Charge carrier traps play a crucial role in all the suggested persistent luminescence mechanisms. One of their main properties is their ‘depth’, the activation energy needed to release a captured charge carrier. Shallow traps (with a depth lower than around 0.4 eV [

94]) are fully emptied at low temperatures, and do not actively take part in processes at room temperature. Very deep traps (around 2 eV or deeper [

10]), on the other hand, require more energy to be emptied than is available at room temperature. Therefore, charge carriers caught by these traps remain there until the material is sufficiently heated. To observe persistent luminescence at room temperature, the traps should have an appropriate activation energy somewhere between these two extremes (a trap depth around 0.65 eV is considered to be optimal [

5]).

In chapter 4, we will see that the nature of the trapped charge carriers (electrons or holes) is still subject of discussion. It is therefore noteworthy that the techniques described in the following paragraphs give an estimate for the trap depth regardless of the charge carrier type.

#### 3.1. Thermoluminescence

The experimental technique of thermoluminescence (TL) was first explored in the beginning of the 20th century [

95]. In this method, a material is initially heated or kept in the dark for a sufficiently long time until all traps are emptied. The material is subsequently cooled to liquid nitrogen or helium temperatures, and fully excited by a (usually white) light source for some time. The excitation is switched off, and the temperature is increased linearly, with a heating rate

β (in K/s). Meanwhile, the optical emission from the sample is measured and plotted against temperature. The curve obtained in this way is usually denounced as ‘glow curve’ [

3,

11]. It is customary to also measure the temperature dependence of the fluorescence (by repeating the measurement under constant excitation), in order to compensate for temperature quenching effects [

3].

For nearly all materials, the glow curve shows one or more broad, often asymmetrical peaks (an example for SrAl

_{2}O

_{4}:Eu

^{2+},Dy

^{3+} is given in

Figure 6). Each peak is believed to originate from a separate trap or trap distribution. Studying the shape and location of these different peaks can provide insight in the different depths and distributions of the traps present in the sample. These studies are especially popular in the field of geology, where thermoluminescence is used as a dating technique [

95], and radiation dosimetry [

96]. Thermoluminescence measurements and glow curves are often neglected in the study of persistent luminescence, but also in this field they can be useful to give information on trap depths and distributions.

**Figure 6.**
The glow curve for SrAl

_{2}O

_{4}:Eu

^{2+},Dy

^{3+} after excitation for different periods of time. From bottom to top: 10, 30, 60 and 120 seconds (Reprinted with permission from [

39]).

**Figure 6.**
The glow curve for SrAl

_{2}O

_{4}:Eu

^{2+},Dy

^{3+} after excitation for different periods of time. From bottom to top: 10, 30, 60 and 120 seconds (Reprinted with permission from [

39]).

Many authors have tried to develop a method to analyze glow curves in a reliable and consistent way. A full discussion of all these efforts and their theoretical details is beyond the scope of this text, but can be found in, for example, [

95]. We will briefly mention those methods that are still used frequently today. The most simple way to estimate trap depths from the location of the glow peak maximum was derived empirically and formulated in 1930 by Urbach [

97]. If

T_{m} is the temperature for which the glow curve reaches a maximum, the related trap depth is approximately:

This equation, despite its simplicity, incorporates an important intuitive result: deeper traps (

i.e., with a higher activation energy

E_{T}) result in glow curve peaks at higher temperature. Indeed, to free charge carriers from deeper traps, a larger thermal energy is required. The trap energy obtained is only approximate, since equation (1) is not based on a theoretical model for the behavior of charge carriers in materials with trap levels. This problem was tackled in a famous series of articles by Randall and Wilkins in 1941 [

98,

99]. They looked at the simplified situation of a host material with a single trap level in the band gap. It is important to note that although they assumed the charge carriers to be electrons, their results are equally valid in the case of holes. According to their theory, the glow intensity

I during heating is found to be proportional to the concentration

n, the frequency factor or ‘escape frequency’

s of the trapped charge carriers and an exponential part containing the trap depth (Boltzmann factor):

where

k is Boltzmann

’s constant. This leads to a transcendental equation for the trap depth:

where

T_{m} is again the location of the glow curve maximum, and

β is the heating rate (in K/s). This linear relationship between glow intensity and trapped charge carrier concentration is generally referred to as “first order kinetics”. This theory assumes that every charge carrier released from a trap recombines in a luminescent center. The possibility of `retrapping

’, when the charge carrier is caught again by a trap and not by a luminescent center, is assumed to be negligible. However, Randall and Wilkins pointed out that certain experimental results suggested similar probabilities for both processes (retrapping and recombination) [

99]. In 1948, Garlick and Gibson (coworkers of Randall and Wilkins) explored this possibility and obtained a “second order kinetics”, with the glow intensity proportional to

n². They found that this assumption yielded better results for several materials [

100]. The effect of the parameters

s,

T,

E_{T} and

β on the shape of the glow curve can be found in [

101], illustrated for both first and second order kinetics.

Two major obstacles arise when applying the Randall-Wilkins or associated methods. Firstly, the frequency factor

s is initially unknown. Very often,

s is approximated or assumed comparable to the vibrational frequency of the lattice. The value of

s does, however, greatly influence the resulting trap depth. Even worse, the frequency factor itself can (and most probably does) depend on temperature, and therefore changes during the course of the thermoluminescence experiment, as pointed out by for example Chen [

102]. The uncertainty on the value of

s is bypassed in the Hoogenstraaten method [

103]. For this, the thermoluminescence experiment is repeated several times for different heating rates

β_{i}. According to equation (3), the exact glow maximum will shift to different temperatures

T_{mi} when the heating rate is varied. For every value of

β_{i}, a similar equation can be written down, and the unknown

s can thus be eliminated by plotting ln(

${T}_{mi}^{2}$/

β_{i})

versus 1/

T_{mi} and fitting these data points with a straight line. The slope of this line reveals the activation energy of the trap [

3].

