vii training course in the physics of correlated electron systems and high-tc superconductors

VII Training Course in the Physics ofCorrelated 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. Devreeseand the Silvera group (Harvard): I. F. Silvera, J. Huang

Theoretische Fysica van de Vaste Stof

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 thereofVII. Spherical electron gas and effective electron-electron interaction

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).

anode

0.2

mm

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

I. Introduction : what are multielectron bubbles ?


edge of the bubble;the liquid helium surface

spherical 2Delectron fluid

or solid

0.2 mm

typical size: 0.1 m - 0.1 mmtypical charge: 103 - 108 e

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.

[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).

The bubble radius

1) The surface tension energy of the helium:E = S where = 3.6×10–4 J/m2 and S = 4R2

This energy becomes smaller at smaller radius.

2) The external pressure applied 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

2

2

3/12

3/4222

2)(3176.0

)(2 dm

N

dR

Ne

dR

NeE

e

Confinement energy of the electron layer; d is the distance between elec-trons and the helium surface.[V. B. Shikin, JETP 27, 39 (1978)]

3) The Coulomb repulsion (and the confinement energy) of the electrons. This always favours expanding the bubble.

Exchange energy [K.W.K. Shung and F.L. Lin, PRB45, 7481 (1992)].

Coulomb repulsion [Lord Rayleigh, Proc. Roy. Soc. London 29, 71 (1879)].

The last two terms are negligible for N>1000.


102

104106

108

N

1010

1014

1012

n (c

m-2)

10-2100

102104

p (mbar)

Rmax = 41/3 R(p=0)pc = –(3/2)4/3 (4/(Ne)2)1/3

Both the pressure and the number of electrons in the bubble control its radius. As a function of the pressure, the surface density N/(4R2) can easily and continuously be varied over four orders of magnitude.


NEW EXPERIMENT

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

A new scheme for creating stable bubbles [1]

superfluid helium

filling line

bellows

tungsten filament

cryostat with a domed roof

window, coated with transparent metal

[1] I. F. Silvera et al. in: “Frontiers of High-Pressure Research II” (eds. H. D. Hochheimer et al., Kluwer Academic Publ. 2001)


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.

Looking at the multielectron bubble: fiber optic illuminator and imaging

Lens (5)

cryostat

Light is fed into an optical fiber which takes it into the cell and illuminates the bubble.


Measuring the oscillation frequencies

referencecapacitor

The window andthe glass piece are coated with a transparant metal(indium tin oxide).

~

coaxflangeFrequency

generatorsuppliesa drivingforce

GE 1615-A capacitance bridge measures dissipated power and capacitance.

I Both the energy dissipation and the visualobservation of the bubble allow to observethe resonant frequencies.


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.

Assume that Ql,m Rb and nl,m n0

and keep the terms up to second order in the nonspherical deformation amplitudes.

1

222 )2(2

14

mmQRS

1

23

3

4

mmQRRV

),(),(1

mmmb YQRR

Shape of the bubble surface:

),(),(0

mmmYnn

Density distribution of the spherical 2DEG:

Deformed bubbles & vibrational modes

S CC dSRVneU ),,(),(


),(,in,

0

in

mm

m

YvR

rV

),(,out,

1

0

out

mm

m

Yv

r

RV

n

Boundary conditions:0),,( ),,( outin RVRV

),(4),,(. ),,(. outin enRR DnDn

),(),(1

mmmb YQRR

Shape of the bubble surface:

),(),(0

mmmYnn

Density distribution of the spherical 2DEG:

Deformed bubbles & vibrational modes


1

*,,

2

1

2

,22

,

6

1

2

,22

3

2223

23

)1(

1

)1(

4

)()1(2

24

3

4

2

mmm

mmphononm

e

mmripplonm

QnNe

nnN

Rm

QQR

R

NeRpR

RRE

)1(

)1(

4

2)2(1

)( 2

3

22

22

3

R

Ne

pR

Rripplon


Ripplons

)1(

)1()(

3

2

Rm

Ne

ephonon

Phonons

Ripplon-phononcoupling*

*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).

J.T., I.F. Silvera, J.T.Devreese, PRL 87, 275301 (2001).


Ripplon-phonon modes:

BUBBLE STABILITY

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

Universiteit Antwerpen UIATheoretische 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).

3

22

16

eNRC

2123

R

l = 1 mode is translation l=2 mode is unstable


The trouble with bubbles… is that they split up!

* 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).

If the l=2 excitation has =0, then it does notcost 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 bubbleswith N/2 electrons, bubbles may be unstable.[M. M. Salomaa & G. A. Williams, Phys. Rev. Lett. 47,1730 (1981)].

?

z

cL

aL

L

cR

aMaR

z2–z1


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 z 2–z1, the other parameters are optimized to minimize the energy, and this provides an energy diagram for fission.


