* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Slide 1
Topological quantum field theory wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Bell test experiments wikipedia , lookup
Electron configuration wikipedia , lookup
Renormalization wikipedia , lookup
Matter wave wikipedia , lookup
Atomic orbital wikipedia , lookup
Quantum dot cellular automaton wikipedia , lookup
Renormalization group wikipedia , lookup
Basil Hiley wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Scalar field theory wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum decoherence wikipedia , lookup
Double-slit experiment wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Wave–particle duality wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Path integral formulation wikipedia , lookup
Probability amplitude wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Density matrix wikipedia , lookup
Quantum field theory wikipedia , lookup
Particle in a box wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Bell's theorem wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum dot wikipedia , lookup
Coherent states wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum fiction wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum computing wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum teleportation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum group wikipedia , lookup
Quantum state wikipedia , lookup
Quantum cognition wikipedia , lookup
Five-Lecture Course on the Basic Physics of Nanoelectromechanical Devices • Lecture 1: Introduction to nanoelectromechanical systems (NEMS) • Lecture 2: Electronics and mechanics on the nanometer scale • Lecture 3: Mechanically assisted single-electronics • Lecture 4: Quantum nano-electro-mechanics • Lecture 5: Superconducting NEM devices Lecture 4: Quantum NanoElectromechanics Outline • Quantum coherence of electrons • Quantum coherence of mechanical displacements • Quantum nanomechanical operations: a) shuttling of single electrons; b) mechanically induced quantum interferrence of electrons Lecture 4: Quantum Nano-Electromechanics 3/29 Quantum Coherence of Electrons p x r, p Classical approach 2 Heisenberg principle in quantum approach Formalization of Heisenberg’s principle: Â operators for physical variables eigenfunctions – quantum states p (r ) Ce i pr (r ) a p p (r ) p quantum state with definite momentum In this state the momentum experiences quantum fluctuations Lecture 4: Quantum Nano-Electromechanics 4/29 Stationary Quantum States Hamiltonian of a single electron: Stationary quantum states: 2 ˆ p Hˆ U (r ); 2m Ĥ E pˆ i r Lecture 4: Quantum Nano-Electromechanics 5/29 Second Quantization • Spatial quantization discrete quantum numbers • Due to quantum tunneling the number of electrons in the body experiences quantum fluctuations and is not an integer • One therefore needs a description that treats the particle number N as a quantum variable Wave function for system of N electrons: N (r ) n Creation and annihilation operators aˆ N (rn ) N 1 (rn ) Nˆ nˆ a a a , a , fermions aˆ N (rn ) N 1 (rn ) Hˆ nˆ a , a , bosons Lecture 4: Quantum Nano-Electromechanics 6/29 Field Operators ˆ (r ) a (r ); nˆ (r ) ˆ (r )ˆ (r ) Hˆ drˆ (r ) Hˆ (r )ˆ (r ) ˆ (r1 ) ,ˆ (r2 ) (r1 r2 ) Density Matrix A Sp{ˆ Aˆ }; ˆ i [ Hˆ , ˆ ] t Louville – von Neumann equation Lecture 4: Quantum Nano-Electromechanics 7/29 Zero-Point Oscillations Consider a classical particle which oscillates in a quadratic potential well. Its equilibrium position, X=0, corresponds to the potential minimum E=min{U(x)}. A quantum particle can not be localized in space. Some “residual oscillations" are left even in the ground states. Such oscillations are called zero point oscillations. Classical motion: d 2x U k m 2 kx dt x m Quantum motion: 1 U ( x) U 0 kx 2 2 p Amplitude of zero-point oscillations: x0 x E ( x) 2 2mx 2 kx 2 2 E( x0 ) min E( x) m Classical vs quantum description: the choice is determined by the parameter x0 d where d is a typical length scale for the problem. “Quantum” when x0 d 1 Lecture 4: Quantum Nano-Electromechanics 8/29 Quantum Nanoelectromechanics of Shuttle Systems x p 2 x 2 x0 M If R( x x) R( x) R( x) then quantum fluctuations of the grain significantly affect nanoelectromechanics. Lecture 4: Quantum Nano-Electromechanics 9/29 Conditions for Quantum Shuttling 2 x0 x0 1. Tunneling length 1 2M Fullerene based SET X0 0.1 Quasiclassical shuttle vibrations. 1 THz 2. Suspended CNT STM L 10 14 d Hz L 2 d 1 nm 108 - 109 Hz for SWNT with L 1μm 9 Lecture 4: Quantum Nano-Electromechanics 10/29 Quantum Harmonic Oscillator pˆ 2 1 H M 2 xˆ 2 2M 2 1 d pˆ i dx 1 En n 2 1 M x2 M 1 2 M 4 | n n H n x exp 2 2 n! Ledder operators M i xˆ 2 M a a pˆ M i ˆ x 2 M pˆ a | n n | n 1 a | n n 1 | n 1 a a 2M xˆ pˆ i H a a 1/ 2 a 2M a [a, a ] 1 Probability densities |ψn(x)|2 for the boundeigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x,and brighter colors represent higher probability densities. Eigenstate of oscillator is a ideal gas of elementary excitations – vibrons, which are a bose particles. Lecture 4: Quantum Nano-Electromechanics 11/29 Quantum Shuttle Instability Quantum vibrations, generated by tunneling electrons, remain undamped and accumulate in a coherent “condensate” of phonons, which is classical shuttle oscillations. e eV thr eE d 2k Shift in oscillator position caused by charging it by a single electron charge. d Phase space trajectory of shuttling. From Ref. (3) References: (1) D. Fedorets et al. Phys. Rev. Lett. 92, 166801 (2004) (2) D. Fedorets, Phys. Rev. B 68, 033106 (2003) (3) T. Novotny et al. Phys. Rev. Lett. 90 256801 (2003) Lecture 4: Quantum Nano-Electromechanics 12/29 Hamiltonian Hˆ Hˆ leads Hˆ dot Hˆ oscillator Hˆ tunneling Hˆ leads k ak a k ,k Hˆ dot q cq cq Ec Nˆ 2 e ( xˆ ) Nˆ ,k Hˆ oscillator Hˆ tunneling pˆ 2 1 M 2 xˆ 2 2M 2 T kq ( xˆ ) ak cq h.c. ,k ,q Nˆ cq cq q Lecture 4: Quantum Nano-Electromechanics 13/29 Theory of Quantum Shuttle x The Hamiltonian: Hˆ Hˆ leads Hˆ dot Hˆ oscillator Hˆ tunneling Hˆ leads k ak a k ,k Dot Lead Lead Time evolution in Schrödinger picture: tˆ (t ) i[ H , ˆ (t )] Total density operator Reduced density operator Hˆ dot q cq cq Ec Nˆ 2 e ( xˆ ) Nˆ ,k Hˆ oscillator Hˆ tunneling pˆ 2 1 M 2 xˆ 2 2M 2 T kq ( xˆ )ak cq H .c. L , R ;k , q TL , R ( xˆ ) T0 e xˆ / 0 ˆ 0 (t ) ˆ (t ) Trleadsˆ t ˆ 0 ( t ) 1 Lecture 4: Quantum Nano-Electromechanics 14/29 Generalized Master Equation ˆ 0 : density matrix operator of the uncharged shuttle ˆ1 : density matrix operator of the charged shuttle At large voltages the equations for ˆ 0 , ˆ1 are local in time: t ˆ 0 i[ H osc e ( xˆ ), ˆ 0 ] { L ( xˆ ), ˆ 0 } R ( xˆ ) ˆ1 R ( xˆ ) L ˆ 0 Free oscillator dynamics Electron tunnelling Dissipation t ˆ1 i[ H osc eExˆ, ˆ1 ] { R ( xˆ ), ˆ1} L ( xˆ ) 0 L ( xˆ ) L ˆ1 L ˆ i xˆ, pˆ , ˆ xˆ, xˆ, ˆ 2 2 Approximation: x0 1, ˆ ˆ 0 ˆ1 : ˆ ˆ 0 ˆ1 : e '( x) M 2 1, Lecture 4: Quantum Nano-Electromechanics 15/29 Shuttle Instability After linearisation in x (using the small parameter xo/l) one finds: x p/M x(t ) Tr ˆ (t ) xˆ p M 2 x p '( x 0)q p(t ) Tr ˆ (t ) pˆ q q 2e x q(t ) 1 Tr ˆ1 (t ) Result: an initial deviation from the equilibrium position grows exponentially if the dissipation is small enough: x(t ) exp( t ); ( thr ) / 2; thr e '( x 0) 2M 2 Lecture 4: Quantum Nano-Electromechanics 16/29 Quantum Shuttle Instability Important conclusion: An electromechanical instability is possible even if the initial displacement of the shuttle is smaller than its de Broglie wavelength and quantum fluctuations of the shuttle position can not be neglected. In this situation one speaks of a quantum shuttle instability. Now once an instability occurs, how can one distinguish between quantum shuttle