Fermi surface of a system with strong valence fluctuations: Evidence for a noninteger count of valence electrons in ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$

We present de Haas-van Alphen (dHvA) measurements on an Eu-based valence-fluctuating system. ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$ exhibits a temperature-dependent, noninteger europium valence with ${\mathrm{Eu}}^{2.8+}$ at low temperatures. The comparison of experimental results from our magnetic-torque experiments in fields up to 32 T and density functional theory band-structure calculations with localized $4f$ electrons shows that the best agreement is reached for a Fermi surface based on a valence of ${\mathrm{Eu}}^{2.8+}$. The calculated quantum-oscillation frequencies for ${\mathrm{Eu}}^{3+}$ instead cannot explain all the experimentally observed frequencies. The effective masses, derived from the temperature dependence of the dHvA oscillation amplitudes, show not only a significant enhancement with masses up to $11m_e$ ($m_e$ being the free electron mass), but also a magnetic-field dependence of the heaviest mass. We attribute the formation of these heavy masses to strong correlation effects resulting from valence fluctuations of $4f$ electrons being strongly hybridized with conduction electrons. The increase of the heavy masses with magnetic field likely results from the onset of the expected field-induced valence crossover that enhances these valence fluctuations but does not alter the Fermi-surface topology in the field range studied.

Figure 1

Phase diagram of $\mathrm{Eu}{\left({\mathrm{Rh}}_{1-x}{\mathrm{Ir}}_x\right)}_2{\mathrm{Si}}_2$ showing the Néel temperatures $T_N\left(x\right)$, the valence transition temperatures $T_{\nu }\left(x\right)$, as well as the position of the valence crossover [maximum in $\rho \left(T\right)]$ observed as a function of the Ir content $x$: Red circles from resistivity, blue triangles from susceptibility. Adapted from Ref. [24]. We add a sketch of the general phase diagram of Eu systems: Dashed line: Second-order paramagnetic-to-AFM phase boundary, solid line: First-order valence transition. The latter ends in a classical critical end point. (Inset) Eu valence in ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$ as a function of temperature. Data from [25, 26, 27].

Figure 2

Spectral distribution of the dHvA frequencies at ${\mathrm{\Theta }}_{110}={84}^{\circ }$, i.e., the magnetic field is rotated by ${84}^{\circ }$ from [001] to [110] of the tetragonal crystal structure, obtained by FFT based on the oscillating part of the signal shown in inset (a); black (red, blue, pink) curves measured at 22 (400, 650, 1000) mK, respectively. [Inset (b)] Effective mass determination for $F_7$ by applying the Lifshitz-Kosevich fit function (red line) to the measured FFT amplitudes.

Figure 3

(a) Angular dependence of the dHvA frequencies for ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$. Black symbols represent experimental data whereas green and red lines are derived from the extremal FS cross sections in the result of an open-core calculation with ${\mathrm{Eu}}^{2.8+}$. Green lines are based on the extremal cross-sections of the FS from band 48 shown in (b), and red lines are based on the FS from band 49 in (c). (d) Same experimental data as in (a) in comparison to calculated frequencies based on trivalent europium in ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$. Here color coding corresponds to the depiction of the calculated FSs from band 48 in (e) and band 49 in (f).

Figure 4

Field dependence of the effective mass of frequency $F_8$ at ${\mathrm{\Theta }}_{110}={84}^{\circ }$. (Inset) Effective masses of $F_6$ and $F_7$ at the same angle.