Quantum physics in quantum dots

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2 Quantum physics in quantum dots Klaus Ensslin Solid State Physics Zürich AFM nanolithography Multi-terminal tunneling Rings and dots Time-resolved charge detection

3 Moore s Law Transistors per chip gate length 100 nm? 10 7 Pentium Pro Pentium micro nano Year

4 Vanishing electrons Electrons per device 10 4 (4M) (16M) (64M) gate length 100 nm (256M) (Transistors per chip) (1G) (4G) (16G) micro nano Year

5 Quantized charge Capacitance of a capacitor: C = Q U = charge voltage Energy to charge the capacitor: E = U dq = 0 Q 0 Q Q C dq = Q2 2C -Q Q voltage U Energy to put one electron (Q=e) on a capacitor with C = 1 nf ( E = As) F = Joule = ev Equivalent to temperature T = 0.1 mk

6 Size of a capacitor capacitance Q C = 0 = 0 area separation = ( 1 μm) 2 1 μm =1016 F equivalent to temperature T = 7 K 1 μm -Q 1 μm 1 μm -> use nanotechnology to 10 make nm a small capacitor decoupled from its environment

7 direct patterning of AlGaAs/GaAs high mobility two-dimensional electron gas (2DEG) below sample surface 2DEG: W. Wegscheider Uni Regensburg Matsumoto et al., APL 68, 34 (1996) Held et al., APL 73, 262 (1998)

8 lateral resolution oxide line 4 Ti film 1μm height (nm) nm writing speed 1μm/s humidity 40 % bias 8V x(nm)

9 oxidation of GaAs - reproducibility

10 quantum dot AFM gallery antidot lattice 4-terminal ring ring + dots quantum point contacts rings, dots + qpc s 1μm 3μm

11 Double layer AFM lithography Lithography on 8nm Ti top gates: Martin Sigrist, Andreas Fuhrer

12 Aharonov-Bohm effect 2 1 = 1 2 = geom. + q h r r A dl conductance becomes a periodic function of magnetic flux

13 AFM defined quantum ring QPC current flow source QPC Kekulé Bull. Soc. Chim. Fr. 3, 98 (1865) -> benzene 300 nm plunger plunger Aharonov & Bohm Phys. Rev. 115, (1959) -> magnetic flux QPC drain QPC Büttiker, Imry, & Landauer Phys. Lett. 96A, (1983) -> persistent currents

14 AB-oscillations in an open ring Magnetoresistance Fourier-Spectrum At T=1.7K up to h/6e l (T) T 1, typical for e - -e - interaction l (1.7K) 3μm ; l (100mK) = 60μm

15 electron rings on different scales Benzene ring: Ring accelerator : Large Electron Positron Collider at CERN in Geneva 8.6km 0.5nm Aharonov-Bohm effect: one flux quantum (h/e) through ring area h e r 2 = 5000 T 1 μm h e r 2 = T

16 Coulomb blockaded quantum ring QPC source QPC 0.02 T 100 mk plunger plunger 0.01 QPC drain QPC E ring (mev)

17 Coulomb blockade E F source drain e 2/2C E F E F source drain e 2/2C E F kt << e 2/2C source drain eu = E - E << e 2 /2C F F -> no current transport discrete level between source E drain F and EF -> coherent resonant tunneling r =100 nm disk: r C = 4 0 r > C =100 af > e 2 /2C = 600 μev 7K

18 QPC source QPC quantum ring plunger plunger 0.6 QPC drain QPC B (T) h/e E (mev) ring V plunger (V)

19 perfect 1D ring in a magnetic field B 0 > E m,l = h2 (m l)2 2 2mr m,l () = 1 2 eil fixed N B = 0 - > H = h2 2 2mr 2 2 E l (m) [h 2 2m * r 0 2 ] energies : E l = h2 2mr 2 l2 wave functions : l () = 1 2 eil m magnetic flux (h/e)

