Finite-temperature spectra and quasiparticle interference in Kondo lattices: From light electrons to coherent heavy quasiparticles

Recent advances in scanning tunneling spectroscopy performed on heavy-fermion metals provide a window onto local electronic properties of composite heavy-electron quasiparticles. Here we theoretically investigate the energy and temperature evolution of single-particle spectra and their quasiparticle interference caused by pointlike impurities in the framework of a periodic Anderson model. By numerically solving dynamical mean-field theory equations, we are able to access all temperatures and to capture the crossover from weakly interacting $c$ and $f$ electrons to fully coherent heavy quasiparticles. Remarkably, this crossover occurs in a dynamical fashion at an energy-dependent crossover temperature. We study in detail the associated Fermi-surface reconstruction and characterize the incoherent regime near the Kondo temperature. Finally, we link our results to current heavy-fermion experiments.

Figure 1

(a) Feynman diagrams that enter the single-impurity $T$ matrix used in our calculation. Bold lines are full (i.e., interacting) propagators ${\widehat{\mathcal{G}}}^0$, the cross denotes impurity scattering with strength ${\widehat{g}}_s$. (b) Sample diagrams contained in (a), where thin lines are now noninteracting propagators, and the wavy line represents the interaction $U$. (c) An interference process between ${\widehat{g}}_s$ and $U$ which is neglected in (a).

Figure 2

Schematic representation of the $T\to 0$ band structure expected for a PAM. $V_{\mathrm{eff}}$ and ${\tilde{\epsilon }}_f$ are the renormalized hybridization and $f$-chemical potential, respectively. ${\Delta }_g$ is the hybridization gap between the lower and upper renormalized bands.

Figure 3

Frequency dependence of the self-energies, $\mathrm{Im}{\Sigma }_f$ and $\mathrm{Im}{\Sigma }_c$, of the $f$ and $c$ electrons, respectively, calculated using DMFT-NRG for the periodic Anderson model with parameter $U=10,{\epsilon }_f=-0.9,{\epsilon }_c=0.4,V=0.55$ at a low temperature of $T=1.8\times {10}^{-6}$. Energies are given in units of the half bandwidth $W$.

Figure 4

(a) $c$ and (b) $f$-electron local spectral functions of the periodic Anderson model, calculated using DMFT-NRG at $T=1.8\times {10}^{-6}$. The different curves correspond to different values of the local interaction $U$; other parameters are as in Fig. 3. Because the chemical potential is kept fixed, the total and individual band fillings vary as follows: $n_{\mathrm{tot}}-n_c-n_f\approx 2.2-0.42-1.78$ ($U=0$), $1.98-0.78-1.2$ ($U=1.5$), $1.52-0.63-0.89$ ($U=3.5$), and $1.44-0.57-0.87$ ($U=10$).

Figure 5

Local spectral functions (a) ${\rho }_c\left(\omega \right)$ and (b) ${\rho }_f\left(\omega \right)$ of the PAM calculated using DMFT-NRG for different temperatures $T$, with model parameters as in Fig. 3. (a1),(b1) Spectra on the full energy range for two temperatures, $T\gg T_{\mathrm{K}}$ and $T\ll T_{\mathrm{K}}$. (a2),(b2) Temperature evolution of ${\rho }_c$ and ${\rho }_f$ shown in a restricted energy range near the Fermi level. (a3),(c3) Low-temperature, low-energy results for ${\rho }_c$ and ${\rho }_f$, with the high-temperature spectra at $T=4.7\times {10}^{-1}$ subtracted. The progressive formation of the hybridization pseudogap (a) and the Abrikosov-Suhl resonance (b) are clearly visible.

Figure 6

Evolution of the electronic spectrum of the PAM, with parameters as in Fig. 3, with temperature: Shown is $(-1/\pi )\mathrm{Im}({\mathcal{G}}_{ff}+{\mathcal{G}}_{cc})(\mathbf{k},\omega )$ as function of $k_x=k_y$ and $\omega$ over (a) the full range of energies and (b) a restricted energy window near the Fermi level. In (a) the dotted horizontal lines mark the renormalized $f$ level (i.e., the lower Hubbard band), while in (b) the dotted lines represent the Fermi level. Further the dashed line in (a1) shows the bare conduction band ${\epsilon }_{\mathbf{k}}$. At low temperatures, the formation of heavy bands is obvious, with the reconstruction happening mainly within the optical gap. In (a5) the dashed lines represent a fit of the intensity at elevated energies to the effective two-band picture using Eq. (22) with $V_{\mathrm{eff}}=0.17,{\tilde{\epsilon }}_f=0.025$, and ${\tilde{\epsilon }}_c=0.47$; for details, see text.

Figure 7

Constant-energy cuts through the electronic spectra of the PAM, with parameters as in Fig. 3. Panels (a) and (c) show $(-1/\pi )\mathrm{Im}{\mathcal{G}}_{cc}(\mathbf{k},\omega )$ in a quarter of the first Brillouin zone, panels (b) and (d) show $(-1/\pi )\mathrm{Im}{\mathcal{G}}_{ff}(\mathbf{k},\omega )$. The temperatures are (a),(b) $T=7.3\times {10}^{-3}$ and (c),(d) $T=1.8\times {10}^{-3}$. Note that the intensity color scale is logarithmic in all cases.

Figure 8

Evolution of the differential conductance $dI/dV$ for small bias voltage with the ratio $t_f/t_c$ for different temperatures $T\gg T_K$, $T\sim T_K$, and $T\ll T_K$. The effect of destructive (constructive) interference for positive [negative] ratios $t_f/t_c$ is visible at low temperatures in panels (b), (d), (f) [(c) and (e)], while the interference is essentially ineffective at high $T$.

Figure 9

(a) Intensity map of the Fourier transform of the spatially inhomogeneous part of the differential tunneling conductance, $|\delta \frac{dI}{dV}(\mathbf{q},\omega )|$, induced by a single pointlike weak scalar impurity (i.e., the QPI signal), along the $q_x=q_y$ direction of the first Brillouin zone. The peculiar behavior near $\omega /W=0.5$ is due to perfect nesting of ${\epsilon }_{\mathbf{k}}$; for details, see text. (b) Closeup near the Fermi level to be compared with the single-electron spectrum in Fig. 6b. The model parameters are as in Fig. 3, the ratio of the tunneling amplitudes is $t_f/t_c=0$.

Figure 10

Constant-energy cuts through (a) the QPI signal, (b) the $c$-electron spectrum, and (c) the $f$-electron spectrum, showing a quarter of the first Brillouin zone for different temperatures at an energy $\omega =0$. The ratio of the tunneling amplitudes is $t_f/t_c=0$. The model parameters are the same as in Fig. 3.

Figure 11

Same as in Fig. 10, but for an energy of $\omega =7.4\times {10}^{-3}$ inside the hybridization gap.

Figure 12

QPI signal at zero bias, $\omega =0$, and different temperatures, now showing the variation with the ratio of the tunneling amplitudes $t_f/t_c$. (a) $t_f/t_c=-0.3$, (b) $t_f/t_c=0.6$, (c) $t_f/t_c=1.2$. The model parameters are the same as in Fig. 3.