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Corso per il dottorato- 2012-13

Nanostrutture e sistemi di bassa dimensionalita' (Michele Cini)

http://people.roma2.infn.it/~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 Bosonizzazione-Separazione di spin e carica- Applicazioni.

5-Effetti di correlazione in 2d: Gas di Fermi in campo magnetico ed effetto Hall quantistico intero e frazionario.

Exact solutions:Bethe ansatz,Ising model,Nagaoka ferromagnetism….

New concepts and specific phenomena: anyons, charge fractionalization,spin-charge separation, QHE

But strong correlation (not only in models, but in reality): exotic behavior, troubles in standard treatments

Small is different: all properties of nanostructures are size dependent

2 2 33 2 on finely dispersed iron

Chemical properties of Fe depend on particle size

N H O NH

Rich phenomenology, many applications

2d, 1d, 0d nano-objects: molecular size in 1, 2 or 3 dimensions

Models of reduced dimensionality are endorsened by quantum mechanics at low temperatures: gaps develop and degrees of freedom are frozen!

Special Methods:Topology plays an important role, bosonization, Bethe Ansatz

Phase transitions strongly depend on dimensionality!

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The colors were achieved by a colloid dispersion of gold nano-particles in glass.

Stained Glass

Gothic window of Notre Dame de Paris

(XIV century)

44

Transverse electromagnetic wave in homogeneous isotropic medium

i

( ( ),0,0) (0, ( ),0) (0,0, )

Simplest case : no and the current ( ( ),0,0)

i t i t

i tnd

E E z e B B z e S E B S

E J J z e

Consider the plane wave going upwards

iMaxwell equations: 4 0 and 0nddivE divB

( , , ) (0, ( ),0)

( ) ( )1y z z y z x x z x y y x z

t z

rotE E E E E E E E z

rotE iE z B zBc c

( , , ) ( ( ),0,0)

1 4 4( ) ( ) ( )

y z z y z x x z x y y x

t z

zrotB B B B B B B B z

r iB z E z J zc

otB E Jc c c

( ) ( )

4( ) ( ) ( )

z

z

iE z B zc

iB z E z J zc c

2

22 2

4( ) ( )ziE z E z J

c c

Putting together the inhomogeneus equations,

5

( ) ( ) ( ), ( ) conductivity.J z E z

Assuming a transverse elecromagnetic wave in homogeneous local medium

2 22

2 2 2

4 4( ) ( ) 1 ( ) ( )zi iE z E z J E z

c c c

22

2

22

In vacuo, Maxwell's equations yield ( ) ( )

Maxwell equations in medium are obtained by ,( )

4( ) 1 ( )

, ref

thedielectric function

raction index, , assuming =1.

zE z E zc

cc

i

cc n nn

How can we model ( )?

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2 2

4( ) ( ) has 2 unknowns, a further condition needed.ziE z E z J

c c

6

Drude’s bold theory of electromagnetic waves in metals vEOM for electrons: , relaxation time

Lorentz force ( )

v

v friction ter m,

dm Fdt

F e E v B

m

m

In constant field, for t>> one finds , mobilityev Em

0 0

0

0 0

0 00

v vOscillating field : .

This is solved by v v yielding1( )v complex oscillating velocity

v .1 1( )

i t i t

i t

i t i t

d mE E e m eE edt

e

m i e eE e

eE Eem im i

00This produces a current ( ) v v .

1i t i tEeJ t ne ne e ne e

m i

22 2 20

0

22

( ) 1 1 4( ) ,( ) 1 1 4 4

4 where plasma frequency.

p

p p

J t ne ne neE t m i i m m

nem

7

Drude dielectric function2

01 2

2 2 2 2

1 22 2 2 2

( )( ) 1 ( ) ( ) 1 ( ) ( )1 ( )

( ) 1 , ( )1 (1 )

p

p p

i iii

8

2 22 2

1 2

2 21 2

Low frequency region: 1, ( ) 1 , ( )

Typically 1, is large negative, 1.

pp

p

2 2

1 22 3intermediate frequency region: 1 , ( ) 1 , ( )

large refractive index, metal reflects

p pp

2 2

1 22 3high frequency region: , ( ) 1 1

transpare

, ( ) 0

meta nl is t.

p pp

9

2

22( ) with ikzE z e kc

2 2

22

41 ,pp

nem

Simple metals have

propagating waves have 2 2 2 2 quadratic plasmon dispersionp c k

3

12 , 1 13.6

5.64 3.51

3.0 9.07

2.0 16.66

ps

s p

s p

s p

Ry Ry eVr

Cs r eV

Au r eV

Al r eV

absorption should be in the UV, width 1/d.

From Blaber et al., J.Chem.Phys (2009) experimental, with g=1/

2 22

2 2

4Maxwell's equations ( ) 1 ( ) ( ) ( ) ( ) implyziE z E z E z

c c

unexplained!

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At surfaces, Surface plasmon polaritons!They are plasmon-photon modes

localized at an interface.

