Statistical physics (2005)

I.2. At any temperature nEc + nEv = 2, in the present approximation in which the total number of electrons is 2. It follows that

I.3. This equation can be reexpressed, noting x = eβµ, y = eβEc and z =

eβEv :

At 300 K, at thermal equilibrium, the probability for the conduction levels to be occupied is thus extremely small.

II.1.

1

fnls = eβ(Enls−µ) + 1

As the energy Enls only depends of n, consequently we will write fnls = fn.

II.2. Noting gn for the degeneracy of state n, one has

n = 1 = gn fn = 2(n + 1)fn

nn

an implicit relation between µ and β.

II.3. If one had µ > Ec, 2f0 would be larger than 1, in contradiction with

n = 1.

II.4. En is an increasing function of n, as is also eβ(En−µ), whence the result. One has f0 < 1/2, and fn+1 < fn, so that :

1

i.e., fn ≤ (n + 1)(n + 2)

deduces that in f1 one has eβ(E1−µ) > 5, in f2 eβ(E2−µ) > 11, which justifies the use of the Maxwell-Boltzmann approximation, i.e.,

Taking into account the validity of this approximation, the implicit relation between µ and β becomes

∞

1 = 2f0 + 2 (n + 1)e−β(En−µ)

n=1

With En = Ec +n ωc, and w = e−β(Ec−µ), this expression can be transformed into

of w. When T increases, β decreases. For g to remain constant [equal to 1/2 from (1)], the fugacity w must decrease when T increases.

Since w = e−β(Ec−µ), ∂w∂T = w kT1 2 (Ec − µ) + β ∂T∂µ . From II.3., µ < E0 = Ec and the first term of the right member is > 0. Consequently, ∂w∂T < 0 implies

∂T∂µ < 0.

II.7. From equation (1),

∞

– when β → 0, (n + 1)e−βn ωc tends to infinity, thus the solution w must

n=1

tend to 0. Then µ tends to −∞. (One finds back the classical limit at high temperature);

– when T → 0, the sum tends to 0 and (1) becomes 12 = w w+ 1 , so that w tends to 1 in this limit : the low temperature limit of µ is then E0 = Ec.

1

II.8. f0 = w−1 + 1 . At low temperature, w tends to 1 and f0 tends to 1/2,

whereas at high temperature w tends to 0 and the Maxwell-Boltzmann statistics is then applicable to the 0 level. Since w is a monotonous function of T , this latter limit must be valid beyond a critical temperature Tc, so that then eβ(E0−µ) 1 (w 1).

allows to conveniently calculate the sum in the right member of (1), which is equal to :

II.10. This condition is equivalent to w < 1/e, and at T = Tc one has w = 1/e. For w = 1/e, (2) becomes

Numerically kB Tc = 0.65 ωc.

II.11. For T > Tc, the Maxwell-Boltzmann statistics is applicable to the

1

fundamental state and consequently to any state. f0 = w−1 + 1 can be approached by w, which allows to write (2) as :

It follows that

III.1. The average number of electrons in a state of spin s of energy E in the valence band is also the Fermi-Dirac distribution function :

hn s is indeed an expression of the Fermi-Dirac statistics, under the condition of writing Et = −E. The holes chemical potential is then given by

µt = −µv .

The results on the electrons in questions II.1. to II.11. apply to the holes (provided this transposition for the energy and for µ).

IV.1. If all recombinations are radiative,

dn = − n dt τr

whence the result : n = n0 e−t/τr with n0 = 1.

IV.2. At T = 0, both f0 and h0 are equal to 1/2, all the other occupation factors being equal to zero.

Consequently, one has :

IV.3. One deduces from the figure τr = 2τ0 = 1 nsec, that is, τ0 = 0.5 nsec.

IV.4. At any temperature one has

Moreover the average numbers of electrons and holes are equal to 1. As the occupation factors are spin-independent, for a given value of s one gets :

264 Solution of the Exercises and Problems

One deduces :

This limit being reached at T = 0, one then finds that at any temperature the radiative lifetime is larger than or equal to its value at T = 0.

