Corso per il dottorato- 2012-13
Nanostrutture e sistemi di bassa dimensionalita'
(Michele Cini)
1-Introduzione: Nanoparticelle metalliche e Mie scattering -Fullereni-Punti quantici-Pozzi quantici- Embedding-Stati a 1 elettrone nel
Grafene e nei Nanotubi di Carbonio -Catene di Heisenberg- Bethe Ansatz-Magnoni.
2-Trasporto quantistico: correnti balistiche- caratteristiche corrente-tensione- Effetti magnetici nei circuiti nanoscopici-Pumping.
3-Ordine e dimensionalita': teoria di Ginzburg Landau delle transizioni di fase- Ferromagnetismo di Weiss - modello di Ising in 1d- Assenza
di transizioni in 1d- Approccio del gruppo di rinormalizzazione per la percolazione e per il modello di Ising - modello di Ising in 2d: transfer
matrix. Fermionizzazione della transfer matrix- Soluzione di Onsager e transizione di fase- Caso a infinite dimensioni. Magnetismo in 2d
nel modello di Hubbard- Teoremi di Lieb -Ferromagnetismo di Nagaoka
4-Effetti di correlazione in 1d: andamenti a legge di potenza nei nanotubi di carbonio: liquido di Luttinger-Tecnica della BosonizzazioneSeparazione di spin e carica- Applicazioni.
5-Effetti di correlazione in 2d: Gas di Fermi in campo magnetico ed effetto Hall quantistico intero e frazionario.
http://people.roma2.infn.it/~cini/
1
2d, 1d, 0d nano-objects: molecular size in 1, 2 or 3 dimensions
Small is different: all properties of nanostructures are size dependent

 2 NH 3 on finely dispersed iron
N 2  3H 2O 

Chemical properties of Fe depend on particle size
Rich phenomenology, many applications
Models of reduced dimensionality are endorsened by quantum mechanics at low
temperatures: gaps develop and degrees of freedom are frozen!
Exact solutions:Bethe ansatz,Ising model,Nagaoka ferromagnetism….
But strong correlation (not only in models, but in reality): exotic
behavior, troubles in standard treatments
Phase transitions strongly depend on dimensionality!
Special Methods:Topology plays an important role, bosonization, Bethe Ansatz
New concepts and specific phenomena: anyons, charge fractionalization,spincharge separation, QHE
Stained Glass
Gothic
window of
Notre
Dame de
Paris
(XIV
century)
The colors were achieved by a colloid dispersion of gold nano-particles in glass.
3
Transverse electromagnetic wave in homogeneous isotropic medium
Consider the plane wave going upwards
E  ( E ( z ), 0, 0)e  it
B  (0, B( z ), 0)e  it  S  E  B  (0, 0, S )
Simplest case : no i nd and the current
E
J  ( J ( z ), 0, 0)e  it
Maxwell equations: divE  4i nd  0 and divB  0
rotE  ( y Ez   z E y ,  z Ex   x Ez ,  x E y   y Ex )  (0,  z E ( z ), 0)
1
i
rotE    t B   z E ( z )  B ( z )
c
c
rotB  ( y Bz   z By ,  z Bx   x Bz ,  x By   y Bx )  ( z B( z ), 0, 0)
1
4
i
4
rotB   t E 
J    z B( z )   E ( z ) 
J ( z)
c
c
c
c
Putting together the inhomogeneus equations,
i
B( z )
c
i
4
  z B( z )   E ( z ) 
J ( z)
c
c
 z E( z) 
 E( z)  
2
z
2
c2
E( z) 
4 i
J
2
c
4
4
 E( z)  
2
z
2
c2
E( z) 
4 i
J has 2 unknowns, a further condition needed.
2
c
Assuming a transverse elecromagnetic wave in homogeneous local medium
J ( z )   ( ) E ( z ),
2
 ( )  conductivity.
2
4

i


 2z E ( z )   2 E ( z )  2 J   2
c
c
c
 4 i

1


(

)
E( z)




In vacuo, Maxwell's equations yield  E ( z )  
2
z
2
c2
E(z)
c2
Maxwell equations in medium are obtained by c 
,
 ( )
4 i
 ( )  1 
 ( ) the dielectric function
2

c
c  , n  refraction index, n   , assuming  =1.
n
How can we model  ( ) ?
5
Drude’s bold theory of electromagnetic waves in metals
dv
mv
F
,   relaxation time
dt

mv
Lorentz force F  e( E  v  B ), 
friction term
EOM for electrons: m

In constant field, for t>> one finds
v   E,
e
 mobility
m
dv
mv
 eE0 e it 
.
dt

yielding
Oscillating field : E  E0 e it 
This is solved by v  v 0 e  it

m
1
m(i  )v 0 e it  eE0 e it  complex oscillating velocity

v0 
eE0
1
m(i  )


e E0
.
m 1  i
e E0
e it .
m 1  i
2
0
J (t ) ne 2
1
ne 2
1 4 ne 2  p
 ( ) 


