Charges and Spins in Quantum Dots

Transcription

1 Charges and Spins in Quantum Dots L.I. Glazman Yale University Chernogolovka 2007

2 Outline Confined (0D) Fermi liquid: Electron-electron interaction and ground state properties of a quantum dot Confined (0D) Fermi liquid: Transport across a quantum dot Kondo effect in quantum dots

3 Bulk Fermi Liquids e h interaction remains weak (Landau 1956)

4 Droplets of a Fermi Liquid small, symmetric descendants of dirty bulk Fermi liquid allow for statistical description Marcus; Chang ~1996 physics of artificial atoms Kouwenhoven+Tarucha ~1996

5 Quantum Chaos and Interactions in Quantum Dots: Energy Scales Single-particle level spacing: L Thouless energy: or Charging energy: Good conductor: (Thouless1972) 2D: (ratios for the ballistic case)

6 Quantum Chaos and Interactions in Quantum Dots: Universal Hamiltonian Single-Particle Part Random Matrix Theory (RMT) limit L are random, repel each other Random eigenfunctions (Porter-Thomas statistics) + GUE, GOE GOE

7 Quantum Chaos and Interactions in Quantum Dots: Universal Hamiltonian of Interactions Random eigenfunctions GUE, GOE GOE Important only in supercond. dots, charging exchange Cooper

8 Quantum Chaos and Interactions in Quantum Dots: Universal Hamiltonian of Interactions No other matrix elements with non-zero averages in the limit finite part, does not fluctuate fluctuates, average is zero variance is

9 Full Form of the Universal Hamiltonian not random random Cooper pairs operator number of electrons spin of the dot Kurlyand, Aleiner, Altshuler,`00; Aleiner, Brouwer, LG (review) 02 quantum dot gate V g 2D electron gas

10 Corrections to the Universal Hamiltonian Leading correction ( only planar dots): does not limit quasiparticle lifetime Blanter, Mirlin, Muzykantskii, 1997 Smaller correction (planar dots, 3D grains): causes transitions between the levels Sivan, Imry, Aronov, 1994; Blanter 1996

11 Equilibrium Properties of a Quantum Dot Charge vs. gate voltage gate quantum dot Electrostatics V g 2DEG P. Lafarge et al (SACLAY) Matveev, LG, Shekhter (theo review) 1994 Matveev, Larkin, 1997 Delft, Ralph Phys Rep (review) 2001

12 Spin States of a Quantum Dot theory: Oreg, Brouwer, Halperin`99; Ullmo, Baranger, LG 2000; Usaj, Barabger, 2002 experiments: Tarucha group (small dots) 2001 Ensslin group (larger dots) 2001 Marcus group (large dots) 2001

13 I Transport through a Quantum Dot: Classical Coulomb Blockade V Vg

14 Classical Coulomb Blockade I V Vg Balance Equations (currents through L and R junctions equal each other):

15 Classical Coulomb Blockade: Linear Conductance Peak width: 1. Equilibration between the tunneling events; 2. Independent tunneling through the junctions (no interference) 3. Small level spacing (we used the Fermi distr. of quasiparticles) P. Joyez et al. (SACLAY) PRL 1997

16 Transport through a Quantum Dot: Resonances I V Vg separate-level resonances fluctuates from level to level (Stone, Alhassid1992) Folk et al 1996

17 Transport through a Quantum Dot: Statistics of Coulomb Blockade Peaks

18 Coulomb Blockade Valleys: Inelastic Transport electron-hole excitation is left behind Linear conductance Averin, Odintsov 1988

19 Coulomb Blockade Valleys: Elastic Transport partial amplitudes are random survives averaging small, fluctuates (Landauer formula) Averin, Nazarov 1990; Aleiner, LG 1996; Cronnenwett et al (exp) 1997

