Momentum-dependent hybridization gap and dispersive in-gap state of the Kondo semiconductor SmB${}_6$

We report the temperature-dependent three-dimensional angle-resolved photoemission spectra of the Kondo semiconductor SmB${}_6$. We found a difference in the temperature dependence of the peaks at the $X$ and $\Gamma$ points, due to hybridization between the Sm $5d$ conduction band and the nearly localized Sm $4f$ state. The peak intensity at the $X$ point has the same temperature dependence as the valence transition below 120 K, while that at the $\Gamma$ point is consistent with the magnetic excitation at $Q=(0.5,0.5,0.5)$ below 30 K. This suggests that the hybridization with the valence transition mainly occurs near the $X$ point, and the initial state of the magnetic excitation is located near the $\Gamma$ point.

Figure 1

(a) Low-energy electron diffraction (LEED) pattern of the SmB${}_6$ $\left(001\right)$ surface with an electron energy of 100 eV. Spots marked by circles represent diffractions of $1\times 1$; other spots represent $1\times 2$ and higher order diffractions. (b) The first Brillouin zone of SmB${}_6$ and high-symmetry points. (c) The photon energies corresponding to the $\Gamma$ and $X$ points in normal emission geometry and the $M$ point in grazing emission geometry. (d1)–(d3) ARPES intensity mapping images and calculated band dispersions (solid lines) on the $M$-$X$ ($h\nu =45.8$ eV), $X$-$\Gamma$ (70.0 eV), and $R$-$M$ (48.2 eV) lines at 10 K. The left-hand images in each panel are the ARPES intensity mappings, which emphasize flat bands, and the right-hand images are the same intensity mappings but divided by angle-integrated photoemission spectra, which emphasize highly dispersive bands.

Figure 2

Temperature dependence of energy distribution curves (EDCs) at the $X$ point [$h\nu =10.6$ eV (a1)] and the $\Gamma$ point [26 eV (a2)], and their spectra divided by the Fermi-Dirac distribution curve [(b1), (b2)]. EDCs were normalized at the binding energy $E_B$ of 120 meV. Successive curves in (b1) and (b2) are offset by 0.3 and 0.4, respectively, for clarity. The dotted lines in (b1) and (b2) are the spectra at 200 K. The temperature-dependent peak intensities relative to the intensity at 200 K of the 15-meV peak at the $X$ point (c1) and of the 20-meV peak at the $\Gamma$ point (c2). The mean valence number evaluated by the x-ray absorption spectroscopy (XAS) (Ref. 9) and the peak intensity of the inelastic neutron scattering (INS) at $Q=(0.5,0.5,0.5)$ (Ref. 11) are also plotted.

Figure 3

(a) Temperature dependence of energy distribution curves (EDCs) divided by the Fermi-Dirac distribution curve in the $X$ (0 Å${}^{-1}$)-$M$ (0.76 Å${}^{-1}$) line, using 10.6-eV photon energy. The hybridization band dispersions are shown by open circles, and the observed dispersive in-gap state is shown by vertical lines. Insets show the second derivative curves of smoothed EDCs at $k_{\left(100\right)}=0.25{\AA }^{-1}$. The marks correspond to the band dispersions. Temperature dependence of the hybridization band in the $X$-$M$ line [$h\nu =10.6$ eV (b)] and the $\Gamma$-$X$ line [$h\nu =26.0$ eV (c)]. The band dispersion derived from the LDA calculation [Fig. 1(d1)] with the energy scale in the right axis is also plotted by a solid line in (b).