A second problem is the unknown ‘order’ of the glow curve under investigation. Some glow peaks yield better fitting results when first order kinetics are assumed, some require second order kinetics for a decent fit. This problem can be avoided by looking only at the low-temperature side of the observed glow peaks. Regardless of the order of the peak, the intensity will be proportional to an exponential Boltzmann factor [

3]:

The estimation of trap depths by fitting the low-temperature end of a glow peak to such an exponential factor is known as the “initial rise” method. In practice, however, it is often very difficult to isolate the initial rising part of a glow peak, making the obtained trap depths less accurate.

Another popular way to avoid the problem of the unknown order was proposed by Chen in 1969 [

102] and is known as ‘general order kinetics’. Chen looked more closely at the shape of the glow peak, by taking into account the full width ω at half maximum and its low-temperature and high-temperature halves τ and δ, respectively. He was able to write down several compact formulas to calculate the trap depth in specific cases. Furthermore, he calculated the coefficients necessary for these formulas for many different values of the trap depth (0.1–1.6 eV) and frequency factor (10

^{5–}10

^{13} s

^{-1}) [

104]. For example, the activation energy for a peak with first order kinetics and a temperature-independent frequency factor is simply given by:

All of the mentioned models are still in use today. To simplify glow curve analysis for researchers, specialized software was developed, such as

TL Glow Curve Analyzer [

105], which makes use of first, second and general order kinetic equations [

106].

The analysis of glow curves has fascinated researchers for many decades, but it should be approached with caution. The results obtained by the different techniques described above are not always comparable. This is illustrated by applying several methods on the same glow curve, as was done by for example [

107] in the case of cerium and copper doped barium sulfide. In this paper, the Urbach, Randall-Wilkins and Chen methods (among others) were compared. The Randall-Wilkins analysis gives the lowest trap depth (around 0.45 eV), Chen

’s method results in a trap approximately 0.05–0.10 eV higher, and Urbach

’s formula estimates a trap depth of about 0.75 eV. This indicates that comparing different trap depths should only be done when both glow curves were studied with the same technique. Another problem that often occurs in practice is the overlap between different peaks, which can make decent analysis almost impossible.

#### 3.2. Other methods

Thermoluminescence is the most common way to estimate trap depths, but some other techniques are known that do not rely on the analysis of glow curves. Bube [

108] noted that, for first order kinetics, the temperature dependence of the afterglow decay constant is given by:

By measuring the decay constant for various temperatures and plotting the results in an Arrhenius diagram, a straight line is obtained whose slope is related to the trap depth. Often, the afterglow decay cannot be described by a single decay constant. In that case, multiple exponentials can be fitted to the decay, and for each of these an appropriate trap depth can be estimated. Although this method is sometimes used in binary sulfides [

107,

109], it is, to the best of our knowledge, never applied in Eu

^{2+}-doped materials.

Another technique was proposed by Nakazawa and is frequently called “transient thermoluminescence” (TTL,TRL) [

110]. While the sample is heated very slowly, it is repeatedly excited by a light source. The intensity of the afterglow is measured at various delay times

t_{d} after the termination of the excitation. When this intensity is plotted against the temperature, the location of the peak

T_{m} depends on the delay time in the following way:

An advantage of the TTL method is that it is unaffected by thermal quenching, as opposed to normal thermoluminescence [

110].

Table 4 shows some approximate values for the best-known persistent luminescent materials, as reported in literature. It should be noted that these are not always exact results. Sometimes the trap is not a single level, but a distribution of energy levels. In that case, a mean value of the trap is cited. All trap levels mentioned in the respective articles are noted, although sometimes they are too shallow or too deep to contribute to the persistent luminescence at room temperature. The numbers in the table demonstrate again that comparing trap depth values estimated with different techniques should be done with caution.

**Table 4.**
Estimated trap depth(s) in some of the best-known persistent luminescent materials.

**Table 4.**
Estimated trap depth(s) in some of the best-known persistent luminescent materials.
Phosphor | Method | Trap depths (eV) | Reference |
---|

SrAl_{2}O_{4}:Eu^{2+},Dy^{3+} | initial rise | 0.55/0.60/0.65/0.75 | [111] |

| Chen | 0.30/0.65/0.95/1.20/1.40 | [112] |

| Hoogenstraaten | 0.65 | [5] |

| TTL | 1.1 | [113] |

CaAl_{2}O_{4}:Eu^{2+},Nd^{3+} | initial rise | 0.55/0.65 | [114] |

Sr_{4}Al_{14}O_{25}:Eu^{2+},Dy^{3+} | Chen | 0.72 | [47] |

| Chen | 0.49 | [46] |

| TTL | 0.91 | [25] |

Sr_{2}MgSi_{2}O_{7}:Eu^{2+},Dy^{3+} | Chen | 0.75 | [51] |

Sr_{2}MgSi_{2}O_{7}:Eu^{2+},Nd^{3+} | Hoogenstraaten | 0.08/0.18/0.29/0.23 | [115] |

Ca_{2}MgSi_{2}O_{7}:Eu^{2+},Tm^{3+} | Chen | 0.56 | [52] |