What is the typical energy of an acoustic phonon of 50GHz ? Deduce the typical time of interaction between an electron and that acoustic phonon

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MASTER of PHYSICS Examination

Lecture : Ultrafast phenomena and optoacoustics (P. Ruello)

Part 1 : WITHOUT DOCUMENT

Heisenberg relation :

1-‐ Localization time :

We consider the two electronic structures depictured below. The grey area corresponds to occupied levels. Give an order of the localization time of an electron in the vicinity of a cation for both electronic structure for the band of 10eV and 0.5ev width.

2-‐ Interaction time between electrons

What is the typical energy of the Coulomb interaction between two electrons ? What is then the typical interaction time

3-‐ Interaction time between an electron and a phonon

What is the typical energy of an optical phonon of 3THz ? Deduce the typical time of interaction between an electron and that optical phonon.

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What is the typical energy of an acoustic phonon of 50GHz ? Deduce the typical time of interaction between an electron and that acoustic phonon.

Which process among electron-‐acoustic phonon and electron–optical phonon is the fastest process ?

4 Femtosecond Pump-‐probe setup

The objective of the question is to describe the principle of the optical pump-‐probe method.

Explain the different elements shown below (fill the boxes for examples).

Part 2 WITH DOCUMENT

Electronic pressure and optoacoustics

1-‐ We consider a free electron gas at 3D (N electrons in a volume V) according to the Sommerfeld model. The metal will be considered as isotropic. The lattice is cubic with a lattice parameter a=0.2nm. Each atom provides 2 conduction electrons. The molar mass is 80g/mol. The longitudinal acoustic sound velocity is 5000m/s.

-‐ give the energy of the free electron

-‐ what is the concentration of conduction electrons in this metal ? -‐ give the wave function form of a free electron

-‐ considering the Born Von Karman boundary limits, give the density of state g(k)

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-‐ Detail the calculation leading to g(E) : g(E)= V

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3-‐ Considering the definition of the pressure P=-‐(dE/dV)N, calculate it for a quantum free electron gas (T=0K). Give an estimate of this pressure.

(question 4 to 7 are independent to previous questions)

A femtosecond laser excites a metal giving rise to a rapid increase of the electron gas pressure (following model of question 3 that we assume to be valid). The pump light penetrates beneath surface over a distance L=10nm according to the profile P=P0exp(-‐

z/L).

4-‐ What is the typical acoustic frequency which is generated by the femtosecond laser ?

5-‐ The maximum electronic pressure on the surface is P0=1GPa. Considering Newton Law, what is the force F imposed to the atoms ? Give the numerical value of F for z=0.

6-‐ Considering this metal has a longitudinal sound velocity of 5000 m/s. Deduce the elastic constant C characterizing the elastic interatomic force. Deduce the lattice parameter change Δx induced by the electronic pressure for atoms on the surface (z=0).

Deduce Δx/a.

7-‐ If we consider the photoelastic effect with Δn=P×(Δx/a), where P is the photoelastic coefficient with P= 1 here, what will be the order of the change of optical reflectivity induced by this photo-‐induced acoustic wave ? Can we detect directly this optical reflectivity change by a photodiode ?

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