Fermi-edge Singularities in Mesoscopic Systems: From Quantum Dots to Graphene

Transcription

1 Fermi-edge Singularities in Mesoscopic Systems: From Quantum Dots to Graphene Martina Hentschel MPIPKS Dresden, Germany Collaboration: Swarnali Bandopadhyay (Postdoc) Georg Röder (PhD student) Paco Guinea (Madrid) -4 (Duke Univ): Harold Baranger, Denis Ullmo Motivation Many-body effects interesting!! interesting Mesoscopic systems Introduction Fermi-edge singularities: Examples, Metallic x-ray edge problem Metallic x-ray edge problem: Expectations when going mesoscopic I. Anderson catastrophe in mesoscopic systems Chaotic vs. integrable systems vs. graphene II. Mesoscopic X-ray Edge Problem Photoabsorption spectra for chaotic quantum dots and nanoparticles: Small is different!

2 Examples : Fermi-edge singularities unexpected behaviour at threshold (Fermi) energies Magnetic-field-induced singularities in spin-dependent tunneling through InAs quantum dots Hapke-Wurst et al. (Haug group, Hannover), PRB Examples : FES in Photoluminescence in Quantum Wells (d) Observation of a many-body edge singularity in quantum-well luminescence spectra Skolnick et at., PRL (987)

3 Examples 3: FES in Photoluminescence in Quantum Wires (d) Large optical singularities of the dim electron gas in semiconductor quantum wires Calleja et at., Solid State Comm. (99) Explanation: Oreg and Finkelstein, PRB (996) Resonance in the Fermi-edge singularity of dim. Systems, physics is similar to the Kondo resonance Examples 4: FES-type resonances in carbon nanotubes (d) Many-body effects in finite metallic carbon nanotubes F. Guinea, PRL (5) 3

4 Examples 5: The x-ray edge problem in metals (3d) enhanced photo-absorption at the Fermi edge inhibited photo-emission at the Fermi edge 77 K Absorption [a. U.] 3 K K.7 K E [e V] from K. Ohtaka and Y. Tanabe, RMP 6, 99 (99) Ł X-ray Edge Problem for Metals 96s: Singularities in Photoabsorption Spectra of Metals at the Fermi Energy Theory: Anderson, Mahan (969), Nozières, DeDominicis et al. (969), Ohtaka and Tanabe (98ies) Experiments: Citrin (and many others), see Review 979 Bulk Metal: X-ray Absorption E F conduction band X-ray Photoabsorption Cross Section A(ω) Expected: ω th ω h ν Absorption E C Observed: A(ω) Peaked edge core level Rounded edge ω th ω reduce size of system: metal metallic nanoparticle quantum well quantum dot Photo-absorption cross section =? 4

5 Ł Many-body response to a sudden perturbation: Anderson orthogonality catastrophe in metals + Ł Response of ~ 3 electrons to sudden perturbation of 3 electrons < ψ i ϕ i > < ~ all M ~ 3 electrons overlap of many-body wave functions < Ψ Φ > Anderson Orthogonality Catastrophe < Ψ Φ > M ε universal many-body response mechanism to a sudden perturbation M P.W. Anderson, PRL (967) Ł Anderson Orthogonality Catastrophe (AOC) X-ray edge problem: causes the rounded edge Mössbauer effect Α(ω) Ł So what about the peaked edge? second, counteracting many-body effect: Mahan-Nozières-DeDominicis (MND) response (Mahan s enhancement, Mahan s exciton) acts only on optically excited channel (if dipole selection rules between core and conduction electrons fulfilled) ω th Α(ω) ω th ω ω depends on symmetry: core electron p-like (L,3 -edge) peaked L-edge s-like (K-edge) rounded K-edge (AOC only) 5

6 β Ł Plug-in: Metallic (bulk) x-ray edge problem (fully understood) Photoabsorption Α(ω) (ω - ω th ) β Power law, exponent β : δl δ = (l + ) lo π π l Α(ω) AOC (all δ l ) MND (δ lo only) Α(ω) ω th ω ω th ω Friedel sum rule: l δ (l + ) l = Z π Mahan, Nozieres, DeDominicis, Ohtaka, Tanabe, : 96ies-98ies Ł Now: Towards the Mesoscopic Regime Interference when interference matters! Microscopic world (quantum mechanical) nm Mesoscopic regime (semiclassical) ~ nm µm Decoherence Macroscopic world (classical). mm System size L syst l ϕ Coherence and Interference Aharonov-Bohm effect Weak localization geometry-dependent Mesoscopic fluctuations (e.g. UCF) Quantum mechanical description discrete energy levels, wave functions l ϕ Magnetoresistance R (B) B= Magnetic field B Chang et al., PRL 994 6

