NIRT: Coherence and Correlation in Electronic Nanostructures (Duke - PDF document

NIRT: Coherence and Correlation in Electronic Nanostructures (Duke University) Start date: July 1, 2001 PIs: Harold Baranger Shailesh Chandrasekharan Weitao Yang post-docs: San-Huang Ke (DFT) James Osborn (QMC) [ → Utah] Gonzalo Usaj (RMT) [ → Yale] grad students: Hong Jiang (DFT) Ribhu Kaul (RMT) Anand Priyadarshee (QMC) undergrads: James Darsie Mike Miller joining soon: Denis Ullmo (Visiting Professor, Orsay) Martina Hentschel (post-doc) Amit Ghosal (post-doc) collaborators: Harsh Mathur (Case Western Reserve) Cyrus Umrigar (Cornell) DFT = density functional theory QMC = quantum Monte Carlo RMT = random matrix theory

1 Motivation I. Quantum Dots confinement ⇒ interference confinement ⇒ interactions ( charging + residual) ⇒ interplay between interactions and interference spin of dot?? → spintronics ... quantum computing ... − 100 − 1000 electrons [Marcus group] II. Molecular Electronics single molecule attached to leads • I-V curve? • non-equilibrium effects? • Kondo effect recently observed! V V g III. Nano– Magnetism, Superconductivity, ... [Ralph and McEuen groups] strong e-e interactions + confinement/disorder • fluctuation of anisotropy energy in magnetic nanoparticles • superconductor-insulator transition disordered metal films • metal-insulator transition in disordered 2DEG

Outline I. Random Matrix Theory Description of Quantum Dots • Coulomb blockade (CB) conductance peaks → fluctuation of ground state energy and spin − • the “universal Hamiltonian” • CB peak spacing distribution agrees with experiment II. Density Functional Theory of Quantum Dots • 2D LSDA calculation for isolated quantum dots 3D calculation with real gate configuration coming soon... • Addition energy for up to 200 electrons – large N matters! • Interaction effects much stronger than expected III. Quantum Monte Carlo Study of Disorder in Superconductivity • modified Hubbard model: localization ⇐ ⇒ strong e-e interactions • Kosterlitz-Thouless in clean; Superconductor-Insulator in dirty • exponential decay of pair correlations for arbitrarily small disorder!

Spacing of Coulomb Blockade Conductance Peaks Gonzalo Usaj, Denis Ullmo, and H.B. • weakly coupled dot ⇒ N is well defined • N → N +1 costs energy ∼ e 2 /C ⇒ blockade. • E N ( V ∗ g )= E N +1 ( V ∗ g ) ⇒ G peaks. [Patel et al. ] GS − eN C g − e ( N + 1) C g E N g = E N +1 C V ∗ C V ∗ g GS • peak position V ∗ g ∝ E N +1 GS − E N GS ⇒ spacing ∝ ( E N +1 − E N GS ) − ( E N GS − E N − 1 = ) GS GS • Fluctuation of spacing ⇔ Fluctuation of GS energy of the dot.

An effective Hamiltonian • only energy levels around the Fermi energy are relevant • chaotic dynamics → RMT for single-particle properties (energy window up to E th ∼ ¯ h/t flight ) • Screening is important ⇒ weak interaction V sc ( r 1 , r 2 )= e 2 C + V TF ( r 1 , r 2 )+ V ( r 1 )∆+ V ( r 2 )∆ ⇓ α,σ c β,σ X α,β + 1 ˆ H α,β,γ,δ c † δ,σ c † n 2 +ˆ � c † � H int = E C ˆ n γ,σ ′ c β,σ ′ c α,σ 2 α,β,σ α,β,γ,δ � � d r 1 d r 2 Ψ ∗ δ ( r 1 )Ψ ∗ d r V ( r )Ψ ∗ H α,β,γ,δ = γ ( r 2 ) V TF ( r 1 − r 2 )Ψ β ( r 2 )Ψ α ( r 1 ) X α,β =∆ α ( r )Ψ β ( r ) • Fluctuations Ψ α ( r ) ⇒ fluctuations H α,β,γ,δ and X α,β √ • Fluctuations are small ⇒ controlled by 1 /k F A (or 1 /g , g = E th / ∆ ) . ⇒ expand ˆ H int in powers of 1 /g [Blanter, Mirlin, Muzykanskii; Aleiner, Brouwer, Glazman; Altshuler]

