Combined tight-binding and configuration interaction study of unfolded electronic structure of G-color center in Si

TL;DR

This work tackles the challenge of enabling efficient light emission from silicon by introducing the G-color center in the emissive B-type configuration. It develops a combined theoretical framework using empirical tight-binding, band unfolding, and configuration interaction to describe the electronic structure and exciton states at the $\Gamma$ point. The results reveal a direct $\Gamma$-point transition in silicon with an energy near $E_g\approx970$ meV, achievable through specific defect-induced modifications and band-offset tuning, along with a very small exciton fine-structure splitting that supports potential entangled-photon emission. The approach highlights the ability to attribute spectral features to particular defect components and demonstrates a practical pathway to engineer silicon-based quantum light sources for communication and computation applications.

Abstract

We have theoretically studied the G-center in bulk silicon material using the empirical tight-binding model for calculations of unfolded band structures with configuration interaction correction for the exciton at $Γ$ point of the Brillouin zone. The G-center in B configuration (emissive) being a candidate structure as the telecom single- and entangled-photon source has two substitutional carbons and one interstitial atom embedded into the bulk in six equally possible configurations. Taking the advantage of the low computation effort of the tight-binding and unfolding approach, it is possible to calculate and analyze the behavior of a variety of the electronic configurations. Our tight-binding model is able to describe not only the behavior of the G-center in the silicon bulk but using the unfolding approach it can also pinpoint the contributions of different elements of the supercell on the final pseudo-band structure. Moreover, the configuration interaction correction with single-particle basis states computed by our unfolded tight-binding model predicts a very small fine-structure splitting of the ground state exciton both for bright and dark doublet in the studied system. That underscores the possibility of the silicon G-center to become a very good emitter of single and entangled photons for quantum communication and computation applications.

Figures (9)

• Figure 1: The structure of the supercell (SC) with the G-center. (a) Six positions of the silicon interstitial atom Si$_\textrm{i}$ (dark blue and violet) with $i$=0-5 and two carbon substitutional atoms C (orange) aligned along the $\langle 111 \rangle$ crystal direction in the Si bulk (light blue). (b) The SC with the G-center and depicted periodic boundary conditions by the black parallelogram. The structure of $5\times 5\times5$ primitive cells (PCs) contains 251 atoms (248 Si, 2 C$_\textrm{sub}$, 1 Si$_\textrm{int}$). (c) Two bonds to nearest neighbors (NNs) from Si$_\textrm{i=0}$ (blue) to C (orange) are depicted by red dashed lines.
• Figure 2: The Hamiltonian for the supercell with the G-center containing 55 atoms (52 Si, 2 C$_\textrm{sub}$, 1 Si$_\textrm{int}$). Red (AC) and yellow (CA) 20x20 off-site ETB matrices represent those where VCA was used and black (AC) and gray (CA) where VCA was not employed. Note that some on-site ETB matrices are also modified (not shown in this figure) due to VCA (those which have at least one NN changed by VCA). On the main diagonal, there are 20x20 on-site ETB matrices for anions (A) and cations (C) where different colors represent different types of atoms; light-blue (Si bulk), orange (C$_\textrm{sub}$) and dark-blue (Si$_\textrm{int}$).
• Figure 3: The determination of the weight $i_{\alpha}$ of the bond (Eq. \ref{['VCA_off']}) for an example between silicon anion and carbon cation with their NNs. Notice that final bond is composed of two off-site ETB parameters with weights $i_\textrm{Si}=0.5625$ and $i_\textrm{C}=0.4375$.
• Figure 4: (a) Folded band structure of the silicon SC composed from $3\times 3\times3$ PCs with $|\mathbf{k}|=|\mathbf{K}+\mathbf{G}_{m}|$. There are only three reciprocal vectors $\mathbf{G}_{m}$ (out of total number of 27 reciprocal vectors) pointing in $\mathbf{X}$ direction ($\mathbf{G}_{0}=[0,0,0],~\mathbf{G}_{1}=[-0.66,0,0],~\mathbf{G}_{2}=[0.66,0,0]$). The weights $W_p^m(\mathbf{K})$ are calculated for each $m$-th reciprocal vector and for all eigenstates $p$ of the SC. The black broken vertical lines in (a) and (b) represent the absolute values of the three reciprocal vectors and red lines represent the edge of the FBZ of the SC. (b) Unfolded band structure, plotted for states with $W_p^m(\mathbf{K})>0.1$ only.
• Figure 5: Energy band gap of the G-center (for i=0 and $N_{SC}=5$) with a dependence on the band offset of the substitutional atom $\Delta {E}^{\rm offs.}_\textrm{sub}$ and the bond length ${d}_{\textrm{s.-i.}}$ where the color bar marks a difference of reciprocal wavevector during the transition (dark balls represent direct transition). Red plane indicates the energy of the G-center of $970$ meV. Panel (a) gives the case without and (b) with adding the band offset of interstitial atom $\Delta {E}^{\rm offs.}_\textrm{int}=0.63$ eV which lowers the maximum of band gap of G-center transition energy.