MASTER « Materials Sciences and Nanotechnologies » Tutorials : Solide state Physics, Quantum Mecahnics , Nanophysics

(Prof.A.Kassiba ) Exercice 1- Surface Cristallographie, reconstruction, relaxation , notation

1. Define what are the surface energy and the surface tension 2. Define briefly what is the meaning of surface relaxation and surface reconstruction 3. What is the main difference between semiconducting surface and metallic surface 4. Recall the energetic principles which monitor the relaxation and reconstruction of

surfaces 5. Recall the model of Smoluchowski and use it to explain the physical phenomena which

contribute to the relaxation of atomic layers close to metallic surface.

6. a- Show that the matrix notation for °− 3033 Rx superstructure on 2D hexagonal

lattice is

− 11

12, when the basic translation vectors make an angle of 120°.

b- How will the matrix change if the basic vectors make an angle of 60°? Exercice 2- Solid state Physics – electronic properties, Drude Model

In the Drude Model, the scattering probability of an electron with a lattice ion is

supposed to be time independent. The probability that an electron is scattered during the period

dt is proportional to τdt

. The time constant τ is relaxation time which characterizes the

features of scattering.

1. Show that a given electron at t=0 didn’t feel any scattering during a coming period t

with a probability τ/te−

2. Show that for the same electron, the probability that a first scattering in the period

between t and t+dt is given by τ

τ/te

dt − .

3. Show that for an electron the average time between two scattering (average time of

flight) is equal to τ . Use the following relation: 1

0

!+

−∞

=∫ naxn

a

ndxex

Let us consider now a gas of free electrons without any collisions between the particles

but submitted to an electric field xuEErr

= .

4. Gives the expression of the velocity )(tv of the electrons and the displacement x(t).

5. Determine the expression of the average velocity v and the average distance d

between 2 collisions of the electrons taking into account the distributions of times of flight.

Application : evaluate d for copper with E=106 V/m , s1410.7.2 −=τ at room temperature.

Exercice 3- Nanophysics: Electronic density of states in 1D nanostructure We consider 1D carbon nanotube with the axis defined by ∆ and the chiral vector

Wr

whose the norm is dW .π=r

; d represents the nanotube diameter. We use L as the total

length of the tube.

Experimental density of states of single carbon nanotubes

The motion of conduction electrons can be considered as free along the tube axis while a confinement is felt by these electrons in the perpendicular direction. The wave-functions can be set as plane waves with two characteristics wave-vectors : //k parallel to ∆ and ⊥k perpendicular

to ∆.

1. Which periodic conditions are satisfied by the wave-functions in perpendicular direction to the tube axis ∆?

2. Show that the wave-vectors ⊥k are quantized and give their expression in function of the parameters of the tube.

3. Prove that the kinetic energies are quantized in the perpendicular direction and give their expressions.

4. Derive the expression of the electronic density of states N(E) and give its qualitative draw

5. Compare this theoretical result with the experimental results obtained on single carbon nanotube. Explain briefly how these measurements were carried out?

Exercice 4- Solid state Physics, Nanophysics - Collectives excitations, Excitons 1. Define the physical composition of excitons in semiconducting materials, 2. What is the difference between Frenkel and Wannier like excitons? 3. Describve briefly without any calculations, how we can determine the energetic diagram of the exciton-like system? 4. The experimental absorption coefficient of 2D nanostructure is shown below as function of the nanostructure width. What are the main features which can be extracted from these results?

Exercice 5 Exercice 4- Solid state Physics, Nanophysics - Collectives excitations, Excitons

: Size inducing shifts of Plasmon frequencies in semiconducting nanodots EEL (electron energy losses) spectroscopy coupled with transmission electron microscopy can investigate a nanocluster thanks to the electron probe size about 2nm. The results obtained on Silicon as bulk or as nanoscale clusters are depicted below. The reduction of the cluster size leads to broader band of the plasmon absorption and to high frequency shifts (M.Mitome et al. J.Appl.Phys.72(2),1992). The main reason of such behaviour is discussed below in term of quantum confinement phenomena which manifests below some critical size.

1. Define precisely the size criterion for the appearance of quantum confinement in semiconducting structure. Concrete examples and estimations are required.

2. When a valence electron of the structure is submitted to an applied electrical field )(ωE and to the atomic force ( oscillator-like ), show that the dielectric function is given

by :

and give the expression of the plasmon energy ppE 0ωh= as function of the density of

electrons (n), their masse (m) and the dielectric permittivity of vacuum )( 0ε .

3. The crude expression derived in question 2, is an approximation and more refined formulae can be obtained by quantum mechanics. The correction to the plasmon frequency is 0. pp f ωω = where f represents the oscillator strength given by:

2

22

2vc

g de

mEf

h=

Bulk Si

10 nm 6.6 nm3.3 nm Cluster

Bulk Si

10 nm 6.6 nm3.3 nm Cluster

20

2

20 )(1)(

ωωωωε

−−= p

In this formulae, Eg represents the energy band gap of the cluster and dvc the dipole element for the transition from valence to conduction band. As the size of the cluster is reduced, the band gap energy increases. To evaluate this change of Eg , we consider a simple model of one exciton (electron-hole) with radius r (size of the cluster).

