R)(L)AlSbInAsGaSbAlSb
Chapter 8 Photon generation for a
8.2 The interaction Hamiltonian and Fermi’s golden rulegolden rule
Figure 8.1 describes schematically the photon emission process under study. Πσ (Πλ) is the label of the initial (final) irreducible representation. These labels indicate the starting and final bands between which the interband transition takes place. |ii and
|fi denote the specific initial and final states, respectively, inside each band. Their corresponding energies areEi andEf. Finally,|γi represents the state of the emitted photon.
To study this process, consider a plane wave interacting with a crystal. The plane wave can be described by just its vector potential in the radiation gauge [10]:
A=A0acos (q·r−ωt), (8.1)
where A is the vector potential, A0 is the amplitude of the wave, a is a unit vector in the direction of A, q is the wavevector of the wave and ω is its frequency. The electric and magnetic fields can be found using the usual relations:
E=−∂A
∂t (8.2)
B=∇ ×A, (8.3)
to describe an elliptical polarization state.
The Hamiltonian of a crystal coupled to an applied electromagnetic plane wave can be written, in the low-intensity limit and dipole approximation (i.e., the photon wavelength much bigger than the primitive cell size), as [11]:
H =H0− e
moP·A, (8.4)
where the dipole magnetic term has been omitted because it is much weaker than the dipole electric term, e is the absolute value of the electron charge,mo is the free electron mass and Pis the momentum operator.
Following Ridley [12], the cosine in Eq. (8.1) can be expanded into a sum of com- plex exponentials and introduced into Eq. (8.4) to yield, by selecting the appropriate terms, the space part of the Hamiltonian describing the photon emission process:
Hνem =− e mo
½~(nν + 1) 2V ²νων
¾1/2
a·P, (8.5)
where ~ is the reduced Planck constant, nν is the number of photons present corre- sponding to the mode ν and having a frequency ων, ²ν is the optical permittivity of that mode and V is the volume of the crystal. The time part of the Hamiltonian leads to the appearance of the Dirac delta factor in energy in Eq. (8.6).
On the other hand, Fermi’s golden rule can be used to find Wi→f, the probability per unit time that an initial state |ii will make a transition to any |fi belonging to a family with dSf degenerate states due to the action of a perturbation Hν em. It states [12]:
Wi→f = 2π
~ |hf|Hν em|ii|2δ(Ei−Ef) dSf. (8.6) Thus, the probability of spontaneous emission into a solid angle dΩ per unit time Wem can be obtained by taking nν = 0 in Eq. (8.5) and plugging this equation into
Eq. (8.6):
Wem = e2 4π²0
4
~ων
|a·Pf i|2 2m2
ηrων2 c3
dΩ
4π, (8.7)
where²0 is the vacuum permittivity,cis the speed of light,ηr is the crystal refractive index and Pf i is the momentum matrix element between the initial and the final states.
The important point about Eq. (8.7) is that the probability of emission is pro- portional to the scalar product of the polarization vector of the plane wave and the matrix element of the momentum operator between the initial and final states. If only transitions between two bands are of interest, as is commonly the case in semi- conductors, all the photons emitted will have the same frequency and the only thing that will change between two emission events will be the |a·Pf i|2 factor. There- fore, the knowledge of that factor is the only thing needed to determine the relative transition rates between some initial and final states belonging to two given bands.
Mathematically, this can be written as:
Wi→f,a
Wi0→f0,a0
= |a·Pf i|2
|a0·Pf0i0|2. (8.8)
8.2.1 The Wigner-Eckart theorem for point groups applied to momentum matrix elements
The expression in Eq. (8.8) can be further simplified by making use of symmetry considerations. The Wigner-Eckart theorem for point groups (see Ref. [13] and ap- pendix B.3) isolates the symmetry effects from other contributions in the matrix elements of tensor operators. In particular, the momentum matrix elements can be written as follows:
Pµ f im = f¯
¯Pµm¯
¯i®
=X
k∈κ
X
l∈λ
hf|κ, ki κ, k¯
¯Pµm¯
¯λ, l®
hλ, l|ii= hκkPµkλiX
k∈κ
X
l∈λ
hf|κ, kihµ, m;λ, l|κ, kihλ, l|ii, (8.9)
Γ7 |l = 1i ⊗ |s= 1/2i such that j = 1/2
Γ8 |j = 3/2i
Table 8.1: Table of selected irreducible representations (irreps) of the zincblende point group Td and their equivalent full rotation group SO(3) irreps.
where the initial and final states have been expanded into the basis states corre- sponding to their irreducible representation (irrep). Greek indices label irreps and latin indices label specific basis states inside an irrep. hκkPµkλi is the so-called re- duced matrix element [14], and will not depend on|iinor|fias long as these are taken as linear combinations of basis states belonging to the Πλ and Πκ bands, respectively.
