Exciton mean free path

Thus we find that, in this limit, the reduction of the exciton mean free path does not affect the radiative broadening RK, as noticed by Agranovitch,152 since ImX and RK enter the reflection amplitude (4.23) additively. This conclusion however does not hold for the emission, as we shall show in Section IV.A.4.b below. 

In some molecular crystals a crossover from coherent excitons (exciton mean free path l A) to incoherent ones ( k, A, Ioffe-Regel criterion) takes place with increasing temperature. We then expect that upon increasing the temperature from very low values, at some threshold temperature the decreasing behavior of the diffusion constant for coherent excitons goes over into an increasing behavior. 

Materials whose dimensions (typically in the 20-80 A range for semiconductor particles) are comparable to the length of the Broglie electron, the wavelength of the Broglie electron, the wavelength of phonons, and the mean free paths of excitons. Size quantization can occur in one, two, or three dimensions and manifests in altered optical, electronic, and chemical properties. 

For the band model phonons are not needed to cause energy migration as in the hopping model and instead exciton-phonon scattering limits the mean free path of the excitons and, thus, inhibits the migration at high temperatures. The time between scattering events for diffusion in the ith direction can be calculated from... 

The fact that the dimensions of the particles approaches, or becomes smaller than, the critical length for certain phenomena (e.g., the de Broglie wavelength for the electron, the mean free path of excitons, the distance required to form a Frank-Reed dislocation loop, thickness of the space-charge layer, etc.). 

It is clear even from a very qualitative consideration, that an exciton with a wavevector k will be scattered by phonons, and an increase of temperature will decrease its mean free path. If an exciton is localized and changes its local position by hopping from one molecule to another, the thermal motions will intensify the exciton diffusion (for more details about this subject see Ch. 14). Studies of the mobility of excitons are very important for biophysics. 

The motion of the exciton wavepacket causes the transport of energy. In order to find the appropriate energy diffusion coefficient we must estimate the mean free path and the mean free time of the wavepackets. This situation is quite similar to that of phonon heat conductivity (see, for example, (12)). 

Here, th is the lifetime of the host excitons without traps. We furthermore make the simplifying assumption that every exciton is captured when it reaches a trap and that the excitation density is constant with time and remains homogeneous. If, in addition, the mean free path of the exciton is smaller than the capture radius R of the traps for excitons, and D is the exciton diffusion coefficient, then we obtain... 

Assuming coherent exciton motion during the coherence time r, the mean free path L can also be estimated I = rv (v is the velocity of the excitons). The velocity V can be obtained from the relation mv = kT, with the effective exciton mass m. 

An entirely new situation arises for a semiconductor with NEA. Here the threshold for photoemission of electrons is the band gap energy, that is, a bulk rather than a surface property, and novel phenomena are to be expected. Tbis is indeed the case, and the most spectacular of these phenomena is certainly the contribution of bulk excitons to the photoelectron yield of diamond surfaces with NEA as first reported by Bandis and Pate [73, 107]. In addition, the depth from which electrons contribute to the yield is no longer limited by the inelastic mean free path of some tens of Angstroms but by the diffusion length of electrons and excitons of the order of micrometers, a fact that is responsible for the near 100% quantum efficiency of NEA diamond surfaces alluded to earlier (Section 10.3). 

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