**2. Thermal and ion beam formation quantum dots**

Obtaining methods of semiconductor quantum dots are questions of great importance in the fields of science and engineering. Numerous advances have been achieved in the synthesis of QDs, among them bulk etching [17], laser pyrolysis [18], gas phase synthesis [19], thermal vaporization [20], and wet chemistry techniques [21]. It should be noted that all of the above methods often require special additives such as surfactants and dopants, as well as high-temperature postprocessing, to stabilize the QDs and control their size.

Another convenient method is the thermal synthesis of quantum dots. As shown in [22–24], the synthesis of silicon QDs in the suboxide matrix takes place according to a reaction, which, depending on the temperature and duration of annealing, can be stopped at any intermediate stage: 2SiOX ! Si QD + SiO2.

The schematic representation of the sample transformation is shown in **Figure 1**.

With ion implantation, direct accumulation of the introduced material with not only the quantum dots formation but also the quantum dots occurrence is possible due to the evolution of defective structures. In particular, there is an innovative method of creating Si quantum dots in SiO2 under pulsed ion beam exposure [28]. In this case, quantum dots are formed as a result of the conversion and clustering of radiation defects: ODC(II) ! E' ! ODC(I) ! Si QDs. During Gd-ion implantation with different doses, Si–O bond softening appeared, and the three main stages of defect evolution were identified: (A) formation of primary oxygen-deficient centers; (B) conversion of defects; and (C) clustering into Si QDs (**Figures 2** and **3**).

By changing the modes of radiation exposure of the SiO2 matrix, we can control the qualitative and quantitative composition of defects and modify the optical properties of the host, including the ultraviolet transmittance and visible luminescence. As seen from **Figure 2**, the radiation defect conversion includes several consequent stages. The implementation of the described conversion mechanism provides the controlled formation of stable quantum dots at various modes of ion

implantation [25]. The resulting Si QDs have a relatively small size of about 3.6 nm, which makes it possible to excite red luminescence due to the strong quantum

*Scheme of the successive stages for oxygen-deficient defect formation and fabrication of silicon quantum dots in SiO2 subjected to ion beam irradiation. (A) Radiation formation of vacancy defects; (B) transformation of*

*Scheme of silicon nanoclusters (Si QD) formation in an initial multilayered structure under high-temperature*

*annealing in nitrogen atmosphere. The RyOz (R—Si, Al, Zr) is a dialectical layer.*

*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots*

*DOI: http://dx.doi.org/10.5772/intechopen.91888*

Described above the thermal synthesis of quantum dots and ion implantation methods are most convenient for stabilizing QDs in a dielectric matrix. Such systems are better suited for studying the mechanisms of excitation and radiative

confinement effect (**Figure 3**).

*defects; (C) clusterization into quantum dots.*

**Figure 2.**

**25**

**Figure 1.**

relaxation of QDs [26–28].

*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots DOI: http://dx.doi.org/10.5772/intechopen.91888*

#### **Figure 1.**

significantly differ in shape and type when using various excitation methods. The PL temperature curves of quantum dots are most often presented in the form of decreasing functions with increasing temperature [4, 5, 7]. At the same time, curves with a clearly defined maximum or a monotone increase in PL intensity is sometimes observed. This is most often characteristic of direct luminescence excitation of confined excitons [9–13]. The energy transfer of emitting nanoparticles through intermediate electronic states of the matrix [4–7] can also lead to an increase in PL intensity and to curves with an extreme shape. Thus, there is a need for a detailed analysis of various energy transfer schemes and types of electronic transitions in order to explain the observed variety of forms of temperature dependences of QD

It should also be said that according to some researchers, the spectral parameters of most luminescent low-scale structures are largely determined by sized and geometric factors [12, 15, 16]. In this case, the shape of the temperature curves of quantum dots PL is also substantially transformed [12]. This suggests that information on the features of the confined excitons in quantum dots can be obtained by analyzing the temperature behavior of photoluminescence. However, the lack of

The purpose of this chapter is to analyze the luminescence temperature dependences under direct and indirect optical excitation of spatially confined excitons in QDs. A generalized analytical description of such functional dependences is also reported.

