VII Training Course in the Physics of Correlated Electron Systems and High-Tc Superconductors

1. VII Training Course in the Physics of Correlated Electron Systems and High-Tc Superconductors Salerno, 14-26 october 2002 Multielectron bubbles properties of a spherical 2D electron gas coupling to ripplons J. Tempere The results reported here were obtained in a collaboration between: TFVS (UIA) : J.T., S. N. Klimin, V. M. Fomin, J. T. Devreese and the Silvera group (Harvard): I. F. Silvera, J. Huang Contents: I. Introduction II. Experiment III. Vibrational modes of the MEB (ripplons, phonons and their coupling) IV. Bubble stability V. Electrons coupling to these vibrational modes VI. Wigner lattice and melting thereof VII. Spherical electron gas and effective electron-electron interaction Theoretische Fysica van de Vaste Stof

2. INTRODUCTION Key papers: A.P. Volodin, M.S. Khaikin, and V.S. Edelman, JETP 26, 543 (1977). U. Albrecht and P. Leiderer, Europhys. Lett. 3, 705 (1987). M. M. Salomaa, and G. A. Williams, Phys. Rev. Lett. 47, 1730 (1981).

3. He vapour + electrons 0.2 mm Liquid He anode I. Introduction : what are multielectron bubbles ? A.P. Volodin, M.S. Khaikin, and V.S. Edelman, JETP 26, 543 (1977). U. Albrecht and P. Leiderer, Europhys. Lett. 3, 705 (1987). Multielecton bubble size: 0.1-100 m charge: 103-108 e Theoretische Fysica van de Vaste Stof

4. Electronic structure of an MEB[1] : In the helium bubble, the electrons form a nanometer thin layer, hugging the helium surface at a distance of the order of a nanometer. Though confined in the radial direction, they are free to move in on the spherical surface. edge of the bubble; the liquid helium surface spherical 2D electron fluid or solid 0.2 mm typical size: 0.1 m- 0.1 mm typical charge: 103 - 108 e [1] M. M. Salomaa, and G. A. Williams, Phys. Rev. Lett. 47, 1730 (1981); K. W. K. Shung and F. L. Lin, Phys. Rev. B 45, 7491 (1992).

5. 3) The Coulomb repulsion (and the confinement energy) of the electrons. This always favours expanding the bubble. Confinement energy of the electron layer; d is the distance between elec-trons and the helium surface. [V. B. Shikin, JETP 27, 39 (1978)] Coulomb repulsion [Lord Rayleigh, Proc. Roy. Soc. London 29, 71 (1879)]. Exchange energy [K.W.K. Shung and F.L. Lin, PRB45, 7481 (1992)]. The bubble radius 1) The surface tensionenergy of the helium: E = Swhere = 3.6×10–4 J/m2andS = 4R2 This energy becomes smaller at smaller radius. 2) The external pressureapplied on the bubble. The helium liquid can be pressurized up to 25 bar before it solidifies and to –9 bar before the liquid cavitates. Positive pressure favours smaller radii, negative pressure expands the bubble. E = –pV where p is the pressure and V is the volume V = (4/3)R3 The last two terms are negligible for N>1000. Theoretische Fysica van de Vaste Stof

6. 1014 1012 n (cm-2) 108 106 104 1010 N 104 102 102 100 p (mbar) 10-2 Both the pressure and the number of electrons in the bubble control its radius. As a function of the pressure, the surface density N/(4R2) can easily and continuously be varied over four orders of magnitude. Rmax = 41/3R(p=0) pc = –(3/2)4/3 (4/(Ne)2)1/3 Theoretische Fysica van de Vaste Stof

7. NEW EXPERIMENT Initial proposal: I. F. Silvera, Bull. Am. Phys. Soc. 46, 1016 (2001).

8. A new scheme for creating stable bubbles [1] window, coated with transparent metal filling line cryostat with a domed roof tungsten filament superfluid helium bellows [1] I. F. Silvera et al. in: “Frontiers of High-Pressure Research II” (eds. H. D. Hochheimer et al., Kluwer Academic Publ. 2001) Theoretische Fysica van de Vaste Stof

9. Looking at the multielectron bubble: fiber optic illuminator and imaging Lens (5) Light is fed into an optical fiber which takes it into the cell and illuminates the bubble. A coherent fiber bundle built up out of 60000 individual fiber strands of 3 m diameter takes the image out of the cryo-stat and into a microscope. cryostat Theoretische Fysica van de Vaste Stof

10. Measuring the oscillation frequencies coax flange Frequency generator supplies a driving force The window and the glass piece are coated with a transparant metal (indium tin oxide). ~ reference capacitor GE 1615-A capacitance bridge measures dissipated power and capacitance. I Both the energy dissipation and the visual observation of the bubble allow to observe the resonant frequencies. Theoretische Fysica van de Vaste Stof

11. VIBRATIONAL MODES Key papers: J. Tempere, I. F. Silvera, J. T. Devreese, Phys. Rev. Lett. 87, 275301 (2001). S. N. Klimin, V. M. Fomin, J. Tempere, J. T. Devreese, I. F. Silvera, submitted to Phys. Rev.

