Quantum Transport in Semiconductor Spintronics

Transcription

1 Quantum Transport in Semiconductor Spintronics Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A.

2 Moore s Law Forever? The observation made in 1965 by Gordon Moore, co-founder of Intel, that the number of transistors per square inch on integrated circuits had doubled every year since the integrated circuit was invented. Moore predicted that this trend would continue for the foreseeable future. In subsequent years, the pace slowed down a bit, but data density has doubled approximately every 18 months, and this is the current definition of Moore's Law, which Moore himself has blessed. Most experts, including Moore himself, expect Moore's Law to hold for at least another two decades.

3 Semiconductors Source Gate Drain Mesoscopic Dimensional Physics L< Reduction, Quantum Hall Size Effects, reduction L< B Conductance Quantization L ϕ Semiconductor Semiconductors: Low carrier density (p, n) Heterostructures Electrical Control of p,n Volatile information Single electron transistor

4 Metallic Ferromagnets Size reduction New Physics GMR L<L sr Superparamagnetism Ferromagnetic metals: Collective Coordinate Permanent information High carrier density Heterostructures Magnetic Control of current Single atom magnet

5 What is Metallic Sp ntron cs?

6 GMR Explained: Spin-Dependent Interface Scattering

7 What is next generation Sp ntron cs? MRAM Spin Qubits Spin Transistors Spintronics promises devices that are: nonvolatile, faster in data processing, with decreased power consumption, increased integration densities, storage and information processing on the same chip.

8 What is Semiconductor Sp ntron cs? Source Gate Drain Das and Datta spin-fet Semiconductor SPINTRONICS: (APL 1990) Merger of semiconductor based and ferromagnet InAs based information technologies High Low Resistance

9 What are the Challenges in Semiconductor Sp ntron cs? Spin injection, Local coherent spin control, Spin Transport Charge is conserved, but spin is not! Gate voltage tunes the Rashba SO coupling in DEG which controls spin orientation by inducing spin precession. Spin dependent transmission controls resistance: Probabilty ( ik x L ik x L x x ) ( ik e e e L e ik L ) Ψ = + + ( ) x x Ψ = 4 cos k k = x m* α ħ R k k L

10 Spin-FET vs. Charge-FET Flipping of electron spin takes much less energy and can be done faster than pushing an electron out of the channel. Changing the orientation of the source and drain electrode magnetization via magnetic field: logic gates whose functionality can be changed on the fly. Experimental realization remains illusive: injection, detection, tuning of SO coupling, ballistic charge transport,

11 Basic Problems of Semiconductor Spintronics Device concepts Interface Physics (spin injection) Spin dynamics, spin decoherence Optical Manipulation New Materials (DMS)

12 Diluted (Para) Magnetic Semiconductors II III IV V VI B C N O E F E G Al Si P S Mn Zn Ga Ge As Se (II,Mn)-VI (Zn,Mn)-Se (Zn,Mn)-S (Cd,Mn)-Te Cd Hg In Sn Sb Te

13 Diluted (Ferro) Magnetic Semiconductors III IV V VI B C N O E F E F E G II Al Si P S III-V (III,Mn)-V Ga-As (Ga, Mn)-As In-As (In, Mn)-As Ga-Sb (Ga, Mn)-Sb Mn Zn Cd Hg Ga In Ge Sn As Sb Se Te

14 Chemistry of II-VI:Mn and III-V:Mn Electronic configuration of Mn: 4s 3d 5 4p 0 Electronic configuration of Ga (III): 4s 3d 10 4p 1 Electronic configuration of Cd (II): 4s 3d 10 4p 0 Mn in III-V: gives magnetic moment and holes Mn in II-VI: gives magnetic moment

15 Why DMS? Spin Injection Heterostructures III-V + (III,Mn)-V Magnetic control of transport Electric control of Magnetism BUT: Working at low temperature Small effects Curie Temperature < 150 Kelvin Y. OHNO et al., Nature 40, 790 (1999) Tanaka, H. Ohno, Higo, Nature PRL 001 (000) Magnetic Light emitting Electric All semiconductor diode Control of Magnetic Spin Ferromagnetism injection Tunnel Junction Large Compatible First electrically TMR (at 4 with Kelvin) GaAs tunable ferromagnet Reversible change of T c

16 History of magnetic semiconductors Material Year T c Transport EuO, EuS <65 K Insulating (II,Mn)VI Semic. PbSnMnTe CdTeMn:N <1.5 K P-type semiconduc. ZnMnTe:X InMnAs K Ins. (Ga,Mn)As K Semic, bad metal (Ga,Sb)Mn, GaP:Mn, GaN:Mn, ZnCrTe (DARPA time) 900 K??

