Strictly one-dimensional behavior emerging from a dispersive two-dimensional system: Implications on metallic nanowires on semiconducting substrates

We studied single and two Su-Schrieffer-Heeger wires on a simple cubic semiconducting substrate. The wire-wire coupling is either perpendicular or diagonal hopping respecting the particle-hole and time-reversal symmetries. The hybridization to the substrate renormalizes the model parameters of the wires toward the hopping parameter of the substrate without changing the basic nature of perpendicular or diagonal coupling and it can mediate effective perpendicular hopping but not diagonal hopping in the absence of direct wire-wire coupling. This justifies the investigation of multi-uniform tight-binding wires with perpendicular or diagonal hopping parameters while neglecting the substrate. Perpendicularly coupled uniform wires reveal an anisotropic two-dimensional band dispersion. Diagonally coupled uniform wires reveal strictly one-dimensional bands parallel to the wires direction if the intrawire hopping parameter is larger than twice the diagonal hopping parameter despite strong dispersion perpendicular to the wires. Otherwise, they reveal strictly one-dimensional bands parallel and perpendicular to the wires direction simultaneously. We established the possibility of realizing strictly one-dimensional properties emerging from dispersive two-dimensional systems if time-reversal and particle-hole symmetries are respected. This can facilitate the investigations of Luttinger liquid in the Au/Ge(001) and Bi/InSb(001) surface reconstructions.

Figure 1

(a) Nearest-neighbor wires on substrate that can be perpendicularly coupled (b) Next-nearest-neighbor wires on substrate that can be diagonally coupled.

Figure 2

Single-particle spectral function for single and two SSH wires with $N_u=500$ and $\delta =-0.3$. (a) Single wire (b) two perpendicularly coupled wires with $t_{\perp }=1.5$. (c) Two diagonally coupled wires with $t_d=1.5$.

Figure 3

Topological phase diagrams of (a) two perpendicularly coupled SSH wires and (b) two diagonally coupled SSH wires.

Figure 4

Energy spectrum of single and two SSH wires with OBCs and $N_u=200$ (a) as a function of $\delta$ for a single wire, (b) as a function of $t_{\perp }$ for two perpendicularly coupled wires with $\delta =-0.3$, and (c) as a function of $t_d$ for two diagonally coupled wires with $\delta =-0.3$.

Figure 5

LDOS as defined in Eq. (34) at one edge of single and two SSH wires with $\delta =-0.3$ and $N_u=200$. (a) LDOS on sites from the same sublattice in first five unit cells at one edge of single SSH wire. (b) LDOS of first two unit cells of two perpendicularly coupled SSH wires with $t_{\perp }=1.5$. (c) LDOS of first two unit cells of two diagonally coupled SSH wires with $t_d=0.5$. Panels (b) and (c) share the same lines key.

Figure 6

Single-particle spectral function for single and two SSH wires on semiconducting substrate with $N_u=500$, $L_y=16$, $L_z=8$, $\delta =-0.3$, and $t_{\text{ws}}=4$. (a) Single wire. (b) Two perpendicularly coupled wires with $t_{\perp }=1.5$. (c) Two perpendicularly coupled wires with $t_{\perp }=0$. (d) Two diagonally coupled SSH wires with $t_d=1.5$. (e) Two diagonally coupled wires with $t_d=0$.

Figure 7

Topological phase diagrams of single and two SSH wires on semiconducting substrate. (a) Single wire. (b) Two perpendicularly coupled wires with $t_{\perp }=2.1$. (c) Two perpendicularly coupled wires with $t_{\perp }=0$. (d) Two diagonally coupled wires with $t_d=1.1$. (e) Two diagonally coupled wires with $t_d=0$.

Figure 8

Energy spectrum of single and two SSH wires on a semiconducting substrate with OBCs, $N_u=200$, $L_y=4$, and $L_z=4$ as a function of $t_{\text{ws}}$ (a) for a single wire with $\delta =-0.3$, (b) for two perpendicularly coupled wires with $t_{\perp }=0$ and $\delta =-0.3$, and (c) for two diagonally coupled wires with $t_d=1.5$ and $\delta =0.3$.

Figure 9

Local density of states as defined in Eq. (34) at one edge of single and two SSH wires on semiconducting substrate with OBCs, $N_u=200$, $L_y=4$, and $L_z=4$. (a) Single wire with $\delta =-0.3$ and $t_{\text{ws}}=4$. (b) Two perpendicularly coupled wires with $t_{\perp }=0$, $\delta =-0.3$, and $t_{\text{ws}}=2$. (c) Two diagonally coupled wires with $t_d=1.5$, $\delta =0.3$, and $t_{\text{ws}}=2$. Panels (b) and (c) share the same lines key.

Figure 10

(a), (b) Energy dispersions given by Eq. (43) of two-dimensional perpendicularly coupled uniform wires with $t_{\perp }=0.3$ and $t_{\perp }=0.7$, respectively. (c), (d) Energy dispersion given by Eq. (44) of two-dimensional diagonally coupled uniform wires with $t_d=0.3$ and $t_d=0.7$, respectively. The intrawire hopping is $t=1$.