Orbitally defined field-induced electronic state in a Kondo lattice

TL;DR

This study reveals a field-induced orbital change in CeRhIn5 that causes a Fermi-surface reconstruction and enhances electron hybridization, leading to a quantum-critical point at high magnetic fields.

Contribution

It demonstrates that the emergent state at B* is due to a change in Ce's 4f orbitals driven by crystal-electric field evolution, revealed through NMR measurements.

Findings

Discontinuous decrease in $^{115}$In Knight shift at B*

Emergent state results from orbital change in Ce's 4f electrons

Quantum-critical point at approximately 50 T

Abstract

CeRhIn$_{5}$ is a Kondo-lattice prototype in which a magnetic field B$\bf{^{\ast}\simeq}$ 30 T induces an abrupt Fermi-surface (FS) reconstruction and pronounced in-plane electrical transport anisotropy all within its antiferromagnetic state. Though the antiferromagnetic order at zero field is well-understood, the origin of an emergent state at B$^{\ast}$ remains unknown due to challenges inherent to probing states microscopically at high fields. Here, we report low-temperature Nuclear Magnetic Resonance (NMR) measurements revealing a discontinuous decrease in the $^{115}$In formal Knight shift, without changes in crystal or magnetic structures, of CeRhIn$_{5}$ at fields spanning B$^{\ast}$. We show that the emergent state above B$^{\ast}$ results from a change in Ce's 4f orbitals that arises from field-induced evolution of crystal-electric field (CEF) energy levels. This change in orbital character enhances hybridisation between the 4f and the conduction electrons (c.e.) that leads ultimately to an itinerant quantum-critical point at B$\bf{_{c0} \simeq}$ 50 T.

Equations

$$B_{local}(1)\simeq \left[1+K_{c.e.}^{\Vert c}(1)\right]B+B_{int}^{\Vert c}(1)$$

$$B_{local}(1)=\left[1+K^{\prime }(1)\right]B,$$

$$B_{local}(2)=\left[1+K^{\prime }(2)\right]B+B_{int,0}^{\Vert c}(2)$$

Full text

Development of the peculiar electronic state above B*∗*** in CeRhIn5 is signaled clearly in quantum oscillations,Jiao et al. (2015) magnetoresistance,Moll et al. (2015); Jiao et al. (2015) magnetostrictionRosa et al. (2019) but not in specific heat.Jiao et al. (2019) The lack of a detectable specific heat anomaly suggests that B∗ may not reflect a well-defined phase transition but a crossover from one state to anotherRosa et al. (2019) where not only the Fermi surface reconstructs from small-to-largeJiao et al. (2015) but also in-plane anisotropy develops in electrical resistivity.Ronning et al. (2017) Qualitatively, these responses could be consistent with a field-induced change in crystal and or magnetic structure from below to above B∗ – a distinctly plausible interpretation that could be tested straightforwardly by a diffraction measurement if B∗ were sufficiently low to be accessible in neutron or x-ray experiments. Even if such measurements could be made at fields to 30 T and higher, experiments point to a more complex picture, with similarities to other correlated electron systems. Electrical resistivity studies reveal a hysteretic transition at B∗ that was interpreted intially to reflect the formation of a density wave, analogous to that found in correlated copper-oxide materials.Moll et al. (2015) More recent studies are even more surprising:Ronning et al. (2017) when an applied field is tipped about 200 from the tetragonal c-axis toward in-plane perpendicular directions, there is a striking inequivalence of electrical resistivity for current flow along each pair of orthogonal crystallographic directions. This unexpected in-plane symmetry breaking is consistent with a proposed strong $XY$ nematic susceptibility that is similar to but distinct from Ising-nematicity that is found in high-$T_{c}$ copper oxide,Ando et al. (2002); Kivelson et al. (1998) iron-pnictideFernandes et al. (2013); Chuang et al. (2010) and correlated ruthenate materials.Borzi et al. (2007)