Obtaining methods of semiconductor quantum dots are questions of great importance in the fields of science and engineering. Numerous advances have been achieved in the synthesis of QDs, among them bulk etching [17], laser pyrolysis [18], gas phase synthesis [19], thermal vaporization [20], and wet chemistry techniques [21]. It should be noted that all of the above methods often require special

Another convenient method is the thermal synthesis of quantum dots. As shown in [22–24], the synthesis of silicon QDs in the suboxide matrix takes place according to a reaction, which, depending on the temperature and duration of annealing, can

The schematic representation of the sample transformation is shown in **Figure 1**. With ion implantation, direct accumulation of the introduced material with not only the quantum dots formation but also the quantum dots occurrence is possible due to the evolution of defective structures. In particular, there is an innovative method of creating Si quantum dots in SiO2 under pulsed ion beam exposure [28]. In this case, quantum dots are formed as a result of the conversion and clustering

oxygen-deficient centers; (B) conversion of defects; and (C) clustering into Si QDs

the qualitative and quantitative composition of defects and modify the optical properties of the host, including the ultraviolet transmittance and visible luminescence. As seen from **Figure 2**, the radiation defect conversion includes several consequent stages. The implementation of the described conversion mechanism provides the controlled formation of stable quantum dots at various modes of ion

By changing the modes of radiation exposure of the SiO2 matrix, we can control

additives such as surfactants and dopants, as well as high-temperature

of radiation defects: ODC(II) ! E' ! ODC(I) ! Si QDs. During Gd-ion implantation with different doses, Si–O bond softening appeared, and the three main stages of defect evolution were identified: (A) formation of primary

systematic research in this field leaves this question open.

**2. Thermal and ion beam formation quantum dots**

postprocessing, to stabilize the QDs and control their size.

be stopped at any intermediate stage: 2SiOX ! Si QD + SiO2.

photoluminescence [4–13].

*Quantum Dots - Fundamental and Applications*

(**Figures 2** and **3**).

**24**

*Scheme of silicon nanoclusters (Si QD) formation in an initial multilayered structure under high-temperature annealing in nitrogen atmosphere. The RyOz (R—Si, Al, Zr) is a dialectical layer.*

#### **Figure 2.**

*Scheme of the successive stages for oxygen-deficient defect formation and fabrication of silicon quantum dots in SiO2 subjected to ion beam irradiation. (A) Radiation formation of vacancy defects; (B) transformation of defects; (C) clusterization into quantum dots.*

implantation [25]. The resulting Si QDs have a relatively small size of about 3.6 nm, which makes it possible to excite red luminescence due to the strong quantum confinement effect (**Figure 3**).

Described above the thermal synthesis of quantum dots and ion implantation methods are most convenient for stabilizing QDs in a dielectric matrix. Such systems are better suited for studying the mechanisms of excitation and radiative relaxation of QDs [26–28].

**Figure 3.**

*Typical luminescence spectra of quantum dots and radiation defects in SiO2 implanted with 30 keV Gd ions at fluences (2.5 1017 and 5 <sup>10</sup><sup>17</sup> cm<sup>2</sup> ). The spectra were obtained at temperature of 8 K and excitation by photons 6.6 eV.*

## **3. Mechanisms of QD excitation and scheme of electronic transitions**

In considering the intracenter relaxation processes for spatially confined excitons in QDs, their spin state should be taken into account. Bound electron–hole pair is a diamagnetic excitation characterized by singlet and triplet levels of energy associated with mutually antiparallel and parallel spins, respectively. In accordance with the well-known Hund's rule, the triplet state is the smallest excited state because the atomic level with lower energy has a full orbital angular momentum or maximum multiplexing [29, 30].

Here *E2*, *E3*, *E5*, *E8*, *E11*, and *E12* are the thermal activation barriers. The frequency factors characterizing the nonradiative transitions 2, 3, 5, 8, 11, and 12 are designated as *p*02, *p*03, *p*05, *p*08, *p*011, and *p*012, respectively; the singlet-triplet radiative transition rate and energy transfer rates corresponding to transitions 7 and 10

*Scheme of electron transitions in QD under direct and indirect excitations. Scheme shows two independent channels of excitation of the quantum dot and three different models of relaxation of excitons described by Eqs. (1)–(3). Optical and nonradiative transitions are indicated by solid and dashed arrows, respectively; the ground and excited singlet states are marked with S0 and S1, respectively; the excited triplet state of the exciton is*

