Spin-Orbit Effects on Exciton Complexes in Diamond

Ultrafine splittings are found in the optical absorption spectra of boron-doped diamond measured with high resolution. An analytical model of an exciton complex is developed, which permits assigning all absorption lines and sizing the interactions among the constituent charges and crystal field. We conclude that the entry of split-off holes in the acceptor-bound exciton fine structure yields two triplets separated by a spin-orbit splitting of 14.3 meV. Our findings thereby resolve a long-standing controversy [R. Sauer et al., Revised fine splitting of excitons in diamond, Phys. Rev. Lett. 84, 4172 (2000).; M. Cardona et al., Comment on “Revised fine splitting of excitons in diamond,”, Phys. Rev. Lett. 86, 3923 (2001).; R. Sauer and K. Thonke, Sauer and Thonke reply, Phys. Rev. Lett. 86, 3924 (2001).], revealing the underlying physics common in diverse semiconductors, including diamond.

Figure 1

(a) Four-body diagram of an acceptor-bound exciton, (b) conventional model describing an acceptor-bound exciton, considering only ${\mathrm{\Gamma }}_8$ acceptor holes, and (c) our model introducing the split-off holes of ${\mathrm{\Gamma }}_7$ symmetry.

Figure 2

(a) Schematic of absorption transitions from acceptor (${\mathrm{\Gamma }}_8$, ${\mathrm{\Gamma }}_7$) to acceptor-bound exciton states. (b) Two-hole states extended for split-off (${\mathrm{\Gamma }}_7$) holes, including (c) hole-hole exchange interaction and (d) crystal-field splitting. ${\mathrm{\Delta }}_{\mathrm{a}}$ and $\mathrm{\Delta }$ denote the spin-orbit splitting in the acceptor and bound exciton states, respectively. The split levels are shown not to scale and grouped in (d) by the triplet system, rather than energy ordering (numbers 1–6) in the measured spectra.

Figure 3

(a) Absorption spectra of bound excitons in boron-doped diamond between 6 and 160 K. The dashed lines represent transitions from the upper acceptor level. (b) Temperature dependence of the Lorentzian width (FWHM) of line number 6 [shaded in (a)]. (c) Temperature shift of line number 6. The solid lines in (b) and (c) represent fit functions (see text).

Figure 4

(a) Higher-resolution absorption spectrum at 2 K (thick black line) fitted with the sum (dotted red line) of 11 Voigt functions, numbers $1^{\prime }$–6 (thin lines). The labels near the respective peaks represent the symmetry of the two-hole states. The brackets and horizontal bars at the top indicate two sets of triplets and the approximate size of the splittings, respectively. (b) Comparison of the experimental and theoretical absorption strengths. Orange bars with dots indicate the areal intensities of numbers 1, 2, and 4 superposed on top of numbers $1^{\prime }$, $2^{\prime }$, and $4^{\prime }$, respectively. Thinner bars indicate intensities of primed peaks. The theory (gray bars) includes the electron-hole exchange effect for numbers $1^{\prime }-2$ and valley-orbit splitting for numbers $5^{\prime }$ and 5 (details in Sec. III of the Supplemental Material [37]).