Physics and Mathematics of the Photoluminescence of Complex Systems

Abstract

The photoluminescence (PL) of thermally evaporated Alq3 thin films has been studied in a few samples annealed and non-annealed and afterwards exposed to the laboratory atmosphere for over six years. It was found that the measured emission intensity decays with a long lifespan and with four different time-spectral behaviors, which imply the existence of four molecular aggregations, or components. In particular, the time behavior of each component follows the trend of a Kohlrausch−Williams−Watt (KWW) function, which is well known in mathematics but without any physical meaning. Here, by introducing the concept of the material clock, the system has been described by a damped harmonic oscillator, which in certain conditions, fulfilled in the present case, allows the expansion of the KWW function in the so-called Prony series. The terms of this series can be attributed to chemical and physical processes that really contribute to the decay, i.e. the degradation, of the Alq3 thin films when interacting with internal and environmental agents. These insights unveiled the usefulness of proper mathematical procedures and properties, such as the monotonicity and the complete monotonicity, for investigating the PL of this ubiquitous organometallic molecule, which possesses one among the highest emission yield. Moreover, this method is also promising for describing the photoluminescent processes of similar organic molecules important both for basic research and optoelectronic applications.

On the application of Mittag-Leffler functions to hyperbolic-type decay of luminescence

“…According to Theorem 2 in [22], the photoluminescence described by (4.1) is a complete monotone function, because the Mittag-Leffler function is a complete monotone function for ν P p0, 1q as proven in [23; 24] and the logarithm is a non-negative function with a complete monotone derivative. As shown in [7], it implies that the corresponding reduced mass is increasing and therefore, the barrier is also increasing, confirming the result obtained in [19; 20]. The increasing barrier implies that the dynamics is ruled by an exclusive damping mechanism.…”

“…In the previous Section the Becquerel decay law was extended to describe a stretched hyperbolic behaviour, but its physical interpretation still needs to be clarified. A strategic approach to frame the physical meaning of photoluminescent empirical functions has been designed in [7], where it was applied to the Kohlrausch-Williams-Watts function. Despite the different function used to model the fluorescence, the methodology and the results are quite general and can be applied to other cases.…”

“…The exponent in (5.3) describes a current-dependent barrier height that is almost zero at the beginning and causes a slower current decay when starts to increase (i.e., the resistance decreases, as reported in [19; 20]). In photoluminescence processes, the role of this increasing barrier can be understood in the light of the reduced (or effective) mass, that can be related to the inertia of the system preventing the photoluminescent emission [7]. The reduced mass can be associated to the presence of the cloud of virtual particles around the emitting centers, that deform the surrounding environment creating a distribution of relaxation times (that can be modeled as traps) and leading to a non-hyperbolic, anomalous behaviour.…”

Anomalous Kinetics of Photoluminescence from Degrading Organic Materials and Its Theoretical Rendition

“…The Kohlrausch-Williams-Watts (KWW) function is commonly observed in the relaxation of physical systems, such as glassy materials, 20,[28][29][30][31] glass-forming liquids, 30 and luminescent decay in solid state materials. [1][2][3][4][5][6][7][8][9][10]16,18,19,[21][22][23][24][25]32 However, although its success in fitting well the known experimental data, the physical meaning and origin of the KWW function is not understood at all. To address these not secondary issues, a mathematical model has been proposed and described in the following three steps.…”

A Novel Fit-Flexible Fluorescence Soft Imager: Tri-Sensing of Intensity, Fall-Time, and Life Profile

“…That author suggested four circuits including resistor, capacitor, diode, and transistor to obtain some degree of flexibility. It is important to note that the fractional KWW system itself can be expanded via (1), also known as the Prony series expansion in physics [34].…”