*T1; the levels of free and self-trapped host matrix excitons are shown as FE and STE, respectively.*

*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots*

*DOI: http://dx.doi.org/10.5772/intechopen.91888*

In the work of [10], an analytical expression is presented that well describes the extreme form of the temperature dependence of PL. It was obtained taking into account the three-level energy scheme, which includes transitions 1–5. Process with an intersystem crossing (transition 2) and a triplet-singlet luminescence (transition 4) are the main stages of the two-stage radiative recombination of the excitons. Herewith the PL intensity will be proportional to the product of the quantum

Given the above and skipping the intermediate stages, the final expression for the model of QDs direct excitation can be written in the following function form:

> Δ*E*ISC *kT* <sup>1</sup> <sup>þ</sup> <sup>δ</sup>*P*<sup>R</sup> exp � *<sup>E</sup>*<sup>Q</sup>

where *I0* is the maximum luminescence intensity achieved at η*ISC* = 1 and η*<sup>R</sup>* = 1. Eq. (1) has the four fitting parameters: δ*P*ISC and Δ*E*ISC (transition 2) and δ*P*<sup>R</sup> and *E*<sup>Q</sup> (transition 4). In expression (1) the physical meaning of the constants (Δ*E*ISC, δ*P*ISC, *E*Q, and δ*P*Q) is quite clear. The Δ*E*ISC and δ*P*ISC parameters characterize the occupation process for T1 triplet state, whereas *E*<sup>Q</sup> and δ*P*<sup>Q</sup> are the parameters of the quenching process for the triplet-singlet PL. As can be seen from Eq. (1), an increase in the intensity of the triplet luminescence in the general process can be caused by the conversion of excitation from the singlet to the triplet state under some definite conditions. However, the shape of the temperature curve of the luminescence can significantly depend on various ratios between the activa-

*kT* �<sup>1</sup>

, (1)

are denoted by *P*4, *P*7, and *P*10, respectively.

**Figure 4.**

*IT* ¼ *I*0ηISCη<sup>R</sup> ¼ *I*<sup>0</sup> 1 þ δ*P*ISC exp

tion energies *E*<sup>2</sup> and *E*3.

**27**

efficiencies ηISC and η<sup>R</sup> for transitions 2 and 4, respectively.

The lifetime of triplet excitations can be several orders of magnitude longer than that of the singlet one, because triplet-singlet radiative transitions are spin-forbidden [30]. The thermally activated character of triplet luminescence after direct excitation of the singlet state is due to the energy barrier between these terms. Therefore, a growth in temperature leads to an increase in the glow intensity [10–13].

A generalized diagram for electronic transitions is shown in **Figure 4**. Direct singlet-singlet PL excitation of the QDs is shown by transition 1. At the same time, transition 6 illustrates indirect PL excitation through the electronic states of the matrix.

Each of the three different models presented in this diagram is individually characterized by a small number of fitting parameters. For better understanding of these analytic calculations, we have made the following special notations:

Δ*EISC* = (*E2*–*E3*) denotes the energy factor of the singlet-triplet intersystem crossing (ISC) relative to the QD excitons.

*EQ* = *E5* is the activation barrier of the QD PL quenching.

*EST* = *E8* is the activation barrier of self-trapping of matrix excitons.

Δ*EOC* = (*E11*–*E12*) is the occupation energy factor of the radiative T1 triplet states of the QD exciton.


*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots DOI: http://dx.doi.org/10.5772/intechopen.91888*

#### **Figure 4.**

**3. Mechanisms of QD excitation and scheme of electronic transitions**

maximum multiplexing [29, 30].

*fluences (2.5 1017 and 5 <sup>10</sup><sup>17</sup> cm<sup>2</sup>*

*Quantum Dots - Fundamental and Applications*

of the QD exciton.

matrix to the QDs.

of the QD exciton.

the QDs.

**26**

excitons in QD.

matrix.

