Physico-Chimie II:

Partie 3 d. Gaz parfait de Fermions

Conduction dans les metaux

P. Damman

Lab Interfaces & Fluides Complexes

1/19

Metaux

Pyrite (sulfure de fer)

I Conducteurs de chaleur

I conducteurs de l’electricite

I faces brillantes

I Ductiles, maleables

2/19

Modele de Drude (⇠1900)

1897 Decouverte de l’electronJ.J. Thomson (tube de Crookes)

IModele idealise : Ions fixes (lourds !), Z electrons devalence mobiles (gaz d’electrons)

I Densite d’e (A masse atomique, ⇢m

masse volum.)

n =

N

V= 6.02 10

23 Z⇢m

A' 10

23e/cm3

Rmq. Tres dense ! Pour un gaz dans les conditions standards,

1019part./cm3

I Un e occupe une sphere de rayon

V

N=

1

n=

4

3

⇡r3s

rs

' 0.1nm

Rmq. Tres dense ! Pour un gaz dans les conditions standards, 1 � 10nm.

3/19

Modele de gaz parfait !

I e libres et independants. Entre 2 collisions, les en’interagissent pas avec les ions et les autres e.

I Les e n’entrent en collision qu’avec les ions (fixes).I L’equilibre thermique est obtenu via les collisions.

Vitesse et Chaleur specifique d’un gaz d’e:Equipartition de l’energie

3

2

kT =

1

2

mv2 v ' 10

7cm/s

C(e)v

=

3

2

Nk

IMPOSSIBLE ! Cv

' CReseau

v

(la contribution des e, C(e)v

⌧ 3Nk)7.12 Specific heats of solids 7 APPLICATIONS OF STATISTICAL THERMODYNAMICS

Solid cp Solid cp

Copper 24.5 Aluminium 24.4Silver 25.5 Tin (white) 26.4Lead 26.4 Sulphur (rhombic) 22.4Zinc 25.4 Carbon (diamond) 6.1

Table 4: Values of cp (joules/mole/degree) for some solids at T = 298◦ K. From Reif.

substance, so the mass m involved in lattice vibrations is comparatively small.Both these facts suggest that the typical lattice vibration frequency of diamond(ω ∼

√

κ/m) is high. In fact, the spacing between the different vibration energylevels (which scales like hω) is sufficiently large in diamond for the vibrationaldegrees of freedom to be largely frozen out at room temperature. This accountsfor the anomalously low heat capacity of diamond in Tab. 4.

Dulong and Petite’s law is essentially a high temperature limit. The molarheat capacity cannot remain a constant as the temperature approaches absolutezero, since, by Eq. (7.120), this would imply S → ∞, which violates the thirdlaw of thermodynamics. We can make a crude model of the behaviour of cV atlow temperatures by assuming that all the normal modes oscillate at the samefrequency, ω, say. This approximation was first employed by Einstein in a paperpublished in 1907. According to Eq. (7.131), the solid acts like a set of 3N

independent oscillators which, making use of Einstein’s approximation, all vibrateat the same frequency. We can use the quantum mechanical result (7.111) for asingle oscillator to write the mean energy of the solid in the form

E = 3 N hω

⎛

⎝

1

2+

1

exp(β hω) − 1

⎞

⎠ . (7.134)

The molar heat capacity is defined

cV =1

ν

⎛

⎝

∂E

∂T

⎞

⎠

V

=1

ν

⎛

⎝

∂E

∂β

⎞

⎠

V

∂β

∂T= −

1

ν k T2

⎛

⎝

∂E

∂β

⎞

⎠

V

, (7.135)

giving

cV = −3 NA hω

k T2

⎡

⎣−exp(β hω) hω

[exp(β hω) − 1]2

⎤

⎦ , (7.136)

151

4/19

Modele de Arnold Sommerfeld (1927)

Deux Records absolus:

- Nomine 81 fois pour le Nobel !- Superviseur de 7 futurs prix Nobel !

