Читать книгу Spectroscopy for Materials Characterization - Группа авторов - Страница 17

In the previous paragraph, we have determined theoretical quantities relating the atomic wavefunctions of the energy levels to spectroscopic observables. In the simplified model of the atom with a single electron, it is considered that this latter can oscillate in a harmonic potential well. The atomic system is then a charged harmonic oscillator. The wavefunctions of this system enable to evaluate the electric dipole matrix element μ ₁₂ and to determine the theoretical integrated absorption reported by (1.52). It can be shown that the expected value of integrated absorption is [5, 11]:

(1.53)

where N _A is Avogadro’s number, c is the speed of light, and m is the electron mass (9.109 ⋅ 10⁻³¹ kg). Equation (1.53) gives a numerical value that can be compared with experimental results. This comparison gives origin to the quantity called oscillator strength and usually given by f:

(1.54)

The oscillator strength is a dimensionless quantity characterizing the transition between the two considered energy levels E ₁ and E ₂. By introducing [5, 11]

(1.55)

it is shown that

(1.56)

Expected values of f are lower equal than unity and on decreasing of the probability of the absorption process decreases too. This feature is linked to the selection rules that highlight those quantum transitions between energy levels giving an electric dipole matrix element different from zero [5].

Another form of the oscillator strength is [8, 15]

(1.57)

which, on the basis of (1.34), (1.35) and (1.44), can be written

(1.58)

Considering that A ₂₁ = 1/τ [13], as stated above, it is shown that

(1.59)

connecting the oscillator strength to the radiative emission lifetime at low temperature. Furthermore, using (1.52) to determine μ ₁₂, Eq. (1.57) becomes

(1.60)

giving

(1.61)

This formula relates the integrated absorption coefficient to the concentration of absorbing centers through the oscillator strength. In particular, given a concentration N _1, the area of the absorption curve is higher at larger oscillator strength. Furthermore, both the oscillator strength and the concentration of absorbing centers can be experimentally determined from this formula once one of the two parameters is known. This result shows the relevance of the absorption measurements and once more the exploitability of this experimental technique to determine microscopic information about the matter.

In general, the relation between absorption and emission processes can be described by

(1.62)

where the absorption effect is reported using (1.5) and (1.8) and differences in absorption and emission frequency ν have been neglected, assuming that one is the inverse process of the other. The above formula shows that the number of emitted photons by a sample of thickness L depends on the number of photons impinging on the sample, I ₀, and the number of them giving absorption effect, as evidenced by the terms in parenthesis. The efficiency of emission is determined by η, which is called (external) radiative quantum efficiency or quantum yield [2, 10, 16]. It is worth noting that in the case of low absorption effect, εCL ≪ 1, αL ≪ 1, the emission is proportional to the concentration of absorbing species. The quantum efficiency of photoluminescence is defined as the ratio of the number of emitted photons to the number of absorbed photons

(1.63)

In the interaction process between radiation and a two‐level atom, in which the absorption drives the electron from the starting state E ₁ to the final state E ₂ at higher energy, as reported in Figure 1.4, not all of the absorbed photons give rise to an emitted photon since other processes could drive the excited electron back to the E ₁ state. Based on the above considerations, it is clear that η ≤ 1. To go deeper into the connection between η and other physical parameters, it is useful to consider the simplified energy level scheme of the two‐level atom reported in Figure 1.4. The absorption process, in which the electron is promoted to the E ₂ state, is represented by the vertical arrow connecting E ₁ and E ₂ states. This process has a rate, probability per unit of time, R _abs. The system in the excited state is out of thermal equilibrium because typically E ₂ − E ₁ > kT. As a consequence, the electron returns to the lower energy level. This relaxation process could occur by the emission of a photon with frequency ν = (E ₂ − E ₁)/h, denoted as radiative process with a rate k _r. Furthermore, the electron could relax without emission of photons, not radiatively, by exchanging energy with its environment with a rate k _nr. The rate equation for the variation of the population N ₁ of the state with energy E ₁ can be written as

(1.64)

where the population of the excited state N ₂ has been introduced. This equation is a balance between absorbed photons and relaxation from the excited state through radiative (emitted photons) and non‐radiative processes. Under stationary equilibrium conditions, there will be no change in the populations with time and it is found that

(1.65)

It is reasonable to assume that the number of absorbed photons is proportional to the number of atoms being excited by the radiation, so it can be written

(1.66)

On the other hand, the number of emitted photons is proportional to the number of atoms relaxing radiatively from the E ₂ state, and it is possible to assume

(1.67)

From the above considerations, it is found that the quantum efficiency is given by

(1.68)

Introducing now the rate equation for the excited state

(1.69)

it is found that after the stationary regime is attained, when the population N ₂ has stabilized, if the radiation field is suddenly removed at the time t ₀ and, as a consequence, R _abs is suddenly put to zero, the rate equation of the excited state becomes

(1.70)

whose solution for t ≥ t ₀ is

(1.71)

N ₂(t ₀) being the stationary population of the excited state. Comparing Eq. (1.71) with (1.23), it is found that

(1.72)

and finally, it is found that

(1.73)

It is worth considering that, typically, the non‐radiative processes are active at high temperature when the two‐level system is coupled to other degrees of freedom of the atomic system. At variance, at very low temperature, k _nr = 0. On this basis, it is found that at T = 0 K (or at low enough temperature)

(1.74)

(1.75)

and a measure of the emission lifetime at low temperature, τ ₀, is a direct estimate of k _r. It is expected that the quantum efficiency decreases on increasing the temperature due to the activation of the non‐radiative relaxation processes [2, 5, 10]. In addition, the intrinsic or natural lifetime, τ ₀ = 1/k _r, can in general be determined at any temperature by the measure of the quantum efficiency and of the emission lifetime using (1.73)

(1.76)

this shows that typically the lifetime decreases on increasing the temperature. In general, the lifetime concept and the quantum efficiency are relevant aspects in the dynamic study of emitting centers and in the characterization of optoelectronic devices. They give useful information on the coupling of the system to its neighborhood.