**Figure 3.**

*photons 6.6 eV.*

In considering the intracenter relaxation processes for spatially confined excitons in QDs, their spin state should be taken into account. Bound electron–hole pair is a diamagnetic excitation characterized by singlet and triplet levels of energy associated with mutually antiparallel and parallel spins, respectively. In accordance with the well-known Hund's rule, the triplet state is the smallest excited state because the atomic level with lower energy has a full orbital angular momentum or

*Typical luminescence spectra of quantum dots and radiation defects in SiO2 implanted with 30 keV Gd ions at*

*). The spectra were obtained at temperature of 8 K and excitation by*

The lifetime of triplet excitations can be several orders of magnitude longer than that of the singlet one, because triplet-singlet radiative transitions are spin-forbidden [30]. The thermally activated character of triplet luminescence after direct excitation of the singlet state is due to the energy barrier between these terms. Therefore, a

A generalized diagram for electronic transitions is shown in **Figure 4**. Direct singlet-singlet PL excitation of the QDs is shown by transition 1. At the same time, transition 6 illustrates indirect PL excitation through the electronic states of the

Each of the three different models presented in this diagram is individually characterized by a small number of fitting parameters. For better understanding of

Δ*EISC* = (*E2*–*E3*) denotes the energy factor of the singlet-triplet intersystem

Δ*EOC* = (*E11*–*E12*) is the occupation energy factor of the radiative T1 triplet states

δ*PISC* = *p03* /*p02* is the intersystem crossing kinetic factor for spatially confined

δ*PT* = *p08* /*P7(10)* is the energy transfer kinetic factor from the excitons of the

δ*PR* = *p05* /*P4* is the kinetic factor for the triplet-singlet radiative transition of

δ*POC* = *p012* /*p011* is the occupation kinetic factor of the radiative T1 triplet states

growth in temperature leads to an increase in the glow intensity [10–13].

these analytic calculations, we have made the following special notations:

*EST* = *E8* is the activation barrier of self-trapping of matrix excitons.

*EQ* = *E5* is the activation barrier of the QD PL quenching.

crossing (ISC) relative to the QD excitons.

*Scheme of electron transitions in QD under direct and indirect excitations. Scheme shows two independent channels of excitation of the quantum dot and three different models of relaxation of excitons described by Eqs. (1)–(3). Optical and nonradiative transitions are indicated by solid and dashed arrows, respectively; the ground and excited singlet states are marked with S0 and S1, respectively; the excited triplet state of the exciton is T1; the levels of free and self-trapped host matrix excitons are shown as FE and STE, respectively.*

Here *E2*, *E3*, *E5*, *E8*, *E11*, and *E12* are the thermal activation barriers. The frequency factors characterizing the nonradiative transitions 2, 3, 5, 8, 11, and 12 are designated as *p*02, *p*03, *p*05, *p*08, *p*011, and *p*012, respectively; the singlet-triplet radiative transition rate and energy transfer rates corresponding to transitions 7 and 10 are denoted by *P*4, *P*7, and *P*10, respectively.

In the work of [10], an analytical expression is presented that well describes the extreme form of the temperature dependence of PL. It was obtained taking into account the three-level energy scheme, which includes transitions 1–5. Process with an intersystem crossing (transition 2) and a triplet-singlet luminescence (transition 4) are the main stages of the two-stage radiative recombination of the excitons. Herewith the PL intensity will be proportional to the product of the quantum efficiencies ηISC and η<sup>R</sup> for transitions 2 and 4, respectively.

Given the above and skipping the intermediate stages, the final expression for the model of QDs direct excitation can be written in the following function form:

$$I\_T = I\_0 \eta\_{\rm ISC} \eta\_{\rm R} = I\_0 \left\{ \left[ \mathbf{1} + \delta P\_{\rm ISC} \exp\left(\frac{\Delta E\_{\rm ISC}}{kT}\right) \right] \left[ \mathbf{1} + \delta P\_{\rm R} \exp\left(-\frac{E\_{\rm Q}}{kT}\right) \right] \right\}^{-1}, \tag{1}$$

where *I0* is the maximum luminescence intensity achieved at η*ISC* = 1 and η*<sup>R</sup>* = 1.