(Doc.: Werner Heisenberg, Wolfgang Pauli, Peter Debye, Hans Bethe et

Post-doc.: Linus Pauling, Isidor I. Rabi (NMR), Max von Laue)

I Adaptation du modele de Drude (1863 - 1906) a la mecaniquequantique

I Gaz d’e libres MAIS statistique de Fermi-Dirac

0 2,5 5 7,5 10 12,5 15

0,25

0,5

0,75

1 Distribution Fermi-Dirac

Maxwell-Boltzmann

I Niveaux d’energie d’un e libre

I Remplissage avec principed’exclusion

I

f(✏) =

1

e�(✏�µ)+ 1

5/19

Energie du Niveau Fondamental T = 0

Fermions massiques (non relativistes) - Facteur 2 pour le spin !!

� =

2⇡

k, ✏(k) =

p2

2m=

~2k2

2m, g(✏) = 4

V

h3⇡(2m)

3/2✏1/230.2 The Fermi gas 341

Fig. 30.2 (a) The Fermi function f(E)defined by eqn 29.26. The thick lineis for T = 0. The step function issmoothed out as the temperature is in-creased (shown as thinner lines). Thetemperatures shown are T = 0, T =0.01µ/kB, T = 0.05µ/kB and T =0.1µ/kB. (b) The density of statesg(E) for a non-interacting fermion gasin three dimensions is proportional toE

1/2. (c) f(E)g(E) for the same tem-peratures as in (a).

At T = 0, the distribution function f(E) is a Heaviside step function,

taking the value 1 for E < µ and 0 for E > µ. This step is smoothed

out as the temperature T increases, as shown in Fig. 30.2(a). The den-

sity of states g(E) for a non-interacting fermion gas in three dimensions

is proportional to E1/2(as shown in eqn 30.4) and this is plotted in

Fig. 30.2(b). The product of f(E)g(E) gives the actual number distri-

bution of fermions, and this is shown in Fig. 30.2(c). The sharp cuto�

you would expect at T = 0 is smoothed over an energy scale kBT around

the chemical potential µ.

The electrons in a metal can be treated as a non-interacting gas of

fermions. Using the number density n of electrons in a metal, one can

calculate the Fermi energy using eqn 30.29, and some example results

are shown in Table 30.1. The Fermi energies are all several eV; convert-

ing each number into a temperature, the so-called Fermi temperatureTF = EF/kB, yields values of several tens of thousands of Kelvin. Thus

the Fermi energy is a large energy scale, and hence for most metals the

Fermi function is close to a step function, at pretty much all tempera-

tures below their melting temperature. In this case, the electrons in a

metal are said to be in the degenerate limit.The pressure of these electrons is given (by using eqns 22.49 and 30.17)

as

p =

2U

3V, (30.30)

f(✏) = 1 pour ✏ < µf(✏) = 0 pour ✏ > µ

Principe d’exclusion - e dans une spherede rayon k

F

N =

Z 1

�1d✏ g(✏)f(✏) =

Zµ

0d✏ g(✏)

= 4

V

h3⇡(2m)

3/2Z

µ

0d✏ ✏1/2

=

8V

3h3⇡(2m)

3/2µ3/2

6/19

I

N =

8V

3h3⇡(2m)

3/2µ3/2=

V

3~3⇡2(2m)

3/2✏3/2F

I A T = 0, Potentiel chimique = energie de Fermi ✏F

(vecteur d’onde de Fermi kF

)

N/V = n =

k3F

3⇡2

avec ✏F

= ~2k2F

/2m = kTF

kF

rayon de la sphere de Fermi

I Fermions : La densite FIXE le potentiel chimique (✏F

) !

I Avec des densites n ⇠ 10

23e/cm3, on calcule TF

⇠ 10

5K(Rmq. La valeur de TF est liee au principe d’exclusion !)

I Vitesse des e ?? pF

= ~kF

= mvF

, avec n ⇠ 10

23e/cm3, ona v

F

⇠ 10

8 cm/s (rmq. c ' 3 10

10cm/s)

7/19

I Energie du niveau fondamental

U0 =

Z 1

�1d✏ ✏ g(✏)f(✏) =

Z✏F

0d✏ ✏ g(✏)

f(✏) = 1 pour ✏ < ✏F

f(✏) = 0 pour ✏ > ✏F

I Re-ecrivons la DOS par unite de volume (g(✏) = g(✏)/V ) a

partir de la densite d’e (n = N/V =

8⇡

3h

3 (2m)