By using a simple model of an effective particle of reduced masse he

he

mm

mmm

+=* , show

that the energy change gE∆ is given by :

2

22

*2

3

rmEg

hπ=∆

4. Interpret the experimental result of the shift of the plasmon energy with the cluster size reduction. Exercice 6 : Solid state physics - Quantum Physics 1D,2D and 3D Electronic density of states of free electrons We consider dEEg )( as the energy level number for an electron by volume unit and in the range between E and E+dE. The function g(E) represents the density of states by volume unit. Each electronic state is characterized by a wavevector k and spin s wit two states.

1. Calculate g(E) for the free electron gas with 1D,2D and 3D ( Use the rule that the number of states in the volume of reciprocal lattice Σd is equal to the number of states in the associated energy range dE; se also the dispersion relation E(k)).

2. Plot the graph of the product )().(' EfEg FD for T=0 and T>0 for 1D,2D ad 3D case. 3. describe the evolution of the chemical potential as function o the temperature for the

different gases.

Exercice 7- Surface Cristallographie, reconstruction, relaxation , notation

The deposition of a material, as thin crystalline layers, leads to relaxation and reconstruction phenomena. 1. Discuss the main characteristics of these phenomena and their importance following the electronic properties of the material. b. Explain the matter of the Smoluchowski effect in metals. c. Why the reconstruction phenomena are very rare in metals compared to the relaxation effects? d. The following picture shows different configurations of carbon atoms (cross) on metallic surfaces (open circle) . Give in Wood and in Matrix notation, the nomenclature of the carbon lattice for each configuration.

.

Exercice 8- Nanophysics – Quantum confinments – Nanodots

II. Electronic density of states, band structure and dimensionality A- Confinement effects in semiconducting nanodot

1. What is the physical meaning of the quantum confinement effect in a

semiconducting nanodot (QD). 2. If the QD exhibits the shape of a cubic box with a dimension R, give the expression

of the band gap variation )( E∆ due to the confinement effect. )( E∆ must be obtained as a function of the length R and the effective mass of an exciton )(µ .

3. The exciton Bohr radius is about few namometers in a QD semiconductor, explain the features of excitons and how they may affect the electronic and optical properties of the material.

4. What is the important information which we can we learn from the picture (d)? 5. What is the main physical parameter which monitors the features of the absorption

curves depicted in the figure (d)?

B- Structure and electronic properties of carbon nanotubes A carbon nanotube (Fig.1), realized by rolling up a graphene sheet, is characterized by the

chiral vector 21 amanWrrr

+= with 21,aarr

the lattice parameter of grapheme.

A Graphene sheet A Carbon nanotube 1.Structure 1.1 By using the above 2D structure of graphene, give the lattice parameters of the

structure, 21,aarr

, in function of the C-C bond length 0a .

1.2. Deduce the expressions of the reciprocal lattice vectors 21,bbrr

1.3. Show that the first Brillouin zone (BZ) is a regular hexagon with the corners

characterized by a distance 033

4

aK

π=r

with respect to the center of the BZ.

W

θθθθ

W

θθθθ

2. Electronic behaviour The electronic structure of a carbon nanotube can be represented in the vicinity of BZ

borders ( vectors Kr

) by the following dispersion relation:

κγ rr

±=2

3)( 01akE

with 1γ the exchange integral between adjacent carbon atoms and κrrr

+= Kk , κr represents an infinitesimal vector.

2.1.By using the boundary conditions satisfied by the electronic wavefunction :

)()()( ).( reWrWrk

Wri

Kk

rrrrrr

rrr

rrr ψφψψ κκ =+=+ +

and the Bloch theorem traduced by )()( . reWrk

Wki

k

rrrr

rr

r ψψ =+ , determine the expression of the

wavevector κr in term of a transverse component )( ⊥κ perpendicular to the tube axis and a

longitudinal component )( //κ along the tube axis.

2.2. Deduce the expression of )( ⊥κ component in function of the nanotube parameters (n,m).

2.3. Give the dispersion relation )(kEr

in the vicinity of the BZ border as function of the above parameters. 2.4. Discuss the electronic properties of the carbon nanotube (semicondutor, metal) following the values of (m,n) parameters. 2.5. Plot qualitatively the dispersion curves in the cases of semiconducting and metallic nanotube. 2.6. Determine the expression of the electronic density of states N(E) in the case of a nanotube with square-like basis with a nanometric dimension (L). 2.7. Plot qualitatively N(E) in parallel with the dispersion curve drawn in 5) and comment the

overall diagram ))(),(( ENkEr

.

Exercise 9: Lattice vibrations Phonons

1. Linear chain, influence of second neighbours A monoatomic linear chain (mass m) with force constants f1 and f2 respectively between first neighbours and second neighbours ( )12 ff < .

