Title. examples of interplay of interactions and interference. Coulomb Blockade as a Probe of Interactions in Nanostructures

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1 Coulomb Blockade as a Probe of Interactions in Nanostructures Title Harold U. Baranger, Duke University Introduction: examples of interplay of interactions and interference We need a tool the Coulomb blockade and how it changes 1. Basic Coulomb Blockade conductance through a nearly isolated system. Mesoscopic Effects in the Coulomb Blockade fluctuations of CB peak heights and spacings 3. Kondo Effect in Quantum Dots

2 OUTLINE Mesoscopic Outline-Summary Effects and Level Quantization in the Coulomb Blockade 1. Evidence for mesoscopic and single level effects. Conductance through a single level 3. Interlude: Statistics of energy levels and wave functions 4. Statistics of peak heights, theory and experiment 5. Spacing between peaks simple theory falls flat on its face 6. A better expression for the ground state energy Strutinsky approach 7. The Universal Hamiltonian 8. Statistics of peak spacings, theory and experiment 9. Fluctuations of elastic co-tunneling more universality!

3 Evidence for single-particle and/or mesoscpic effects in CB CB data: Marcus Variation in peak heights [C. Marcus group, Harvard]

4 Peak Height Varies with Temperature! Height vs T peak height ~ constant peak height increases at lower temperature!! [Foxman, et al.; Kastner group, MIT `94]

5 Peak Height Varies with Magnetic Field! Height vs B Bottom: conductance in grey for 1 CB peak as a function of B and Vg Top: Gpeak as a function of B (along line highlighted in bottom) Sensitivity to weak magnetic field Interference effect! [Folk, et al.; Marcus group, Stanford `96]

6 Non-linear Transport Probes Single Particle States Non-lin. schem. Apply a bias voltage between the two leads connected to the dot: L lead dot R lead tunneling spectroscopy of individual quantum states [After D. Ralph]

7 Discrete States in Metallic Nanoparticles Discrete: metal peaks caused by individual levels [D. Ralph, et al., Harvard `97]

8 [Johnson, et al., Delft `9] [Tarucha & Kouwenhoven groups] CB peaks vs Vg for 3 different Vsd Discrete: semic. Discrete States in Semiconductor Quantum Dots

9 Energy Scales E C : the charging Energy energy scales : the mean single-particle level spacing Estimate the ratio: Using ν dot ~EF/N, ν dot L d e E C ~ κ L ( for spin; in d dimensions; κ is dielectric constant) (rs is electron gas intereaction param.) Level spacing is small compared to charging energy (except for 1d which, as always, is special)

10 Energy Scales (cont.) Γ: the Energy width of the scales single-particle () levels Γ is related to the conductance of the tunnel junction: Consider a grain attached by a single junction to a lead Suddenly apply a bias V ev/ occupied levels cross the Fermi level Escape time, τ esc, gives the width of the level: τ esc ~ ħ/γ I Comp. Tech. V e ev e Γ ~ ~ 4 π e G τ esc h α 4 π h Γ α = Eth: ~time for particle to explore the whole system ballistic: E th ~ h v F / L diffusive: E th ~ hd / L E th 1 g a semiclassical parameter h E th 1 4 N (ballistic d)

11 Coulomb Blockade: Classical vs. Quantum Class. vs. qm µ I 1 µ V gate Classical: e position: constant spacing = C peak height: constant given by series resistors Quantum: G peak G 1 G G + G position: single particle energies and residual interactions spacing fluctuations peak height: coupling from quantum state ψ in dot to lead height fluctuations = 1

12 CB Conductance Through a Single Level Previously, used a continuous G single ν dot requires lev T >. Now consider: Γ α < T < thermally activated transport (for peak!) Again, close to the degeneracy point N0*=1/, consider only charge states Electrostatic energy difference: Use rate equations for 1 level in dot! Have to keep track of spin now Ps Convert sum to integration (note: sum only over lead states!): Use: [Following Pustilnik & Glazman,`04]

13 CB Conductance Through a Single Level (cont.) Comments: G single () 1. The peak maximum is not at N0=N0 *, but rather off by ~T/EC. The lineshape is asymmetric 3. Peak height is approximately note the similarity to classical height with G Γ G/G T/ [Beenakker, van Houten, Staring] [Beenakker,`91]

14 Peak Height Statistics Heights (1) What do we know about the ΓL,R0?? The width of each level will be different depending on how it s wave function overlaps with the states in the leads. Point contact connection between lead and dot: r α φ ( t n n Γ, α ) α, n = πν α t α, n φ n ( r α ) r tunnel from lead α to point rα in the dot Need to know about the wave functions (and levels) in the dot Mesoscopic perspective: instead of trying to find Gpeak for a particular precise situation, let s look at the statistics of Gpeak.