IV.5. For the electrons ωc 100 meV, whereas ωv 15 meV for the holes. Tc is thus of the order of 780 K for the electrons and 117 K for the holes. Expression (2) shows that when kB T ω, w is of the order of 1, which indeed constitutes the zero temperature approximation. The MaxwellBoltzmann approximation is valid for any hole state if T > Tc = 117 K, from II.10.

IV.6. One thus has f0 = 1/2 and all the other fn’s are equal to zero, whereas from (3),

h0 = (1 − e−β ωv )2 2

One then deduces that the only active transition is associated to the levels |001 and |00 − 1 (like at zero temperature) and that

IV.7. This model allows one to qualitatively understand the increase of the radiative lifetime with temperature, due to the redistribution of the hole probability of presence on the excited levels. This is indeed observed above 250 K, a temperature range in which a decrease of the lifetime is observed in the experiment.

1

V.1. The probability per unit time of radiative recombination is τr . This

means that the variation of the average number of electrons per unit time, due to these radiative recombinations, is given by

d n = − n dt τr

and a radiative yield η equal to the ratio between the number of radiative recombinations per unit time and the total number of recombinations per unit time :

V.3. When only radiative recombinations are present, τnr → ∞ and the radiative yield η is equal to 1, independently of the value of the radiative lifetime. Here the yield strongly decreases above 200 K, whereas it is constant for T < 150 K. This implies that a nonradiative recombination mechanism becomes active for T > 200 K.

Correlatively, this is consistent with the decrease of the total lifetime τ in the same temperature range, whereas the results of question IV.6. show that the radiative lifetime should increase with temperature.

1

In this expression τnr = γ (fn,l,s + hn,l,s). At su ciently high tem-

n≥n0 ,l,s

perature, around 350 K, τ ≈ 0.5 nsec whereas τr > 2 nsec. In this regime,

the nonradiative recombinations determine τ , and η ≈ τnr . Besides, only the

τr

hole levels of index n0 > 0 are occupied, they satisfy a Maxwell-Boltzmann statistics. This allows to write, using (3),

266 Solution of the Exercises and Problems

Taking an orbital degeneracy equal to n0 + 1 for each excited level, one has

≈ 2γτ0(n0 + 1)e−βn0 ωv

In this temperature range ln(η) = constant + n0 ωv . kT

The slope of η(T ) provides an approximation for n0 : between 200 and 350 K

Problem 2002 : Physical Foundations

of Spintronics

Part I : Quantum Mechanics

I.1. A state having a plane wave e ·r for space part is an eigenstate of the

ik

momentum operator, with the eigenvalue k. The e ect of the hamiltonian on this type of state gives :

I.2.

(a)For fixed k and φ, one is restricted to a two-dimensional problem, only on the spin variables, and the hamiltonian is equal to

to satisfy the periodical limit conditions : this requires ki = 2πni/Li, with ni positive, negative or null integer. As for the two vectors |χ± ,

they constitute a basis of the spin space. Thus the vectors |Ψk,± are an

ˆ

orthonormal eigenbasis of the hamiltonian H0.

I.3.

(a)The development of the initial spin state on the pair of eigenstates |χ± is

(b) One deduces that the electron spin state at time t is characterized by

After development, one finds

a±(t) = e−i 2k2 t / 2m a±(0) cos(ωt) − ia (0)e iφ sin(ωt) .

+2 sin(2ωt) Im a+(0)a−(0)e−iφ

One then deduces the result announced in the text.

I.5.

(a) The new term in the hamiltonian, arising from the magnetic field, commutes with the momentum operator. One can thus always look for ei-

vector providing the spin state. The hamiltonian determining the evolution of this spin is written in the |± z basis :

(b)One verifies that the eigenenergies of the above matrix are those given in the text.

Part II : Statistical Physics

II.1. For α = 0, one has E± = 2k2/(2m). The number of quantum states in a given spin state, with a wave vector of coordinates in the range (kx to kx +dkx ; ky to ky + dky ) is equal to

II.2.

(a) The dispersion law in this branch is written

E+(k) = 2(k + k0)2 − 0 2m

This corresponds to a portion of parabola, plotted in Fig.1. The spectrum of the allowed energies is [0, ∞[.

Dispersion relations

Fig. 1 : Typical behavior of the dispersion laws E±(k) in either branch.

The corresponding curve is plotted in 2.