1. Write the Hamiltonian for the nth atom of the chain and deduce the equation of its

motion. 2. Deduce the dispersion relation )(kω by using the relation )(

0)( tknain eutu ω−=

3. for f2>0, determine the condition that )(kω shows a minimum in the first zone of Brillouin

4. for f2<0, (repulsive interaction), find the condition that the chain will be stable. 5. Draw )(kω for both cases (3, 4).

Exrcice Solid state Physics (refernce EPFL – 2010 – pr.H.Brune) Linear lattice vibrations Part I Lets consider a linear chain of equidistants ions with a period a and possessing alternatively the mass m for ions at positions 2na and the mass M at positions (2n+1) a. The ions are submitted to a force constant f between first neighbours.

1. Write the equations of motions for the two kind of ions. By using solutions as )2(

2 )( nkatin Aetu −= ω and ))12((

12 )( kantin Aetu +−

+ = ω , determine the dispersion relation

as:

kamMMm

fMm

fk 22

2 sin41111

)( −

+±

+=ω

2. Draw the curve of )(kω in the first brillouin zone. Gives the expression of )0(ω 3. Evaluate )0(ω for NaCl with f=40N/m

Part II Consider the case where the ions m have charge +e and those M with charge –e. A transversal electric field is applied such as )(

0kxti

z eEE −= ω of electromagnetic (EM) sinusoidal wave with

pulsation ω which propagates along the chain. 1. Evaluate the wavelength of EM wave associated to )0(ω of NaCl for the optical

branch. 2. Show that this wavelength is quite large compared to the lattice parameters a=2.8A°. 3. Describe the displacement of ions under EM wave and comment that this wave will

excite the optical transversal vibration modes of low wavevectors. 4. The propagating term of the electric field is very low (k=0), deduce the equations of

motions under the electric field and the interaction between first neighbours. (consider for the transverse modes the same force constant f used for longitudinal modes).

5. By using the form tin Aeu ω=2 and ti

n Beu ω=+12 , deduire the expressions A and B for

the amplitude of motions along z direction as function of m, M, ω and

)11

(2Mm

fT +=ω .

6. Gives the expression of the dipole moment in the z direction )(tp z and produced by two neighbouring ions.

7. The chain is a part of a solide which contain N ion groups (m,M) by volume unit. The polarization by volume unit P for the ionic crystal under the action of the electric field taking into account the electronic polarizabilities Mm and αα of each kind of ions is

)()( tNpENP zMm ++=rr

αα

8. Give the evolution of the relative dielectric constant )(ωε of the solid in function of

the pulsation ω . Explore the situation of 0=ω (static) and for ∞→ω . 9. Show that the dielectric function can be put on the form:

2

1

)(

−

−+= ∞

∞

T

s

ωω

εεεωε

What is the behaviour when Tωω →

Exrcice 10 Bose-Einstein distribution and specific heat of solids 1. The Bose-Einstein distribution gives the average number of phonons in the mode defined by k,s:

1

1)(,

−

=Tk

ksk

B

s

e

n ωh

a- Determine the behaviour of the function 2/1, +skn at high temperature

b- Draw 2/1, +skn as function of s

BTkx

ωh=

2. Specific heat of Bose condensate A gas of free bosons can undergo a phase transition at critical temperature TC, below which a Bose condensate can be formed. The existence of this phase was predicted by Bose and Einstein before its experimental demonstration. In 2001, the Nobel prize was attributed to Eric Cornell, Carl Wiemann and Wolfgang Ketterle for proving its existence experimentally. This phase corresponds to a majority occupation of the fundamental state (E=0). The number of occupation is given by the Bose-Einstein statistics for T<Tc:

1

1

−

=Tk

E

Be

n

The density of states g(E) of free bosons gas is derived in the same way as for electron gas. The general formula for particle of spin S is given by:

Em

SEg2/3

22

2

4

1)12()(

+=hπ

Write the expression of the density of energy u of Bose condensate and derive the expression

for the specific heat T

ucv ∂

∂=

Uses: )2/5(4

3

10

2/3

ςπ=−∫

∞

dxe

xx

where Zeta Rienman function is 34.1)2/5( =ς

Exercice 11 : Solid state Physics - Electronic band structure of crystalline surface In a crystalline surface with a cubic like elementary (parameter a), the electrons weekly bonded to atoms can be considered as submitted to a perturbation potential given by :

=a

y

a

xAyxV

ππ 2cos.

2cos),(

1. Define the reciprocal lattice vectors 1br

et 2br

. 2. Draw the first and second Brillouin zones 3. Gives the Fourier development of the potential in term of reciprocal lattice vectors. 4. Discuss the degeneracy of energy levels in the first Brillouin zone. 5. Show that at first order of perturbation, the correction of the energy by the periodic

potential are different than zero for the wave-vectors at the boundaries of Brillouin zone. Calculates the corrections in these cases.

6. Calculate the correction of energy at the second order of perturbation for the wave-

vectors inside the Brillouin zone and gives the expression of the energy )(kEr

for a given wave-vector.

7. In the vicinity of the centre of Brillouin zone, determines the effective mass of electrons in the crystallographic directions [10] et [11].