17 Magnetic Semiconductors Summary types (ferro and para) Compatible with semiconductor technology Main Issue: Increase T c

18 Evading Ferromagnets: Mesoscopic Intrinsic Spin Hall Effect Generator of pure spin currents in electron- or hole-doped semiconductors with Spin-Orbit Interaction: I = I + I = 0 I I ħ G s H = 0 e V1 V 4 I = I I

19 Evading Ferromagnets: Macroscopic Intrinsic Spin Hall Effect in p-type Semiconductors d x = d t ħ d k d t 1 ε n ( k ) d k + B ħ k d t d x = e E + B ( x ) d t n ( k ) Bn( k) = An( k), An α( k) = i nk nk k i d d k j = e f n ( ) n ( ) d k v k π n ( ) z jy e σh = = E 8π x ˆ ħ 5 H k ( ) m γ γ = + γ k S 1 Murakami, Nagaosa, and Zhang, Science 301, 1348 (003).

20 What are the Challenges in Mesoscopic Sp ntron cs? Mesoscopic rings can modulate even unpolarized currents [Nitta, Meijer, Takayanagi, Appl. Phys. Lett. 75, 695 (1999)] L< L Ψ H H φ o s e 1 ( 1 cos ( ) cos ( )) e G= π n n+ π n n+ ( 1 cos [ πqr sin γ π(1 cos γ) ]), QR tan γ h + + = + = h

21 Spin Transport Through DEG px + py α β + V conf ( x, y ) + V disorder ( x, y ) + p y x p x y + p x x p y y ( σ σ ) ( σ σ ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ * m ħ ħ 1 ( ) ˆ ρ ˆ ˆ inject = Σ Σ = 1+ σx = ρ P = 1 inject inject ˆ ρ d e te c t P d e te c t ˆ ρ d e te c t < 1

22 Spin Detection in Transport-Based Schemes P = I I + I I P G + G G G = = G + G + G + G? P T T = = T + T + T + T? P T + T T + T = σ + σ = T T e T = T + T + T + T, G = T h?

23 Pure Quantum States A vector in Hilbert space: Ψ e iφ Ψ H Pure state density matrix projection operator: [ ] ˆ = Ψ Ψ ˆ = ˆ; S = Tr ˆ ln ˆ = 0 ρ ρ ρ ρ ρ

24 Separable vs. Entangled Pure States Pure separable states: Ψ = a b H H AB A B Pure entangled states: d d r A B Ψ = a b = p u v H H AB i A j B k k A k B A B i= 1 j= 1 k= 1 Signature of entanglement: Schmidt rank { } r m in d, d A B A B

25 Mixed Quantum States (Proper) Proper Mixtures: ˆ = p α Ψ α Ψ α ˆ ˆ α βhˆ βhˆ ˆ ρeq = e / Tr e ρ ρ ρ Examples: Spin density matrix ρ ρ ˆ 1 ρ 1 ˆ s = = + ρ ρ P ( P σ ) Spin P olarization Completely unpolarized current 1 1 Iˆ ˆ ρ = + = s P = 0

26 Spin Density Matrix: Tips and Tricks ρ ρ 1 1 ( ˆ) 1 Pz Px ip + + y ˆ ρ 1, Tr ˆ ˆ s σ = = + P = P= ρσ s ρ ρ Px ipy 1 P z Any density matrix has to satisfy: s s s s ˆ = ˆ Tr ˆ = 1 Σ ˆ Σ 0 ρ ρ ρ ρ ˆ ρ Bloch vector for specifies a pure spin state: Eigenvalues of the spin density matrix: s = 1 1 g1 = + P + P + P g = P + P + P ( ) ( ) 1 x y z, 1 x y z ˆ ρ s θ iφ θ Σ = c o s e + s in e iφ

27 Pure vs. Mixed: Quantum Interference Effects 1 1+ i 1 1+ i 1 1+ i ρ ρ ˆ ρ = i = = i 1 ρ ρ ρ incoherent s 1 = + ( ) iα iβ e + e coherent ρs = u u, u = ˆ σ = 0 z ˆ σ x = 0 σˆ = cos ( α β ) x