12. Deformed bubbles & vibrational modes Shape of the bubble surface: Density distribution of the spherical 2DEG: Assume that Ql,m  Rb and nl,m  n0 and keep the terms up to second order in the nonspherical deformation ampli- tudes. Theoretische Fysica van de Vaste Stof

13. Boundary conditions: Deformed bubbles & vibrational modes Shape of the bubble surface: Density distribution of the spherical 2DEG: Theoretische Fysica van de Vaste Stof

14. Ripplons Ripplon-phonon coupling* Phonons J.T., I.F. Silvera, J.T.Devreese, PRL 87, 275301 (2001). *For electrons on a flat surface, ripplon-phonon coupling was described in D.S.Fisher, B.I.Halperin and P.M. Platzman, Phys. Rev. Lett. 42, 798 (1979). Theoretische Fysica van de Vaste Stof

15. Ripplon-phonon modes: Theoretische Fysica van de Vaste Stof

16. BUBBLE STABILITY Key paper: J. Tempere, I. F. Silvera, J. T. Devreese, accepted for publication in Phys. Rev. B.

17. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof The trouble with bubbles… Note that at zero pressure, – the radius is given by the “Coulomb radius” – and hence the ripplon frequency simplifies to * Negative pressures stabilize the bubble, in the sense that all frequencies > 0 * An increasing positive pressure drives all modes unstable one by one. For mode l, the critical pressure is p = (l–2)/(2R). l = 1 mode is translation l=2 mode is unstable

18. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof The trouble with bubbles… is that they split up! If the l=2 excitation has =0, then it does not cost energy to make small l=2 oscillations. But since the energy of a bubble with N elec- trons is larger than the energy of two bubbles with N/2 electrons, bubbles may be unstable. [M. M. Salomaa & G. A. Williams, Phys. Rev. Lett. 47, 1730 (1981)]. ? * Negative pressures stabilize the bubble, in the sense that all frequencies > 0 * An increasing positive pressure drives all modes unstable one by one. For mode l, the critical pressure is p = (l–2)/(2R).

19. z2–z1 aL aR aM cL cR  L z Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof Fissioning of a multielectron bubble We apply the Bohr model for fissioning nuclei to the fissioning of multielectron bubbles. This model assumes that the shape of the bubble is constructed out of three quadratic forms (ellipsoids or hyperboloids), smoothly knit together at their edges. Fixing the total length z2–z1, the other parameters are optimized to minimize the energy, and this provides an energy diagram for fission.

20. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof Fissioning of a multielectron bubble The shape of the bubble is described, in cylindrical coordinates, by From the eleven parameters, six can be eliminated using continuity and continuous derivatives where the different sections meet. The energy of a given configuration is given by :

21. C D Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof B A E F

22. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof

23. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof Bubbles are stabilized against fissioning by an energy barrier: the intermediate shapes in going from one branch to another are higher in energy.

24. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof

25. Universiteit Antwerpen UIA Theoretische Fysica van de Vaste Stof

26. ELECTRONS INTERACTING WITH THE VIBRATIONAL MODES J. Tempere, I. F. Silvera, J. T. Devreese, Phys. Rev. Lett. 87, 275301 (2001). S. N. Klimin, V. M. Fomin, J. Tempere, J. T. Devreese, I. F. Silvera, submitted to Phys. Rev.

27. surface tension An electron on a helium surface: A person on a trampoline: inside bubble surface tension gravity e– electric field liquid 4He outside bubble Electron-ripplon coupling: The electric field acting on the electron, perpendicular to the He surface, consists of: 1. The field of the image charge: weak (  1) but also present for e– on a flat He surface, 2. The field induced by the other electrons on the spherical surface (strong).  The dimpling effect (the coupling between the electron and the surface deformation or ripplons) in MEBs is stronger than that for electrons on a flat helium surface. Theoretische Fysica van de Vaste Stof

28. d r Fratini-Quémerais lattice potential[1]: The potential felt by an electron in a 2D electron solid with lattice parameter d if found by treating the wigner solid around the electron as a homogeneous charge distribution with a circular hole of radius d. 0 Near the origin (the lattice site) this potential is quadratic, and has a character-istic frequency, FQ = [e2/(40med3)]1/2, of the order of THz. [1] S. Fratini and P. Quémerais, Eur. Phys. Journal B 14, 99 (2000). -0.5 VFQ(r) (in units e2/d2) -1 -1.5 -2 0 0.5 1 1.5 2 r/d Theoretische Fysica van de Vaste Stof

29. Ripplopolarons Coulomb lattice potential: ~ THz frequency Ripplons: ~ GHz/MHz frequency A product ansatz can be made for the wave function of the ripplopolaron, separating the (rapid) electron wave function and the ripplon wave function: Theoretische Fysica van de Vaste Stof