CEI model ( g → ∞ ) Large N limit ⇒ �H α,β,γ,δ � = J ′ δ α,δ δ β,γ + J S δ α,γ δ β,δ dominates ⇒ � n − N ) 2 − J S � S 2 H CEI = ǫ α ˆ n α,σ + E C (ˆ α,σ [Kurland, Aleiner,Altshuler] • J S < ∼ ∆ / 2 ⇒ main reason CI model is wrong ∆ 1 • Interplay of J S and { ǫ α } ⇒ S = 1 if ∆ 1 − 2 J S < 0 2 even Probability Density odd 1.5 0.99 1 1 0.79 1 P(S) 0.5 0.5 0.5 0.21 0.01 0 0 0 0 -0.5 -0.5 0 0 0.5 0.5 1 1 1.5 1.5 0 0 0.5 0.5 1 1 1.5 1.5 S spacing [ ∆] r s ∼ 1 ⇒ J s ≈ 0 . 3∆ [Ullmo, Baranger]

5 Experiment Theory 4 k B T = 0.3 ∆ "Even" Probability density 2.0 r.m.s. = 0.235 ∆ 3 Probability Distribution P(s) 1.5 2 1.0 even odd 1 0.5 "Odd" 0.0 -0.5 -0.5 0 0 0.5 0.5 1 1 -1.0 -0.5 0.0 0.5 1.0 Spacing ( ∆ ) spacing [ ∆] [Patel et al. ; Ong et al. ] Theory includes leading corrections to CEI model + temperature • “scrambling”: added electron changes mean-£eld potential and so { ǫ i } • “gate effect”: gate voltage which increases N also changes interference • residual interactions: fluctuations of diagonal terms in H α,β,γ,δ • k B T : compare to ∆ − 2 J s − → important! Note: even/odd effect, no δ -function, large spacing tail in even

T � = 0 PSD: including corrections 2 2 2 2 k B T = 0.1 ∆ k B T = 0.3 ∆ Probability density Probability density r.m.s. = 0.286 ∆ r.m.s. = 0.235 ∆ 1.5 1.5 1.5 1.5 1 1 1 1 even odd 0.5 0.5 0.5 0.5 0 0 0 0 -0.5 -0.5 0 0 0.5 0.5 1 1 1.5 1.5 -0.5 -0.5 0 0 0.5 0.5 1 1 1.5 1.5 spacing [ ∆] spacing [ ∆] • Notice double peak structure at low T • At k B T ∼ 0 . 3∆ , temperature effect dominates ⇒ lower T experiments are needed!

Density Functional Study of CB in Quantum Dots Hong Jiang, Weitao Yang, and H.B. 2D Model System: � x 4 � b + by 4 + 2 λx 2 y 2 + γ ( x 2 y − xy 2 ) V ( x, y ) = a 0.8 Calculated NNS distribution Wigner GOE distribution 0.6 20 Distribution 10 0.4 0 0.2 -10 -20 0 0 1 2 3 4 Spacing -20 -10 0 10 20 a = 1 . 0 × 10 − 4 , Nearest neighbouring spacing distribution for b = π 4 single-particle spectrum in 1 symmetry class. λ = − 0 . 6 , γ = 0 . 0

Computational Techniques DFT-Kohn-Sham Method for e-e interaction � � 2 m ∗ ∇ 2 + e 2 | r − r ′ | d r ′ + δ E xc [ ρ,ζ ] h 2 ρ ( r ′ ) − ¯ � ψ σ i ( r ) = ǫ σ i ψ σ + V ext ( r ) i ( r ) κ δρ σ ( r ) • 2D Local-spin density approximation (LSDA) and Tanatar-Ceperley parameterized form for exchange-correlation functional; • Fast-sine transform for kinetic energy operator, and real space representation of wave functions; • Preconditioned conjugate-gradient method for Kohn-Sham equations. • Fast-Fourier transform method for Hartree potential 3