vibrations and classical shuttle vibrations? More generally: How can one detect quantum mechanical displacements? This question is important since the detection sensitivity of modern devices is approaching the quantum limit, where classical nanoelectromechanics is not valid and the principle for detection should be changed. Lecture 4: Quantum Nano-Electromechanics 17/29 How to Detect Nanomechanical Vibrations in the Quantum Limit? Try new principles for sensing the quantum displacements! Consider the transport of electrons through a suspended, vibrating carbon nanotube beam in a transverse magnetic field H. What will the effect of H be on the conductance? Shekhter R.I. et al. PRL 97(15): Art.No.156801 (2006). Lecture 4: Quantum Nano-Electromechanics 18/29 Aharonov-Bohm Effect The particle wave, incidenting the device from the left splits at the left end of the device. In accordance with the superposition principle the wave function at the right end will be given by: (b) 1 (b) 2 (b) A0 (b)exp i1 (1 exp i(2 1) (b) i (b) A0 (b) exp{i i } i i i ( x) 2 1 2 1 ; 0 2 dl A L ds H ; 0 S e c dl A Li 2 c (flux quantum) e The probability for the particle transition through the device is given by: 2 2 2 2 | T | | A0 | 1 cos 2 sin 2 0 0 Lecture 4: Quantum Nano-Electromechanics 19/29 Classical and Quantum Vibrations In the classical regime the SWNT fluctuations u(x,t) follow well defined trajectories. In the quantum regime the SWNT zero-point fluctuations (not drawn to scale) smear out the position of the tube. Lecture 4: Quantum Nano-Electromechanics Quantum Nanomechanical Interferometer Classical interferometer Quantum nanomechanical interferometer 20/29 Electronic Propagation Through Swinging Polaronic States 21 Lecture 4: Quantum Nano-Electromechanics 22/29 Model H L, R H H e H m ie 3 H e d r { (r ) 2m r c 2 T L,R H a , a , 2 A (r ) U y u ( x, z ) (r ) (r )} 2 2 1 EI u ( x ) 2 2 H m L dx ˆ 2 2 2 x 2 L (r ) aqq (r ) q ˆ i d du 22 Lecture 4: Quantum Nano-Electromechanics 23/29 Coupling to the Fundamental Bending Mode Only one vibration mode is taken into account uˆ ( x) x0u0 ( x) (bˆ bˆ) 2 ; x0 2 2 L 0 EI 1 4 CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other. Lecture 4: Quantum Nano-Electromechanics 24/29 Tunneling through Virtual Electronic States on CNT • • Strong longitudinal quantization of electrons on CNT No resonance tunneling though the quantized levels (only virtual localization of electrons on CNT is possible) Effective Hamiltonian H a , a , , gHLY0 / 20 ˆ ˆ i ( bˆ b b (Teff e bˆ ) 2 Y0 2M 0 L a ,R a ,L h.c) , Teff TLTR* E Lecture 4: Quantum Nano-Electromechanics 25/29 Calculation of the Electrical Current I G0 P (n) | n |e i ( bˆ bˆ ) | n l |2 n 0 l n d f ( ) 1 f ( l ) f ( ) 1 f ( l l G r G0 r P ( n ) | n | e i ( b ˆ bˆ ) l ) | n |2 n 0 P(n) | n | e n 0 l 1 i ( bˆ bˆ ) | n l |2 2l kT exp(l ) 1 kT Lecture 4: Quantum Nano-Electromechanics 26/29 Vibron-Assisted Tunneling through Suspended Nanowire • Tunneling through vibrating nanowire is performed in both elastic and inelastic channels. • Due to Pauli-principle, some of the inelastic channels are excluded. • Resonant tunneling at small energies of electrons is reduced. • Current reduction becomes independent of both temperature and bias voltage. Lecture 4: Quantum Nano-Electromechanics 27/29 Linear Conductance Vibrational system is in equilibrium 2 G exp , G0 0 2 G 1 1 , G0 6 0 kT 4 gLx0 H , kT kT 1 1 0 h e For a 1 μm long SWNT at T = 30 mK and H ≈ 20 - 40 T a relative conductance change is of about 1-3%, which corresponds to a magnetocurrent of 0.1-0.3 pA. Lecture 4: Quantum Nano-Electromechanics Quantum Nanomechanical Magnetoresistor R.I. Shekhter et al., PRL 97, 156801 (2006) 28/29 Lecture 4: Quantum Nano-Electromechanics 29/29 Magnetic Field Dependent Offset Current I I 0 (V ) I ; I I 0 (V ) I I 0 eV 0 I 2 eV 0 2 eV0 0 V0 V 2 30 Lecture 4: Quantum