The symbol hµ, m;λ, l|κ, ki represents the complex conjugate of the Clebsch-Gordan coefficient for the point group under consideration.
8.2.2 Application to III-V zincblendes
To exemplify this abstract formalism, consider a transition between the conduction band and the valence band of a III-V zincblende at the zone center. For most of the III-Vs, the conduction band edge is described by the Γ6 irrep of theTd point group;
while the valence band edge is described by the Γ8 irrep, where the Koster-Dimmock- Wheeler-Statz (KDWS) [15] notation is being used. The treatment of Td is simplified by the fact that some of their irreps can be identified with full rotation group irreps according to Table 8.1. Assume also that the electron in the conduction band has spin up in the z axis and it can go to either the heavy hole state |3/2,+3/2i or the light hole |3/2,−1/2i, and the photon is being emitted along the z direction. The
relative transition rates will be given by the ratio of the momentum matrix elements:
Ws↑→|3/2,+3/2i Ws↑→|3/2,−1/2i =
¯
¯
¯
¯
¯
h3/2kPΓ5ksiX
k∈Γ8
X
l∈Γ6
h3/2,+3/2|3/2, kih1,+1; 1/2, l|3/2, kih1/2, l|1/2,+1/2i
¯
¯
¯
¯
¯
2
¯
¯
¯
¯
¯
h3/2kPΓ5ksiX
k∈Γ8
X
l∈Γ6
h3/2,−1/2|3/2, kih1,−1; 1/2, l|3/2, kih1/2, l|1/2,+1/2i
¯
¯
¯
¯
¯
2.
(8.10) The reduced matrix elements in Eq. (8.10) cancel out and, using the orthonor- mality of the basis kets, one is left with the simple expression:
Ws↑→|3/2,+3/2i
Ws↑→|3/2,−1/2i
= |h1,+1; 1/2,+1/2|3/2,+3/2i|2
|h1,−1; 1/2,+1/2|3/2,−1/2i|2 = 1
1/3 = 3, (8.11) where the Clebsch-Gordan coefficients are for the full rotation group, and can be found in any standard quantum mechanics book.
8.2.3 Complications following the path of Fermi’s golden rule
In the previous two subsections it has been shown how to find ratios between the probabilities that a photon be emitted from a transition into a given band using Fermi’s golden rule. This might seem good enough to achieve the goal of generat- ing single events using a random number generator. However, in the derivation of Eq. (8.11), the formalism has left discretionary choices that make it unsuitable for event generation. Some of the questions that arise are
• What set of final states must be chosen?
In Sec. 8.2.2, it was assumed that the initial state transitioned to either
|3/2,+3/2i or |3/2,−1/2i, where the quantization axis for the final sates was chosen to point along the z axis. However, had another quantization axis been chosen, the matrix element Pf i would have taken another value and the re-
• What polarization of the emitted photon must be chosen?
Also in Sec. 8.2.2, it was implicitly assumed that the polarization vector a was parallel to Pf i. The polarization of the emitted photon is given by the vector a [cf. Eq. (8.1)]. In principle, one might think that the polarization of the emitted photon should be independent of the choice of polarization basis vectors. But in this formalism, falling back to the example in Sec. 8.2.2 and considering emission along the z axis; the following choice of polarization modes: a1 = ˆx, a2 = ˆy with ˆx (ˆy) a unit vector along the x (y) axis, would have generated photons with linear polarizations only. This is clearly unacceptable because it is known that when the electron spin points along thez axis and light is emitted along that axis too, a circular polarization of 50 % is expected [16].
In principle, one could overcome these difficulties by adopting the prescription that any basis set spanning the final state subspace can be chosen, and then pick a polarization that is in the same direction as P⊥f i, where P⊥ is the component of P perpendicular to the direction of emission. Alas, this is anad hocprescription, which does not shed any insight on the nature of the radiative process. But this prescription yields, indeed, the correct results. In the next section a more natural and univocal way of obtaining a prescription is shown.