Unusual scarcity in the optical absorption of metallic quantum-dot nanorings described by the extended Hubbard model

We report results of a systematic analysis of optical absorption in finite metallic quantum-dot nanorings containing a variable number of electrons described by the extended Hubbard model. Despite the very strong electron correlations, the number of significant spectral lines is astonishingly small, and the optical spectra can still be rationalized within a simple framework relying upon the single-particle picture. The main effect of correlations is to split the optical transitions that are degenerate within the single-particle description and to give rise to numerous avoided crossings (anticrossings). Unusually, the latter often involve more than two states. For closed-shell systems, the optical spectrum is practically monochromatic. We also present results of an approximate scheme, based on the Landau idea to describe low excitations in an interacting electron system. It consists of diagonalizing the Hamiltonian in the configuration interaction truncated to include dressed particle-hole excitations derived from the exact ground state. This method provides a good description of the overall optical absorption, including that of the diabatic state corresponding to the bright transition at avoided crossings. The scarcity in the optical spectra reported here points toward a hidden dynamical quasisymmetry in the optical absorption of finite nanorings described within the extended Hubbard model, generalizing thereby our recent finding [I. Bâldea and L. S. Cederbaum, Phys. Rev. B 75, 125323 (2007)]. In addition, we report a qualitative difference between the optical gap and the charge gap occurring at quarter filling.

Figure 1

(Color online) Curve in the ($U/t_0$, $V/t_0$) plane accessible by varying the interdot spacing in the range $1<D/\left(2R\right)<2$ in assemblies of QDs of silver, showing that broad parameter ranges can be explored. Notice the logarithmic scale on both axes.

Figure 2

(Color online) Momentum distribution in the ground state of six (circles) and ten (triangles) half-filled QDs for several values of interdot spacing $d$ given in the legend. The values of $d$ that increase downward for $k/\pi <0.5$ are given in the legend. The lines are guides for the eyes (Ref. 31).

Figure 3

(Color online) Absorption frequency and intensity of the relevant spectral line for six-QD nanorings at half-filling. For small $d$ the exact results (represented by points) are well approximated by those of the perturbation theory (p.t.) (see Sec. 4), whereas for large $d$ the absorption frequency approaches the asymptotical limit $U\text{−}V$ (see main text). Concerning the curves denoted by $hl\text{−}ph$ and $ph$, see Sec. 7.

Figure 4

(Color online) Absorption frequency and intensity of the relevant spectral line for ten-QD nanorings at half-filling. The meaning of the various curves is indicated in the legends and explained in the caption of Fig. 3.

Figure 5

(Color online) [(a) and (c)] Energies and (b) spectral intensities of optical absorption for six electrons on ten QDs in the ground $\left({{}^{1}A}_{1g}\right)$ state. Only three excited $E_{1u}$ states are important, which are shown by the solid, dashed, and dashed-dotted lines. Note that except around $d\approx 1.87$, where an avoided crossing involving all these three states occurs, practically only one optical transition possesses a significant intensity. Because of the small energy splitting at the avoided crossing, the measured spectrum is practically monochromatic and has a smoothly varying intensity in the whole $d$ range, which is well approximated by the results of the $ph$ approximation (see Sec. 7). In (c), the $ph$ curve for energy has been shifted downward by 0.7 meV. For small $d$, the exact results are well approximated by those of the perturbation theory (p.t.) (see Sec. 4).

Figure 6

(Color online) Results for optical absorption of five electrons on six QDs. The superscripts indicate the energy ordering of the excited states with significant spectral intensity. The solid lines represent the exact results. In (a)–(d), the points represent the results of perturbation theory (p.t.), while in (e)–(h), they are the results of the ph method (see Sec. 7). In (g), the diabatic $ph$ curve approximating the exact energies of $E_{2g}^5$ and $E_{2g}^6$ in the regions where these two are bright (for $d\lesssim 1.56$ and $d\gtrsim 1.56$, respectively) has been shifted downward by 5.5 meV.

Figure 7

(Color online) Energies and spectral intensities of optical absorption for five electrons on ten QDs in the ground $\left(E_{1u}\right)$ state. The solid lines represent the exact results. The superscripts indicate the energy ordering of the excited states with a significant spectral intensity. In (a) and (b), the points are the results of the p.t. approximation (see Sec. 4). In (c), (d), and (e), the points represent the results of the $ph$ approximation (see Sec. 7).

Figure 8

(Color online) Exact results for [(a) and (b)] energies and (c) spectral intensities of optical absorption for seven electrons on ten QDs in the ground $\left(E_{2g}\right)$ state. The superscripts indicate the energy ordering of the excited states with a significant spectral intensity.

Figure 9

(Color online) Results for optical absorption of nine electrons on ten QDs obtained exactly (solid lines) and by the $ph$ method (see Sec. 7). The latter are depicted by points [for clarity, in (e) and (f), they are joined by the thin lines]. The superscript indicates the energy ordering of the exact excited states with a significant spectral intensity.

Figure 10

(Color online) Exact results for energies and spectral intensities of optical absorption for four (upper panels) and eight electrons (lower panels) on ten QDs in the triplet ground $\left({{}^{3}A}_{3g}\right)$ state. The superscripts indicate the energy ordering of the excited states with a significant spectral intensity.

Figure 11

(Color online) (a) Optical gap $\epsilon$ for various numbers of electrons $\left(N_e\right)$ and QDs $\left(N\right)$. These numbers are indicated in the legend along with the gap value for free electrons. In addition, because the lowest optically active excitation changes the symmetry for seven electrons on ten QDs, its symmetry is also given in parentheses. For three electrons on ten QDs, the ground state switches beyond $d=1.47$ from doublet $\left(S=1/2\right)$ to quartet $\left(S=3/2\right)$, and the free gap of the latter case is also indicated in parentheses. (b) Optical gap ${\epsilon }_{N_e,N}$ and charge gap ${\Delta }_{N_e,N}$ for several values of $N_e$ and $N$. The charge gap for five electrons on ten QDs does not tend to zero as for other cases away from half-filling for physical reasons explained in the main text.