Wigner crystallization in mesoscopic semiconductor structures

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1 Wigner crystallization in mesoscopic semiconductor structures Michael Bonitz Rostock University In collaboration with: V. Golubnichy, A. Filinov, Yu. Lozovik (Moscow) Talk given at ISG, Forschungszentrum Jülich, 7. März 2002

2 Outline 1. Introduction: strong correlations in semiconductors, Wigner crystallization 2. Mesoscopic Wigner crystals in open/semiconductor quantum dots (N= ) 3. Wigner crystallization in heterostructures 4. Single-electron control of crystallization and mobility 5. Summary and outlook

3 Coulomb correlations in semiconductors Coulomb interaction between electrons / holes U ( r) = ± e Coupling strength: Ratio of interaction energy/kinetic energy Classical system: Quantum degeneracy: Γ U( r) / k B d χ nλ T quantum system: E F - Fermi energy s 2 / ε r r U / E r / a λ - electron quantum wave length, d - dimension F ab - exciton Bohr radius B Weakly coupled e-h plasma (Fermi gas) r s = 1 χ = 1 Strong correlations Γ = 1 Weakly coupled classical e-h plasma E R - Exciton binding energy M. Bonitz, Quantum Kinetic Theory, Teubner 1998

4 Strong Coulomb correlations Γ U / k B T E F r U / E < r > / a s - Fermi energy F B ab - exciton Bohr radius Expected behavior: rs << 1 Fermi gas r s = 1 rs rs ~ 1 > 1 Fermi liquid, Electron-hole liquid Excitons, biexcitons Γ = 1 rs 37 Electron Wigner crystal Non-perturbative theories needed Example: Ga As -Greens functions: ladder diagrams - Quantum Molecular Dynamics -Path integral Monte Carlo

5 Quantum Monte Carlo Simulations Punkte: Elektron- (Loch-) hochdichte, partiell delokalisierte Phase Aufenthaltswahrscheinlichkeit (Elektronen- Loch-Tropfen) V. Filinov, W. Hoyer, Rs=2.1 S.W. Koch, and M. Bonitz 2001

6 Partiell ionisiertes e-h-plasma Exzitonen Biexzitonen Trionen, Cluster,... Rs=4.2

7 Rs=8.6

8 Wigner crystal Ground state of the electron gas in metals E. Wigner, Physical Review 46, 1002 (1934):! exchange and correlation energy of the electron gas If the electrons had no kinetic energy, they would settle in configurations which correspond to the absolute minima of the potential energy. These are close-packed lattice configurations, with energies very near to that of the body-centered lattice...

9 Wigner crystals In general:! lattice state of Coulomb systems (electrons, ions) Observed/predicted Wigner crystals 1. Electrons on the surface of liquid helium droplets - Experiment: Grimes/Adams (1979), - Theory: Fisher, Halperin, Platzman (1979) 2. 2D electrons in heterojunctions in semiconductors in strong B-field Experiment: Andrei et al Theoretical predictions: 2D/3D electron gas Monte Carlo simulations, Ceperley et al. 4. Ion crystals: - observed experimentally in classical dusty plasmas - predicted in dense white and brown dwarf stars

10 2D Wigner crystals Quantum Wigner transition (T=0, B=0): n W = [ ] c 2 c r πa, r 37 ( T = 0) s B s! experimentally not yet observed B-field: Magnetic ordering γ = VM k T B F = ν 1, V = hω, ω = M c c eb * m c! Observed in 2D semiconductor heterostructures See e.g. Andrei, Williams, Glattli and Deville, In: Physics of Low-Dimensional Semiconductor Structures, Eds. Butcher, March, Tosi, Plenum Press 1993

11 Wigner crystals in 2D e-gas n W * m ε 2 a 2 B

12 Experiments on 2D Wigner crystals!evidence for solid: Low-frequency shear mode ω t ω l,t ~ plasmon/acoustic mode (B=0) liquid solid B=0 B>0 Andrei et al., PRL 60, 2765 (1988)

13 Mesoscopic Systems Examples: Mendeleyev table, nuclear matter, clusters, electrons in quantum dots,... Peculiarities: - small particle numbers, N<100, no thermodynamic limit - Macroscopic averages are N-dependent - Large fluctuations around average values - Big influence of symmetry and boundary effects Energy per particle and addition energy Related to chemical potential µ = ε / N System: electrons in spherical 2-dimensional harmonic trap

14 Mesoscopic Systems, contd. Structurally sensitive quantity: electron addition energy change - Corresponds to derivative of chemical potential µ 2 2 / N = ε / N - Sensitive to shell closure effects, ( magic clusters, unusual stability) - Directly measurable in quantum dots: capacitance measurements at electron addition System: electrons in 2-dimensional harmonic trap potential

15 Mesoscopic electron clusters Question: possibility of Wigner crystallization? Is there a metal-insulator transition? Model: 2D electron clusters in spherical trap, N= T=const, B=0 Ground state result: - Shell structure, - pronounced symmetry effects - Strong N-dependence Away from ground state, problem: simultaneous account of very strong correlations, quantum and spin effects at finite temperature Hartree-Fock methods not applicable Path integral Monte Carlo

16 Thermodynamics of correlated quantum systems N-particle density operator: ρ = exp( βh ) r i r1 r2 r N {,,..., } i i i M. Bonitz (Ed.), Introduction to Computational Methods..., Rinton Press 2002

17 High-temperature approximation Hamiltonian: kinetic+potential energy (non-commutative!) Hˆ = Kˆ + Vˆ Factorization (Trotter-formula) + corrections Exact result in the limit M 2 o, o ~ 1/ M Final Result: Quantum exchange (sum over permutations) reduced to determinant

18 Thermodynamics of correlated quantum systems (contd.) Snapshots of closed electron paths spin-less particles With exchange included 5 Electrons in 2D simulation box, Path ends labeled by thick dots

Wigner crystallization of electron clusters Wigner crystal: U Coulomb / E Kin 30.