7 Mesoscopic transport properties Geometry and dynamics of the system when Quantum Chaos comes in! chaotic mixed regular µm Marcus Lab (Harvard) µm A.Bäcker (TU Dresden) M.H. universal Differences between bulk and mesoscopic-coherent systems? Geometry dependence of mesoscopic many-body signatures? Excursion: Classical Billiards trajectory of particle (billiard ball) regular, predictable for long times integrable systems e.g. billiard tables chaotic systems e.g. generic mesoscopic devices extreme sensitivity to initial conditions: trajectories irregular after few reflections, unpredictable for long times A.Bäcker, nlin.cd/46 7

8 Possible (and feasible) Experimental Realizations Many-body response of a mesoscopic system to a perturbation x-ray photoabsorption with metallic nanoparticles: becoming feasible now double quantum dots: constriction µ -wave photoabsorption via impurity states in semiconductor heterostructures Quantum Dot Array (diam.~ nm) etched from heterostructure Chang Lab (Purdue/Duke) GaAs DEG and impurities Control Experiment bulk-like : extended DEG with impurities (already performed?) I. Anderson Orthogonality Catastrophe A: Chaotic vs. integrable systems B: Graphene M.H., D. Ullmo, and H. U. Baranger, PRL 93, 7687 (4), M.H., D. Ullmo, and H. U. Baranger, PRB 7, 353 (5), M.H., G. Röder, and D. Ullmo, Prog. Theor. Phys. Suppl. 66, (7), M.H., D. Ullmo, and H. U. Baranger, to appear in PRB, M.H. and P. Guinea, PRB 76, 547 (7), G. Röder, S. Bandopadhyay, M.H, in prep. 8

9 κ κ Ł Model: AOC for a localized (rank-) perturbation N chaotic or regular levels subject to perturbation unperturbed: perturbed: Ĥ Ĥ + Vˆ c { { k,, k }... }... Fluctuations = localized (rank-) perturbation N v c k k' * + k (rc ) k' (rc ) c k c k' K. Ohtaka, Y. Tanabe, series of PRBs 98ies, RMP 99 for bulk i) assume { ε k } GOE / GUE distribution { ϕ k (r c ) } Porter-Thomas distribution i.e. Random Matrix Theory for chaotic quantum dots, nanoparticles I. Aleiner, K. Matveev, PRL 998 ii) consider specific regular systems with well defined, but non-universal, solutions Ł Description of coherent-chaotic Nanosystems Many mesoscopic samples are chaotic systems. Angular momentum is not a good quantum number. Quantum chaotic systems Random Matrix Theory Hamiltonian = random matrix with Gaussian distributed entries energy levels = eigenvalues of random matrix wave functions = eigenvectors of random matrix statistics depends on time reversal symmetry (TRS) P(s) d Wigner surmise 3 4 n.n. energy spacing s P(A).5.5 Porter-Thomas distribution GUE (no TRS) GOE (with TRS) 3 4 wave function intensity A 9

10 φ λ κ ε Ψ Φ ε λ δ λ ε π ε Perturbed energies λ κ from { ε k }: k k (rc ) = k N v c Example: ε k equidistant, φ k constant = bulklike case = M f(λ) = Σ k /(λ ε k ) Phase shift δ o : N i= j= M + filled empty ε λ ε j i )( j i ) ( j i )( j i ) ( λ o λ λ ( i d Anderson overlap (between perturbed and unperturbed many-body ground states) ε ε ε ε 3 ε 4 ε 5 d λ λ λ δ o λ 3 λ 4 λ 5 i ) = v c large v c small π arctan N- j M- i v d c l.h.s. r.h.s. d E F can easily compute distribution of Anderson overlaps (in chaotic case: using a Metropolis algorithm as tool) {ε } {λ } I. Anderson Orthogonality Catastrophe A: Chaotic vs. integrable systems B: Graphene M.H., D. Ullmo, and H. U. Baranger, PRL 93, 7687 (4), M.H., D. Ullmo, and H. U. Baranger, PRB 7, 353 (5), M.H., G. Röder, and D. Ullmo, Prog. Theor. Phys. Suppl. 66, (7), M.H., D. Ullmo, and H. U. Baranger, accepted in PRB, arxiv:76.6 (7) G. Röder, S. Bandopadhyay, M.H, in prep.

