Coulomb-Blockade and Quantum Critical Points in Quantum Dots

Transcription

1 Coulomb-Blockade and Quantum Critical Points in Quantum Dots Frithjof B Anders Institut für theoretische Physik, Universität Bremen, Germany funded by the NIC Jülich Collaborators: Theory: Experiment: E. Lebanon 1, A. Schiller 2 R. Potok, D. Goldhaber-Gordon 1 Rutgers University, USA Stanford University, USA 2 Hebrew University, Israel 30. July 2004, SCES 2004, Karlsruhe p.1/13

2 Quantum Critical Point and Non-Fermi Liquid T Fermi Liquid x c Magnetic Phase SDW x 30. July 2004, SCES 2004, Karlsruhe p.2/13

3 Quantum Critical Point and Non-Fermi Liquid T Fermi Liquid NFL example: x c bulk Magnetic Phase SDW x CeCu 6 x Ag x MnSi, CeCu 2 Si July 2004, SCES 2004, Karlsruhe p.2/13

4 Quantum Critical Point and Non-Fermi Liquid T NFL T NFL Fermi Liquid Fermi Liquid Fermi Liquid example: x c bulk Magnetic Phase SDW x x c quantum impurity CeCu 6 x Ag x MnSi, CeCu 2 Si 2 Candidate: coupled quantum dot x 30. July 2004, SCES 2004, Karlsruhe p.2/13

5 Contents 1. Introduction Quantum dot and Coulomb blockade Low energy decription of quantum dots Lifting of Coulomb blockade: Kondo effect 2. Coupled Quantum dot Dynamical channel symmetry Quantum critical point and non-fermi liquid 3. Results Spin susceptibilty: indicator for NFL Entropy near the quantum critical point Scaling of the zero bias conductance 4. Summary 30. July 2004, SCES 2004, Karlsruhe p.3/13

6 Nano-Devices: Coulomb-Blockade in Quantum-Dots Ĥ c = e2 2C 0 ˆN 2 B + V g ˆQB = E c ( ˆNB N g ) 2 + E(Vg ) Goldhaber-Gordon, Nature 98 electron in a box plus charging energy: E c = e2 2C July 2004, SCES 2004, Karlsruhe p.4/13

7 Nano-Devices: Coulomb-Blockade in Quantum-Dots Ĥ c = e2 2C 0 ˆN 2 B + V g ˆQB = E c ( ˆNB N g ) 2 + E(Vg ) Goldhaber-Gordon, Nature 98 E c k B T : classical regime 3 <Q>/e Q = C 0 V g 2 1 1/2 3/2 5/2 N g =VC 0 /e 30. July 2004, SCES 2004, Karlsruhe p.4/13

8 Nano-Devices: Coulomb-Blockade in Quantum-Dots Ĥ c = e2 2C 0 ˆN 2 B + V g ˆQB = E c ( ˆNB N g ) 2 + E(Vg ) Goldhaber-Gordon, Nature 98 E c k B T : classical regime E box N 3 <Q>/e Q = C 0 V g E box 2 E c k B T : quantum regime Coulomb blockade regime: N g n N 1 N g =VC 0 /e Conduction peaks: N g n /2 3/2 5/2 30. July 2004, SCES 2004, Karlsruhe p.4/13

9 Nano-Devices: Coulomb-Blockade in Quantum-Dots Goldhaber-Gordon, Nature 98 E c k B T : classical regime E box N 3 <Q>/e Q = C 0 V g E box 2 E c k B T : quantum regime Coulomb blockade regime: N g n N 1 N g =VC 0 /e Conduction peaks: N g n /2 3/2 5/2 30. July 2004, SCES 2004, Karlsruhe p.4/13

10 Low Energy Description of Quantum Dots H = ɛ kσ c c kσl kσl + ( ) 2 ɛ mσ c mσb c mσb + E c ˆNB N g kσ mσ + ) t B (c 0σL c 0σB + h.c σ level spacing E B t 2 πρ L = Γ L quasi-continuous levels = quantum box: free-electron gas +Ĥc level spacing E B Γ L discrete levels = effective single-channel Anderson model 30. July 2004, SCES 2004, Karlsruhe p.5/13

11 Lifting of Coulomb blockade: Kondo Effect v v v E i E f J > 0: T K D e 1/(Jρ) ω, T < T K : universality zero bias peak oldhaber-gordon, Nature 1998, Kouvenhoven, Science July 2004, SCES 2004, Karlsruhe p.6/13

