Multi-Phonon Emission

8.5 Multi-Phonon Emission

Since $E′$ centers upon hole capture may undergo structural relaxation [148], a description of the restructuring process of the concerned defect center is required. By using first-principles density function theory (DFT), Schanovsky et al. thoroughly investigated the hole capture process of various point defects embedded in an orthorhombic alpha-quarz supercell structure of $72$ atoms [149, 150, 151]. During such a hole capture the electronic and vibrational state of the defect system change at the same time, leading to a so-called vibronic transition. This vibronic transition can be modeled using the Born-Oppenheimer approximation and the Franck-Condon (FC) principle [152, 153]: Due to the different masses, electrons and holes only take femto-seconds to switch their states, while defect centers respond with a factor of 100 slower ( $1e − 15s$ versus $1e− 13s$ ). As a consequence, the electrons are able to immediately follow the potential of the defect centers, i.e. they are always in equilibrium compared to the defect centers. On the contrary, the structure of the defect does not change during an electronic transtion, which is illustrated by the vertical transition arrow between two different electronic states Fig. 8.6 (left). In this figure the total potential energy of a defect $Etot$ is modeled as a quantum harmonic oscillator featuring the eigenenergy levels of the defect’s vibronic states $En = ℏω(n + 1∕2)$ with $n > 1,2,...$ . According to the Franck-Condon principle, a change of the electronic state, i.e. when moving from one to another harmonic oscillator, at the same time causes a change of the vibrational state and with it a change of the equilibrium position of the defect center [154, 155, 141]. This is known as electron-phonon coupling³ . Mathematically, the vibronic transition from the electronic state $1$ to state $2$ can be derived from Fermi’s golden rule

2π k1α→2β = ---|⟨η2βϕ2|V ′|ϕ1η1α⟩|2δ(E2 β − E1 α), ℏ

(8.15)

where the first index denotes the electronic state and the second index the vibrational state of the electronic $|ϕa⟩$ and vibrational $|ηab⟩$ wave functions. After [152, 149], the matrix element of the transition rate in (8.15) can be split into the electronic matrix element represented by a WKB tunneling term and the Franck-Condon overlap factor $2 |⟨η2β|η1α⟩|$ . To consider all possible transitions, the overlap factor has to be calculated for each initial and final state combination, followed by thermally averaging over all intial vibrational states⁴ and then summing over all final vibrational states [156, 149]. Where the initial and the final total energies are equal [153, 125, 157], Dirac peaks are obtained whose contour line is called line-shape function (LSF), which describes the broadening of the absorption spectra. Multiplying the WKB term, which approximates the electronic matix element, and the LSF finally yields the transition rate

2π- ′ 2 ∑ 2 k1α→2 β = ℏ |⟨ϕ2|V |ϕ1⟩ avgα |⟨η2β|η1α⟩| δ(E2β − E1α). ◟----◝◜-----◞◟----β--------◝◜-------------◞ WKB LSF

(8.16)

  8.5.1 Approximation of the Vibronic Transition
  8.5.2 Radiative Multi-Phonon Emission
  8.5.3 Non-Radiative Multi-Phonon Theory

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