Currents through Quantum dots. Bhaskaran Muralidharan Dept. of Electrical Engineering, Indian institute of technology Bombay HRI Allahabad 29/02/2016

Transcription

1 Currents through Quantum dots Bhaskaran Muralidharan Dept. of Electrical Engineering, Indian institute of technology Bombay HRI Allahabad 9/0/016

2 ingle pins: An exciting frontier Read-Out of spins Elzerman et.al., (004) Initailization of spins Ono et.al., cience, (00) pin decoherence Koppens et.al., cience, (005) Coherent Manipulation Koppens et.al., Nature, (006)

3 γ N a = N b a a b b γ Preview K. Ono et al. PRL 04 EXPERIMENT γ N a a a N b b b γ Na Nb Na Nb THEORY THEORY Fin tructure Dual Resonance Un-identical baths Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 3

4 Outline pin Correlation effects in Quantum-dot transport pin Blockade Transport: Multiple NDR pin Blockade Transport: Role of Non-equilibrium cattering Processes Hysteretic behavior x K. Ono et al. cience 0 K. Ono et al. PRL 04

5 Introducing Fock pace Transport

6 Regimes of Transport Σ s µ 1 H + U µ Σ 1 Σ U-> elf Consistent field U CF = U < n > CF Regime Works for N Γ U N ^N many electron levels N N Fock space approach CWJ Beenakker (1991) CB Regime Works for Γ << U

7 Let us try to introduce the Fock space View Point: elf Consistent Field Σ s µ 1 µ H + U Σ 1 Σ ingle particle view point: ε U < n > NEGF-CF γ L ε γ R U

8 A small pocket of transport problems? Coulomb Blockade effects pin Correlation effects x Park et al, nature 0 pin Correlation coupled to Hot-scatterers Ono et al, cience 0 Bi-stability

9 Fock space picture ε γ L ε γ R N=1 γ L γ R 0 N=0 ε +U N= ε,u ε ε N=1 0 N=0

10 How Coulomb Blockade transport works ε ε +U ε ε +U ε ε +U N = CB ε ε N =1 0 N = 0 Correlations matter even for a minimal model! NEGF-CF Muralidharan et al., PRB06

11 Generalized Viewpoint N = n +1 ε tr1 ε tr { P} i N N N = n N = n N N ε tr ε tr1 Given a bias point a set of Fock states are probabilistically distributed!! These may be viewed as transition Energies in the one-particle picture

12 Fock space master equations { N + 1, j} { N, i} { N 1, j} dp N, i dt Fock space probability distribution + R P R P = j { P N, i } N, i N ± 1, j N, i N ± 1, j N, i N ± 1, j 0 j With scattering { N, i } { N, j} dp N, i dt + R P R P = j N, i N ± 1, j N, i N ± 1, j N, i N ± 1, j j

13 Part II pin Correlation effects in Quantum-dot transport pin Blockade Transport: Multiple NDR pin Blockade Transport: Role of Non-equilibrium cattering Processes Hysteretic behavior x K. Ono et al. cience 0 K. Ono et al. PRL 04

14 Coulomb Blockade v/s pin Blockade ε +U ε ε ε µ R Coulomb Blockade itself does not differentiate the spin degree pin Degree of freedom results in zero current ε +U ε ε X ε ε X X Finite Current Flows pin Degree of freedom + Coulomb Blockade pin Blockade!

15 pin Blockade regime in Double Quantum Dots: ε 1 ε No Ferromagnetic Contacts! What is the Blockade mechanism? But why does current flow at all? What is the mechanism for NDR Current Blockade!

16 NDR: Conventional Viewpoint µ L µ µ R L x I a) b) V L. Esaki, RTD Phenomenon 197 No band edges in our case! What makes pin blockade NDR novel? NEED FOCK PACE VIEWPOINT

17 NDR due to dark states ε BA ε CA ε BA I a) b) V B N = n C R = γ * M * f C A AC C A N = n 0 M = C H A CA Transition Rates reflect on the symmetry properties R C A RD ~ 1 τ CA

18 NDR from the dark state model τ τ + τ R CA τ > τ + τ R CA R BA R BA L AB L AB τ τ + τ L CA L BA R AB Muralidharan and Datta, PRB07

19 pin Blockade regime in Double Quantum Dots: ε 1 ε ε 1 ε + U X No Ferromagnetic Contacts! What is the Blockade mechanism? But why does current flow at all? ε Mechanism for NDR Explains Current Blockade!

