Coulomb matrix elements for the impact ionization process in nanocrystals: An envelope function approach

Abstract

We propose a method for calculating Coulomb matrix elements between exciton and biexciton states in semiconductor nanocrystals based on the envelope function formalism. We show that such a calculation requires proper treatment of the Bloch parts of the carrier wave functions which, in the leading order, leads to spin selection rules identical to those holding for optical interband transitions. Compared to the usual (intraband) Coulomb couplings, the resulting matrix elements are additionally scaled by the ratio of the lattice constant to the nanocrystal radius. As a result, the Coulomb coupling between exciton and biexciton states scale as 1/R 2 . We present also some statistical estimates of the distribution of the coupling magnitudes and energies of the coupled states The number of biexciton states coupled to exciton states form a certain energy range shows a power-law scaling with the ratio of the coupling magnitude to the energy separation. We estimate also the degree of mixing between exciton and biexciton states. The amount of biexciton admixture to exciton states at least 1 eV above the multiple exciton generation threshold can reach 80% but varies strongly with the nanocrystal size.

Interband Coulomb coupling in narrow-gap semiconductor nanocrystals:k·ptheory

“…Mesoscopic and microscopic contributions to the inter-band Coulomb coupling are studied in detail by generalizing the previous results 23 to the realistic model of wave functions including the band mixing. The relatively low computational cost of the k · p method allows us to find coupled pairs of X-BX states in a very broad energy window and to study the distribution of the magnitudes of the matrix elements vs. the energies of the coupled configurations in order to build reliable statistics.…”

“…This can occur when absorbing a photon is followed by creation of an electron-hole pair (an exciton) which then relaxes into an energetically lower state and the excess energy is used to create a second electron-hole pair (thus creating a biexciton state) [7][8][9][10][11]19,20 . The same process can also occur coherently via a superposition of single-and bi-exciton states, or with the single exciton state playing the role of a virtual intermediate state [13][14][15][21][22][23] . In any case, this process is mediated by Coulomb scattering between electron states in different bands which does not conserve the number of electron-hole pairs.…”

“…We will refer to this contribution as "mesoscopic" as the relevant matrix element involves only the envelope parts of the wave functions (similarly to the common electron-electron or electron-hole interactions computed in the usual way in a nanostructure). Second, the coupling between such configurations can appear directly when one takes the Bloch part of the wave function into account 23 . This is, in turn, formally similar to the k · p calculation of electron-hole exchange coupling in a quantum dot 28 .…”

“…order of a/R as compared to the mesoscopic one 23 (where a is the lattice constant and R is the nanocrystal radius). In the absence of band mixing, the leading order term in the expansion of the Coulomb coupling is proportional to a/R.…”

“…In any case, keeping terms up to the first order in a/R in the microscopic part and the zeroth order in the mesoscopic part is sufficient to capture both contributions to the leading order. Estimates obtained using a simplified, single-band model of wave functions 23 yield values of the microscopic part of the matrix element up to several meV or a few tens of meV and the resulting degree of mixing between singleexciton and biexciton states is on the order of 0.1, which suggests that this contribution is not negligible.…”