Nano-electromechanics Quantum Suppression of Electronic Backscattering in a 1-D Channel G.Sonne, L.Y.Gorelik, R.I.Shekhter, M.Jonson, Europhys.Lett. (2008), (in press); Speaker: Professor Robert Shekhter, Gothenburg University 2009 31/41 31 Lecture 4: Quantum Nano-electromechanics CNT with an Enclosed Fullerene Shallow harmonic potential for the fullerene position vibrations along CNT Strong quantum fluctuations of the fullerene position, comparable to Fermi wave vector of electrons. x0 2M Speaker: Professor Robert Shekhter, Gothenburg University 2009 32/41 32 Lecture 4: Quantum Nano-electromechanics Electronic Transport through 1-D Channel • • • Weak interaction with the impurity Only a small fraction of electrons scatter back. Elastic and inelastic backscattering channels I. Left y Right No backscattering Left and right side reservoirs inject electrons with energy distribution according to Hibbs with different chemical potentials Left V I0 ; RQ 4e R G0 h 1 Q 2 Right py Speaker: Professor Robert Shekhter, Gothenburg University 2009 33/41 33 Lecture 4: Quantum Nano-electromechanics II. Backflow Current I I 0 I back I back dN e dt back r dN f Right ( p ) 1 f Left (q ) P(n) dt back h p ,q n ,l n Vback n l 2 Backscattering potential R† ( y ) a †p e ipy p L† ( y ) aq†e iqy Hˆ back r R† ( y ) L† ( y ) ( y X ) h.c. 2 ikF Xˆ † ˆ H back r a p aq e h.c. p ,q q Vback re 2 ikF Xˆ h.c. Speaker: Professor Robert Shekhter, Gothenburg University 2009 34/41 34 Lecture 4: Quantum Nano-electromechanics Backscattering of Electrons due to the Presence of Fullerene n The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target. non movable Wback Wn Wback n However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions . Speaker: Professor Robert Shekhter, Gothenburg University 2009 35/41 35 Lecture 4: Quantum Nano-electromechanics Pauli Restrictions on Allowed Transitions through Oscillator Bias potential across tube selects allowed transitions through oscillator as fermionic nature of electrons has to be considered. × Speaker: Professor Robert Shekhter, Gothenburg University 2009 36/41 36 Lecture 4: Quantum Nano-electromechanics Excess Current I excess I G0V (1 r ) G0 r lim eV e αħω=4εFm/M, excess current independent of confining potential, voltage and temperature. Scales as ratio of masses in the system. Speaker: Professor Robert Shekhter, Gothenburg University 2009 37/41 37 Model H L, R H H e H m T L,R ie 3 H e d r { (r ) 2m r c 2 Hm L 2 L 2 H a , a , 2 A (r ) U y u ( x, z ) (r ) (r )} 1 2 2 u ( x) dx ( x) EI 2 2 x 38 Backscattering of Electrons on Vibrating Nanowire. n The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target. W W W back n non movable back n However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions . 39 Pauli Restrictions on Allowed Transitions Through Vibrating Nanowire The applied bias voltage selects the allowed inelastic transitions through vibrating nanowire as fermionic nature of electrons has to be considered. × 40 4-5 41 4-6 42 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University 2009 43/41 43 Lecture 4: Quantum Nano-electromechanics General conclusion from the course: Mesoscopic effects in electronic system and quantum dynamics of mechanical displacements qualitatively modify principles of NEM operations bringing new functionality which is now determined by quantum mechnaical phase and discrete charges of electrons. Speaker: Professor Robert Shekhter, Gothenburg University 2009 44/41 44 Lecture 4: Quantum Nano-Electromechanics 45/29 Magnetic Field Dependent Tunneling iSˆ ˆ iSˆ e He In order to proceed it is convenient to make the unitary transformation x ˆ ( r ) eH 3 ˆ S i d r uˆ ( x)ˆ (r ) i dxuˆ ( x) ˆ (r )ˆ (r ) y c 0 2 2 Hˆ el dxˆ ( x) ˆ ( x) 2 2m x 2 x 2 1 ˆ eH EI u ( x ) 2 Hˆ mech dx [ˆ ( x) dxˆ ( x)ˆ ( x)] 2 2 c 2 x 0 eH Tˆl ( H ) T exp i r c L 2 0 dxuˆ ( x) Tˆl (0) h.c. r