu( x)( b + b) ; Y= L CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other

23 Tunneling Through Virtual Electronic States on CNT Effective Hamiltolian Strong longitudinal quantization of electrons on CNT No resonance tunneling though the quantized levels (only virtual localization of electrons on CNT is possible) hω H = a a + b b + ( T e a a + hc. ) ( iφ b + b ) + εα α, σ α, σ eff α, R α, L ασ, αα, Φ = ghly / Φ Y = h Mω L T eff = TT L * R E μ

24 Calculation of the Electric Current n= l= n + iφ ( b + b) ( ) ( ) h h I= G Pn () ne n+ l dε fl()1 ε fr( ε l ω) fr()1 ε fl( ε l ω) ( + i b b) i ( b+ Φ + Φ + b) G = Pn ( ) n e n + Pn ( ) n e n+ l G n= n= l= 1 lhω kt exp( lhω ) 1 kt

25 Linear Conductance Vibrational system is in equilibrium G G Φ hω exp, >> 1 Φ kt G 1 Φ hω hω 1, << 1 G 6 Φ kt kt Φ= 4 π gly H, Φ = hc e For a 1 μm long SWNT at T = 3 mk and H - 4 T a relative conductance change is of about 1-3%, which corresponds to a magneto-current of.1-.3 pa.

26 Quantum Nano-mechanical Magneto-resistor

27 Transfer of Electric Charge by Swinging Electrons

28 Conclusions NEM coupling may result in electro-mechanical instability and quantum shuttling of electric charge Shuttling of spin-polarized electrons offers spintronic manipulations of mechanical nanovibrations Interferometry of quantum nanovibrations is possible

29

30 Shot noise spectroscopy of the electronic spin-flip

31 Spin Blockade and Coulomb Blockade of a Single-electronic Shuttling V/ x +V/ B E Dot B: magnetic field E: electric field Flipping of the electronic spin blocks tunneling of an electron out of dot (spin blockade) Then tunneling of electrons into the dot Is also blocked due to Coulomb blockade Current flow is totally controlled by Electronic spin-flip relaxation As a result the low frequency noise is Largely enhanced

32 Shot Noise and Spin-Flip Spectroscopy a) mn n m H b) I(t) t Due to the effect of spin-dependent tunneling, the current is controlled by electronic spin flip. Giant shot noise is expected to be a spectroscopic tool for study of electronic spin-flips

33 Formulation of the Electronic Transport Problem At high enough voltages transport can be vied as incoherent events of the electronic tunneling in and out of moving dot The probability to find a non-occupied electronic levels as well as ones to find the spin-up or spin-down levels to be occupied are governed by rate question = ( ; ; ) T = Lt ( ) t ΓL() t ΓR() t Lt ( ) = ΓL () t ΓR () t γ γ γ γ

34 Electronic Tunneling Events (S.Hershfield et al. Phys.Rev.B47, 1967,1993.) A probability for an electronic (dot right electrode) tunneling event is given by P() t Δ t =Γ () t () t Δt J() t () t Δt A probability to have two tunneling events at times t and t R P( t, t) ΔtΔ t = J( t ) g(, t, t) J( t) ( t) ΔtΔt A conditional probability g to have dot in a state spin-up at time t if at time t it was unoccupied is expressed in terms of evolution operator U(t,t) of the rate equation: g(, t, t) U ( t, t) t (, ) exp Utt = T dτ L( τ ) t

35 Shot Noise Sn ( n) dd τ τ SnT ( + τ, nt+ τ) T Introducing an averaging over periods function S(n -n) St t = e Pt t Pt Pt + e t t Pt (,) (,) () () δ( ) () ( ) One gets a noise spectral function S(ω) iωn S( ω ) = e S( n) n

36 Results of Exact Calculations Electric current: J = sh( Γ ) ( Γ ) + ( Γ ) e T 3ch sh thγ Fano factor of the shot noise: F( ω) S( ω) e J = 4 C + C Sin ω + Sin ω 1 1 ( ) ( ) 4 B + BSin ω + Sin ω ( ) ( )

37 [ ( ) ( ) ] B = shγ sh Γ ch Γ + γ + shγ chγ 1 B = ch( Γ + γ) ch( Γ γ) chγ + sh γ 1 ( ) ( ) ( ) 3 C 1 = shγ sh Γ ch γsh Γ + γ + shγch Γ + shγch γ ( ) γ ( ) ( γ ) C = sh Γ + sh ch Γ sh Γ + Γ Γτ; γ γt T- period of shuttling τ - contact time

38 Giant Superpoissonic Noise At γ<<γ the low frequency noise reaches the gigantic value F( ω) F ( ) ( δω γ) = F F nt Γ γ 1 () = ( ω = π ) = 1 Spectroscopy of Electronic Spin-flip

39 Linear Conductance Vibrational system is in equilibrium G G Φ hω exp, >> 1 Φ kt G 1 Φ hω hω 1, << 1 G 3 Φ kt kt c Φ= glyh, Φ = h e ΔG At L= k = T= mk and H= Tesla we estimate G μ ; ω 1 sec ; 1 5 5%

40 Conclusions Electronic and mechanical degrees of freedom of nanometer-scale structures can be coupled. Mechanically assisted transport penomena. Sensing of quantum NEM performance.

41 Shuttling of Spin-Polarized Electrons Mechanism: Quantum transmission of electrons through single energy level on vibrating dot. H - - ω; Ω=μH/ђ The quantum evolution can be viewed as a sequence of two alternating events: (i) Tunneling: Transfer of electron between left or right lead and dot (time for tunneling τ π/ω) (i) Free evolution: On-site electronic state is disconnected from the leads. In this stage the spin of the electron precesses in the external magnetic field H. Natural scale for H: ΔH ђω/μ; (Ω/ω=1/). But this is not the only scale here! L.Y.Gorelik et al., Phys.Rev.,B71, 3537 (5)

42 Spectroscopy of nanomechanical vibrations I ΔH δh1 δh 1 ΔH H δh 1 ΔH = Γ / μ = hω / μ Study of magnetoresistance makes it possible DC spectroscopy of nanomechanical vibrations Giant magnetoresistance: δh 1 Oe for Gohm tunnel resistance

43 Shot Noise Spin-Flip Spectroscopy a) b) mn I(t) n H m t ( Ie) Γ+Γ ΓΓ ΓΓ F ΓΓ Γ H + + d Γ +Γ Γ H + + d Γn p n n p n p p n p γ n p H = ( μh) ( Δ ) + ( Γp ) d ( ) p γ Γ Γ α β αβ, n p = Γ +Γ Due to the giant effect of spin-dependent tunneling, the current is controlled by electronic spin flip. Giant shot noise is expected to be a spectroscopic tool for study of electronic spin-flips

44 Part Quantum Nanoelectromechanics