Electronic structure of $d$-wave superconducting quantum wires

We present analytical and numerical results for the electronic spectra of wires of a $d$-wave superconductor on a square lattice. The spectra of Andreev and other quasiparticle states, as well as the spatial and particle-hole structures of their wave functions, depend on interference effects caused by the presence of the surfaces and are qualitatively different for half-filled wires with even or odd number of chains. For half-filled wires with an odd number of chains $N$ at (110) orientation, spectra consist of $N$ doubly degenerate branches. By contrast, for even $N$ wires, these levels are split, and all quasiparticle states, even the ones lying above the maximal gap, have the characteristic properties of Andreev bound states. These Andreev states above the gap can be interpreted as a consequence of an infinite sequence of Andreev reflections experienced by quasiparticles along their trajectories bounded by the surfaces of the wire. Our microscopic results for the local density of states display atomic-scale Friedel oscillations due to the presence of the surfaces, which should be observable by scanning tunneling microscopy. For narrow wires the self-consistent treatment of the order parameter is found to play a crucial role. In particular, we find that for small wire widths the finite geometry may drive strong fluctuations or even stabilize exotic quasi-one-dimensional pair states with spin-triplet character.

Figure 1

(210) surface or wire. Two lines of impurities (closed squares) are needed to isolate the lattice sites (closed circles) with nearest-neighbor hopping.

Figure 2

Local density of states $\rho (x,\omega )$ for a (100) surface. Normal state (upper panel) and superconducting state (lower panel), $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$ for various distances $n=x∕d$ from surface; ${\Delta }_{\max }=0.4t,\mu =0$.

Figure 3

Local density of states $\rho (x,\omega )$ in superconducting state vs $\omega ∕{\Delta }_{\max }$ for the case of one (110) surface, ${\Delta }_0=0.2t,\mu =0$. Left: chains an odd distance $n=x∕d$ from surface at $x=0$. Right: chains an even distance from $x=0$.

Figure 4

Amplitude of Andreev state vs distance $n=x∕d$ from (110) surface for ${\Delta }_0=0.2t,\mu =0$.

Figure 5

Local density of states $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$ for a closed (210) nontransparent surface at half filling model and ${\Delta }_0=0.2t,\mu =0$. Each curve corresponds to a chain located at a distance $x∕d$ to the surface.

Figure 6

Local density of states $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$ for a (100) wire of width $N=4$ (left panels) and $N=5$ (right panels), using ${\Delta }_0=0.2t,\mu =0$.

Figure 7

Geometry of (110) wire (sites are filled circles) of width $N$ bounded by two impurity lines (filled squares). Wire is infinite in both $y$ and $-y$ directions.

Figure 8

Local density of states for a (110) wire of width $N=5$, with ${\Delta }_0=0.2t,\mu =0$. Left: normal state, $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$; right: superconducting state, $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$.

Figure 9

LDOS for a (110) wire of width $N=4$ with ${\Delta }_0=0.2t,\mu =0$. Left: normal state, $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$; right: superconducting state, $\rho (x,\omega )$ vs $\omega ∕{\Delta }_{\max }$.

Figure 10

Quasiparticle spectra and the LDOS for superconducting half-filled (110) wire with $N=11$. (a) Dispersive modes ($k_{y}d$ vs $\omega ∕t$) for $N=11,{\Delta }_0=0.2t$, and $\mu =0$. (b) The blowup of the dispersive spectra near the edge of the Brillouin zone. (c) The LDOS for energies less than ${\Delta }_{\max }:n=2$ (solid line) and $n=1$ (dashed line), where a broadening $\delta =0.005t$ has been used.

Figure 11

Quasiparticle spectra and the LDOS for superconducting half-filled (110) wire with $N=10$. (a) Dispersive modes ($k_{y}d$ vs $\omega ∕t$) for $N=10,{\Delta }_0=0.2t,\mu =0$. (b) The blowup of the spectra near the edge of the Brillouin zone. (c) The LDOS for energies less than or equal to ${\Delta }_{\max }:n=2$ (solid line), $n=1$ (dashed line), where a broadening $\delta =0.005t$ has been used.

Figure 12

The weight of the zero-energy peak in the LDOS, taken at the first layer, as a function on odd (open circles) and even (filled circles) $N$.

Figure 13

The lowest energy branches for $N=24$,25,26. The parameters are $t=2.5,\mu =0.2t=0.5,{\Delta }_0=0.2t=0.5$, and ${\Delta }_{\max }=2{\Delta }_0=1$.

Figure 14

Local density of states for a (110) wire with $N=10$ (left column) and $N=11$ (right column); $\mu =0$ (upper panels), $\mu =0.2t$ (lower panels). On each panel the different curves represent various chains.

Figure 15

Narrow wires with a nearest-neighbor interaction strength of $V=1.1575t$, which gives rise to a gap value of ${\Delta }_0=0.2t$ for the bulk system at $\mu =0$. Lower panels: phase diagram of the wires with even width (c) and with odd width (d). White space denotes $\Delta =0$, i.e., the normal state. Upper panels: corresponding amplitudes of the $d_{x^2-y^2}$ and $is$-order parameters displaying the value in the center of the wire for the $d_{x^2-y^2}$-wave case and the value at the edge of the wire in the $s$-wave case. These are the positions where the largest values of the respective order parameters are expected in the usual $d_{x^2-y^2}+is$ state.

Figure 16

Quasi-one-dimensional triplet superconducting state for $N=8$ and $\mu =0$. (a) Bond gap values ${\Delta }_{nn+1}^{\pm }$ [for definition see Eq. (58)] across the wire. (b) Illustration of the one-dimensional structure of the superconducting correlations for an $N=6\text{−}\left(110\right)$ wire at $\mu =0$. (c) Density of states starting from the outermost layer of the wire up to the middle layer of the wire, where for illustrational purposes each layer has been shifted by an additional offset of $0.3t$.

Figure 17

Evolution of the $d_{x^2-y^2}$-wave order parameter with increasing wire width for two different interaction strengths $V=1.1575t$ (upper panels) and $V=1.7682t$ (lower panels), which give rise to a gap value of ${\Delta }_0=0.2t$ and ${\Delta }_0=0.4t$ for the bulk system at $\mu =0$, respectively. The first column displays the amplitude of the $d_{x^2-y^2}$ order parameter as a function of the wire width for two different chemical potentials $\mu ∕t=0$ and $\mu ∕t=0.4$. The other two columns show the variation of the $d_{x^2-y^2}$- and $is$ component of the order parameter across the wire for $\mu ∕t=0$ (center column) and $\mu ∕t=0.4$ (right column). For comparison the reduced magnitude of the bulk order parameter at $\mu ∕t=0.4$ is displayed with dashed lines in the right panels.

Figure 18

Mapping of the $N=2$ wire with (110) orientation to the 1D chain with lattice constant $d$. Note that ${\Delta }_2=-{\Delta }_1$ corresponds to a triplet pairing state with $S_z=0$.

Figure 19

Phase diagram for 1D chain as a function of chemical potential $\mu ∕t$ and nearest-neighbor interaction $V∕t$.

Figure 20

Density of states for the $N=10$ wire (left panels) and the $N=11$ wire (right panels). Upper panels show the non-self-consistent results assuming a constant $d_{x^2-y^2}$ order parameter of ${\Delta }_0=0.2t$, whereas the lower panels display the density of states resulting from the self-consistently determined values of the $d$-wave order parameter, whose variation throughout the wire is depicted in the insets. All panels show the density of states starting from the outermost layer of the wire up to the middle layer of the wire, where for each layer towards the center of the wire an additional offset of $0.3t$ has been used.