Thermalisation time

The measurement of the mean lifetimes of positrons in matter has been one of the cornerstones of positron science over the past half-century. The lifetime of a positron in matter—gas, liquid or solid—will depend on the electronic environment in which it finds itself, and this in turn tells us much about the submicroscopic nature of the material. In condensed matter a positron will approach thermal energies within about lps, so that measured lifetimes are essentially those of a thermal positron in the material under study. In some gaseous environments—particularly in the noble gases—the time taken for a positron to come to thermal equilibrium with its surroundings is much longer—10°-102 ns—and this thermalisation time has to be taken into account in the analysis of time spectra. 

Efficient MEG also requires strong coupling between exciton and biexciton states relative to YP Wc/ >> Yi- Fulfdling this condition satisfies the obvious requirement that the exciton thermalisation time should be longer than the coupling time between the optically created exciton and the asymmetric biexciton state (Nozik, 2002). 

The assumption of drift or diffusive transport is, of course, based on complete thermalisation within the absorber. This assumption appears to hold for most devices treated in this chapter with absorber thicknesses in excess of 100 A. Taking a free carrier mobility of 100 cm s a thermalisation time of 1 ps and an electric field... 

Table 4. Thermalisation times for electrons in gases at 1 atmosphere and 1 torr pressure.

The values of Ec for some metals are shown in Table 4.14. The thermalisation time 1r in the energy region Ec < E < Em where excitations of conduction electrons play a dominant role may be calculated from Equation 4.45 with the substitution... 

The integration of Equation 4.41 with Equation 4.45 over the range k T < E < Ec yields the phonon scattering thermalisation time tph ... 

Figure 4.30 Temperature dependence of the phonon thermalisation time fph for some metals [90]...

Thermalisation times r and fph are collected in Table 4.14. Values of fph correspond to room temperature T = 20 °C). The total thermalization time... 

One can immediately conclude that thermalisation times are very short compared to positron bulk lifetimes tb. The fact that positrons reach thermal energies very quickly is important for the application of the positron method. Clearly only thermalised positrons annihilate. The momentum of positrons is thus very small compared to the momentum of the electrons with which they annihilate. It is evident from Table 4.14 that positron scattering off phonons occupies more than 50% of the thermalisation time. Therefore, the positron spends most of the slowing-down process with an energy slightly above the thermal energy. This fact is important for the possible trapping of non-thermalised positrons. 

In Equation 4.60, the deformation potential constant has been eliminated using Equation 4.55. By integrating Equation 4.55 between ea = ik- T/2 and S2 ei, we obtain the following relation for the thermalisation time associated with the emission of thermal phonons ... 

At lower altitudes where significant concentrations of ozone exist, O- ions are generated by dissociative attachment [reaction (10b)]. These electron attachment processes and the laboratory techniques used to determine their rate coefficients were reviewed some time ago by Phelps1 S4 In the stratosphere and troposphere, negative ions can also be generated by dissociative attachment reactions of thermalised electrons with pollutants1 s5,1561 such as the freons e.g. 

The Raman scattering strength of E,(LO) in the vicinity of the fundamental bandgap has been investigated in resonant Raman scattering as a function of temperature between 77 K and 870 K [35], Studies of photocarrier thermalisation have been performed by time resolved Raman spectroscopy [36],... 

Figure 3.4 illustrates two lifetime spectra collected by methods similar to those outlined above, (a) exhibits the non-exponential shoulder region associated with the annihilation of non-thermalised positrons. After thermalisation (essentially at time zero for condensed matter) the spectra are sums of exponential components associated with each decay mode, and a background component B, A] = 2 A, exp(-Ajt,) + B. For long lifetime components (> Ins) each X can be extracted by non-linear least squares fitting. For short X values characteristic of condensed matter, however, a... 

The solvated electron is a transient chemical species which exists in many solvents. The domain of existence ofthe solvated electron starts with the solvation time ofthe precursor and ends with the time required to complete reactions with other molecules or ions present in the medium. Due to the importance of water in physics, chemistry and biochemistry, the solvated electron in water has attracted much interest in order to determine its structure and excited states. The solvated electrons in other solvents are less quantitatively known, and much remains to be done, particularly with the theory. Likewise, although ultrafast dynamics ofthe excess electron in liquid water and in a few alcohols have been extensively studied over the past two decades, many questions concerning the mechanisms of localisation, solvation, and thermalisation ofthe electron still remain. Quantum and molecular dynamics simulations are necessary to unravel the structure ofthe solvated electron in many solvents and to better understand its properties. 

The optical excitation of electron-hole pairs represents a non-equilibrium state. The subsequent relaxation processes from the initial state includes both carrier-carrier interactions and coupling to the bath phonons. In some treatments, there is a distinction made between carrier-carrier and carrier-phonon interactions in which the latter is referred to as thermalisation. A two-temperature model is invoked in that the carrier-carrier scattering leads to a statistical distribution that can be described by an elevated electronic temperature, relative to the temperature characterising the lattice phonons (Schoenlein et al, 1987 Schmuttenmaer et al, 1996). This two-temperature model is valid only if the carrier-carrier energy redistribution occurs on time scales much faster (>10 times) than relaxation into phonons. This distinction has limited value when there is not a sufficient separation in time scale to make a two-temperature model applicable. The main emphasis in this section is on the dynamics of the energy distribution of the carriers as this is most relevant to energy storage applications. 

The main message from this class of experiments is that the details of the surface do affect the carrier relaxation. In the presence of surface defects associated with conventional surface preparation, the carrier relaxation in the surface region is exceptionally fast relative to bulk processes (10-100 fs dynamics). As can be seen by comparing the dynamics shown in Fig. 2.9, the effect of the surface is to increase the rate of relaxation and thermalisation. The asymmetry, more anharmonic character to the surface modes and increased mixing of states at defect sites all conspire to speed up the relaxation processes. With proper attention to surface structure, it is possible to intervene in the relaxation process and achieve carrier and phonon scattering rates that approach bulk processes. In this limit, 200 fs to picosecond dynamics define the operative time scales. 

With sufficiendy thin absorber layers, particnlarly with quantum-dot absorbers, it also appears possible to harvest hot carriers and reduce the thermalisation loss in solar cells. This idea was pursned at an early time by Cooper et al. (1983). Hot electron transfer has been demonstrated in an electrochemical arrangement, and the possibility of hot-carrier transfer across solid interfaces is starting to be realised. Furthermore, the associated phenomenon of mnltiple exciton generation from a single hot electron-hole pair created by a single photon within a semicondnctor qnantnm dot has now been realised, as Art Nozik discnsses in Chapter 3. 

Time of flight measurements can yield useful information in a similar vein. The work of Comsa et al. (1980), for instance shows a shift of D desorption from Pd(100) from a Maxwell-Boltzmann, surface thermalised desorbing flux to one with fast Dj molecules emerging from the surface with a narrow distribution of energies after sulphur is deposited on the surface... 

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