3/2✏3/2F

)

g(✏) = 4

⇡

h3(2m)

3/2✏1/2=

3

2

n

✏F

✓✏

✏F

◆1/2

I Avec la DOS/unite de volume - densite d’energie

u0 =

Z✏F

0d✏ ✏ g(✏) =

3

2

n

✏3/2F

Z✏F

0d✏ ✏3/2

=

3

5

n ✏F

=

3

5

n kTF

8/19

Energie pour des T > 0 30.2 The Fermi gas 341

Fig. 30.2 (a) The Fermi function f(E)defined by eqn 29.26. The thick lineis for T = 0. The step function issmoothed out as the temperature is in-creased (shown as thinner lines). Thetemperatures shown are T = 0, T =0.01µ/kB, T = 0.05µ/kB and T =0.1µ/kB. (b) The density of statesg(E) for a non-interacting fermion gasin three dimensions is proportional toE

1/2. (c) f(E)g(E) for the same tem-peratures as in (a).

At T = 0, the distribution function f(E) is a Heaviside step function,

taking the value 1 for E < µ and 0 for E > µ. This step is smoothed

out as the temperature T increases, as shown in Fig. 30.2(a). The den-

sity of states g(E) for a non-interacting fermion gas in three dimensions

is proportional to E1/2(as shown in eqn 30.4) and this is plotted in

Fig. 30.2(b). The product of f(E)g(E) gives the actual number distri-

bution of fermions, and this is shown in Fig. 30.2(c). The sharp cuto�

you would expect at T = 0 is smoothed over an energy scale kBT around

the chemical potential µ.

The electrons in a metal can be treated as a non-interacting gas of

fermions. Using the number density n of electrons in a metal, one can

calculate the Fermi energy using eqn 30.29, and some example results

are shown in Table 30.1. The Fermi energies are all several eV; convert-

ing each number into a temperature, the so-called Fermi temperatureTF = EF/kB, yields values of several tens of thousands of Kelvin. Thus

the Fermi energy is a large energy scale, and hence for most metals the

Fermi function is close to a step function, at pretty much all tempera-

tures below their melting temperature. In this case, the electrons in a

metal are said to be in the degenerate limit.The pressure of these electrons is given (by using eqns 22.49 and 30.17)

as

p =

2U

3V, (30.30)

n =

Z 1

�1d✏ g(✏)f(✏)

u =

Z 1

�1d✏ ✏ g(✏)f(✏)

avec

f(✏) =

1

e�(✏�µ)+ 1

g(✏) =

3

2

n

✏F

✓✏

✏F

◆1/2

Expansion de Sommerfeld autour de ✏F9/19

Expansion de Sommerfeld

I On doit calculerR 1

�1 d✏H(✏) f(✏) avec H(✏) = ✏ g(✏) ⇠ ✏3/2

I Introduisons la fonction

K(✏) =

Z✏

�1du H(u) H(✏) =

✓dK

d✏

◆

✏

I ce qui nous permet d’integrer par parties (Rmq. g(✏) = 0 8✏ < 0)Z 1

�1d✏

dK(✏)

d✏f(✏) = [K(✏) f(✏)]1�1 �

Z 1

�1d✏K(✏)

df(✏)

d✏

=

Z 1

�1d✏K(✏)

✓�df(✏)

d✏

◆

-7,5 -5 -2,5 0 2,5 5 7,5

0,25

0,5

0,75

1F_Fermi

dF_Fermi/dE

0 0,5 1 1,5 2

0,5

1

1,5

2

2,5

3

H(E) = dK/dE

K(E)

10/19

0 5 10 15 20 25 30

0,8

1,6

2,4

3,2

4

-df(e)/de

K(e)

energie

La fonction df(✏)/d✏ ne varie qu’autour de µ,! Develop. Taylor de K(✏)

K(✏) = K(µ) +

X

n

(✏ � µ)

n

n!

dnK

d✏n

= K(µ) + (✏ � µ)K 0(µ) +

(✏ � µ)

2

2

K 00(µ) + ...