15 Interlude: Random Matrix Theory (RMT) One of the (two) big ideas from RMT Quantum intro Chaos: The quantum properties of a classically chaotic system are statistically universal and the same as those of a random matrix. [the Bohigas-Giannoni-Schmidt conjecture, `84] Assume quantum dot or nanoparticle is irregular (ballistic or diffusive) motion classically chaotic use RMT for quantum properties Gaussian ensembles for the Hamiltonian: β=1 real symmetric matrix, time-reversal symmetry GOE β= complex hermitian matrix, broken time-reversal symmetry GUE Caution: some properties are system specific; use RMT for those that are independent of ensemble and so expected to be universal.

16 Interlude: RMT properties Energy level statistics: RMT lots of correlations! properties close levels repel { n } ε P(s) s ε ε n +1 n (Wigner surmise) Wave function (eigenvector) statistics: 1. Different eigenvectors Comp. are uncorrelated Tech. as N.. Components of the same eigenvector are uniformly distributed under the constraint of normalization: By integration, find the distribution of / i γ ψ i ψ (Porter-Thomas distribution)

17 Interlude: RMT in chaotic systems Nearest neighbor spacing distribution in a chaotic system RMT- chaotic Solid: RMT Wigner surmise Dashed: Poisson statistics (random) Expect to be valid for levels within a window of size E th (Bohigas, Giannoni, and Schmidt, `84)

18 Interlude: RMT in diffusive systems Disordered rectangular quantum dot (onsite disorder Anderson model) RMT- diffusive [Miller, Ullmo, HUB, `05] (caution: doesn t work well for all quantities )

19 Interlude: Spatial Correlation of ψ beyond RMT Need to know correlation RMTof wave spatial function in position psi space (interactions!) RMT: each element ψ i is independent (except for normalization) Need to go beyond RMT: Random Plane Waves ψ ( ψ * r ) = N 1 ikn ˆ a e α N α α = 1 r r Distribution of ň α and a α are uniform and ( r ) ψ ( r ) = independent (Berry, `77) r r 1 e N α 1 N ikn ˆ ( r r ) 1 ikn ˆ ( r r ) 1 r r a e α = α α F ( A A = r r Fermi Surf. ) r d F ( r ) = J 0 ( k F r r r ) r r A ψ ( ) ψ ( )

20 Peak Height Statistics (cont.) Heights () r Γ α φ, n n ( α ) We know the distribution of Γ, ie. Porter-Thomas distribution Assume Γ at the two contacts are independent Distribution of G peak follows from integration extract the dimensionless fluctuating part: α Γ Γ L 0 ( Γ + Γ 0) L 0 Γ R 0 R Can generalize to non-point contacts, multiple channels, effects of periodic orbits, temperature [Jalabert, Stone, and Alhassid, `9]

21 Peak Height Statistics (cont.) Plot of the distribution, compared to numerics for a ballistic billiard (the Robnik billiard) Heights (3) B=0 B 0 [Bruus & Stone, `94] Experiment: CB conductance peaks in a semiconductor quantum dot, temperatures B=0 Note how 3 peaks vanish at the lowest T no coupling!! [Chang, et al. `96]

22 Peak Height Statistics (cont.): Experiments Heights (4) [Marcus, et al. `96] [Chang, et al. `96] Simplest theory works well! More detailed study reveals a few problems but in quite good shape.

23 Spacing Between Peaks in G(Vg) Remember from the last lecture: Spacing (1) * g E gr N + 1) E gr ( ) V ( N E gr spacing E E N + 1) + E ( N 1) E ( N gr gr ( gr gr ) Try model Hamiltonian from last lecture (Constant Interaction Model CI): This does not work! Theoretically: No reason to think all contributions at the scale D are captured by the single particle physics. What about the interactions? Is E C all there is to the story even at that fine scale?? Experimentally:

24 Peak Spacings: Simplest Model and Experiment Standard up-down filling of the orbitals yields S=0 for N even S=1/ for N odd Const. Int.: Three groups studied Marcus, Sivan, and Ensslin. Nobody agrees with Constant Interaction model. [Ong, et al., Marcus group, unpublished]

25 Toward a Better E gnd : The Strutinsky Approach* Suppose one has a fairly Strutinsky good description 1 of some phenomenon at a classical (or macroscopic) level. How can one add quantum interference effects to leading order? macroscopic smooth quantities (ie. smooth DOS, ) quantum interference fluctuating or oscillating quantities Connect the two with a semiclassical expansion. Our small parameter: To be concrete: Comp. 1 1 Tech. 1 ~ ~ k F L N g smooth description: generalized Thomas-Fermi (GTF) includes charging, no interference, DOS and n(r) completely smooth mesoscopic description: density functional theory (DFT) adds interference *Brack, et al. Rev Mod Phys, 197 [D. Ullmo and HUB, `01]

26 Strutinsky (cont.): Semiclassical DFT Find Strutinsky assuming is small Solve the DFT equations order-by-oder in δn; need to go to nd order. In TF, the kinetic energy is a local density functional: self-consistent eq.: n GTF (r) d r = N

27 Strutinsky (cont.) In DFT, use Kohn-Sham theory Strutinsky to relate the kinetic 3 energy to orbitals: Big unknown: exchange-correlation functional (includes interacting part of kinetic energy) First, quantize in the TF potential: Note: is small but larger than δn because of screening!