11 Ψ Φ λ ε λ ε Ł Reference System: bulk-like situation: equidistant level, amplitudes const. Anderson Elektronen Elektronen Überlapp Störungsstärke Bandfüllung Fluctuations in a mesoscopic system:.5 - Bandfüllung M = N i= j= M + filled empty ( λ j i )( ε j i ) ( λ j i )( ε j i ) circular quantum dot with hard walls (Georg Röder), r c =.77 Störungsstärke -.5 Bandfüllung -.5 Bandfüllung Origin of mesoscopic fluctuations: Disk Effective perturbation strength depends on v c on wave function amplitudes at r c, mostly on those near E F = zeros of (Bessel) wave function at Fermi energy eff. pert. = no AOC Ł Result: mesoscopic fluctuations originate at the Fermi energy

12 Distribution of AOC overlaps chaotic systems integrable systems (disk) all at half filling B=, v c /d=-.5, N B finite, N=, v c /d P( ) v c /d = -.5, M = N/ N= N=5 N=5 N= N=5 N= b CUE double-peak structure due to lifted degeneracies P( ) COE (B=) Significant differences between chaotic and integrable systems (esp. probability for overlap= for finite B) Reason: mostly amplitude statistics Parabolic quantum dots (Swarnali Bandopadhyay) realistic description of systems with fewer electrons energy levels are highly degenerate (lifted in magnetic field B) E B ~ B / Ł Shell effects d.8 Overlap M/N - two energy scales: intershell (d) and intrashell (w c = f(b)) spacing - two perturbation scales: v c /d and w c (B) scaling: only dependence is on v c /w c for very small v c /w c <.5 AOC overlap drops whenever a new shell is started: due to degeneracy

13 I. Anderson Orthogonality Catastrophe A: Chaotic vs. integrable systems B: Graphene M.H. and P. Guinea, PRB 76, 547 (7) Graphene exists! 3

14 Atomic Structure of Graphene Ł Vanishing DOS at Dirac Points Thanks to Eduardo Mucciolo for this slide. AOC in Graphene Bulk-like Elektronen - e-.5 band filling Anderson overlap - Elektronen.5 - Graphene Störungsstärke Störungsstärke e- Störungsstärke scaled pertubationstörungsstärke strength Störungsstärke Mesoscopic.5 band filling - filling band filling Dirac point region unchanged when increasing N 4

15 Reason: DOS vanishes at Dirac Point no particles available to react to the perturbation Anderson overlap scaled pertubation strength Störungsstärke Anderson Overlap at Dirac point next to Dirac point -.5 filling band filling Cluster size N Result holds for different perturbation strengths at DP next to DP Overlap at DP Overlap at DP Scaled perturbation Overlap.. # particles / Cluster size N power law recovered very close to DP or in the presence of zero-energy states (edge states, midgap states due to impurities) Ł The presence or absence of zero-energy states significantly influences AOC as well as Kondo physics. (dependence on gate voltages, edge structure, and cleanness of sample) 5

16 II. Mesoscopic X-ray Edge Problem Mesoscopic Photoabsorption Spectra for Chaotic Nanosystems M.H., D. Ullmo, and H. U. Baranger, PRL 93, 7687 (4) M.H., D. Ullmo, and H. U. Baranger, PRB 7, 353 (5) M.H., D. Ullmo, and H. U. Baranger, accepted in PRB, arxiv:76.6 (7) Ł Towards Mesoscopic Absorption Spectra Fermi golden rule approach K. Ohtaka, Y. Tanabe: Series of papers, PRB 98ies, Rev. Mod. Phys. 6, 99 (99) N ω A( ) = Ψ ˆ Φ ( E - E ) with ˆ + π F D c δ F i D = wcj [ c j cc + h.c.] j= F j γ E F M δ F i µ {ε} {λ} h ω core core c o direct process replacement shake-up ~ w cj ~ Σ µ w cµ µγ repl. direct AOC MND 6

