Entanglement in Spintronic Quantum Transport

Transcription

1 BNL May 2003, Upton, L.I. 1 Entanglement in Spintronic Quantum Transport Branislav K. Nikolić Dept. of Physics and Astronomy, University of Delaware, Newark, DE bnikolic

2 BNL May 2003, Upton, L.I. 2 What is Sp ntron cs?

3 BNL May 2003, Upton, L.I. 3 Generations of Sp ntron cs Devices? Spintronics that works: GMR in FM/Metal/FM (magnetic read heads). Spintronics about to hit the market: TMR in FM/Insulator/FM (nonvolatile MRAM as a universal memory solution which is dense, fast and has unlimited read and write endurance). Spintronics at basic research level: Manipulation and amplification of spin-polarized currents in FM/Semiconductor/FM (spin transistors, Hall effect MRAM). Speculative Spintronics: Spin as a qubit (two level system) in solidstate based quantum computing. Exploitation of, promises novel devices: nonvolatile, faster in data processing, with decreased power consumption, increased integration densities, storage and communication on the same chip.

4 BNL May 2003, Upton, L.I. 4 What is Semiconductor Sp ntron cs? 1990: Datta-Das spin-fet in narrow-gap (InAs) semiconductors. Gate voltage controls the Rashba SO coupling in 2DEG that induces spin precession: k 1 + k 2, θ = 2m αl h 2 Spin-FET vs. charge-fet: Flipping electron spin takes much less energy and can be done much faster than pushing an electron out of the channel. Changing the orientation of the source or drain with a magnetic field would introduce an additional type of control that is not possible with a conventional FET: Logic gates whose functions can be changed on the fly Experimental realization remains illusive : injection, detection, tuning of SO?

5 BNL May 2003, Upton, L.I. 5 Four Noble Truths About Quantum States E. Schrödinger, Naturwissenscahften 23, 807, 823, 844 (1935). Superposition: A quantum state is described by a linear superpositions of the basis states. Interference: The result of measurement depends on the relative phases of the amplitudes in the superpositions. Entanglement: Complete information about the state of the whole system does not imply complete information about its parts: EPR ± = 0 1 ± GHZ ± = ± , Bell ± = 0 0 ± 1 1 2, Also: quantum states in superconductors, strongly correlated systems, at quantum criticality, in quantum computers,... Nonclonability and uncertainty: An unknown quantum state can be neither cloned nor observed without being disturbed No Quantum Copier Machine U QCM : Ψ orig φ 0 Ψ orig Ψ orig.

6 BNL May 2003, Upton, L.I. 6 What is Entanglement (Vesränkung)? From Founding Fathers to Quantum Information Science Einstein-Podolsky-Rosen: An entangled wave function does not describe the physical reality in a complete way. Einstein: Spooky action at a distance. Schrödinger: The best possible knowledge of the whole does not include the best possible knowledge of its parts. J. Bell:... a correlation that is stronger than any classical correlation. A. Peres:... a trick that quantum magicians use to produce phenomena that cannot be imitated by classical magicians. C. Bennet:... a resource that enables quantum teleportation. P. Shor:... a global structure of the wave function that allows for faster algorithms. A. Ekert:... a tool for secure communication. Horodecki:... the need for first application of positive maps in physics.

7 BNL May 2003, Upton, L.I. 7 Pure Quantum States Pure States in the Hilbert space H: Ψ or projector ( pure state density operator ) ˆρ = Ψ Ψ ˆρ 2 =ˆρ Pure Separable States in the composite Hilbert space H A H B : Ψ AB = a A b B Pure Entangled States in H A H B : Ψ AB = d A d B c ij a i A b j B = r wk u k A v k B i=1 j=1 k=1 d A =dim(h A )andd B =dim(h B ). Signature of entanglement: Schmidt rank r min{d A,d B }.