I df(✏)/d✏ est une fonction paire, on ne conserve que les termes pairs(et H = dK/d✏)Z 1

�1d✏H(✏) f(✏) =

Zµ

�1du H(u)+

Z 1

�1d✏

(✏ � µ)

2

2

H 0(µ)

✓�df(✏)

d✏

◆

I Changement de variable (✏ � µ)/kT = X(faire apparaıtre une

Rdefinie)

Z 1

�1d✏H(✏)f(✏) =

Z µ

�1du H(u) + k2T 2H 0(µ)

Z 1

�1dX

X2

2

✓�df(X)

dX

◆

=

Z µ

�1du H(u) +

⇡2

6k2T 2H 0(µ) + ...

11/19

Calcul de l’energie (µ ' ✏F )

H(✏) = ✏ g(✏) = ✏3

2

n

✏F

✓✏

✏F

◆1/2

u =

Z 1

�1d✏ ✏ g(✏) f(✏) =

Zµ

�1du ✏g(✏) +

⇡2

6

k2T 2(✏g(✏))0

µ

+ ...

=

3

5

n ✏F

+

⇡2

6

k2T 2 3

2

g(✏F

) + ...

u = u0 +

3⇡2

8

k2T 2 n

✏F

La chaleur specifique electronique (c(e)V

= du/dT , !! / unite devolume !!)

c(e)V

=

3⇡2

4

k2Tn

✏F

=

3⇡2

4

nk

✓T

TF

◆

12/19

Chaleur specifique des metaux

c(e)V

=

3⇡2

4

nk

✓T

TF

◆

Evolution lineaire avec T ET negligeable pour des temperaturesusuelles (T

F

⇠ 10

5 K)Chaleur specifique totale - elect. + reseau : c

v

= �T + AT 3

ou bien cv

/T = � + AT 2

8.14 White-dwarf stars 8 QUANTUM STATISTICS

Figure 11: The low temperature heat capacity of potassium, plotted as CV/T versus T2. FromC. Kittel, and H. Kroemer, Themal physics (W.H. Freeman & co., New York NY, 1980).

Hence,cV

T= γ + A T2. (8.118)

If follows that a plot of cV/T versus T2 should yield a straight line whose intercepton the vertical axis gives the coefficient γ. Figure 11 shows such a plot. The factthat a good straight line is obtained verifies that the temperature dependence ofthe heat capacity predicted by Eq. (8.117) is indeed correct.

8.14 White-dwarf stars

A main-sequence hydrogen-burning star, such as the Sun, is maintained in equi-librium via the balance of the gravitational attraction tending to make it collapse,and the thermal pressure tending to make it expand. Of course, the thermal en-ergy of the star is generated by nuclear reactions occurring deep inside its core.Eventually, however, the star will run out of burnable fuel, and, therefore, start tocollapse, as it radiates away its remaining thermal energy. What is the ultimatefate of such a star?

A burnt-out star is basically a gas of electrons and ions. As the star collapses,its density increases, so the mean separation between its constituent particlesdecreases. Eventually, the mean separation becomes of order the de Brogliewavelength of the electrons, and the electron gas becomes degenerate. Note,

195

Avec � ⇠ 10

�3J/mol K2, on a c(e)v

⇠ 0.3J/mol K(⌧ 25J/mol K)

13/19

Approche ”qualitative”

Copyright Oxford University Press 2006 v1.0 -

7.4 Non-interacting bosons and fermions 143

This is called the Bose–Einstein distribution (Fig. 7.4)

�n�BE =

1

e

�(��µ) � 1

. (7.44)

The Bose–Einstein distribution describes the filling of single-particle

eigenstates by non-interacting bosons. For states with low occupancies,

where �n� ⌧ 1, �n�BE � e

��(��µ), and the boson populations correspond

to what we would guess naively from the Boltzmann distribution.

19The

19We will derive this from Maxwell–Boltzmann statistics in Section 7.5.

condition for low occupancies is �k

� µ � kB

T , which usually arises at

high temperatures

20(where the particles are distributed among a larger

20This may seem at odds with the for-mula, but as T gets large µ gets largeand negative even faster. This hap-pens (at fixed total number of parti-cles) because more states at high tem-peratures are available for occupation,so the pressure µ needed to keep themfilled decreases.

number of states). Notice also that �n�BE ! � as µ ! �k

since the

denominator vanishes (and becomes negative for µ > �k

); systems of

non-interacting bosons always have µ less than or equal to the lowest of

the single-particle energy eigenvalues.

