Fishing Bosons
in the depths of Fermi Sea
Giorgio Benedek
Università di Milano-Bicocca
http://www2.mater.unimib.it/utenti/benedek/
Pavia, 6 March 2014
from a collaboration with:
J. Peter Toennies
Marco Bernasconi
Davide Campi
Pedro M. Echenique
Evgueni V. Chulkov
Irina Sklydneva
Klaus-Peter Bohnen
Rolf Heid
Vasse Chis
Condensed matter:
the Fermion & Boson zoo
 Fermions:
- electrons, holes, protons, neutrons,
- neutral atoms (A = odd)
 Bosons:
- photons
- Cooper pairs
- neutral atoms (A =even)
- Elementary excitations (and their quanta)
- e-h pairs, excitons
- phonons
- plasmons
- magnons
- rotons
- polaritons
- plasmarons
Welcome to the Fermi Sea
Otto Stern (Sohrau 1888 – Berkeley 1969)
Nobel Laureate 1943
Otto Stern, O.R. Frisch, I. Estermann
(Hamburg, 1929-1933).
He
2
4.542
[Å] 

1
k[Å ]
Ei [meV]
k  (K , k z ) k f  ki
2
K  G G 
(m, n)
a
a
NaCl(001)
Supersonic nozzle beam sources
J. P. Toennies: HUGO (MPI-SF, Goettingen)
Angular distributions
Diffraction
Inelastic processes:
- inelastic bound state resonances
- kinematical focussing
kf
d 2( 1 )

| 1  n(E ) | F fi  ImG(ΔE )  Fif
dE f dΩ f
kiz
Manson and Celli (1971)
GB (GF formulation, 1973)
u*Q (0)u Q (0)
G (ΔE )  Qv
E f -Ei  Qv  i 0 
displacements of the SURFACE atoms (layer index = 0)
…to a slab of
Nz layers
Surface phonons 2: from
one monolayer…
Time-of-Flight spectra
Longitudinal
resonance
Rayleigh
wave
U. Harten, J.P. Toennies
and Ch. Wöll (1983-85)
The bones and the skin!
Questions:
1) Why the longitudinal resonance is so soft?
2) Why is it observed at all?
3) Why is it found in ALL metals?
Giorgio, Vittorio & Peter
Bibi
V. Chis, B. Hellsing, G. Benedek, M. Bernasconi, E. V. Chulkov, and J. P. Toennies
“Large Surface Charge-density Oscillations Induced by Subsurface Phonon Resonances”
Phys. Rev. Letters, 101, 206102 (2008)
DFPT + SCDO for Cu(111)
Phonon-induced surface
charge-density oscillations
Why so many phonons?
Milano
(Bernasconi, GB)
Göttingen
(JPT)
DIPC
(Chulkov)
Karlsruhe
(Bohnen, Heid)
The quantum sonar effect
Bi(111)
Pb(111)
Theory: DFPT (mixed plane + spherical wave basis)
for a 5 or 7 ML film on a rigid substrate
Pb/Cu(111)
Surface charge density oscillations of the topmost modes at Q = 0
5 ML Pb/rigid substrate
Almost identical SCDO’s
for two completely
different modes:
just as found in HAS
experiments!
HAS perceives
underground phonons
(5 layers deep)
via e-p interaction !
HAS scattering intensities
kf
2
d 2( 1 )

[1  nBE (E )] Qv Kn  V fi (Kn, Q )  (E   Qv )
dE f dΩ f
k i
 V (r,t)  A  n(r, t )
f  n Kn,Q (r ) i  A n '
the non-diagonal elements of the electron
density matrix act as effective inelastic
scattering potential
f  K n (r ) K +Q n' (r ) i
EKn  EK Q n '   Q
g nn ' (K , K + Q ; )
electron-phonon interaction matrix
 (Q,  ) 
0 (Q,  )
1  (4 e2 / Q2 )  0 (Q,  )
electronic susceptibility
mode-selected e-p coupling lambda
 
Kn
n'
g nn ' (K , K  Q ; ) f  K n (r ) K Q n' (r ) i
2

1
N ( EF )  Q3  Q I ( Q )
2
d 2(1)
 f (E ) N ( E F )Qv Qv  (E   Qv )
dE f dΩ f
a slowly varying function
HAS from metal surfaces and thin films can measure the
mode-selected electron-phonon coupling constants !
Persistent SC in Pb/Si(111)
16 ML down to 1 !
S. Qin, J. Kim, Q. Niu, and C.-K. Shih,
Science 324,1314 (2009).
T. Zhang, P. Cheng, W.-J. Li, Y.-J. Sun, G. Wang,
X.-G. Zhu, K. He, L. Wang, X. Ma, X. Chen, Y. Wang,
Y. Liu, H.-Q. Lin, J.F. J ia, and Q.-K. Xue,
Nature Physics 6, 104-108 (2010).
Superconductivity in Pb/Si(111) ultra-thin films
Theory predicts
also the drop of
total  and Tc
below 4 ML !
The interface mode is
the culprit for SC!
1
Acoustic Surface Plasmons (ASP) observed by HAS in Cu(111)!
ASP0
ASP
Band structure of graphene
Dirac massless fermions
Dirac massive fermions
Graphene / Ru(0001)0
HAS: Daniel Farias (Madrid)
DIRAC?
p  ( K  q), c   K / 2m
E ( p)   p 2c 2  m2 c 4
m
1