17 Ł Averaged Photoabsorption Cross Section K-edge Α(ω).5.5 direct total (direct+repl.+shake up) naive bare (norm.) shake up slightly peaked K-edge L-edge Α(ω) ( ) / d (ω ω th )/d largely peaked L-edge - replacement processes near E F and through bound state (L-edge) dominate - shake-up: one-pair processes dominate the contribution - spectator spin: AOC only Ł Dependence on perturbation strength K-edge (mesoscopic) Α(ω) ω ω th / d small perturbation (v c /d=-.3): metallic nanoparticles large perturbation (v c /d=-): quantum dots L-edge (mesoscopic and bulk-like) Α(ω) 4 3 N=, M=5, CUE v c = - v c = -.3 bare ω ω th / d Special thanks for discussion to Prof. Ohtaka 7

18 ψ Ł Averaged Photoabsorption cross section: Mesoscopic vs. Bulk-like v c /d = -, COE 3 L-edge mesoscopic & bulk-like Α(ω).3 K-edge mesoscopic K-edge bulk-like (ω ω th ) / d Rounded K-edge goes into a (slightly) peaked K-edge as the system becomes coherent M. Hentschel, D. Ullmo, and H.U. Baranger, PRL 93, 7687 (4) Ł Reason: Dipole matrix element wcj = φc r E ψ j loc. bulk-like r r ik j e = Jl ( r r kr ) ψ j l ilθ e partial wave decomposition of Bloch wave, l conserved mesoscopic- chaotic j = ar r r k j k, k = k j j j r r ik j e random superposition of plane waves, l not conserved ψ j f ({ε k },{λ k }) ψ j Porter-Thomas K-edge (s-like φ c ) w cj ψ j = L-edge (p-like φ c ) w cj ψ j 8

19 Photoabsorption: Dependence on electron number (M=N/) K-edge (mesoscopic) Α(ω) norm..9 weak perturbation (v c /d=-.3) (metallic nanoparticles).8 N=, M=.7 N=, M=5 N=5, M=5.6 N=4, M=.5 power law bulk filled: bulklike band filling ω ω th +.5d / N d Α(ω) norm..5.5 strong perturbation (v c /d=-) (quantum dots) band filling ω ω ω th ω +.5d / N d L-edge (mesoscopic) Α(ω) norm band filling ω ω th +.5d / N d norm. Α(ω) 3* 4 * band filling ω ω ω th ω +.5d / N d What to expect in real measurements? Measuring individual photoabsorption spectra of, e.g., four different quantum dots (a) A(ω) (c) A(ω) num. photo-absorption direct/repl. processes direct/repl. with shake-up exp. resolution d/ (ω ω th ) / d x-ray energy wrt st signal (b) A(ω) (d) A(ω) (ω ω th ) / d 9

20 Ł Mesoscopic fluctuations in A(ω) = f (fluctuations of, repl, shake-up, w cj ) P(A(ω) / Α(ω) ).5.5 v c /d = -, COE ω=ω th ω=ω th + d ω=ω th + d P.Th. K-edge w jc ~ ψ j large Porter-Thomas like fluctuations overwhelm overlap correlations and dominate fluctuations of A(ω) A(ω) / Α(ω) P(A(ω) / Α(ω) ) A(ω) / Α(ω) L-edge w jc ~ ψ j narrowly distributed Symmetry: replacement through bound state acts like a ground state overlap with δ F = π δ F, resulting in highly peaked edge Thanks to Paco Guinea (Madrid), Harold Baranger (Duke), Denis Ullmo (Paris), Eduardo Mucciolo (UCF Orlando) Mesoscopic Systems Group at MPIPKS Dresden Guest Scientists Swarnali Bandopadhyay Sebastien Burdin Tae-Yoon Kwon Jeong-Bo Shim Grigory Tkatchov PhD students Georg Röder Rainer Bedrich External Funding: DFG Emmy Noether Group DFG Forschergruppe 76

21 Delay time τ Second Research Topic of the group: Quantum Chaos in Optical Microcavities n =, n =3, n =6 B A D C A B D C Wave vector k A B C D A B C D C B A D A B C D sin χ - φ φ φ φ 4 6 AOC in mesoscopic systems pert. strength Störungsstärke Summary and Conclusion Bulk-like Mesoscopic Graphene Elektronen -.5 band filling Störungsstärke -.5 band filling Photoabsorption in chaotic mesoscopic systems -.5 band filling Anderson overlap.5 photoabsorption A(ω) K-edge mesoscopic K-edge metallic x-ray energy (ω ω th )/d GaAs Quantum Dot Array (diam.~ nm) etched from heterostructure DEG and impurities