28 Mixed Quantum States (Improper) Improper Mixture Quantum state of an entangled subsystem (Landau 197): ˆ ρ Experimentally Non-classical correlations: d A Ψ = a d B AB i A j B i = 1 j = 1 = Tr Ψ Ψ, ˆ ρ = Tr Ψ Ψ A B AB AB B A AB AB ˆ ρ = Ψ Ψ ˆ ρ ˆ ρ AB Oˆ = Tr ˆ ρoˆ = Tr ˆ ρ Oˆ Oˆ Oˆ Oˆ Oˆ A A A A A A B A B AB A b B

29 Classification of States in Bipartite Composite Quantum Systems Partial States ρ ˆA ρ ˆB both pure one pure, one not both not pure Total State pure nonpure yes no no yes yes yes ˆρ ρ ˆA ρ ˆB ˆρ pure, pure, pure pure, nonpure, nonpure nonpure, nonpure, pure nonpure, nonpure, nonpure Correlated no no yes yes Uncorrelated yes yes no yes ˆ A THEOREOM: If describes a pure state, then (i.e., a A B pure reduced density operator must be a factor of the total density operator). ρ ˆ ρ = ˆ ρ ˆ ρ

30 Coherence of Spin Quantum States Purity: Tr ˆ spin s spin = = ς ρ ς 1 for pure states Von Neumann Entropy: [ ˆ ˆ ] S ρ ρ S spin s s spin 1 ς = 1 + P spin ( ) = Tr log = 0 for pure states S spin = ( 1 + P ) log ( ) ( ) ( ) 1 1 log 1 + P P P

31 Quantifying Entanglement Entangled pure bipartite states: Schmidt rank: r > 1 S S = Tr ˆ log ˆ = Tr ˆ log ˆ > 0 [ ρ ρ ] [ ρ ρ ] von Neumann A A B B linear Ψ = ˆ ρ > 1 Tr A 0, Ψ Given the two states which one is more entangled? 1 AB Mixed bipartite states: Entanglement of formation (concurrence), distillable entanglement, relative entropy of entanglement. For multipartite systems there is no complete classification of entanglement measures. AB

32 Spin Decoherence Ψ = α e + α ρ ˆs Decoherence = entanglement to the environment: + 1 * α+ α+ α e e1 = * α α+ e1 e α e * eq α αβ T α 0 T1 ρ 0 * eq α β β 0 β 0 ρ Dephasing: ˆ ρ N 1 = Σ Σ s i i N i= 1 T *

33 Genuine Decoherence The fundamental decoherence mechanism is pure entanglement to the environment without any dynamical change of component states: e e 1 ( α ) + + α e α+ e1 + α e e e Localy, coherence is lost The components still exist, but can no longer interfere, since the required phase relations are delocalized (this process has no analog in classical physics): ρ ρ e e1 0

34 False Decoherence Coherence is trivially lost if one of the required components dissapears. For example, relaxation process where decays into state. dρ 1 = ρ e e 1 dt T1 e e dρ 1 1 = iωρ dt T Similar trivial effect would results from dynamics generated by appropriate Hamiltonian: 1 1 n > or collapse of wave function during a measurement procedure. n ρ T = T 1

35 Fake Decoherence Destruction of quantum-interference effects often arises from some averaging procedure: The ensemble either consists of members undergoing the same unitary evolution but with different initial states or an ensemble of identically prepared states subject to slightly different Hamiltonians. In both cases the fundamental dynamics of a single system is unitary so there is no decoherence from a microscopic point of view. N θ 1 iφ j cos sin * N θ e 1 T < T j 1 α 0 = ˆ + ρ = Σi Σ i = N N i= 1 1 iφ j θ sinθ e sin 0 α j= 1 vs. ϒ = Σ Σ Σ 1 n

36 Phase Randomization by Noise Pure phase damping induced solely by elastic collision (no energy exchange) by stohastic forces or noise d ρ = i [ ω + δω ( t) ] ρ dt t iω t ρ ( t) = ρ (0) exp iω t i dt δω ( t ) ρ ( t) = ρ (0) e e 0 δω ( t) = 0, δω ( t) δω ( t ) = γ cδ ( t t ) Random kicks ensemble of unitary evolutions yields a non-unitary evolution of the density matrix: ϕ( x) ϕ( x) e ipx ip( x x ) ρ( x, x ) ( x x ) ρ( x, x ) ρ( x, x ) e 1 exp ( x, x ) t ρ 4λ λ p P( p) = e λ π γ t c

