Ultraviolet probing of quantum crossbars

Ultraviolet scattering on quantum crossbars (QCB’s) is an effective tool for probing QCB spectral properties, leading to excitation of QCB plasmon(s). Experimentally, such a process corresponds to sharp peaks in the frequency dependence of the differential scattering cross section. The peak frequency strongly depends on the direction of the scattered light. As a result, the one-dimensional to two-dimensional crossover can be observed in the scattering spectrum. It manifests itself as a splitting of single lines into multiplets (mostly doublets). The splitting magnitude increases with interaction in QCB crosses, while the peak amplitudes decrease with electron-electron interaction within a QCB constituent.

Figure 1

The scattering process geometry (all notations are explained in the text).

Figure 2

Part of the reciprocal space. The small square $Q_j∕Q_0⩽0.5$ is the quarter of the first QCB BZ. High-symmetry lines are parallel to the coordinate axes. Resonant lines are parallel to BZ diagonals. The arcs coming over the point $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, correspond to different wave numbers of the excited plasmons: $\mid Q∕Q_0\mid =0.3$; 0.5; 0.6; 0.7; 0. 8.

Figure 3

(a) Second-order diagram describing light scattering on QCB. Solid lines correspond to fermions, whereas dashed lines are related to photons. The vertices are described by the interaction Hamiltonian (7). (b) Effective photon-plasmon interaction. The QCB excitation is denoted by wavy line.

Figure 4

Positions of the scattering peaks for $\mid K∕Q_0\mid =0.3$. The doublet in the resonance direction (point $B_1$ in Fig. 2) is well pronounced.

Figure 5

Positions of the scattering peaks for $\mid K∕Q_0\mid =0.5$. Two doublets appear at the BZ boundaries (points $C_1$ and $D_1$ in Fig. 2).

Figure 6

Positions of the scattering peaks for $\mid K∕Q_0\mid =0.6$. Two doublets at the BZ boundaries (points $C_2$ and $D_2$) in Fig. 2 are shifted from the high-symmetry directions.

Figure 7

Positions of the scattering peaks for $\mid K∕Q_0\mid =\sqrt{2}∕2$. Resonance triplet corresponding to point $B_4$ in Fig. 2 (two of four frequencies remain degenerate in our approximation). In ir experiments only one of the triplet components is visible.

Figure 8

Positions of the scattering peaks for $\mid K∕Q_0\mid =0.8$. Two pairs of doublets appear corresponding to excitation of two pairs of plasmons in the two arrays (points $E$ and $F$ in Fig. 2).

Figure 9

QCB inverse space. The first BZ is bounded by a square box.

Figure 10

Wave vectors involved in the resonance interaction. Resonant lines are displayed by solid lines.

Figure 11

The low-energy part of the band structure of a metallic nanotube. Lines 1, 3 (2, 4) correspond to excitations with $p=+1$ $(p=-1)$. Lines 1, 2 (3, 4) correspond to excitations with orbital quantum number $m=0$ $(m=\pm 1)$.