21

21Chemical potential is like a pressurepushing atoms into the system. Whenthe river level gets up to the height ofthe fields, your farm gets flooded.

Notice that the average excitation �n�qho of the quantum harmonic

oscillator (eqn 7.25) is given by the Bose–Einstein distribution (eqn 7.44)

with µ = 0. We will use this in Exercise 7.2 to argue that one can treat

excitations inside harmonic oscillators (vibrations) as particles obeying

Bose statistics (phonons).

0 1 2Energy ε/µ

0

1

f(ε)

∆ε ∼ kBT

T = 0Small T

Fig. 7.5 The Fermi distributionf(�) of eqn 7.48. At low temperatures,states below µ are occupied, statesabove µ are unoccupied, and stateswithin around kBT of µ are partiallyoccupied.

Fermions. For fermions, only nk

= 0 and nk

= 1 are allowed. The

single-state fermion grand partition function is

�

fermionk

=

1X

nk=0

e

��(�k�µ)nk= 1 + e

��(�k�µ), (7.45)

so the total fermion grand partition function is

�NIfermion =

�

k

�1 + e

��(�k�µ)�

. (7.46)

For summing over only two states, it is hardly worthwhile to work

through the grand free energy to calculate the expected number of par-

ticles in a state:

�nk

� =

�1nk=0 n

k

exp(��(�k

� µ)nk

)

�1nk=0 exp(��(�

k

� µ)nk

)

=

e

��(�k�µ)

1 + e

��(�k�µ)=

1

e

�(�k�µ)+ 1

,

(7.47)

leading us to the Fermi–Dirac distribution

f(�) = �n�FD =

1

e

�(��µ)+ 1

, (7.48)

where f(�) is also known as the Fermi function (Fig. 7.5). Again, when

the mean occupancy of state �k

is low, it is approximately given by

the Boltzmann probability distribution, e

��(��µ). Here the chemical

potential can be either greater than or less than any given eigenenergy

�k

. Indeed, at low temperatures the chemical potential µ separates filled

states �k

< µ from empty states �k

> µ; only states within roughly kB

Tof µ are partially filled.

I cv

' 3/2 neff

kn

eff

= nombre d’electrons ”actifs”

I considerons neff

= n(T/TF

)

cv

' 3/2 nk

✓T

TF

◆

Comportement Haute temperature(impossible a observer !, T

F

⇠ 10

5K)

14/19

Conductivite electrique des metaux

I Force, champ electro-magnetique

F = mdv

dt= ~dk

dt= �eE

I integration (B = 0)

v = �eE⌧/m

avec ⌧ le temps entre 2 collisions (di↵usion)

I Flux d’e - loi d’Ohm (avec n densite d’e)

j = nev = ne2⌧E/m = E/⇢

⇢ est la resistivite electrique

I La resistance est definie par les collisions des e avec ... quoi ?

15/19

Matthiessen’s rule

⇢ = ⇢i

+ ⇢phonon

(T )

Di↵usion des e par les defauts etles phonons.

Pas toujours vrai ! voir supraconductivite.

16/19

Bilan du modele du gaz d’electrons libres

I Electrons libres: Chal. Specifique, conductivite, susceptibilitemagnetique, ...

I !!!! Existence de Conducteurs, semiconducteurs et isolantsA T = 1K, �

cond

⇠ 10

�10ohm cm et �isol

⇠ 10

22ohm cm

I Bandes d’energie et band-gap(interaction fonction d’onde e - ions)

I Amelioration: Role des ions (potentiel periodique)

17/19

Potentiel periodique - U(x) = U cos 2⇡x/a

I Fonction d’onde = exp ikxavec k = ±n⇡/a

I reflexion sur le plan de BraggOndes stationnaires + = exp ikx + exp �ikx = 2 cos kx � = exp ikx � exp �ikx = 2i sin kx

I Densite d’e (| ±|2)⇢+ ⇠ cos

2 kx - Favorable⇢� ⇠ sin

2 kx - Defavorable

I Band gap Eg

= amplitude du potentiel

18/19

Concept similaire applique a la lumiere - Photonique

Ceci est une autre histoire !

19/19