19 Wigner crystallization of electron clusters Wigner crystal: U Coulomb / E Kin ( coupling parameter Γ, rs ) Variation of temperature or density (confinement) Result: two crystal phases with intra-shell / inter-shell ordering (RO/OO) True finite electron extension probability to find individual electrons red=0! pink=max Density increase: increasing wave function overlap OO crystal! RO crystal! Wigner molecule (...! Fermi liquid )

20 Mesoscopic Electron crystals in the news - Phys. Rev. Focus, April Wissenschaft Online, April Spektrum der Wissenschaft, Mai Sciences et Avenir, Juni APS Picture of the Week - Norddeutsche Neuste Nachrichten Frankfurter Allgemeine, August 2001

21 Phase diagram of the mesoscopic Wigner crystal Temperature Classical liquid Particle number Quantum liquid (Wigner molecule) Confinement strength RM - Radial melting, OM Orientational melting A. Filinov, M. Bonitz, and Yu. Lozovik, Phys. Rev. Lett. 86, 3851 (2001)

22 Crystal melting and mobility increase At the melting point: drastic increase of angular and radial distance fluctuations Electron delocalization, mobility and conductivity increase Relative distance fluctuations Orientational melting Radial melting

B = 19 0 Large distance: 2 independent crystals d in units of interparticle

23 Coulomb Bilayers Two mesoscopic 2D clusters at fixed distance d - New effect: influence of inter-layer correlations on ground state, crystal N T 1 = N2 = = 0, B = 19 0 Large distance: 2 independent crystals d in units of interparticle distance! Change of crystal symmetry with d Small distance: one single combined crystal

24 Symmetry of bilayer crystals

25 Melting of Bilayer crystals Distance and radial fluctuations: Three types of particle ordering: in layer radial and orientational ordering and inter-layer ordering Relative distance fluctuations Melting temperature vs. d Inter-layer Intra-layer Layer locking Peeters et al., Kalman et al., A. Filinov, M. Bonitz, Yu. Lozovik, Contrib. Plasma Phys. 41, 357 (2001)

26 Particle number dependence of crystal stability N Γ0 Γ r r so rsr Magic 10 Nonmagic 11 Magic 12 Magic Nonmagic

27 Interesting Applications Crystallization/melting without changing density and temperature:! by addition/removal of single electron T = 1/ Γ switch between conducting/insulating behavior Novel single electron transistor 1/ 2 n = r s Bonitz et al., Microelectronic Engineering, accepted

28 Nonequilibrium behavior Molecular dynamics simulations of mesoscopic electron clusters - real-time melting behavior - spectrum of eigenmodes, in particular inter-shell rotation -N-dependence of orientational melting: Γ 20 >> Γ > Γ! consider OM OM 19 - dependence on initial conditions

Dusty plasmas Alternative approach to strong correlations:!

Melzer and co-workers (Univ. Kiel) Dust particles: Q=(5000...10000) e d ~ 0.

discharge in plasma chamber Vertical E-field compensates gravity!

29 Dusty plasmas Alternative approach to strong correlations:!highly charged particles Experiments of A. Piel, A. Melzer and co-workers (Univ. Kiel) Dust particles: Q=( ) e d ~ mm melamin-formaldehyd spheres, electron microscope picture Charging in HF discharge in plasma chamber Vertical E-field compensates gravity! dust particles float Background gas pressure determines dust particle density and friction Web page

pressure increase Wigner crystal Crystal

30 Wigner crystallization in dusty plasmas Experiments of A. Piel, A. Melzer and co-workers (Univ. Kiel) "Increase Coulomb coupling by gas pressure increase Wigner crystal Crystal melting Liquid Web page

Kiel) Two-dimensional mesoscopic plasma crystals Tangential

31 Response to external excitation Experiments of A. Piel, A. Melzer and co-workers (Univ. Kiel) Two-dimensional mesoscopic plasma crystals Tangential excitation by 2 laser pulses Γ 20 OM >> Γ > Γ 19 OM N=19 N=20 Web page

32 Summary and Outlook I. Wigner crystallization in few-electron quantum dots - two crystal phases: radial/orientational ordering, strong N-dependence - melting by temperature/density change - melting by electron addition/removal II. Single-electron controlled insulator-metal transition Animations III. Outlook: crystals in heterostructures, B-field, exciton crystal?

Structure Formation in Strongly Correlated Coulomb Systems

Structure Formation in Strongly Correlated Coulomb Systems M. Bonitz 1, V. Filinov, A. Filinov 1, V. Golubnychiy 1, P. Ludwig 1,3, H. Baumgartner 1, P.R. Levashov, V.E. Fortov, and H. Fehske 4 1 Institut