28 Strutinsky (cont.): Results First Order Strutinsky 4 osc where mean single particle energy included in GTF. Very natural: quantize in the Thomas-Fermi potential! Has been used extensively in atomic and nuclear physics before justification by Strutinsky, see e.g. Maria Mayer 1941 who was studying the stability of heavy atoms. But not good for us: this is in the same spirit as the constant interaction model, ie. add single-particle levels to the classical potential energy. How can going to higher order possibly help??

29 Strutinsky (cont.): Results Second Order Strutinsky 5 Try to get rid of δn in favor of Can show: Also very natural! The ripples in the bare density interact via the screened interaction [D. Ullmo and HUB, `01]

30 Final Egnd

31 Statistics of Residual Interactions Statistics Mij δ-function interaction: Interplay of constant M, N with fluctuating

32 Constant Exchange and Interaction Model Consider large dot limit: kfl CEI or equivalently g Combine constant M s and N s into a single exchange parameter: Also known as the universal Hamiltonian [Kurland, Aleiner, Altshuler, `00] Filling of orbitals is not up-down: Probability of gnd. state spin, rs=1.3 [Brouwer Oreg & Halperin; Baranger Ullmo & Glazman]

33 Beyond Universality: 1/g Corrections Beyond Univ. Mij and Nij

34 CB Peak Spacings: Temperature Experiment Theory Spacings: T

35 The Universal Hamiltonian Start with bare Hamiltonian: Uham 1 Separate H into two parts based on semiclassical parameter 1/g: (1) Symmetry argument for the universal part, Suppose the low-energy single particle properties are described by RMT, ie. diffusive or irregular ballistic system. universal part of H should not depend on the basis construct it from the 3 operators which are invariant under RMT [Aleiner, Altshuler, and Glazman]

36 The Universal Hamiltonian (cont.) Uham charging exchange pairing Pairing: (a) time-reversed states doesn t exist for β= (b) even for b=1, it is small because of renormalization in the Cooper channel which makes Λ small Neglect it! (has been considered ) () Average matrix elements of the Hamiltonian using RMT ψ * ( r ) ψ ( r ) = r r 1 e N α 1 N ikn ˆ ( r r ) 1 ikn ˆ ( r r ) 1 r r a e α = α α F ( A A = r r Fermi Surf. ) for states within E th [Aleiner, Altshuler, and Glazman]

37 The Universal Hamiltonian (cont.) (3) Screening Uham 3 Can t use the bare interaction in above (divergence)! Carry out RPA calculation in a finite system Use screened interaction in expression for ES (natural ) Arrive at the same point for Egnd as in Strutinsky! [Aleiner, Altshuler, and Glazman]

38 Fluctuations of Elastic Cotunneling Flucts El-cotun 1 G e * sign( el ν L ν R t t h + ε Comp. Ln Rn E ε Tech. n C n n ) r α n φ ( t, n α r First, find the average. RMT the φ n ( α ) are uncorrelated in n and r t Ln t Rn keep only diagonal terms yesterday! ) G G 1 G + G E el L R = + * * 4 π e / h C N 0 N 0 N N Second, find the variance: sum of many random terms will it self average?..

39 Flucts of Elastic Cotunneling: Calculation of <G > G el Flucts sign( ε El-cotun 1 n ) sign( ε m ) sign( ε i ) sign( ε j ) n, m, k, j e ν h ε L ν R T n, m, i, j E C + ε n E C + ε m E C + ε i E C + j * * * * * * * * n, m, k, j t Ln t Rn t Lm t Rm t Li t Ri t Lj t Rj t Ln t Lm t Li t Lj t Rn t Rm t Ri t Rj T = Uncorrelated amplitudes must pair up to form intensities. n, m, k, j = t L t R { δ + δ δ δ δ } T + δ δ nm ij nj im β1 ni mj G el h e ν L ν R t L t R Time-reversal symmetry effect! 1 ( 4 β ) n, m E C + ε n E C + ε m As for mean, convert to integral, keep E c terms in the sum ( ) G el = 4 β G el δ G = G β el el 1 [Aleiner & Glazman]

40 Flucts of Elastic Cotunneling: Experiment El-cotunn Expts GaAs lateral quantum dot Unresolved puzzle with regard to Bc [Marcus group, Harvard]

41 OUTLINE Summary Mesoscopic Outline-Summary Effects and Level Quantization in the Coulomb Blockade 1. Evidence for mesoscopic and single level effects. Conductance through a single level 3. Interlude: Statistics of energy levels and wave functions 4. Statistics of peak heights, theory and experiment 5. Spacing between peaks simple theory falls flat on its face 6. A better expression for the ground state energy Strutinsky approach 7. The Universal Hamiltonian 8. Statistics of peak spacings, theory and experiment 9. Fluctuations of elastic co-tunneling more universality!

42 THE END

43 Title Template

44 Temperature Dependence of CB Spacing Distribution Further T [G. Usaj]