Doping-induced band shifts and Lifshitz transitions in heavy-fermion metals

For some heavy-fermion compounds, it has been suggested that a Fermi-surface-changing Lifshitz transition, which can be driven, e.g., by varying an applied magnetic field, occurs inside the heavy-fermion regime. Here we discuss, based on microscopic calculations, how the location of such a transition can be influenced by carrier doping. While doping rigidly shifts the bands in a free-electron system, strong electronic correlations drastically modify the picture. We therefore systematically study the doping-induced energetic shift of band features in heavy Fermi liquids, with the surprising result that the actual shift is determined by the interplay of heavy and additional light bands crossing the Fermi level: doped carriers tend to populate heavy and light bands equally, despite the fact that the latter contribute a small density of states of excitations only. This invalidates naive estimates of the band shift based on the low-temperature specific heat of the heavy Fermi liquid. We discuss applications of our results.

Figure 1

The bare band dispersion ${\epsilon }_{\mathbf{k}}$ as defined in Eq. (7) in the first Brillouin zone. Notice the weakly dispersing piece near $\mathbf{k}=(\pi ,0)$ and $(0,\pi )$.

Figure 2

(a,b) Zero-temperature heavy-fermion band structure with a shallow pocket, as obtained within the slave-boson mean-field approach: (a) shows the full dispersion of the two hybridized bands along a path in the 2D Brillouin zone for different values of the chemical potential $\mu$, (b) is a zoom near the Fermi level. The depth of the shallow Fermi pocket is $E_m\equiv E_{\mathbf{k}=(2.75,0)}^{+}$ whose doping evolution will be discussed in the paper. The height of the pocket defines a second scale $E_M\equiv E_{\mathbf{k}=(\pi ,0)}^{+}$; the vHs at $(\pi ,0)$, $(0,\pi )$ produces a peak in the DOS. (c, d) Corresponding total DOS (per spin) of quasiparticles, ${\rho }_{\mathrm{tot}}={\rho }_c+{\rho }_{\tilde{f}}$. The bare band dispersion is as specified in Eq. (7); furthermore, ${\epsilon }_f/W=-9.65$, $\mu /W=0.575$ (respectively, $\mu /W=0.499$ for the dashed dispersion and $\mu /W=0.651$ for the dot-dashed one), and $V/W=1.5$, which results in a filling of $n_{\mathrm{tot}}=2.33$ (respectively, $n_{\mathrm{tot}}=2.23$ and $n_{\mathrm{tot}}=2.45$) and a quasiparticle renormalization of $1/Z\simeq 99$ (respectively, $1/Z\simeq 86$ and $1/Z\simeq 120$). Momentum integrals were performed with ${1600}^2$ $\mathbf{k}$ points, and a Lorentzian broadening of ${10}^{-5}W$ was employed for the DOS—this smears the logarithmic divergence of the DOS at the vHs.

Figure 3

Evolution of the mass renormalization $Z^{-1}=m^{*}/m$ with the hybridization $V$, as obtained from the slave-boson approximation. Here, ${\epsilon }_f/W=-9.65$ is kept fixed and $\mu$ is adjusted to ${\mu }_0\left(V\right)$ such that $|E_m|=|E_M|$; see text for details. $Z\left(V\right)$ is dominated by the exponential dependence[3] of the single-impurity Kondo temperature $T_{\mathrm{K}}^{\left(1\right)}$ on $V$.

Figure 4

Slave-boson results for (a) the band fillings $\langle n_c\rangle$ and $\langle n_f\rangle$, (b) the energy scale $E_m$ measuring the pocket depth, and (c) the total quasiparticle DOS at the Fermi level normalized by the quasiparticle weight, ${\rho }_{\mathrm{tot}}\left(0\right)Z$, as function of the chemical potential, defined via $\mu ={\mu }_0\left(V\right)+\Delta \mu$, where $|E_m|=|E_M|$ for $\Delta \mu =0$. Here, ${\epsilon }_f/W=-9.65$, $V/W=1.5$ are kept fixed, and ${\mu }_0\left(V\right)/W\simeq 0.569$. The interval $-0.33<\Delta \mu /W<-0.26$, where $\langle n_c\rangle$, $\langle n_f\rangle$ are constant, corresponds to the Kondo insulator. Comparing panel (c) with the energy-dependent DOS at fixed $\mu ={\mu }_0\left(V\right)$ shown in Fig. 2d, illustrates again that a rigid-band picture is by no means appropriate.

Figure 5

(a) Dimensionless parameters $P$ and $Q$, defined in Eq. (8), as a function of the hybridization $V$, with ${\epsilon }_f/W=-9.65$ and $\mu ={\mu }_0\left(V\right)$ as before. $P$ and $Q$ characterize the changes of $E_m$ and $\langle n\rangle$ with $\mu$, respectively, and appear strongly suppressed with respect to the noninteracting reference value 1. (b) Rescaled parameters $P/Z$ and $Q/Z$, which are now of order unity.

Figure 6

Evolution of the ratio $P/Q$, defined in Eq. (9), with the addition of an increasing number of light bands according to Eq. (10). ${\rho }_l$ scales with the number of light bands crossing the Fermi level. The heavy-band parameters are given in the legend of Fig. 5, with $V/W=1.5$, and ${\rho }_c$ refers to the $c$-electron part of the DOS per spin ${\rho }_{\mathrm{tot},h}$ of the two-band system described by the PAM with the same heavy-fermion band parameters.