Fissioning of a multielectron bubble

The shape of the bubble is described, in cylindrical coordinates, by

22

2

12

for )(

for )(

for )(

)(

zzzfzba

zzzfzba

zzzfzba

z

bRRR

baMMM

aLLL

From the eleven parameters, six can be eliminated using continuity and continuousderivatives where the different sections meet. The energy of a given configuration is given by :

CEpVSE

dzdz

dzS

z

z

2

1)(2

1

)(

0

22

1

zz

z

ddzV

|'|

)]('[)](['

433

2

rrrr

zzdd

eEC


A B

C

D

E

F


Bubbles are stabilized against fissioning by an energy barrier: the intermediate shapes in going from one branch to another are higher in energy.

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.

gravity

surface tension

A person on a trampoline:

liquid 4Heoutside bubble

electricfield

e–

surface tension insidebubble

An electron on a helium surface:

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.


Electron-ripplon coupling:

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.

222

02

2

)(

4)sgn()(

)(

4||

2)(

rd

rdKrdrd

rd

rdErd

d

erVFQ

0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2

0

0.5

1

1.5

2

r/d

VF

Q(r

) (i

n un

its

e2 /d2 )

Near the origin (the lattice site) this potential is quadratic, and has a character-istic frequency, FQ = [e2/(40med3)]1/2, of the order of THz.

[1] S. Fratini and P. Quémerais, Eur. Phys. Journal B 14, 99 (2000).

d

r


Fratini-Quémerais lattice potential[1]:

Ripplopolarons

mmrmmrm

mmm

FQe

QYQYe

QQM

rVm

pH

,,ˆ,,ˆ

*,

,

2,

2ripplon

2,

2

ˆ)(ˆ)(2

||

|ˆ|)(|ˆ|2

)ˆ(2

ˆˆ

E

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:

)}2/(exp{1

with 22

ripplon

aa

e

e


mmmm

meFQe

eee

QQQM

QM

rVm

pH

,

2

,,2ripplon

2,

2,

2ripplon

2

ˆ)(|ˆ|2

2

)()ˆ(

2

ˆˆ

The product ansatz allows to write the Hamiltonian as:

electron energy termenergy reduction throughthe presence of the dimple

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.

ermem YM

eQ

)(

)(

||ˆ,2

ripplon,

E



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 ?

MELTING OF THE WIGNER LATTICE

S. N. Klimin, V. M. Fomin, J. Tempere, J. T. Devreese, I. F. Silvera, submitted to Phys. Rev.

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].

Lindemann melting criterion[1]:

[1] F. A. Lindemann, Phys. Z. 11, 609 (1910).[2] C. C. Grimes and A. Adams, Phys. Rev. Lett. 42, 795 (1979).


Motion out of the lattice site can be increased through:

Decreasing lattice parameter (pressurizing) or, equivalently, increasing zero-point motion: quantum melting

Increasing temperature: classical melting

Electron (m, r)

Fictitious particle (M, R)

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)

000

1S

SSFF


Jensen-Feynman approach 1 2

In the JF-approach, we can calculate<r2>, <(R-r)2>, <Rcms

2>

THE ELECTRON GAS ON A SPHERE

Key paper:J. Tempere, I. F. Silvera, J. T. Devreese, Phys. Rev. B 65, 195418 (2002).

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):

2

2

2

2

)1(

2

RmE

YEYRm

em

mmme

l

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

LF = 3

22

)1( 12/

Rm

LLENL

e

FFFF

Since R N 2/3, the Fermi energy is proportional to N –1/3 and decreases with increasing N. The surface density, N/(4R2), is also proportional to N –1/3 .


In calculations, the angular momentum takes on the role that the momentum hasfor a flat 2DEG:

For example, the polarisation ‘bubble’ diagram can be calculated and used to derivethe RPA dynamical structure factor:


),(

'

'||)','(),(

ˆ,|',';,0,|0,';0,

)12(4

)1'2)(12(ˆ

ML

L

LMLmm

cMLmmL

Lc

0 0.5 1 1.5 2 2.5 3lLF

0

10

20

30

40

wHmc-1 L

0 6´10- 3

LF=20, R=0.210909The 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.

Single particleexcitations (anelectron from inside the Fermisphere is excitedto a higher energylevel).

Plasmon branch(collective excitation)

J.T., I.F. Silvera, J.T.Devreese, Phys. Rev. B 65, 195418 (2002).


Dynamical structure factor of the spherical 2DEG

Plasmon branch is a discrete setof excitations, and has a lowestfrequency which is not zero.


Plasmons on a spherical surface


Effective electron-electron interaction in the MEB electron gas

),()','(),()','(),(),(,',',

int ˆˆˆˆ );,()(ˆmmnjmnjm

njmm

ccccnjVH

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:

12

1

2

)()(

2);,(

2

22

2

eff

bR

e

iMmV

with

38

)2/1(||

bReM E

From the effective interaction to a BCS type interaction

22

2

eff

22

eff

12

1

4)0;,1( and

2

12

1

2)0;,(

gR

emV

MR

emV

b

b

(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).


),,(),,(),',(),',(',int ˆˆˆˆ~ˆ

mLmLmLmLmm

L

LmFFFF

F

F

ccccVH

22

'|]|,2max[

22', ),',,())(1()(

~jF

L

mmjFFmm MjmmLCLfLfV

F

“BCS-like”

vii training course in the physics of correlated electron systems and high-tc superconductors

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