20 E (mev) ring energy spectrum E l (m) [h 2 2m * r 2 0 ] fixed N B (T) 0.3 m magnetic flux (h/e) E l = h2 l m * r / = I 2 0 l 0 Experiment: l = 8; I l 22nA 13

21 perfect ring: m,l () = 1 2 eil -> probability density uniformly spread over the ring -> cannot explain oscillations of Coulomb peak amplitude imperfect ring imperfection at position = 0 * m,l () = 1 sin ( m( 0) ) 0 E m,l (h 2 2mr 2 ) magnetic flux (h/e)

22 perfect ring symmetry breaking asymmetric ring with finite width 0 > 0 1 cos(2) ( )

23 energy levels and wave functions energy flux quanta through ring

24 How to measure resistances U I I V two-terminal measurement of a classical resistor four-terminal measurement of a classical resistor -> elimination of contact resistances

25 How to measure resistances Quantum dot in the Coulomb blockade regime: high impedance device U I two-terminal measurement of a quantum dot What about more than two terminals? How to differentiate between contacts and quantum dot?

26 multi-terminal quantum dot lithographic size: nm 2 electronic size: nm 2 LG 2 1 LG 1 PG LG 4 charging energy: E C 0.5 mev 2 mean level spacing: 35 ev 3 LG 3 4 electronic temperature: k B T10 ev 1 m Renaud Leturcq & Davy Graf

27 multi-terminal quantum dot Experimental set-up lead 2 lead 3 lead 1 I/V bias (e 2 /h) V LG4 (V)

28 multi-terminal quantum dot conductance matrix I 1 I 2 I 3 G 11 G 12 G 13 = G 21 G 22 G 23 G 31 G 32 G 33 sum rules current conservation: I i = 0 G ij = 0 3 i=1 V 1 = V 2 = V 3 I i = 0 V 1 V 2 V 3 conductance G ij (e 2 /h) 3 G ij = 0 j= lead 2 lead 3 lead V LG4 (V)

29 measurement set-up apply voltage to one terminal measure current in three terminals Kirchhoff rules G 11 G 12 G 13 G 21 G 22 G 23 = 1 G G 31 G 32 G 1 + G 2 + G 3 33 sequential tunneling: G n = e2 4kT G ij : three-terminal conductance G l : lead conductance G 1 (G 2 + G 3 ) G 1 G 2 G 1 G 3 G 1 G 2 G 2 (G 1 + G 3 ) G 2 G 3 G 3 G 3 G 3 G 2 G 3 (G 1 + G 2 ) S D n n 1 cosh V n 2 G G V G 2kT ( )

30 individual tunnel couplings Lead conductances G k (e 2 /h) Weak coupling regime V LG4 (V) independent fluctuations Tunneling rate ħ (ev) strong overlap weak overlap ~50 n Individual coupling to the leads extend of the wave function in the dot in the vicinity of the leads. F

31 Spins in Coupled Quantum Dots for Quantum Computation magnetic field gradients by current wire D. Loss & D. DiVincenzo, PRA 57 (1998) 120 n.n. exchange local Zeeman each dot has different g-factor ->individually addressable via ESR

32 spin as a qubit one spin 1/2 particle is a natural qubit two spin 1/2 particles: singlet state: 1 2 triplet states:, ( ) 1 2 (entangled) ( + ), Spin coherence times have been shown to be much longer than charge coherence times, up to 100 μs

33 Spin qubits in quantum dots General qubit state: two-level system = cos ei sin 2 1 Possible realizations employing quantum dots: charge qubit spin qubit Zeeman

34 semi-circular dot with charge readout I dot (pa) G QPC (e 2 /h) dg/dv gate (a.u.) N N+1 N+2 V gate V gate gate detector QPC gate gate S D 2 μm Roland Schleser Elisabeth Ruh Thomas Ihn V gate (V) See also Gardelis et al, PRB67, (2003), Elzermann et al., Phys. Rev. B 67, (R), (2003)