20 Dark tate model: Double Quantum Dots: ε 1 ε ε 1 ε ε 1 ε N= T ε TB B N=1 ε B ε B Under pecial Conditions Triplet tate Can be Dark!

21 Results Theory Experiment Ono et. al., science 0 Muralidharan and Datta, PRB 07 Muralidharan et.al., JCEL 08

22 Pauli Blockade: A Broader Perspective X ε 1 ε Off state ε 1 ε Permits manipulation of single Electron spin detected by a current Measurement! Host Nuclei can also assist!

23 Part III pin Correlation effects in Quantum-dot transport pin Blockade Transport: Multiple NDR pin Blockade Transport: Role of Non-equilibrium cattering Processes Hysteretic behavior x K. Ono et al. cience 0 K. Ono et al. PRL 04

24 γ N a = N b a a b b γ Preview K. Ono et al. PRL 04 EXPERIMENT γ N a a a N b b b γ Na Nb Na Nb THEORY THEORY Fin tructure Dual Resonance Un-identical baths Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 4

25 pin-blockade Toy Model ingle QD with single nuclear bath pin-down polarized right contact Blockade lifted by pin-flip transitions γ ε + ε N γ + Ĥ HF + 0 5

26 Hyperfine mediation B app Apply B field externally pin-flip at the cost of nuclearflop N 6

27 Analysis: Fermi Golden rule Z-component Mean Field Approximation F I = average nuclear z-polarization Electron dynamics:!! X-Y component Fermi s Golden Rule to give spin-flip rate Nuclear spin dynamics: 7

28 No Overhauser field T B Energy of T Decreases under applied magnetic field No effect on energy of 8

29 With Overhauser field B T Bapp Bov Overhauser field:! It can either oppose the resonance or aid it via negative or positive feedback! Feedback: Origin of Hysteresis!! 9

30 Including Overhauser field T B Bapp Bov Negative feedback from gµb to J eff F I during forward sweep pseudo-linear build-up of F I Hysteresis: resonance breaking Positive feedback during reverse sweep rapid rise of F I 30

31 Recap K. Ono et al. PRL 04 EXPERIMENT THEORY THEORY Fin tructure Dual Resonance Un-identical baths Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 31

32 Two Dots, Two Nuclear Baths γ U aa ε a / a a a U ab U b b t ε b / b b b γ Na Nb 3

33 Double Resonance due to Two-Electron states Two-electron states at B = 0 Three = 1 states (T) Three = 0 states () Triplets are blocking states TWO REONANCE: T +1 1 A 1/ A +1/ B 1/ B + 1/ T 1 T 0 T +1 1 N=1 N= B app T -1 T + and T - move in opposite directions under B and F I. 33

34 Double Resonance: Electronic tructure 1 γ U aa ε a / a a a U ab U b b t ε b / b b b γ Na Nb 0 /1 ab + β0 /1 ab + ξ0 /1 aa + δ0 / 1 bb Identical Nuclear baths 0/1 β0/1 0 ( 1,1) (,0) /( 0, ) No Difference Overhauser field 0/1 β0/1 Unlike Nuclear Baths Difference Overhauser field 34

35 Electronic tructure: Continued 1 = 0 /1 ab + β 0 /1 ab + ξ 0 /1 aa + δ 0 /1 bb T 0 = ( ab + ab ) / T + 1 = ab 0 T 1 = ab 0/1 β0/1 Identical Nuclear baths No Difference Overhauser field 0/1 β0/1 Unlike Nuclear Baths Difference Overhauser field 35

36 Two-Dot Two-Bath Hamiltonian Z-component Mean Field Approximation F = average nuclear z-polarization Electron dynamics:!! X-Y component Fermi s Golden Rule to give spin-flip rate Nuclear spin dynamics: 36

37 Fermi s Golden rule: One-bath variable vs two-bath variables γ γ a a b b γ N γ a a b b Na Nb 37

38 Identical Nuclear Baths & Near-imultaneous Resonances Novelty of two nuclear baths: Matrix elements between singlet and triplet elements non-zero simply due to incoherent addition between the two baths. Difference Overhauser field NOT required!! Two dragged resonances Each contributes a triangular current traces uperposition: flat topped hysteretic behavior 38

