Emerging coherence with unified energy, temperature, and lifetime scale in heavy fermion YbRh${}_2$Si${}_2$

We present ultrahigh resolution angle-resolved photoemission data of the prototypical Kondo lattice system YbRh${}_2$Si${}_2$, using a 7 eV laser photon source. Detailed temperature dependent measurements reveal the development of coherent states, in the form of sharp, weakly dispersing peaks at the lowest energy of single-electron spectra, below a characteristic temperature. The characteristic temperature, obtained from angle-resolved photoemission, is intriguingly of exactly the same scale as the energy and the lifetime of the coherent state.

Figure 1

Angle-resolved photoemission spectra of YbRh${}_2$Si${}_2$ measured with a 7 eV laser. (a) ARPES data were taken along the $\Gamma$-X direction of the projected body centered tetragonal Brillouin zone at $T=12$ K, with a geometry shown in the schematic diagram. Circularly polarized light was used and the cut direction is depicted as a strip. Bands are labeled as 1–4 in (a). Energy distribution curves from the same data set as in (a) are shown in (b). The red (dark gray) dashed lines are added to guide the eyes to bands 1–3. The red (dark gray) dots follow band 4. The blue (gray) dashed line shows the position of the binding energy of the flat part in band 4.

Figure 2

Energy vs momentum dispersion from second derivatives of the intensity map and EDC fits. (a) The second derivative of ARPES data shown in Fig. 1 to enhance the visibility of weak features away from the $\Gamma$ point, particularly for $k>0.05$ Å${}^{-1}$. The white dashed box indicates the blown-up region shown in (c), where the dispersion relation extracted from EDC fits of Fig. 1b, as exemplified in (b), is presented. The fits are made with two Lorentzian peaks added by an exponential background to account for the tail from the high binding energy peak, which is then multiplied by the Fermi function and convoluted by a Gaussian to include the experimental resolution. Samples of the fit are shown in (b) for $k=-0.10$, $-0.05$, and 0.02 Å${}^{-1}$. In (c), blue (gray) and green (light gray) shades are added as guides to the eyes for the flat and dispersing features, respectively. Blue (gray) lines show the best fit of the dispersion relation using the slave boson treatment of the periodic Anderson model (Ref. 16). (d) A schematic diagram explaining the energy-momentum dispersion relation of the $f$-$d$ hybridized state (red curves) close to $E_{\mathrm{F}}$ under a periodic Anderson model (see, for example, Refs. 4 and 15), with a schematic representation of the spectral weight of states of $f$ character. The thicker part of the curves implies a higher intensity in the spectral function. $E_f^{\prime }$ denotes the energy level of the renormalized $f$ state and $\epsilon {\left(k\right)}_d$ (black dashed line) is the bare dispersion of the original $d$ band. Note that the position of the $E_f^{\prime }$ lies below $E_{\mathrm{F}}$ following the electron-hole symmetry of Ce and Yb systems.

Figure 3

Temp-erature dependence of the ARPES spectra along the $\Gamma$-X direction. (a)–(d) Intensity plots at $T=12$, 30, 45, and 110 K. The color scale is the same as in Fig. 1e, 1f, 1g, 1h EDCs corresponding to (a)–(d). The temperature was cycled from low to high and back to low temperatures to verify that changes in the spectra are not due to a trivial surface degradation.

Figure 4

Detailed analysis on temperature dependent ARPES spectra. (a) EDCs at the same $k$ point, $k\sim 0.03$ Å${}^{-1}$, the onset of the flat part of the dispersion in Fig. 2, at different temperatures. (b) The spectra in (a) divided by a Fermi-Dirac function convolved with a Gaussian of width comparable to the experimental resolution to remove the thermal broadening contribution in the spectral change. The panel shows a wide range data set. The inset focuses on the most interesting temperature and energy ranges. (c) The relative quasiparticle spectral weight calculated by integrating the spectral weight only for the energy window for the quasiparticle peak ($-15$ meV and above) and divided by the weight from the high temperature value of the $T=110$ K data. The measurement was repeated with three different samples of YbRh${}_2$Si${}_2$ with slightly different temperatures, and the results are highly reproducible as shown in the figure. $T_{\mathrm{ARPES}}^{*}$ is indicated as a red shade.