20 Mesoscopic Fluctuations of Elastic Transport Odd valley: looks like Anderson Impurity Model Kondo effect

21 Summary: Valleys: Peaks:

22 Small quantum dots (~ 500 nm) M. Kastner, Physics Today (1993) E.B. Foxman et al., PRB (1993) conductance (e 2 /h) gate voltage (mv)

23 Even smaller quantum dots (~ 200 nm) 1998 D. Goldhaber-Gordon et al. (MIT-Weizmann) S.M. Cronenwett et al. (TU Delft) J. Schmid et al. Stuttgart) 200 nm van der Wiel et al. (2000)

24 Low-T Conductance Anomaly 15 mk 800 mk conductance (e 2 /h) Activation conductance theory fails qualitatively in every other valley T(K)

25 Anomalous low-temperature behavior ( ) G ln E T C for N = odd all possible electronic configurations have S 0 T- dependence N = even: normal (decrease at T 0) N = odd: anomalous (increase at T 0) S 0 ( ) G ln E T C =? Kondo physics

26 Anomalous behavior of metallic resistivity de Haas et al. (1934) resistivity of a clean Au sample expected saturation

27 Kondo effect Jun Kondo (1964) local spin density of conduction electrons magnetic impurity correction to resistivity grows at T 0 A problem: how to deal with singularities at T 0?

28 The Origin of Exchange Interaction ε F tunneling electron gas t Anderson impurity model strong on-cite repulsion: impurity level is singly occupied: N = 1 ( N ) 2 H = 1, N = n + n d U t = 0: doubly degenerate ground state Ψ = electron gas Ψ = electron gas Finite t : tunneling exchange example: virtual state H exchange conduction electrons 2 J t U > ( s S) = J impurity 0

29 Electron Scattering in the Perturbation Theory Scattering amplitude in the Born approximation: Kondo (1964) correction:

30 Electron Scattering in the Leading-Log Approx Logarithmic scaling, Abrikosov (1965); log RG, Anderson (1970): Laborde, SSCom. 71 electron loc. spin

31 Kondo singlet local spin density of conduction electrons magnetic impurity Cartoon: 2 spins-1/2 GS is a singlet for antiferromagnetic exchange (J > 0) unlike the cartoon, the conduction electrons are delocalized

32 Kondo singlet - favors for J > 0 local spin density of conduction electrons magnetic impurity screening cloud ground state: characteristic energy scale: Kondo temperature small interaction radius Kondo effect = lifting of the ground state degeneracy

33 Digression: : Resonant tunneling r ε 0 t T ( ε ) 2 = t = Γ 2 ( ε ε ) Γ (Breit - Wigner) transmission coefficient: 2 ( ) = sin δ ( ε ) T ε scattering phase shift T ( ε ) Γ T ( ε ) = 1 δ ( ε ) = π 2 δ ( ε ) Γ ε 0 ε ε 0 ε

34 localized level Resonant tunneling ε 0 density of states: ρ( ε) Γ ε ε 0 ρ ( ε ) resonance at ε = ε F localized level is half-occupied δ ( ε ) ( ) 1 δ ε = π ε ε 0 Γ ε ε F ρ( ε) ε N ε F ( ) = ρ ω dω = 1 2 ground state expectation value T π 1 = = max N = 1/ 2 2 ( ε ) = δ ( ε ) ρ ( ε ) F F F

35 Friedel Sum Rule N and δ(ε F ) are related! electrons: 1 N = δ ε F + δ ε F π ( ) ( ) spinless fermions: 1 N = δ ε F π ( ) = ( ) N = 2 δ ε δ ε = π F F ( ) no resonance Anderson impurity: t ε F ε 0 ε ε but ε ε 0 < F F 0 < U impurity level is singly occupied: N = 1 δ ε + δ ε = π ( ) ( ) F ( ) ( ) singlet ground state δ ε = δ ε = π 2 F F F T 2 ( ε ) δ ( ε ) F = sin = 1 F resonance!