8 BNL May 2003, Upton, L.I. 8 Proper vs Improper Mixtures Proper Mixtures: ˆρ = α p α Ψ α Ψ α ˆρ 2 ˆρ Examples: Thermal Equilibrium States: ˆρ = 1 Z e βĥ The most general state of spin- 1 2 : ˆρ s = 1 2 (1 + p ˆ σ) 0 p 1isspin polarization! Standard Unpolarized current: ˆρ s = = Î s /2. Improper Mixtures Quantum state of entangled subsystem (Landau 1927): ˆρ A =Tr B Ψ AB Ψ or ˆρ B =Tr A Ψ AB Ψ ˆρ = Ψ AB Ψ ˆρ A ˆρ B Correlations: O A =Tr[ˆρÔA] =Tr A [ˆρ A Ô A ] O A O B O A O B If ˆρ A is pure than it must be a factor of the total density operator.

9 BNL May 2003, Upton, L.I. 9 What are the Measures of Entanglement? Von Neumann entropy: S = Tr ˆρ A log 2 ˆρ A = Tr ˆρ B log 2 ˆρ B. Pure states: S = 0 and Schmidt rank r =1. Entangled states: S>0 and Schmidt rank r>1. Given the two states Ψ 1 AB, Ψ 1 AB, which one is more entangled? Under local unitary transformations ÛA ÛB entanglement remains unchanged. Local operations and classical communication (LOCC) cannot increase entanglement. Entanglement of formation (concurrence), distillable entanglement, and relative entropy of entanglement as a measure of nonlocal quantum correlations in mixed bipartite states. For more than two particles there is no complete classification of entanglement.

10 BNL May 2003, Upton, L.I. 10 Spin Decoherence vs Spin Relaxation Bloch B T1 > Sphere > T 2 α + β > > Decoherence entanglement to environment: Ψ = α + e 1 + α e 2 ˆρ s = ( α + 2 α + α e 2 e 1 α α + e 1 e 2 α 2 ) α 2 αβ α β β 2 T 2 α β 2 T 1 ρeq 0 0 ρ eq

11 BNL May 2003, Upton, L.I. 11 What is Mesoscopic Sp ntron cs? 0 λ F l e ξ Lφ L 2 L 1 ballistic diffusive localized L mesoscopic (phase coherent) spin coherent spin polarized Combined Orbital and Spin states reside in H = H o H s. Macroscopics: ˆρ =ˆρ o ˆρ o. Mesoscopics ˆρ = Ψ Ψ ˆρ s + Spintronics ˆρ =ˆρ o σ σ =Mesoscopic Spintronics: ˆρ 2 =ˆρ (e.g., ˆρ = Ψ Ψ σ σ )

12 BNL May 2003, Upton, L.I. 12 Two-probe Device: a Theorist s View Leads inject pure states: Ψ nσ = k y k x σ via Ohmic contacts. t L t C t F z 2DEG F y x FERROMAGNET 2DEG E F Energy Γ

13 BNL May 2003, Upton, L.I. 13 What is Rashba Interaction? All spin-orbit couplings arise from the Dirac equation expanded in v/c : Ĥ so = h/(2m 0 c 2 ) V (ˆ σ ˆp). B y E S Ψ Ε θ k E ± (k x )=E n + h2 2m k 2 x ± αk x x S Ψ Ε+ The confining potential V (z) for a 2D electron gas is asymmetric along the z-axis: E = V. ĤR so = α R / h (ˆ σ ˆp) z = B R (p) ˆ σ, Highest achieved value: α R evm in InAs.

14 BNL May 2003, Upton, L.I. 14 Hamiltonian Approach to Sp ntron cs Ĥ = Ĥo + Ĥo + ( Ĥso Orbital part: Ĥ o = m ε m m m + ) m,n t mn m n Î s hopping: t mn = te 2πiφ mn, m =(m x,m y ), φ mn = φ nm so that the flux through a given loop S is Φ S = m,n S φ mn in units h/e. disorder: ε m [ W/2,W/2], or 1 2W RH <t<1, or φ mn [0, 2π). Spin part (Zeeman term): Ĥ s = Î o µˆ σ B, µ = g µ B /2. SO part: Rashba (t R so = α R/2a) + Dresselhaus (t D so = η/2a) Ĥ so = α R h 2a 2 t (ˆv x ˆσ y ˆv y ˆσ x ) η h 2a 2 t (ˆv x ˆσ x ˆv y ˆσ y ). Eigenstates σ of the spin operator hˆ σ â/2[ˆ σ =(ˆσ x, ˆσ y, ˆσ z )andσ =, ] acting in H s, together with m H o define a basis m σ H.