Eq. (1) has the four fitting parameters: δ*P*ISC and Δ*E*ISC (transition 2) and δ*P*<sup>R</sup> and *E*<sup>Q</sup> (transition 4). In expression (1) the physical meaning of the constants (Δ*E*ISC, δ*P*ISC, *E*Q, and δ*P*Q) is quite clear. The Δ*E*ISC and δ*P*ISC parameters characterize the occupation process for T1 triplet state, whereas *E*<sup>Q</sup> and δ*P*<sup>Q</sup> are the parameters of the quenching process for the triplet-singlet PL. As can be seen from Eq. (1), an increase in the intensity of the triplet luminescence in the general process can be caused by the conversion of excitation from the singlet to the triplet state under some definite conditions. However, the shape of the temperature curve of the luminescence can significantly depend on various ratios between the activation energies *E*<sup>2</sup> and *E*3.

Other triplet PL relaxation schemes can be considered in the same way. The energy transfer of free matrix excitons (FE) to quantum dots is confirmed experimentally [4, 5, 13]. In other words, the absorption of high-energy light quanta with the formation of free excitons (**Figure 4**, transition 6) can excite luminescence of QDs. The direct transition of the excitation to the triplet state T1 (transition 7) illustrates the simplest scheme of this process. The indicated option is similar to the direct excitation of QDs described by a two-stage process [10, 13]. The expression for the intensity of triplet luminescence (Eq. (2)) can be written assuming that transition 7 occurs without a barrier, and the formation of a self-trapped exciton (transition 8) is a competing process and also leads to luminescence (transition 9).

As can be seen from **Figure 1**, radiation transition 4 with fitting parameters (δ*P*<sup>R</sup> и *E*Q) is a common step for all considered models. Thus, the constant values of the parameters δ*P*<sup>R</sup> и *E*<sup>Q</sup> for transitions 4 under direct or indirect excitation will

significantly reduce the arbitrariness of approximation by Eqs. (1)–(3).

*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots*

*DOI: http://dx.doi.org/10.5772/intechopen.91888*

ence of parameters from Eq. (1) on the form of the *I*T(*T*) dependence.

**4.1 Triplet luminescence quenching**

**factors**

**Figure 5.**

**29**

**4. Dependence of PL temperature behavior on energy and kinetic**

As was mentioned in previous sections, the dependence of the PL intensity on temperature *I*T(*T*) can be described by a decreasing or increasing function or a curve with an extremum. According to Eq. (1), the form of *I*T(*T*) function depends on the relations between the kinetic and activation parameters of luminescent centers. Thus, if the certain conditions determining the form of function will be known, one can associate the shape of *I*T(*T*) curve with the energy and vibrational characteristics of the emission center. In the framework of two-step PL mechanism, the shape of the curves *I*T(*T*) is determined by the multipliers ηISC(*T*) and ηR(*T*). **Figure 5** shows the calculated functions ηISC(*T*) and ηR(*T*) and their multiplication. The curve (3) reproduces qualitatively the temperature dependence *I*T(*T*) normalized to the number of photons incident on the surface of the sample. The next step is to discuss the influ-

In case both temperature-dependent factors of Eq. (1) decrease with an increasing of temperature, the *I*T(*T*) function is characterized by a negative slope, wherein the quantum efficiency ηR(*T*) is decreasing (see **Figure 5**, curve 1) within the range from 1 to 1/(1 + δ*P*R). It has a minimum which depends on the ratio *p*05/*P*4. If δ *P*<sup>R</sup> ! 0, then η<sup>T</sup> will weakly depend on the temperature, keeping a value that is very close to 1 over the entire temperature range. However, the situation δ*P*<sup>R</sup> > 1 is most often realized, since the radiative triplet-singlet transitions are spin forbidden [30], and their rate *P*<sup>4</sup> is less than the frequency factor *p*<sup>05</sup> for the nonradiative relaxation. The ratio between activation barriers *E*<sup>2</sup> and *E*<sup>3</sup> determines the form of ηISC(*T*) function, which decreases ranging from 1 to 1/(1+ δ*P*ISC) if Δ*E*ISC < 0. In this case,

*Calculated temperature dependences of the quantum efficiency factors for different processes: (1) triplet-singlet*