36 From Scattering to Transport ikf x e ikf x re ikf x te reservoir leads reservoir dot V transmission coefficient I µ L ev µ R ε I = T = T = T µ L dp p e e e ( p ) d ( ) ( F ) ev h ε ε ε ε p h h µ R µ R < ε p < µ L velocity Landauer formula: 2 T = T =1 (resonance) G = 2e h conductance quantum: 2 e = + h ( T T ) G ( 25 Ω) 2 e h k 1

37 Transport in the Kondo regime Isolated dot: doubly-degenerate ground state Dot in contact with leads: Kondo singlet = = Ψ Kondo = 1 2 ( ) scattering phase shifts Conductance: Friedel sum rule: N, δ = π δ = π N 2 2 e e G = T + T = + h h 2 2 ( ) ( sin δ sin δ ) ground state expectation values number of electrons on the dot N = Ψ Nˆ Ψ N = N 2 Kondo Kondo =

38 Transport in the Kondo regime 2 2 e e G = T + T = + h h δ 2 2 ( ) ( sin δ sin δ ) = δ = π N / 2 G 2 2e h T = 0 2 2e G = π N h 2 sin 2 ( ) 0 odd even odd N 0 odd N: G = 2e 2 /h - perfect transmission even N: G = 0 - perfect blockade

39 Effect of a Magnetic Field What is necessary for the Kondo effect to occur? degeneracy interaction electron gas lifted by a magnetic field Zeeman energy: E E = B = gµ H B N 2 2e N G h Control parameter: B/T K ( B T ) 2 1 K Thermal fluctuations have similar effect: 0 T K B π 2 16 [ ln ( B T )] K 2 conductance (e 2 /h) T(K)

40 Other local degeneracies more Kondo effects δ E energy z S = 1 z S = 0 B =δe degeneracy 0 B = gµ H B V (mv) V g (V) even odd field 0 T 0.68 T Kondo effect recovers at B = δe and even electron number! 1.36 T quantum dot: 2 nm-thick nanotube bundle 2.04 T 300 nm J. Nygård, D.H. Cobden, P.E. Lindelof, Nature 408, 342 (2000) review: M. Pustilnik et. al. cond-mat/ ; LNP 579, 3 (2001)

41 Summary: Kondo effect in quantum dots Strong effect: lifting of the Coulomb blockade at low T conductance (e 2 /h) gate voltage (mv) Ubiquity in nanostructures: E δ E z S = 1 z S = 0 B = δ E 0 B interaction degeneracy electron gas always present can be tuned

42 Other stuff: Quantum Impurity systems Kondo effects: S>1/2, Multi-channel, SU(4), out-of-equilibrium Kondo in a carbon nanotube quantum dot (G. Finkelstein et al, 2006): 4 states, 2 channels

43 Other stuff: Evolution of a single electron spin trapped in a dot (GaAs: hyperfine fields) Precession of a single electron spin in the hyperfine field (C.M. Marcus, A. Yacoby, et al 2007)

44 Other stuff: Inelastic electron scattering mediated by magnetic impurity 1. Simplest inelastic process in a toy model only two electrons in the band, el.1 el.2 imp T-matrix: Born 2 nd order

45 Inelastic electron scattering Energy transferred in the collision: Scattering cross-section: 2. Full 2 nd order perturbation theory result Total cross-section averaged over :

46 Experiments on Energy Relaxation: Cu Experimental layout: Results: Cu wire

47 Experiments on Energy Relaxation: Ag Silver Cu

48 Energy Relaxation in Ag, Cu, and Au wires F. Pierre et al., JLTP 118, 437 (2000) and NATO Proceedings (cond-mat/ ) 1 Ag Cu f(e) U=400 µv U=100 µv 0 L=20µm D=200cm 2 /s -1 0 E/eU 1 L=5 µm D=93 cm 2 /s Au 0 L=5 µm D=130 cm 2 /s -1 0 E/eU

49 Energy Relaxation in Cu and Au Wires: Spins Rule! 2001 Au Cu Anthore, 2003