Extinction of quasiparticle interference in underdoped cuprates with coexisting order

Scanning tunneling spectroscopy (STS) measurements [Y. Kohsaka et al., Nature (London) 454, 1072 (2008)] have shown that dispersing quasiparticle interference (QPI) peaks in Fourier-transformed conductance maps disappear as the bias voltage exceeds a certain threshold corresponding to the coincidence of the contour of constant quasiparticle energy with the period-doubled (e.g., antiferromagnetic) zone boundary. Here we show that this may be caused by coexisting order present in the $d$-wave superconducting phase. We show explicitly how QPI peaks are extinguished in the situation with coexisting long-range spin-density wave order and discuss the connection with the more realistic case where short-range order is created by quenched disorder. Since it is the localized QPI peaks rather than the underlying antinodal states themselves which are destroyed at a critical bias, our proposal resolves a conflict between STS and photoemission spectroscopy regarding the nature of these states. We also study the momentum-summed density of states in the coexisting phase and show how the competing order produces a kink inside the “V”-shaped $d$-wave superconducting gap in agreement with recent STS measurements [J. W. Alldredge et al., Nat. Phys. 4, 319 (2008)].

Figure 1

(Color online) Contours of constant quasiparticle energy for the pure $d$-wave superconductor with $\Delta =0.3t$ at energies $\omega /\Delta =0.17,0.33,0.50,0.66,0.83,0.90$. Also shown are the underlying normal-state Fermi surface, the RBZ boundary, and the seven distinct nonzero wave vectors ${\mathbf{q}}_i$ connecting the banana tips.

Figure 2

(Color online) QPI maps vs $q_x/\pi$ and $q_y/\pi$ in the dSC phase in the presence of single pointlike potential scatterer of strength $V=0.1t$. Energies $\omega$ are as shown in the figure titles. Dispersing spotlike features corresponding to some of the octet vectors ${\mathbf{q}}_i$ are easy to identify. For example, we have circled the intrabanana ${\mathbf{q}}_7$ peak (see Fig. 1).

Figure 3

QPI line cuts along the [(a) and (b)] (110) and [(c) and (d)] (100) directions for the pure dSC phase. Panels (a) and (c) [(b) and (d)] correspond to positive (negative) energies at $\left|\omega \right|/\Delta =0.08,0.17,0.25,0.33,0.5,0.66,0.83,0.9$ (bottom to top). For clarity the (110) [(100)] curves are displaced by 0.075 (0.05).

Figure 4

(Color online) Spectral weight $A\left(\mathbf{k},\omega =0\right)$ in the pure SDW phase with (a) $M=0.5t$ and (b) $M=t$. Lower two panels (c) and (d) show CCEs at the same energies as in Fig. 1 for (c) $M=0.5t$ and (d) $M=t$.

Figure 5

(Color online) Same as Fig. 2 but for the pure SDW phase with $M=0.5t$. The results are shown for the same energies as in Fig. 2 (with $\Delta =0.3t$).

Figure 6

(Color online) Contours of constant energy for the coexisting phase with $\Delta =0.3t$ and $M=0.5t$ plotted at the same energies $\omega$ as in Fig. 1. The contours reach the RBZ boundary when $\omega ={\Delta }_0=0.61\Delta$ for $M=0.5t$.

Figure 7

(Color online) QPI maps vs $q_x/\pi$ and $q_y/\pi$ in the coexisting $\text{SDW}+\text{dSC}$ phase ($\Delta =0.3t$ and $M=0.5t$) in the presence of single pointlike nonmagnetic impurity. This figure can be directly compared to Fig. 2. Note the disappearance of the spotlike feature corresponding to octet vector ${\mathbf{q}}_7$ above the critical energy ${\Delta }_0=0.61\Delta$.

Figure 8

(Color online) Density of states $N\left(\omega \right)$ in the coexisting $\text{SDW}+\text{dSC}$ phase with (a) $M=0.4t$, (b) $M=0.8t$, (c) $M=t$, and (d) $M$ averaged. In all panels the dashed (black) line shows the DOS in the pure dSC phase whereas the solid (blue) line displays the DOS in the $\text{SDW}+\text{dSC}$ phase. In (d) the dash-dotted (red) line shows the $M$-averaged result for a case with a linear inelastic-scattering rate $\alpha \left|\omega \right|$ ($\alpha =0.15t$, c.f. Ref. 67) used instead of $\eta$ in Eq. (14), resulting in a smoother DOS kink.

Figure 9

(Color online) CCEs for the coexisting phase at (a) $\omega /\Delta =0.17$ and (b) $\omega /\Delta =0.66$ at $M=0$ (red), $M=0.75t$ (blue), and $M=t$ (black).

Figure 10

(Color online) QPI maps at $\omega /\Delta =0.17$ (left column) and $\omega /\Delta =0.66$ (right column) for the dSC phase (top), coexisting phase with $M=0.5t$ (middle), and $M$-averaged case (bottom). The middle and bottom panels are all plotted on the same false color scale.

Figure 11

(Color online) QPI line cuts at the same energies as in Figs. 3 along the [(a) and (b)] (110) and [(c) and (d)] (100) directions for the coexisting $\text{SDW}+\text{dSC}$ phase with $\Delta =0.3t$ and $M$ averaged over a flat distribution with $M∊\left[0,t\right]$. The line cuts at energies above the crossover $\omega ={\Delta }_0$ are indicated by red dashed lines. The curves are displaced similarly to Fig. 3.