*radiative transitions; (2) intersystem crossing; (3) two-step process ηISC(T)*�*ηR(T).*

$$I\_T ' = I\_0 \eta\_{\rm IT} \eta\_{\rm R} = I\_0 \left\{ \left[ \mathbf{1} + \delta \mathbf{P}\_{\rm T} \exp\left( -\frac{E\_{\rm ST}}{kT} \right) \right] \left[ \mathbf{1} + \delta \mathbf{P}\_{\rm R} \exp\left( -\frac{E\_{\rm Q}}{kT} \right) \right] \right\}^{-1},\tag{2}$$

where η<sup>R</sup> and η<sup>T</sup> are quantum efficiencies characterizing radiative triplet-singlet transitions and energy transfer from FE to QDs, respectively. Eq. (2) has four fitting parameters: δ*P*<sup>R</sup> and *E*<sup>Q</sup> (transition 4) and δ*P*<sup>T</sup> and Δ*E*ST (transition 7), similar to Eq. (1). At the same time, the parameters (δ*P*<sup>R</sup> and *E*Q) of nonradiative decay of excitons are independent of the excitation method, as can be seen from Eqs. (1) and (2). An analysis of the data for direct excitation of PL (**Figure 4**, transitions 4 and 5) allows one to determine these two parameters from independent measurements and then use them in the analysis of indirect excitation processes. Using this approach, one can significantly improve the procedure for approximating the experimental temperature dependences of PL and reduce the number of variable parameters.

At the same time, the large difference between the energy levels of excitons in QDs and free excitons in a dielectric matrix should be taken into account. In this regard, Eq. (2) is approximate and describes only a simplified scheme of luminescence. In fact, transition 7 cannot be considered elementary, and energy from the FE is transferred to the high-energy states (HES) of the QDs (see **Figure 4**, transition 10). Further, successive thermal transitions from HES lead to the filling of the lower triplet state T1. The indicated energy transfer sequence cannot be considered barrierfree, and triplet-singlet luminescence should be described by a multistage model.

Let us consider the generalized transition 11 with effective parameters: activation energy *E*<sup>11</sup> and frequency factor *p*<sup>011</sup> as a sequence of thermal transitions HES ! ... ! T1. A generalized transition 12 with the corresponding effective parameters *E*<sup>12</sup> and *p*<sup>012</sup> will be considered a sequence of competing processes. The energy transfer rate *P*<sup>10</sup> to highly excited states will be assumed to be independent of temperature, and the thermal activation barrier of such a transfer is equal to zero. Then the triplet-singlet PL will be described by the following three-stage model:

$$\begin{split} I\_{T}" &= I\_{0} \eta\_{\text{T}} \eta\_{\text{oc}} \eta\_{\text{R}} \\ &= I\_{0} \left\{ \left[ \mathbf{1} + \delta \mathbf{P}\_{\text{T}} \exp\left( -\frac{E\_{\text{ST}}}{kT} \right) \right] \left[ \mathbf{1} + \delta \mathbf{P}\_{\text{oc}} \exp\left( \frac{\Delta E\_{\text{oc}}}{kT} \right) \right] \left[ \mathbf{1} + \delta \mathbf{P}\_{\text{R}} \exp\left( -\frac{E\_{\text{Q}}}{kT} \right) \right] \right\}^{-1} \end{split} \tag{3}$$

The latter is taking into account the quantum efficiencies (ηT, ηoc, ηR) for energy transfer, triplet state occupation, and radiative triplet-singlet transitions, respectively.

Eq. (3) is the variant of Eq. (2), where by taking into account an additional step there are six parameters, corresponding to transition 10 (δ*P*<sup>T</sup> и Δ*E*ST), transition 11 (δ*P*oc и Δ*E*oc), and transition 4 (δ*P*<sup>R</sup> и *E*Q). The energy transfer from the matrix to the quantum dot is denoted identically in Eqs. (2) and (3) and is described by the parameters δ*P*<sup>T</sup> and Δ*E*ST. However, it should be clarified that in Eq. (2) quantum energy transfer efficiency η<sup>T</sup> characterizes transition 7, while in Eq. (3)—transition 10.

*Temperature Effects in the Photoluminescence of Semiconductor Quantum Dots DOI: http://dx.doi.org/10.5772/intechopen.91888*

As can be seen from **Figure 1**, radiation transition 4 with fitting parameters (δ*P*<sup>R</sup> и *E*Q) is a common step for all considered models. Thus, the constant values of the parameters δ*P*<sup>R</sup> и *E*<sup>Q</sup> for transitions 4 under direct or indirect excitation will significantly reduce the arbitrariness of approximation by Eqs. (1)–(3).