m | 1  TK / 2U K |
m  m
m  m m

 1019
m
mP
back to solid
(K ) 2
TK 
2m
m  mP 
hc
hG
, 2a   P 
G
c3
m m
c (m  m )  G  
P
Δ m hc
VG (r ) 
mP r
Planck lattice
2
h 2 mh  me
Veh (r )  
 0.1 eV
2
r
4am
at r = a
Conclusions:
 HAS can measure deep sub-surface phonons in metal films: a complete
spectroscopy (not accessible to other probes such as EELS)
 HAS can directly measure the mode-selected electron-phonon coupling
in metals: a fundamental information
a) for the theory of 2D superconductivity
b) for the theory of IETS (STS) intensities
c) for understanding phonon-assisted surface reactions, etc.
d) chiral symmetry break: graphene, topological insulators,...
 HAS can measure acoustic surface plasmons
 New trends: Bi(111), and TIs: Sb(111), Bi2Se3 ,...
 New extraordinary possibilities:
 TU Graz
 3He spin-echo spectroscopy
new adventures with Otto Stern’s
invention, a new life for HAS !
Pavia Milano R.do
The Cavendish He3 SpinEcho Apparatus
Parameter
Value
Total scattering angle
44.4 degrees
3He Angular Resolution
0.1 degree
Nominal beam energy
8 meV
Measured beam intensity
1e14 atoms/second
Beam diameter at target
2 mm
Energy resolution (QE peak width)
20 neV
Scattering chamber base pressure
2e-10 mbar
Sample manipulator
6 axis, titanium
Sample manipulator resolution
0.003 degrees
Sample heating
Radiation / E-beam
Sample cooling
Liquid Nitrogen or Helium
Sample temperature range
55 K - >1200 K
Exploiting the old paradox:
- impact EELS doesn’t see valence electrons!
- neutral atoms interact inelastically via valence electrons!!
- phonons via electron-phonon interaction
- acoustic surface plasmons
- surface excitons in insulators
(with keV neutrals: H. Winter et al)
- with 3He spin echo: slow dynamics (diffusion)
magnetic excitations (?)
- plasmarons (topological insulators, graphene...)
The Multipole Expansion (ME) Method
C.S. Jayanthi, H. Bilz, W. Kress and G. Benedek, Phys. Rev. Letters 59, 795 (1987)
(after an idea of Phil Allen for the superconducting phonon anomalies of Nb)
E  Eion  F[n(r)]   v ion (r)n(r)d 3r
v ion (r)  l v l (r  rl  u(l ))
nl r    C l Y r  rl 
C (l)  C0, (l)  c (l)
Equilibrium:
 E  c l   0
 Γ, l , 
E  Eo 
1
 R l , l u l u l 
2 l ,l 
1





[
T
l
,
l


T





 l '  , l ] u l c l  

l ,l 
2
1
  l ,l j H   ' l , l  ' c l c l  ' .
2

2 E
 2 Eion
 2 vl ( r - rl )
3
R (l , l ' ) 

  ll '  d r n( r )
u (l )u (l ' ) u (l )u (l ' )
r r
ion
 R
(l , l ' )  R0el, (l , l ' ) .
T  l , l    2 E  u l   c l 
1
3
d
r    E  n r  u l   Y r  rl 

V
  d 3r   v I r   u l   Y r  rl  ,

H ' l , l  '   2 E  c l   c ' l  '



1
3 3
2

d
r
d
r

E  nr  nr Y r  rl Y r  rl  '  ,

2
V
Stefano Baroni
Density-Functional Perturbation Theory vs. Multipole expansion
occ
R (l , l ' )  2
el
vk
occ
 2 vion ( r )
 vk  vion ( r )
 vk
 vk  2
 vk  c.c.
u (l ) u (l ' )
vk u (l ) u (l ' )
k Kohn-Sham wave functions:
occ
  vk vk n(r)
vk
occ

vk
occ

vk
2
 2 vion ( r )

vion ( r )
 vk
 vk   d 3r n( r )
 R0el
u (l ) u (l ' )
u (l ) u (l ' )
 vk  vion ( r )
 n( r )  vion ( r )
 vk  c.c.   d 3r
u (l ) u (l ' )
u (l ) u (l ' )
  d 3r d 3r '
 vion ( r )
 v (r' )
 (r, r ' ) ion
u (l )
 u ( l ' )
  TH 1T 
Adiabatic condition
c   H 1T u
Secular equation
2
ion
el
1 


M Q
u
Q
ν

(
R

R

TH
T ) uQν 

0
Non-local dielectric response (susceptibility)
1
3
3

H 
' (l , l  ' )    d r d rY (r  rl )  (r, r)Y' (r  rl ' ' ).
Adiabatic dynamic electron density oscillations
 n(r,t )   d 3r   (r, r)l  v ion (r) /  rl  u(l , t )