37 Spin Relaxation

38 Purification of Transported Quantum States Ψ H o H s Macroscopics: Mesoscopics: ˆ ρ = ˆ ρ ˆ ρ o ˆ ρ = Ψ Ψ ρ ˆ ˆ e.g., s + Spintronics: ( ) ρ = ρ Φ Φ σ σ ˆs ˆ ρ = ρ σ σ ˆo

39 Two Probe Spintronic Devices: A Theoretician View Ferrom agnet InA s z y x Leads inject pure states: ( y ikn x n ) e ± Φ σ

40 Rashba SO Interaction Spin-Orbit Interactions: Hˆ so = ħ V p m c 0 ( ˆ σ ˆ) The confining potential for D electrons is asymmetric along the z-axis: E = V ˆ R αr ˆ ˆ ˆ 11 Hso σ p BR ( p) σ α = = R evm ħ z ( ) ( ) ħ E ( k ) E k m α ± = + ± x n * x

41 Quantum Spin-Charge Transport with Lattice Hamiltonian Ĥ = H ˆ + H ˆ o so m ε m m m + t Iˆ mn m n s m,n ε m W W, αrħ ˆ ˆ ˆ ˆ a t ( v σ v σ ) x y y x

42 Landauer Formula for Spintronic Quantum Transport Conductance of spin-degenerate electrons at T=0: e e G = E E T E h = Tr t( F ) t ( F ) n F h n ( ) Conductance Matrix for spin-polarized quantum transport: t t G G e ij, ij, e Tn T n G = = G G h = ij t t h n Tn T n ij, ij, t = Im Σˆ Iˆ G Im Σˆ Iˆ, Gˆ = E Hˆ Σˆ Iˆ R R R L s 1N L s s 1 Spin resolved transport measurements: s injected G scollected

43 What is Quantum Transmissivity? ( a) P( T) ( b) P( T) = = G 1 3/ GQ T 1 T G 1 G T 1 T Q T G dt P( T) TdT, S(0) dt P( T) T(1 T), G dt P( T) ( T ) NS 0 0 0

44 Entanglement in Scattering States Transmission matrix also hides the entanglement of spin and orbital quantum states in the right lead: in n, σ out = tn n, σ σ n σ Ho H s n, σ out * ˆ n σ t n n, σ σ t n n, σ σ n, n, σ, σ n n out * ˆ n σ s = tn n, σ σ t n n, n,, ρ = σ σ ρ σ σ σ σ σ σ No need for master equations that approximate (Markovian or non-markovian) exact quantum evolution: ( ) ˆ ρ ˆ ( ) ˆ ρ Tr ˆ s = o U t,0 (0) U t, 0

45 Spin Density Matrix of Detected Current Phase versus amplitude of matrix elements: ˆ ρ * M n out e / h n n, n n, n n, ˆ current ρ t t t = s = n G G n, n = 1 * t t t n n, n n, n n, Measuring mixed state of spin-½ particle: G G Px = G + G M ˆ / * Tr e h P = ˆ ρcurrent σ Py = Re t t nn, nn, G G n, n 1 = + M e / h * Pz = Im t t nn, nn, G + G n, n = 1 t- +

46 Decoherence in Ballistic Nanowires

47 Decoherence in Multichannel Ballistic Nanowires M=10 Channel Wire

48 D yakonov-perel in unbounded disordered system

49 D yakonov-pererl in ballistic confined system Chang, Mal shukov, and Chao, cond-mat/04051

50 Spin Density Within Nanowire M = 10, tsor = 0.03 M = 30, tsor = 0.03

51 Semiclassical Picture: Spin Wave Packet

52 Spin Diffusion in Nanowires

53 Quantum Corrections to Spin Diffusion

54 Non-Ballistic Spin-FET Can spin coherence survive spin-independent charge scattering in spin-fet operation?

55 Spin-Interference AC Ring Device: Current

56 Spin-Interference AC Ring Device: Coherence

57 Summary: Open Questions Extracting current spin density matrix from the Landauer transmission matrix of quantum transport: Follow the same steps employed in Quantum Information Science when studying the entanglement in bipartite quantum systems. Decoherence + Dephasing in coupling of transported spin to the zero-temperature environment To retain fully coherent spin quantum states in operating Spin-FET or Spintronic Rings, the spin has to be transported through a single channel semiconductor Can we utilize interference effects with partially coherent spin states?