Statistical physics (2005)

This formula allows one to calculate the di erence between an integral and a sum over n, for a = nε.

Constants

e 2.718, π 3.1416

Riemann zeta function :

Exercise 2000 : Electrostatic Screening

I. Let us consider N free fermions of mass me and spin 1/2, without any mutual interaction, contained in a macroscopic volume Ω maintained at temperature T .

I.1. Recall the equation relating N/Ω to the chemical potential µ of these

fermions through the density of states in k (do not calculate the integral).

I.2. In the high temperature and low density limit recall the approximate integral expression relating N/Ω to µ (now calculate the integral).

I.3. The same fermions, at high temperature and small density, are now submitted to a potential energy V (r). Express the total energy of one such particle

versus k. Give the approximate expression relating N to µ and V (r).

I.4. Deduce that the volume density of fermions is given by

n(r) = n0 exp(−βV (r)) ,

where β = 1/kB T and n0 is a density that will be expressed as an integral.

I.5. Show that at very small potential energy, such that βV (r) 1 for anyr, then n0 = N/Ω.

II. From now on, we consider an illuminated semiconductor, of volume Ω, maintained at temperature T . It contains N electrons, of charge −e, in its conduction band together with P = N holes (missing electrons) in its valence band. The holes will be taken as fermions, of spin 1/2, mass mh and charge +e. We assume that the high temperature limit is realized.

Assume that a fixed point charge Q is present at the coordinates origin, which, if alone, would create an electrostatic potential :

Q

Φext(r) = 4πε0εrr

where εr is the relative dielectric constant of the considered medium.

In fact, the external charge Q induces inhomogeneous charge densities −en(r) and +ep(r) inside both fermion gases. These induced charges in turn generate electrostatic potentials, which modify the densities of mobile charges, and so on.

You are going to solve this problem by a self-consistent method in the following way: let Φtot(r) be the total electrostatic potential, acting on the particles.

II.1. Express the potential energies, associated with Φtot(r), to which the electrons and the holes are respectively subjected. Deduce from I. the expressions n(r) and p(r) of the corresponding densities of mobile particles.

II.2. The electrostatic potential Φtot(r) is a solution of the Poisson equation, in which the charge density includes all the charges of the problem. Admit that the charge density Qδ(r) is associated with the point charge Q located at r, where δ(r) is the Dirac delta distribution.

Show that, in the high temperature limit and to the lowest order in Φtot(r), the total electrostatic potential is solution of

Φtot(r) Qδ(r) ∆Φtot(r) = λ2 −

where λ is a constant that will be expressed versus N/Ω and T . What is the unit of λ ?

II.3. Estimate λ for N/Ω = 1022 m−3, T = 300 K, εr = 12.4.

II.4. The solution of this equation is

Φtot(r) = Q exp(−r/λ) 4πε0εr r

Indicate the physical implications of this result in a few sentences.

Exercise 2001 : Magnetic Susceptibility of a “Quasi-1D”

I. Consider an assembly of N magnetic ions at temperature T inside a volume Ω. These ions are independent and distinguishable. The electronic structure of an ion consists in a fundamental non-degenerate level |0 , of energy ε0, and a doubly degenerate excited state of energy ε0 + λ. Under the e ect of a static and uniform magnetic field B, the fundamental level remains at the energy ε0 while the excited level splits into two sublevels |+ and |− of respective energies ε0 + λ + γB and ε0 + λ − γB.

I.1. Without any calculation, plot the location of the states |0 , |+ and |− versus the applied magnetic field, in the following special cases :

a)λ = 0 , kB T γB

b)λ kB T γB

c)kB T λ γB

d)γB λ kB T

In each case indicate which states are occupied at temperature T .

I.2. Calculate the system free energy, assuming that the only degrees of freedom are those, of electronic origin, related to the three levels |0 , |+ and |− .

∂F

Deduce the algebraic expressions of the system magnetization M = −∂B and of its susceptibility χ = limB→0 ΩMB (Ω is the volume).

I.3. On the expressions of the magnetization and of the susceptibility obtained

in I.2. only discuss the limit cases a), b), c), d) of question I.1 (take γB = kB T

x). Verify that the obtained results are in agreement with the qualitative considerations of I.1.

