Bridging $k$ and $q$ space in the cuprates: Comparing angle-resolved photoemission and STM results

A critical comparison is made between the angle-resolved-photoemission-spectroscopy- (ARPES-) derived spectral function and scanning tunneling microscopy (STM) studies of Friedel-like oscillations in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }\left(\mathrm{Bi}2212\right)$. The data can be made approximately consistent, provided that (a) the elastic scattering seen in ARPES is predominantly small-angle scattering and (b) the “peak” feature seen in ARPES is really a dispersive “bright spot,” smeared into a line by limited energy resolution; these are the bright spots which control the quasiparticle interferences. However, there is no indication of bilayer splitting in the STM data.

Figure 1

Experimental ARPES spectral weight at the Fermi level for the sample in the normal state at $100\mathrm{K}$. Lines show a tight-binding fit.

Figure 2

$q$ maps constructed directly from convolution of measured ARPES spectral functions for a series of frequencies $\omega$ from 2 to $12\mathrm{meV}$.

Figure 3

Model ARPES spectral functions, at $\omega ∕\Delta =0.2$ (a), 0.5 (b), 0.8 (c), and 1.0 (d).

Figure 4

$q$ maps for $\omega =0.5\Delta$, ${\Sigma }_1=0.003\mathrm{eV}$: $-{\overline{\chi }}_2^{\prime }$ (a), $-{\overline{\chi }}^{\prime }$ (b), $-{\overline{\chi }}^{\prime }+{\overline{\chi }}^{″}$ (c), and $-{\overline{\chi }}^{\prime }-{\overline{\chi }}^{″}$ (d).

Figure 5

Series of calculated STM $q$ maps corresponding to pair breaking model of Fig. 3, for $\omega ∕\Delta =$ (a) 0.2, (b) 0.5, (c) 0.8, and (d) 1.0. Circles (squares) $=$ calculated positions of convolution intensity corresponding to the most intense (second most intense) peaks in ARPES intensity. Note that in this and subsequent $q$ maps, the origin has been shifted with respect to Figs. 2, 6.

Figure 6

Calculated ARPES (a) and STM $q$ maps (b), assuming limited energy resolution $\omega =0–0.5\Delta$.

Figure 7

Pairbreaking parameter $u$ vs $\omega ∕{\Delta }_0$ for ${\zeta }_2=0.1$ (a), 0.3 (b), 0.6 (c), 0.9 (d). Solid lines $=$ real(u), dashed lines $=\mathrm{imag}\left(\mathrm{u}\right)$, dotted lines $=\omega ∕{\Delta }_0$.

Figure 8

Model ARPES spectral functions, at $\omega ∕{\Delta }_0=0.5$, for ${\Sigma }_2^{el}=0$ (a), 1 (b), 2 (c), 4 (d), 8 (e), and $16\mathrm{meV}$ (f) $[{\Sigma }_1^{el}=30\mathrm{meV}]$.

Figure 9

Model ARPES spectral functions, at $\omega ∕{\Delta }_0=0.2$ (a), 0.5 (b), 0.8 (c), and 1.0 (d) $[{\Sigma }_I^{el}=0]$.

Figure 10

Spectral weight $A$ along contours $\omega =E_{\vec{k}}$ (inset) for $\omega ∕{\Delta }_0=0.5$.

Figure 11

Velocities along constant-energy contours: (a) $v_F$ along the nodal direction $k_x=k_y$, (b) $v_{\Delta }$ at the bright spots (dashed and dotted lines refer to two bilayer-split bands), and (c) ratio $v_F∕v_{\Delta }$. Calculations assume ${\Delta }_0=15\mathrm{meV}$, ${\Sigma }_I^{el}=0$.