vii training course in the physics of correlated electron systems and high-tc superconductors
DESCRIPTION
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:. - PowerPoint PPT PresentationTRANSCRIPT
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 ?
Theoretische Fysica van de Vaste Stof
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.
Theoretische Fysica van de Vaste Stof
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.
Theoretische Fysica van de Vaste Stof
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)
Theoretische Fysica van de Vaste Stof
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.
Theoretische Fysica van de Vaste Stof
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.
Theoretische Fysica van de Vaste Stof
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 ampli- tudes.
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 ),,(),(
Theoretische Fysica van de Vaste Stof
),(,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
Theoretische Fysica van de Vaste Stof
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
Theoretische Fysica van de Vaste Stof
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).
Theoretische Fysica van de Vaste Stof
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
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
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
Universiteit Antwerpen UIATheoretische 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 z 2–z1, the other parameters are optimized to minimize the energy, and this provides an energy diagram for fission.
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
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
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
A B
C
D
E
F
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
Universiteit Antwerpen UIATheoretische 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.
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
Universiteit Antwerpen UIATheoretische Fysica van de Vaste Stof
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.
Theoretische Fysica van de Vaste Stof
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
Theoretische Fysica van de Vaste Stof
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
Theoretische Fysica van de Vaste Stof
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
Theoretische Fysica van de Vaste Stof
Theoretische Fysica van de Vaste Stof
Theoretische Fysica van de Vaste Stof
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).
Theoretische Fysica van de Vaste Stof
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
Theoretische Fysica van de Vaste Stof
Jensen-Feynman approach 1 2
In the JF-approach, we can calculate<r2>, <(R-r)2>, <Rcms
2>
Theoretische Fysica van de Vaste Stof
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 .
Theoretische Fysica van de Vaste Stof
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:
Theoretische Fysica van de Vaste Stof
),(
'
'||)','(),(
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).
Theoretische Fysica van de Vaste Stof
Dynamical structure factor of the spherical 2DEG
Plasmon branch is a discrete setof excitations, and has a lowestfrequency which is not zero.
Theoretische Fysica van de Vaste Stof
Plasmons on a spherical surface
Theoretische Fysica van de Vaste Stof
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).
Theoretische Fysica van de Vaste Stof
),,(),,(),',(),',(',int ˆˆˆˆ~ˆ
mLmLmLmLmm
L
LmFFFF
F
F
ccccVH
22
'|]|,2max[
22', ),',,())(1()(
~jF
L
mmjFFmm MjmmLCLfLfV
F
“BCS-like”