Consider vacuum for z>0 with =1 and metal with for z<0, wave propagating along x.

i

nothing depen

Look for solutions with:

( ( , ), ds on y0, ( , )) (0, ( , ),0)

0 0

x

i t i tx z y

nd

iq

E E x z E x z e B B x z e

J

iMaxwell equations: 4 0 and 0nddivE divB

( , , ) ( , ,0)

1y z y z z x x z x y y x y z z x x z

y z x x z

rotE E E E E E E E E E

BrotE i B E Ec t

( , , ) ( ,0, )

( )(letting 1)

( )

y z y z z x x z x y y x z y x y

z y x

x y y z

rotB B B B B B B B B

B i EErotB cB iqB i Ec t

X

z

1vacuum

metal

wave

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We must solve:

( )

( )

y z x z

z y x

y z

i B E iqE

B i E

iqB i E

2

2 2

Apply to the second, ( ) and get from the first.

Rearranging, ( ) ( ) .

Substitute = from the third and get:

z z y z x z x z x y

y z y z

yz

B i E E E i B

B B q E

qBE

2 2 2

2 2 2

( ) in metal

( ) in vacuoy z y

y z y

B q B

B q B

12

0

2 22 2

2 2

[ ( ) ( ) ]

outside, 0, 1

localized excitatio

inside, 0, ( )

nz zyB B z e z e

z q z qc c

g g

g g

2

2

0

0

( ) , 0

, 0

z

z y x x

z

ci B eB i E E z

ci B e z

g

g

g

g

Next, we find the electric field, which is also localized:

X

z

1vacuum

metal

wave

22 2

2[ ] 0z yq Bc

22 2

2[ ( ) ] 0z yq Bc

2 2

22

41 ,pp

nem

Continuity condition at z 0 gg

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Continuity condition at z=0 dispersion law

2

2

22 2 2

2

2 22 2 2

2 2

and with 1 <0 ω ,( )

Link between q and ω : ( ) with

requires 0

( ) ( ) ( ) .

belowpp

qc

q qc c

gg

g g g

2 2 2 2 2

2 2 2 2 2

2 2

2

22 2 2 2 2

2

This condition can be rewritten [ (c q - )+c q ]=0,as one can see immediately,

so thecondition is: ( ) (c q - )+c q 0;

with Drude 1 1 wecan solve for ( ) :(

( -1

)

(1 )(c q - )+c q

)

p p

p

really

qi

4 2 2 2 2 2 2 20 2 0 biquadratic equationp pc q c q

2 2 2 2 4 4 41( ) [ 2 4 ]2sp p pq c q c q

the other sign is not acceptable because it gives 2

2that implies 1 0.pP

y= sp

p p

cqx

6 1

8 1

/ 1 10 .

This is 10 so in most of the BZ 2

p

psp

cq for q cm

cma

This is not yet suitable for the nanoclusters.

photon-like

polariton

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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

Scattering of light by a spherical metal particle: Rayleigh approximation

2 22 2

2 2

1( ) 1 into ( ) ( )2

p nSet n In

22

2 2get ( ) ( )3

. .3

p

p

p

I

plasma resonance at This is about right

4 62 2

2 2

2 1 1 cos( )2 2 2scatt in

n dI In R

Blue is scattered much more than red.

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Mie scattering

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".

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

32

22 2

1 2

182

m

m

g

1 2

volume density (very small),dielectric constant of medium

dielectric constant of metal,=wave length radius a of particle

m

i

2

1 1 0 2

Re0

0

Resonance : 2 0. ,

24.39 ( ), 6.34 ( ), 7.7( )

( . . , 6,11 (2011))

pm

psonance

m

If

Al Cu AuT J Antosiewicz et al Plasmonics

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nonspherical shapes

size distributions, distance distributions, interparticle multipolar interactions

matrix interactions

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

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1 15 2

321 , ( 20 , 85)Fme k L L size for Al Angstrom partecles

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.

How does the resonance shift from (small sphere) to (plane)?3 2p p

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U. kreibig, Journal de Physique Colloque C2 (1977)

::

:

Broadening of plasma resonanceClassically RQuantum mechanically Kawabata Kubo theory

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Quantum Dots and wires

Quantum dots are semiconductor “nanoparticles” (e.g. CdSe , ZnS)

Optical and electrical properties that are different in character to those of the corresponding bulk material.

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)

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By the size one can control the band gap and so the color. Larger dots give a redder fluorescence spectrum.

smaller dots

larger

dots

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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

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.

SET= single electron tunneling

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Quantum wells

produced by MBE: several monolayers of semiconductor over a host crfystal by molecular beams

, ,

AB-CDalloy:linear dependenceof gap and( ) (1 )

so,onecan choose the gap, makesuperlattices with peri

latticeconstant( ) (1

odically modulated gap, etcGaAs often used with AlAs (samel

)gap gap AB gap AB CDCD a x xa xE x x x E aE

1

attice parameter a):AlAs actsas a barrier,and onecan make x xGa Al As

Type I Quantum wells

GaAs AlAsAlAs

electrons and holes are confined

Type II Quantum wellselectrons are confined, lowest hole energy in host

InAs

GaSb GaSb

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Modulation dopingBy 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.

2( )bound states

x yj

j

m mN E E E

1-body Density of states per spin in QW

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Confinenment is a particle-in-a box problem but walls are finite and masses are different: say, m inside and outside.Matching conditions:

Although band gaps are known, determining the conduction and valence band offsets theoretically and experimentally is not easy.

1, ( )( )

dcontinuous continuous grants continuity equationm z dz

Type I Quantum wells

GaAs AlAsAlAs

electrons and holes are confined

Type II Quantum wellselectrons are confined, lowest hole energy in host

InAs

GaSb GaSb

Confined excitons get distorted and have a position-dependent binding energy.

A more microscopic approach is based on embedding techniques