15 BNL May 2003, Upton, L.I. 15 Landauer Formula for Quantum Sp ntron c Transport Conductance of spin-degenerate electrons at zero temperature: G = 2e2 h Tr t(e F )t (E F ) Zero-temperature Conductance Matrix of spin-polarized electrons: ( ) ( ) ( G = t = 2 G G = e2 t ij, 2 t ij, 2 G h t ij, 2 t ij, 2 ij Im ˆΣ L Îs Ĝr 1N Im ˆΣ R Îs G = e2 h n T n T n T n T n ) Example: t ij, = t 2(i 1)+1,2(j 1)+1, i,j =1,..., M. Spin-resolved experiments measure: s i G s c Real-space spin-space Green functions Ĝ r nm,σσ = n,σ Ĝ r m,σ : Ĝ r =[EÎ o Î s Ĥ ˆΣ r Î s ] 1.

16 BNL May 2003, Upton, L.I. 16 What is Quantum Transmissivity? Landauer formula recast: G = 2e2 h M n=1 T n(e F ). Distribution function P(T) (a) N= (b) 10 1 N= N= Transmission T Linear statistics A = M n=1 a(t n): G 1 P (T ) T ; Shot noise S(0) P (T ) T (1 T ); Proximity conductance G NS 1 P (T ) T/(2 T )2 0

17 BNL May 2003, Upton, L.I. 17 Full Transmissivity

18 BNL May 2003, Upton, L.I. 18 Partial Transmissivity

19 BNL May 2003, Upton, L.I. 19 Ballistic Transmissivity

20 BNL May 2003, Upton, L.I. 20 Entanglement in the Scattering States Transmission matrix t encodes entanglement of spin and orbital states! in p, σ out = p σ t p p,σ σ p σ To each of the outgoing scattering states assign a density operator: ˆρ pσ out = p p σ σ t pp,σσ t pp,σσ p p σ σ Partial tracing leads to reduced density operator for spin subsystem: ˆρ s pσ out = qσ σ t pq,σσ t pq,σσ σ σ No need for phenomenological master equation approximation to ˆρ s =Tr o [Û(t, 0)ˆρ(0)Û (t, 0)] Measurable properties of the improper mixture spin quantum state: p =Tr[ˆρ sˆ σ] S(ˆρ s )= 1 2 (1 + p )log 2[ 1 2 (1 + p )] 1 2 (1 p )log 2[ 1 2 (1 p )]

21 BNL May 2003, Upton, L.I. 21 Entanglement in 2-channel SFET: Disorder Disorder (l =30a/W 2 )

22 BNL May 2003, Upton, L.I channel SFET Again: Tunnel Barrier Interface Scattering on the tunnel barrier at the Fe-N contact

23 BNL May 2003, Upton, L.I. 23 Entanglement in Multichannel Spin-FET Increasing disorder impedes the Dyakonov-Perel decoherence mechanism!

24 BNL May 2003, Upton, L.I. 24 Entanglement Due to Rashba Velocity Mismatch Von Neumann Entropy vs Spin Polarization as a measure of spin entanglement in a clean system with no tunnel barriers!

25 BNL May 2003, Upton, L.I. 25 C O N C L U S I O N S Future Directions The entanglement of spin and orbital degrees of freedom in spintronic devices, due to any kind of scattering + spin-orbit interaction has identical structure to the entanglement of two particles (bipartite systems) studied in Quantum Information Science. The spin-polarization has to be extracted from the improper mixture the strict description of the quantum state of a spin subsystem. Quantum dynamics of improper mixture can be formulated exactly via quantum transport scattering techniques without resorting to phenomenological master equations. The key issue for spin-fet operation in quasi-1d semiconducting structures is to devise a scheme utilizing mixed instead of pure quantum states of spin. FUTURE: What are the mesoscopic fluctuations of entanglement?