35 time-resolved detector signal dg/dv gate (a.u.) time (s) close tunnel barriers -> electron transport one-by-one pinch-off one tunnel barrier completely: - one-off time is a measure for the tunnel rate on and off the quantum dot - one-off probability is a measure for the state occupation -> Fermi distribution dot source drain E F

36 Fermi-Dirac distribution 0 gate voltage fit: Fermi distribution distribution extracted from data example sweeps E (mev) time (s) -> T ~ 150 mk f (E) -0.5

37 Spectroscopy of source electronic states pg source quantum dot drain k B T_ drain E C G S N-1 k B T E C DE GD G SD (10-3 e 2 /h) N N+1 E C (+) V PG (mv)

38 Quantum point source contact as a charge detector pg G QPC k B T_ drain E C 2e 2 /h N-1 G SD (10-3 e 2 /h) N N+1 E C (+) V P M. Field et al., Phys. Rev. Lett. 70, 1311 (1993) V PG (mv)

39 A few electron source quantum dot pg drain M. Sigrist

40 Detection of single electron transport Quantum point contact as a charge detector Low bias voltage on the quantum dot source quantum dot drain T e = 350 mk k B T

41 Low bias - thermal noise : effective dot-lead tunnel coupling E: energy difference between Fermi level of the lead and electrochemical potential of the dot R. Schleser et al., Appl. Phys. Lett. 85, 2005 (2004) L. M. K. Vandersypen et al., Appl. Phys. Lett. 85, 4394 (2004)

42 Determination of the individual tunneling rates Exponential distribution of waiting times for independent events S =< in >, D =< out > N N+1

43 Measuring the current by counting electrons N N+1 Count number n of electrons entering the dot within a time t 0 : I = e<n>/t 0 Max. current = few fa (bandwidth = 30 khz) BUT no absolute limitation for low current and noise measurements we show here: I few aa, S I A 2 /Hz

44 Histogram of current fluctuations maximum: mean current width: fluctuations, noise

45 Histogram of current fluctuations Poisson distribution for asymmetric coupling Sub-Poisson distribution for symmetric coupling Theory: Hershfield et al., PRB 47, 1967 (1993) Bagrets & Nazarov, PRB 67, (2003)

46 Current fluctuations vs. asymmetry Reduction of the second and third moments for symmetric coupling asymmetric barriers a=1 symmetric barriers a=0 Theory: Hershfield et al., PRB 47, 1967 (1993) Bagrets & Nazarov, PRB 67, (2003)

47 Current fluctuations vs. asymmetry Reduction of the second and third moments for symmetric coupling width - noise asymmetry Theory: Hershfield et al., PRB 47, 1967 (1993) Bagrets & Nazarov, PRB 67, (2003)

48 Time-resolved electron transport - small current level (< atto-amperes) - low noise levels (S I A 2 /Hz) - higher correlations in current are accessible -> correlations, interactions and entanglement in quantum dots bandwidth 20 khz

49 Aharonov-Bohm with cotunneling Co-tunneling Electrons are injected from the right lead They pass through either the upper or lower arm The interference take place in the left QD

50 Waves The double slit experiment source Light A. Tonomura et al., American Journal of Physics (1989) double slit screen

51 Waves The double slit experiment source Particles double slit screen

52 Double slit experiment <-> Aharonov Bohm Simon Gustavsson Matthias Studer

53 Aharonov-Bohm oscillations counts / s B-Field [mt] huge visibility! >90% little decoherence - > due to long dwell time in the collecting dot? requires the couplings of upper and lower arm to be well symmetrized

54 Temperature dependence AB amplitude stable below T=400mK Destruction most likely due to thermal broadening

55 Future directions from quantum devices to quantum circuits DD DD non-equilibrium quantum mechanics -> time dependent experiments, MHz - GHz QPC detection of entanglement in solid state quantum systems -> non-classical (microwave) radiation Combination of spatial and temporal resolution novel quantum materials graphene, nanowires

56 Simon Gustavsson thanks Renaud Leturcq Thomas Ihn Martin Sigrist Andreas Fuhrer

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