, 0 



E (t )
m 1  i 1  i
m
4
m
4
This produces a current J (t )  nev  nev 0e  it  ne
4 ne 2
 
m
2
p
where  p  plasma frequency.
6
Drude dielectric function
 p2
0
 ( ) 
  ( )  1 
 1 ( )  i 2 ( )
i
1  i
 (  )

 p2 2
 p2 2
1 ( )  1 
,  2 ( ) 
1   2 2
 (1   2 2 )
 ( )
 ( )  1  i

7
2 2


p
2 2
Low frequency region:  1, 1 ( )  1   p ,  2 ( ) 

Typically  p2 2 1, 1 is large negative,  2 1.
intermediate frequency region: 1

 p2
 p2
 p , 1 ( )  1  2 ,  2 ( )  3


large refractive index, metal reflects
 p2
 p2
high frequency region:    p , 1 ( )  1  2  1,  2 ( )  3  0


metal is transparent.
8
2 


Maxwell's equations  E( z)   2 1 
 ()  E( z)   2  () E( z) imply
c 

c

2
z
 E( z)
e
ikz
with k 
Simple metals have
propagating waves have
p 
2
2
2
c
2
  
 p2
4 ne 2
2
    1  2 ,  p 

m
 2   p2  c2 k 2  quadratic plasmon dispersion
12
Ry, 1Ry  13.6 eV
3
rs
Cs rs  5.64
 p  3.51 eV
Au rs  3.0
 p  9.07 eV
Al
4 i
rs  2.0
 p  16.66 eV
From Blaber et al., J.Chem.Phys (2009) experimental, with g=1/
absorption should be in the UV, width 1/d.
unexplained!
9
At surfaces, Surface plasmon polaritons!
They are plasmon-photon modes
z
1
vacuum
localized at an interface.
Consider vacuum for z>0 with =1
and metal with  for z<0, wave
propagating along x.
X
wave
metal

Look for solutions with:  x  iq
E  ( Ex ( x, z ), 0, Ez ( x, z ))eit
i nd  0
B  (0, By ( x, z ), 0)e it
nothing depends on y
J 0
Maxwell equations: divE  4i nd  0 and divB  0
rotE  ( y Ez   y Ez ,  z Ex   x Ez ,  x E y   y Ex )  ( y Ez ,  z Ex   x Ez , 0)
1 B
  i By   z Ex   x Ez
c t
rotB  ( y Bz   y Bz ,  z Bx   x Bz ,  x By   y Bx )  ( z By , 0,  x By )
rotE  
  z By  i ( ) Ex
rotB  
 (letting c  1) 
c t
 x By  iqBy  i ( ) Ez
 E
10
We must solve:
i By   z Ex  iqEz
 z By  i ( ) Ex
iqBy  i ( ) Ez
Apply  z to the second,   z 2 By  i ( )  z Ex and get  z Ex   z Ex  i By from the first.
Rearranging,   2 ( ) By   2z By  q ( ) Ez .
Substitute Ez =
qBy

from the third and get:
 2 By  ( 2z  q 2 ) By
in metal
 2 By  ( 2z  q 2 ) By
in vacuo
11
[ 2z  q 2 
2
c2
 p2
4 ne 2
2
    1  2 ,  p 

m
]By  0
z
1
vacuum
[ 2z  q 2   ( )

c
 By  B0 [ ( z )e g z   ( z )eg z ]
X
wave
metal
2
2

] By  0
localized excitation
outside, z  0,   1  g  q 
2
2
c
inside, z  0,
2
 g  q   ( )
2
2
c2
Next, we find the electric field, which is also localized:
gc
i
B0 e g z

 z By  i ( ) Ex  Ex 
,z 0
gc
i
B0 eg z , z  0

2
2
Continuity condition at z  0
g 
g

12
Continuity condition at z=0  dispersion law
 p2
g
g 
requires   0 and with     1  2 <0  ω below  p ,
 ( )

Link between q and ω : (g )  g
2
(q 
2
2
c2
) ( )  q   ( )
2
2
2
c2
2
with
g  q 
2
2
c2
.
This condition can be rewritten
( -1)[   (c 2q 2 - 2 )+c 2q 2 ]=0,
as one can see immediately,
 ( ) (c 2q 2 - 2 )+c 2q 2  0;
so the condition really is:
with Drude
 p2
 p2
    1 
 1  2 we can solve for  (q) :
i

 (  )

 p2 2 2 2 2 2
(1  2 )(c q - )+c q  0   4   p2  2c 2 q 2   2  c 2 q 2 p2  0 biquadratic equation

1
2
sp2 (q)  [ p2  2c 2 q 2   p4  4c 4 q 4 ]
the other sign is not acceptable because it gives
 p2
  P that implies     1  2  0.13

x
cq
p
sp
y=
p
photon-like
polariton
cq /  p  1 for q  106 cm1.
This is

a
 10 cm so in most of the BZ sp 
8
1
This is not yet suitable for the nanoclusters.
p
2
Scattering of light by a spherical metal particle: Rayleigh approximation
Rayleigh scattering describes the elastic scattering of light by spheres which are
much smaller than the wavelength of light. The intensity of the scattered radiation is given by
2
 2   n  1   d 
 I in 
 
