Quantum physics in quantum dots Klaus Ensslin Solid State Physics Zürich AFM nanolithography Multi-terminal tunneling Rings and dots Time-resolved charge detection
Moore s Law Transistors per chip 10 9 10 8 gate length 100 nm? 10 7 Pentium Pro 80786 10 6 10 5 8086 80286 80386 80486 Pentium micro nano 10 4 4004 8080 10 3 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year
Vanishing electrons Electrons per device 10 4 (4M) 10 3 10 2 10 1 (16M) (64M) gate length 100 nm (256M) (Transistors per chip) (1G) (4G) (16G) 10 0 10-1 micro nano 1985 1990 1995 2000 2005 2010 2015 2020 Year
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 = 1.61019 As) 2 210 9 F =1.310 29 Joule = 810 9 ev Equivalent to temperature T = 0.1 mk
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
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)
lateral resolution oxide line 4 Ti film 1μm height (nm) 3 2 1 35 nm writing speed 1μm/s humidity 40 % bias 8V 0 600 800 1000 1200 1400 x(nm)
oxidation of GaAs - reproducibility
quantum dot AFM gallery antidot lattice 4-terminal ring ring + dots quantum point contacts rings, dots + qpc s 1μm 3μm
Double layer AFM lithography Lithography on 8nm Ti top gates: Martin Sigrist, Andreas Fuhrer
Aharonov-Bohm effect 2 1 = 1 2 = geom. + q h r r A dl conductance becomes a periodic function of magnetic flux
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, 485-491 (1959) -> magnetic flux QPC drain QPC Büttiker, Imry, & Landauer Phys. Lett. 96A, 365-367 (1983) -> persistent currents
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
electron rings on different scales Benzene ring: Ring accelerator : Large Electron Positron Collider at CERN in Geneva 8.6km 0.5nm 10 13 Aharonov-Bohm effect: one flux quantum (h/e) through ring area h e r 2 = 5000 T 1 μm h e r 2 = 7 1023 T
Coulomb blockaded quantum ring QPC source QPC 0.02 T 100 mk plunger plunger 0.01 QPC drain QPC 0.00 0 0.4 0.8 1.2 E ring (mev)
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
QPC source QPC quantum ring plunger plunger 0.6 QPC drain QPC 0.4 0.02 B (T) 0.2 0 h/e 0.01-0.2 0.00 0 0.4 0.8 1.2 E (mev) ring -0.4 0.1 0.2 0.3 V plunger (V)
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)
E (mev) ring 1.2 1.0 energy spectrum E l (m) [h 2 2m * r 2 0 ] fixed N 0.4 0 0.2 0 0.1 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
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 ) 6 5 4 3 2 1 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 magnetic flux (h/e)
perfect ring symmetry breaking asymmetric ring with finite width 0 > 0 1 cos(2) ( )
energy levels and wave functions energy flux quanta through ring
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
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?
multi-terminal quantum dot lithographic size: 600450 nm 2 electronic size: 400250 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
multi-terminal quantum dot Experimental set-up 0.04 0.02 0-0.02-0.04 lead 2 lead 3 lead 1 I/V bias (e 2 /h) 0.04 0.02 0-0.02-0.04 0.02 0-0.02-0.24-0.22-0.20-0.18-0.16 V LG4 (V)
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=1 0.04 0.02 0-0.02-0.04 0.04 0.02 0-0.02-0.04 0.02 0-0.02 lead 2 lead 3 lead 1-0.24-0.22-0.20-0.18-0.16 V LG4 (V)
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 ) 1 + 1 S D n n 1 cosh V n 2 G G V G 2kT ( )
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
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
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
Spin qubits in quantum dots General qubit state: two-level system = cos 2 0 + ei sin 2 1 Possible realizations employing quantum dots: charge qubit spin qubit Zeeman
semi-circular dot with charge readout I dot (pa) G QPC (e 2 /h) 8 6 4 2 0 1.6 1.5 1.4 1.3 1.2 dg/dv gate (a.u.) 6 5 4 N N+1 N+2 V gate V gate gate detector QPC gate gate S D 2 μm Roland Schleser Elisabeth Ruh Thomas Ihn -0.1-0.08-0.06-0.04-0.02 0 V gate (V) See also Gardelis et al, PRB67, 073302 (2003), Elzermann et al., Phys. Rev. B 67, 161308 (R), (2003)
time-resolved detector signal dg/dv gate (a.u.) 0 1 2 4 6 8 10 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
Fermi-Dirac distribution 0 gate voltage 1 0 1 0 1 0 1 0 1 fit: Fermi distribution distribution extracted from data example sweeps 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4 E (mev) 0 1 2 6 10 time (s) -> T ~ 150 mk 0 0.2 0.4 0.6 0.8 1 f (E) -0.5
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)
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)
A few electron source quantum dot pg drain M. Sigrist
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
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)
Determination of the individual tunneling rates Exponential distribution of waiting times for independent events S =< in >, D =< out > N N+1
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 10-35 A 2 /Hz
Histogram of current fluctuations maximum: mean current width: fluctuations, noise
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, 085316 (2003)
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, 085316 (2003)
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, 085316 (2003)
Time-resolved electron transport - small current level (< atto-amperes) - low noise levels (S I 10-35 A 2 /Hz) - higher correlations in current are accessible -> correlations, interactions and entanglement in quantum dots bandwidth 20 khz
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
Waves The double slit experiment source Light A. Tonomura et al., American Journal of Physics 57 117-120 (1989) double slit screen
Waves The double slit experiment source Particles double slit screen
Double slit experiment <-> Aharonov Bohm Simon Gustavsson Matthias Studer
Aharonov-Bohm oscillations counts / s 1 00 50 0-400 -200 0 200 400 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
Temperature dependence AB amplitude stable below T=400mK Destruction most likely due to thermal broadening
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
Simon Gustavsson thanks Renaud Leturcq Thomas Ihn Martin Sigrist Andreas Fuhrer