I.4. The figure below represents the variation versus temperature of the paramagnetic susceptibility of the organic “quasi-one-dimensional” compound [HN(C2H5)3](TCNQ)2. Using the results of the present exercise can you qualitatively interpret the shape of this curve?

Susceptibility

Temperature (K)

Fig. 1: Magnetic susceptibility of the quasi-one-dimensional organic compound versus temperature.

II. Let a paramagnetic ion havea total spin S. You will admit that, in presence

of a static and uniform magnetic field B, the levels of this ion take the energies

εl = lγB, where −S ≤ l ≤ S, with l = −S, (−S + 1), (−S + 2), . . . + S.

Consider an ion at thermal equilibrium at the given temperature T . Calculate

its free energy f and its average magnetic moment m = −∂B∂f .

Write the expressions of the magnetization in the following limit cases : high and low magnetic fields; high and low temperatures. In each situation give a brief physical comment.

Exercise 2002 : Entropies of the HC Molecule

1.Let a three-dimensional ideal gas be made of N monoatomic molecules of mass m, confined inside the volume Ω and at temperature T . Without demonstration recall the expressions of its entropy S3D and of its chemical potential µ3D.

2.For a two-dimensional ideal gas of N molecules, mobile on a surface A, of the same chemical species as in 1., give the expressions, versus A, m and T , of the partition function Z2D, the entropy S2D and the chemical potential µ2D. To obtain these physical parameters, follow the same approach as in the course.

Which of these parameters determines whether a molecule from the ideal gas of question 1. will spontaneously adsorb onto surface A ?

3.Let us now assume that each of the N molecules from the gas is adsorbed on surface A. What is, with respect to the gas phase, the variation per mole of translation entropy if the molecules now constitute a mobile adsorbed film on A ?

4.Numerically calculate the value of S3D (translation entropy) for one mole of HC , of molar mass M=36.5, at 300K under the atmospheric pressure. Calculate S2D (translation entropy on a plane) for one mole of such a gas, adsorbed on a porous silicon surface : assume that, due to the surface corrugation, the average distance between molecules on the surface is equal to 1/30 of the distance in the gas phase considered above.

5.In fact, in addition to its translation degrees of freedom, the HC molecule possesses rotation degrees of freedom. Write the general expression of the partition function of N molecules of this gas versus ztr and zrot, the translation and rotation partition functions for a single molecule, accounting for the fact that the molecules are indistinguishable. The expressions of ztr and zrot will be calculated in the next questions.

6.In this question one considers a molecule constrained to rotate in a plane around a fixed axis. Its inertia moment with respect to this axis is IA. The

rotation angle φ varies from 0 to 2π, the classical momentum associated with φ is pφ. The hamiltonian for one molecule corresponding to this motion is given by

h = p2φ

2IA

The di erential elementary volume of the corresponding phase space is dφ dpφ/h, where h is the Planck constant. Calculate the partition function

z1rot.

7. In the case of the linear HC molecule, of inertia moment I, free to rotate in space, the rotation hamiltonian for a single molecule is written, in spherical coordinates,

Here θ is the polar angle (0 ≤ θ ≤ π), φ the azimuthal angle (0 ≤ φ ≤ 2π) ; pθ (respectively pφ) is the momentum associated with θ (respectively φ). The volume element of the phase space is now equal to dφ dpφ dθ dpθ/h2. Calculate the corresponding partition function z2rot.

Show that the dependence in temperature and in inertia moment obtained for z2rot is the expected one, when this expression of the rotation energy is taken into account. Verify that the expression found for z2rot is in agreement with the one given in the course for a linear molecule in the classical approximation (be careful, the HC molecule is not symmetrical).

8. Numerically calculate the rotation entropy by mole at 300K of a HC mole.

Take I = 2.7 × 10−40 g · cm 2.

Problem 2001 : Quantum Boxes

and Optoelectronics

For more clarity, comments are written in italics and questions in standard characters.

Semiconducting quantum boxes are much studied nowadays, because of their possible applications in optoelectronics. The aim of the present problem is to study the rate of spontaneous emission and the radiative e ciency in quantum boxes : these are two important physical parameters for the elaboration of emission devices (quantum–boxes lasers). We will particularly focus on the experimental dependence of these parameters.