Data Analysis: Fitting Raw data and fitting Smooth part removed. 0.02 0.19 Polynomial Fitting Raw Data 0.17 0.01 Addition Energy Addition Energy 0.15 0.13 0 0.11 0.09 −0.01 0.07 0.05 −0.02 0 50 100 150 200 0 50 100 150 200 Electron Number Electron Number 6

N=20-80 N=80-140 N=140-200 1.5 (a) (b) (c) P(S) P(S) P(S) 0.5 0.5 0.5 Symmetric 1 Probability Density 0 0 0 0 1 2 0 1 2 0 1 2 0.5 0 1.5 (d) (e) (f) P(S) P(S) P(S) 0.5 0.5 0.5 Asymmetric 1 0 0 0 0 1 2 0 1 2 0 1 2 0.5 0 -1 0 1 2 -1 0 1 2 -1 0 1 2 Spacing

• 2D LSDA model of isolated quantum dot with chaotic external potential • Statistical properties of addition energy for up to 200 electrons • Electron number matters! • Symmetry matters! Why are interactions so strong??

Disorder Induced Superconductor-Insulator Transition James Osborn, Shailesh Chandrasekharan, and H.B. A Modified Hubbard Model: � j,σ c i,σ ) ( n i + n j − 1)( n i + n j − 3) + 2 � S i · � S j + 2 � J i · � σ ( c † i,σ c j,σ + c † H = � � J j <i,j> n i, ↑ − 1 n i, ↓ − 1 n j, ↑ − 1 n j, ↓ − 1 � � � � � � � � � − 4 2 2 2 2 n i, ↑ − 1 n i, ↓ − 1 � � � � � � � + U − µ i n i 2 2 i with n i = n i, ↑ + n i, ↓ and i = 1 i = ( − 1) i c † i = ( − 1) i c i, ↓ c i, ↑ , i, ↑ c † J + J − J 3 i, ↓ , 2 ( n i, ↑ + n i, ↓ − 1) µ i = µ + Wr i with − 1 < r i < 1. Disorder: • Same symmetries as standard Hubbard model: SU(2) spin ; SU(2) charge broken by µ to U(1) number • Shows s-wave superconductivity for U ≤ 0 (except at half-filling where antiferromagnetism is stable) • Not free for U = 0. ⇒ Localization ← → Strong e-e Interactions =

Superfluid Density vs. Disorder β = 5 8 7 L = 8 L = 16 6 L = 24 L = 32 5 2 π ρ s / T 4 3 2 1 0 0 1 2 3 4 disorder (W) ρ s ≡ T ( W x / 2) 2 + ( W y / 2) 2 � � 4 where W x is the total number of particles winding around the boundary in the x direction In the clean case, √ √ 2 π T ρ s = 2 + A coth( A log( L/L 0 )) for T < T c with A = 0 at T = T c .

Pair Susceptibility vs. Length β = 5 10000 W = 0 W = 0.05 W = 0.1 pair susceptibility 1000 100 8 16 32 64 128 L � β 1 e − ( β − t ) H (∆ † + ∆) e − tH (∆ † + ∆) � � � χ L = 0 d t Tr with ∆ ≡ x c x, ↓ c x, ↑ ZV In the clean case for T < T c , χ L ∝ L 2 − η ( T ) with η ( T c ) = 1 / 4 and η (0) = 0

Pair Susceptibility vs. Disorder β = 5 15000 15000 L = 128 10000 10000 L = 96 L = 64 L = 32 10000 10000 L = 16 L = 8 5000 5000 1000 1000 pair susceptibility 0 0 0 0 0.1 0.2 0.3 0.4 0.5 100 100 10 10 0 0 1 1 2 2 3 3 4 4 disorder