  2

n

2

 
 2 
4
I scatt
6
 1  cos( ) 2 


2
2
R


Blue is scattered much more than red.
2

n2  1 2
p
2
Set n   ( )  1  2 into I ( ) ( 2
)

n 2
get I ( ) (
p2
 p  3
2
2
)2
 plasma resonance at  
p
3
. This is about right.
16
Mie scattering
Mie in 1908 solved Maxwell’s equations for the
scattering of a plane-wave in a medium on a sphere
with refractive index n. Absorption coefficient
g
18

3
2
m
2
2
 2 m  1    22
  volume density (very small),
 m  dielectric constant of medium
  1  i 2  dielectric constant of metal,
 =wave length
radius a of particle
In Freiburg, during the Nazi dictatorship,
Mie was member of the university
opposition of the so-called "Freiburger
Kreis" (Freiburg circle) and one of the
participants of the original "Freiburger
Konzil".
 p2
Resonance : 2 m  1  0. If 1   0  2 ,

p
Re sonance 
 0  2 m
 0  4.39 ( Al ),  6.34 (Cu ),  7.7( Au )
(T .J . Antosiewicz et al , Plasmonics 6,11 (2011))
17
How does the resonance shift from  
p
3
(small sphere) to  
p
2
(plane)?
For d<< the dipole plasmon dominates.
With increasing d the quadrupole term acquires importance, and it leads to a
higher resonance frequency; then higher multipoles enter. For a large sphere one
gets the plane response.
Important Complications
Quantum size effects See kawabata and Kubo J. Phys. Soc. Japan
(1966) ; M.Cini and P.Ascarelli J. Phys. C (1974): the dielectric constant
of small Ag particles is semiconductor-like
1  1 
32me2 k F

5 2
L2 , L  size ( for Al 20 Angstrom partecles, 1
85)
nonspherical shapes
size distributions, distance distributions, interparticle multipolar
interactions
matrix interactions
18
Broadening of plasma resonance :
Classically :  R
Quantum mechanically : Kawabata  Kubo theory
U. kreibig, Journal de Physique Colloque C2 (1977)
19
Quantum Dots and wires
Quantum dots are semiconductor “nanoparticles” (e.g. CdSe , ZnS)
Sizes range from 2 to 10 nanometers in diameter (about the width of 50 atoms)
They are produced by molecular beam epitaxy or by lithographic techniques
(lithography is based on covering a plate with chemicals such that the
image is produced by a chemical reaction)
Optical and electrical properties that are different in character to those of the
corresponding bulk material.
20
smaller
dots
larger
dots
By the size one can control the band gap and so the color. Larger dots
give a redder fluorescence spectrum.
21
The energy spectrum of a quantum dot can be engineered by controlling size and
shape. One can also tailor the strength of the confinement potential. Also, one
can connect quantum dots by tunnel barriers to conducting leads.
One can also order arrays of quantum dots by electrochemical techniques
SET= single electron tunneling
Applications to electronics (single-electron transistor, showing the Coulomb blockade
effect) and qbits for quantum computers are also envisaged. Also photovoltaic
devices, LED, photodetectors have been built.
22
Quantum wells
produced by MBE: several monolayers of semiconductor over a host crfystal by
molecular beams
AB-CD alloy:linear dependence of gap and lattice constant
Egap ( x)  xEgap , AB  (1  x) Egap ,CD
a( x)  xa AB  (1  x)aCD
so, one can choose the gap, makesuperlattices with
periodically modulated gap, etc
GaAs often used with AlAs (same lattice parameter a):
AlAs acts as a barrier,and one can make Gax Al1 x As
Type I Quantum wells
electrons and holes are confined
AlAs
GaAs
Type II Quantum wells
electrons are confined, lowest hole energy in host
AlAs
GaSb
GaSb
InAs
23
1-body Density of states per spin in QW
N (E) 
mx my

2
bound states

E  Ej 
j
Modulation doping
By adding donors in semiconductors one introduces conduction electrons
(wanted) and scattering centres (unwanted).
Modulation doping: donors added to host, outside the QW give electrons to
QW but with very little scattering.
24
Type I Quantum wells
electrons and holes are confined
AlAs
GaAs
Type II Quantum wells
electrons are confined, lowest hole energy in host
AlAs
GaSb
GaSb
InAs
Although band gaps are known, determining the conduction and valence
band offsets theoretically and experimentally is not easy.
Confined excitons get distorted and have a position-dependent
binding energy.
Confinenment is a particle-in-a box problem but walls are finite and
masses are different: say, m inside and  outside.
Matching conditions:
 continuous,
1 d
continuous
m( z ) dz
( grants continuity equation)
A more microscopic approach is based on embedding techniques
25