39 uperposition N = a) 1 0 T +1 T 1 N =1 b 1/ 39

40 Un-identical Nuclear Baths: Difference Overhauser field / T = 0/1 ab + β0/1 ab + ξ0/1 aa + δ 0/ 1 bb Fin structure at the two ends of the hysteretic sweep as noted in the experiments Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 40

41 Results EXPERIMENT Fin tructure THEORY THEORY Ideal Dual Resonance Un-identical baths Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 41

42 ummary Key Points: One Dot toy example Hyperfine interaction Hamiltonian Dragged resonance due to Overhauser Field Triangular current trace Double Dot, Two-Bath: Buddhiraju and B Muralidharan, JPCM, 6, 48530, (014) 4

43 y z x B L B R B + = Density matrix formalism: pin dynamics ( ) ( ) ( ) + = + Γ = + + Γ = + = + = = = = τ ε ε π ε ε τ τ ρ ρ ρ ρ ρ ρ B m m p m p J q J E E f U E E f de m p B U f f B m m p m p J q dt d i r q r R L r q x y z, ',,, ˆ ˆ ˆ ) ( 1 ) ( ˆ, ) ( ) ( 1, ˆ ˆ ˆ,, ε +U ε ε,u Γ << k B T ( ) + = τ B m m p m p J q J r q, ˆ ˆ ˆ Injection Relaxation Precession

44 ( ), ˆ ˆ ˆ,, = + = R L r q B m m p m p J q dt d τ QD pin dynamics v/s TT L T L, R T, µ ε +U ε L m L p, R R m p, + Γ = ε ε π E E f U E E f de m p B ) ( 1 ) ( ˆ ' Precession Damping Injection Relaxation Precession

45 L m L p, R R m p, y z x B L B R B + = Thermoelectrically induced spin precession! Pure spin current due to spin precession! Y Tserkovnyak et.al., PRL (00) ( ) + = τ B m m p m p J q J r q, ˆ ˆ ˆ Injection Relaxation Precession pin precession-spin current B Muralidharan and M Grifoni, PRB (013)

46 pin batteries/maxwell s demon

47 Heat Engines with demons N 1 + N 1 = 0 E1 + E + E0 = 0 E1 µ 1N1 E µ N + T T + E T N 1 + N 1 = 0 E1 + E + E E1 µ 1N1 + T 0 = 0 E µ T N Δ 0 0 =klnw Reservoir/Bath at T0 Out of Equilibrium ystem Demon ource E 0 Channel Drain ource E 0 Channel Drain E, N E, N 1 1 E, N 1 1 E, N 1 1N E 1 µ N E µ 1 1N E 1 µ N E µ

48 Examples of nano-device demons Out of Equilibrium ystem ource E 0 Channel E1, N E, N 1 Drain 1 1N E 1 µ N E µ

49 nano-device demons + info battery 0 0 Δ = Δ Δ Δ T E F tot Δ + = + + = + T N E T N E E E E N N µ µ v/s ENERGY INFORMATION

50 nano-device demons + info battery H = J 1 D U u u+d d d+u

51 nano-device demons + info battery µ 1 µ i = 0 f = Nk ln Δ = Nk ln ΔW NkT ln i = 0 f = Nk ln Δ = Nk ln Current tops to flow eventually! Where does the energy come from?! tate of the Demons

52 W Connection with Maxwell s Demon Does it no violate any known laws or Common sense?! If not, what is the catch???! Demon exorcism by zilard (199)! TH Qin Qout Reservoirs at T1 and T TC

53 Connection with Maxwell s demon Δ tot ΔF 0 = ΔE TΔ 0 Δ tot ΔF 0 = ΔE TΔ 0 Energy Information

54 Connection with Maxwell s demon/landauer principle µ 1 Δµ µ u+d ΔW = final initial d+u dn u Δµ i = 0 f = k lnw = Nk Δ = Nk ln ΔW NkT ln E Erase NkT ln ln

55 Connection with Maxwell s demon/landauer principle tate of the Demons Discharging Randomizing the bit charging Erasure Δ = Nk ln ΔW NkT ln E Erase NkT ln

56 Nano-spin-energy group CHARGE pin Energy

57 Acknowledgments iddharth Buddhiraju (student IITB) Prof. upriyo Datta (Purdue University) Prof. Milena Grifoni (University of Regensburg, Germany) THANK YOU FOR YOUR ATTENTION!