12 Coupled Quantum dots: Effective Quantum Impurity H = α=l,b kσ + α=l,b ɛ αkσ c αkσ c ( ) 2 αkσ + E d n d σ + Un d n d + E c ˆNb N g t α σ σ ) (c α0σ d σ + d σc α0σ 30. July 2004, SCES 2004, Karlsruhe p.7/13

13 Coupled Quantum dots: Effective Quantum Impurity H = α=l,b kσ +t L σ ɛ αkσ c αkσ c ( ) 2 αkσ + E d n d σ + Un d n d + E c ˆN Ng } σ {c L0σd σ + d σc L0σ + t B (N + c B0σd σ + d σc B0σ N ) σ Matveev 91, König and Schöller PRL 98, Lebanon, Schiller, FBA., PRB 2003 Decoupling of charge DOF in the Q-Box: ˆN = n n n, N + = n + 1 n n n solved using Wilson s NRG 30. July 2004, SCES 2004, Karlsruhe p.7/13

14 Dynamical Channel Symmetry leads SET E c < T T 0 leads SET E > T c Dynamic generation of two-channels: 2 Coulomb Energy: H c = E c ˆNB N g leads SET E c > T Coulomb-blockade blocks electron transfer from quantum box to the lead effective two-channel Kondo model is formed for J L eff = J B eff Oreg and Goldhaber-Gordon, PRL 90, (2003) 30. July 2004, SCES 2004, Karlsruhe p.8/13

15 Two Channel Kondo Effect SET channel conservation law: H int = J α SET s α Sdot with J > 0 - T K *χ(t) magnetic susceptibility single channel model two channel model T/T K ω, T < T K : universality but for T 0: no global singlet χ(t ), γ(t ) log(t/t K ) = non-fermi liquid 30. July 2004, SCES 2004, Karlsruhe p.9/13

16 Spin Susceptibility as Indicator for NFL Fermi Liquid: χ(t 0) = const. χ(t 0) 1/T K non-fermi liquid χ loc (T ) log(t/t K ) quantum-critical point: t c = t c B /tc L 30. July 2004, SCES 2004, Karlsruhe p.10/13

17 Spin Susceptibility as Indicator for NFL T K χ loc T K U=10, E c =1, Γ 0 =πρt L 2 = E d E d =-5 E d =-2 E d =-1 E d =0 E d =0.25 Fermi Liquid: χ(t 0) = const. χ(t 0) 1/T K non-fermi liquid χ loc (T ) log(t/t K ) quantum-critical point: t c = t c B /tc L T/T K Definition of T K : χ mag (T ) = 1 20T K log(t/t K ) + b 30. July 2004, SCES 2004, Karlsruhe p.10/13

18 Entropy of the Coupled Quantum dots S loc in [k B log(2)] S loc in [k B log(2)] (a) (b) T/ T/(t B /t L -t c )^2 E c SC Fixed point instable: S QCP = 1 2 log(2) T low (t B/t L t c ) 2 T K QCP separates two FL regimes Parameter: D = 10; ɛ d = 1; U = 2; E C = 0.01 Γ L = t 2 L πρ(0) = July 2004, SCES 2004, Karlsruhe p.11/13

19 Zero-Bias Conductance: Asymmetric Model Units: G 0 = 2e2 h U = 10E c E d = 0 4t 2 l1 t2 l2 (t 2 l1 + t2 l2 )2 30. July 2004, SCES 2004, Karlsruhe p.12/13

20 Zero-Bias Conductance: Symmetric Model Units: U = 10E c E d = U/2 G 0 = 2e2 h 4t 2 l1 t2 l2 (t 2 l1 + t2 l2 )2 30. July 2004, SCES 2004, Karlsruhe p.12/13

21 Zero-Bias Conductance: Scaling Curve E D =-5,U=10,E c =1,N= Conductance G(x)/G t B <t c t B >t c 4x x=t*t K /(t box -t c ) 2*1.06 Symmetric model Scaling QCP: two FL T low = (t B t c ) 2 T K G > (x) ( T T low ) 2 G < (x) 1 ( T T low ) July 2004, SCES 2004, Karlsruhe p.12/13

22 Summary A line of NFL fixed points is found in in coupled quantum dot devices These fixed points are characterized by the number of particles in the box and the ratio of the tunneling rates t B /t L They govern the crossover from one Fermi-liquid to another Charge and spin degrees of freedom are usually entangled Phase Diagram: QC Line t B / t L FL Low Conductance High Conductance FL 1 1/2 0 0 Charge 2CK 1 N g 30. July 2004, SCES 2004, Karlsruhe p.13/13