30. The product ansatz allows to write the Hamiltonian as: energy reduction through the presence of the dimple electron energy term Now the ripplon part of the Hamiltonian is that of a displaced harmonic oscillator: the new equilibrium position of the surface has a dimple underneath the electron. Theoretische Fysica van de Vaste Stof

31. Theoretische Fysica van de Vaste Stof

32. The ripplons and the Coulomb lattice potential give rise to a ripplopolaron Wigner crystal. * curving up a triangular lattice onto a spherical surface leads to interesting topological defects [P. Lenz and D. R. Nelson, Phys. Rev. Lett. 87, 125703 (2001)]. * under what conditions will this crystal form (what is the melting surface) ? * what are the differences with an electron wigner crystal ? Theoretische Fysica van de Vaste Stof

33. MELTING OF THE WIGNER LATTICE S. N. Klimin, V. M. Fomin, J. Tempere, J. T. Devreese, I. F. Silvera, submitted to Phys. Rev.

34. Motion out of the lattice site can be increased through: Increasing temperature: classical melting Decreasing lattice parameter (pressurizing) or, equivalently, increasing zero-point motion: quantum melting Lindemann melting criterion[1]: A lattice will melt when the objects (atoms, electrons, molecules,…) residing on the lattice sites travel, on average, more than a critical distance out of their lattice site. For electrons on a flat surface, Grimes and Adams observed classical melting of an electron Wigner lattice when the electrons travel more than 13% of the distance between the the lattice points[2]. [1] F. A. Lindemann, Phys. Z. 11, 609 (1910). [2] C. C. Grimes and A. Adams, Phys. Rev. Lett. 42, 795 (1979). Theoretische Fysica van de Vaste Stof

35. Electron (m, r) Fictitious particle (M, R) Jensen-Feynman approach 12 The free energy F of a ripplopolaron in a MEB is calculated using the Jensen-Feynman variational principle F0 is the free energy for a model system S (S0) is the “action” functional for the ripplopolaron (the model system)  = 1/(kBT) In the JF-approach, we can calculate <r2>, <(R-r)2>, <Rcms2> Theoretische Fysica van de Vaste Stof

36. Theoretische Fysica van de Vaste Stof

37. THE ELECTRON GAS ON A SPHERE Key paper: J. Tempere, I. F. Silvera, J. T. Devreese, Phys. Rev. B 65, 195418 (2002).

38. The electron gas in the MEB Thusfar, the Wigner crystallized phase of the electrons on the MEB has been discussed. We found that this electron solid can melt into an electron liquid, and as the surface density is decreased this may become an electron gas. A useful set of eigenfunctions for the spherical electron gas are of course the spherical harmonics (the eigenfunctions of the non-interacting electron gas): l LF = 3 The single-particle levels are charac-terized by quantum numbers l,m and fill up a Fermi sphere in angular momentum space. m -4 -3 -2 -1 0 1 2 3 Since R  N 2/3, the Fermi energy is proportional to N –1/3 and decreases with increasing N. The surface density, N/(4R2), is also proportional to N –1/3 . Theoretische Fysica van de Vaste Stof

39. In calculations, the angular momentum takes on the role that the momentum has for a flat 2DEG: For example, the polarisation ‘bubble’ diagram can be calculated and used to derive the RPA dynamical structure factor: Theoretische Fysica van de Vaste Stof

40. Plasmon branch (collective excitation) Single particle excitations (an electron from inside the Fermi sphere is excited to a higher energy level). Dynamical structure factor of the spherical 2DEG The dynamical structure factor as a function of frequency and angular momentum is related to the probabily to create an excitation with given angular momentum and frequency. J.T., I.F. Silvera, J.T.Devreese, Phys. Rev. B 65, 195418 (2002). Theoretische Fysica van de Vaste Stof

41. Plasmons on a spherical surface Plasmon branch is a discrete set of excitations, and has a lowest frequency which is not zero. Theoretische Fysica van de Vaste Stof

42. Effective electron-electron interaction in the MEB electron gas To lowest order in the Feynman diagrams, the effective electron-electron interaction is a sum of the ripplon-mediated electron-electron interaction and the Coulomb interaction: with Theoretische Fysica van de Vaste Stof

43. From the effective interaction to a BCS type interaction (1) The effective interaction is attractive for small energy transfers (< l) and for small angular momentum transfers (l < 60 for N=104 electron MEB) (2) The effective attraction can only take place between electrons in the same angular momentum level, since the splitting of the angular momentum (single particle) levels turns out to be larger than l in MEBs. This also means that when the highest level is full or empty, no attractive interactions take place. (3) The Clebsh-Gordan coefficients will suppress the scattering amplitudes except for pairs of electrons with opposite m (z-component of angular momentum). “BCS-like” Theoretische Fysica van de Vaste Stof