Double-Gated Nanohelix as a Novel Tunable Binary Superlattice

Abstract

We theoretically investigate the problem of an electron confined to a nanohelix between two parallel gates modelled as charged wires. The double-gated nanohelix system is a binary superlattice with properties highly sensitive to the gate voltages. In particular, the band structure exhibits energy band crossings for certain combinations of gate voltages, which could lead to quasi-relativistic Dirac-like phenomena. Our analysis for optical transitions induced by linearly and circularly polarized light suggests that a double-gated nanohelix can be used for versatile optoelectronic applications.

Keywords: Binary superlattice; Energy bands crossings; Nanohelix; Non-simply-connected nanostructures; Superlattice.

Fig. 1

Diagram of the system’s geometry and parameters from both face-on and lengthways perspectives. R is the helix radius, and d₁ and d₂ are the distances of the charged wires from the helix axis with charge densities λ₁ and λ₂, respectively. The spatial coordinate φ describes the angular position on the helix from face-on and is related to the z-coordinate via φ=2πz/p with p the helix’s pitch. A transverse electric field E_⊥ is applied parallel to the y-axis

Fig. 2

The four possible superlattice potential configurations with the unit cells highlighted in blue (defined in terms of the dimensionless parameters, see Eq. 3 for the corresponding requirements of the physical parameters, and all with B_g=0.2). a A unary superlattice with degenerate minima and maxima in the unit cell (A_g=C_⊥=0). b–d Binary superlattices formed from either b an asymmetric DQW with differing minima and internal reflection symmetry about either minima due to degenerate maxima (A_g=0.1, C_⊥=0), c a symmetric DQW with degenerate minima only (A_g=0, C_⊥=0.1), or d an asymmetric DQW with differing minima and maxima (A_g=C_⊥=0.1)

Fig. 3

Band structure for double-gated nanohelix system for various combinations of the dimensionless parameters (with B_g=0.4 fixed throughout): a Solid blue (dashed red) plots A_g=0 & C_⊥=0 (A_g=0.2 & C_⊥=0), the inset additionally plots the behaviour of the bottom two subbands subject to the transverse electric field with A_g=0 & C_⊥=0.2 as the dot-dashed green curve. b Solid blue (dashed red) plots A_g=0.63 & C_⊥=0 (A_g=0.8 & C_⊥=0) where the blue curve depicts the first incident of resonance (see text) with energy bands crossing at the centre of the Brillouin zone, the inset compares the behaviour of the bottom excited two subbands with the case where A_g=0.63 & C_⊥=0.2 as the dot-dashed green curve. c Solid blue (dashed red) plots A_g=1.26 & C_⊥=0 (A_g=1.5 & C_⊥=0) where the blue curve depicts the second incident of resonance with energy gaps closing at the edge of the Brillouin for higher bands. d The third resonance and higher subband minigaps close at the centre, with solid blue (dashed red) being A_g=1.9 & C_⊥=0 (A_g=2.2 & C_⊥=0). The unit cell shapes are sketched, n enumerates the bands, and the axis of the insets are the same as the main graphs

Fig. 4

a Band gap size between ground and first subbands as a function of A_g (C_⊥) plotted as dashed red (dot-dashed green), here B_g=0.25. The diagrams indicate the influence of the different perturbations on the isolated DQW unit cell and eigenstates. b–c Band gap size between first and second subbands indicated via a 2D density plot as a function of; b A_g and B_g for C_⊥=0, and c A_g and C_⊥ with fixed B_g=0.25. b Adjacent isoenergy contour lines indicate a difference of 0.17, with zero gap given by the dot-dashed red line for $A_g=\sqrt{B_g}$, while c the difference is 0.13 with zero gap at the centre of the smallest semi-circle contour (0.5,0). The diagrams sketch the isolated DQW and eigenstates. Hybridization does not occur between the s-like and p-like resonant localized individual well states in b, but does in c due to the electric field changing one barrier with respect to the other

Fig. 5

Square of the dimensionless momentum operator matrix element between the gth and fth subbands in the first Brillouin zone as a function of the dimensionless wave vector q of the electrons photoexcited by linearly z-polarized radiation and for a variety of parameter combinations spanning the first incident of resonance. The different blue curves keep A_g=0.5 and C_⊥=0 fixed and vary B_g=0.1, 0.2, and 0.3 corresponding to dot-dashed, dashed, and solid. The different red curves keep B_g=0.3 and C_⊥=0 fixed while varying A_g=0.05, 0.1 and 0.3 as dot-dashed, dashed, and solid, while the dotted blue (dotted red) plots the limiting case A_g=0.5 & B_g→0 (A_g→0 & B_g=0.3). a Transitions between the ground and first bands. The inset plots the behaviour for fixed A_g=0.5 and changing B_g crossing the resonant condition at B_g=0.25 (see text) in a reduced q-range, ranging from upper blue B_g=0.245, lower blue B_g=0.249, upper purple B_g=0.251, to lower purple B_g=0.255. The dashed green curves are for small non-zero transverse field C_⊥=0.05 ranging from B_g=0.245 (upper curve) to B_g=0.255 (lower curve) in increments of 0.05. b Plots transitions between the ground and second bands, the inset plots the behaviour close to resonance when A_g=0.5; blue is B_g=0.249, purple is B_g=0.251, and dark green is at resonance with C_⊥=0.05. c Plots transitions between the first and second bands, the parameters for the inset are the same as those in (b)

Fig. 6

Square of the dimensionless momentum operator matrix element between the gth and fth subbands in the first Brillouin zone as a function of the dimensionless wave vector q of the electrons photoexcited by right-handed circularly polarized radiation |T_x+iT_y|² and for a variety of parameter combinations spanning the first incident of resonance. a The blue curves denote transitions between the ground and first band while the red curves denote transitions between the ground and second band, both with the following parameters: A_g=0.5 and B_g=0.3 for solid curves, A_g=0.5 and B_g=0.1 for dashed curves, A_g=0.3 and B_g=0.3 for dot-dashed curves, and A_g=0.01 and B_g=0.3 for dotted curves (as A_g→0 the maximum of the 0→2 increases rapidly as it approaches q=− 1/2). The inset plots the behaviour as B_g is tuned through resonance for A_g=0.5; dotted is B_g=0.24, dot-dashed is B_g=0.25, and dashed is B_g=0.26. The solid purple (orange) curve denotes transitions between the ground and first (second) band at resonance with C_⊥=0.05 applied. b Plots transitions between the first and second bands. The different blue curves keep A_g=0.5 fixed and vary B_g=0, 0.2, and 0.3 corresponding to dotted, dot-dashed, and solid. The different red curves keep B_g=0.3 fixed while varying A_g=0.05, 0.1, and 0.3 as dotted, dot-dashed, and solid. We have omitted plots for C_⊥≠0 here as it yields no great qualitative change to the matrix elements