Evidence for all the changes in electronic properties at B∗ and their weak coupling to the crystal latticeRonning et al. (2017); Rosa et al. (2019) appears only within the magnetically ordered state of CeRhIn5. In this limit, partially Kondo-compensated Ce moments order below $T_{\rm N}$ = 3.8 K in a spin-spiral structure with ordering wave-vector $\bm{Q}=(0.5,0.5,0.297)$ and moments in the tetragonal plane.Fobes et al. (2018) This structure, however, is unstable against modest applied pressureAso et al. (2009) or in-plane applied magnetic field.Takeuchi et al. (2001); Raymond et al. (2007); Fobes et al. (2018) The near degeneracy of accessible orders in CeRhIn5 supports the possibility that a field of 30 T could change the nature of magnetism at B∗, but with little change in entropy or susceptibility. What might underlie the emergence of the new electronic state above B∗ and a change in magnetic character, if this indeed happens, are fundamental questions raised by recent discoveries in CeRhIn5 and are relevant more broadly to the physics of a Kondo lattice.

With its sensitivity to local spin, charge and lattice degrees of freedom,Bauer et al. (2011); Seo et al. (2014) NMR is a powerful tool to probe the evolution of complex electronic states in correlated electron materials at very high magnetic fields.Wu et al. (2011); Sakai et al. (2014); Berthier et al. (2017) Figure 1 presents the 115In NMR spectra ($I$ = 9/2) from two inequivalent sites of our CeRhIn5 single crystal with $B$ applied along the $c$-axis at 0.5 K below $T_{\text{N}}(B)$. Each Ce atom is surrounded by four tetragonally coordinated In(1) and eight In(2) atoms with local orthorhombic symmetry (Fig. 1d and Supplemental Material). At low fields (Fig. 1a), there are 9 equally-separated transitions associated with In(1) NMR. In contrast, the lower relative intensities of the In(2) NMR signal are a consequence of spectral broadening due to a distribution of internal fields arising from an oscillating hyperfine (internal) field $B_{int}^{\|c}(2)$ associated with $c$-axis incommensuration of the spin-spiral magnetic structureCurro et al. (2000); Curro (2006) (see Figure 1d and Supplemental Material).

At low-fields and well below $T_{\rm N}$, a hyperfine field of $B_{int}^{\bot c}(1)=0.17$ T lies in the Ce-In(1) ($ab$-) plane and rotates between the adjacent layers with the incommensurate pitch of the magnetic structure shown in Fig. 1d.Curro et al. (2000); Curro (2006) At higher fields with $B$ applied along the $c$-axis, $B_{int}^{\bot c}(1)$ can be neglected ($B\gg B_{int}^{\bot c}(1)$). The magnetic field along the $c$-axis induces a canting of the Ce local momentTakeuchi et al. (2001) (Figure 1d) leading to extra internal fields $B_{int}^{\|c}(1)$ and $B_{int}^{\|c}(2)$ at both In(1) and In(2) sites, respectively. Therefore the local field at In(1) can be modeled as:

[TABLE]

For the Kondo-lattice CeRhIn5, the first term in Eq. (1) is associated with a contribution from itinerant quasiparticles and the second term with the internal field at In(1) due to the out-of-plane Ce-moment component. This internal field component $B_{int}^{\|c}(1)=A_{ord}^{\|c}\mu_{Ce}\cos(\beta/2)$, where $A_{ord}^{\|c}$ is the diagonal $c$-component of the hyperfine coupling tensor from the ordered local moments and $\beta$ is the angle of the conical spin structure (see Fig. 1d). This term is proportional to $B$ due to the Zeeman interaction. Therefore, the local internal field $B_{local}(1)$ at In(1) sites is:

[TABLE]

with $K^{\prime}(1)$ defining the formal In(1) Knight shift that bears contributions from both local and itinerant spin susceptibilities. In the case of In(2), the hyperfine field resulting from the in-plane ordered Ce moments follows the oscillatory non-commensurate character of the magnetic structure, $B_{int,0}^{\|c}(2)=B_{i0}$ cos$(2\pi Q_{z}z)$ (Fig 1d). The out-of-plane contribution of the Ce moments for the hyperfine field at the In(2) site lies in the $c$-directionKambe et al. (2007); Ohama et al. (2005); Tokunaga et al. (2011) and is also proportional to the external field due to the Zeeman interaction. Therefore the local field at an In(2) site can be defined in terms of a formal Knight shift, $K^{\prime}(2)$:

[TABLE]

As indicated by solid vertical (gray) arrows in Figs. 1a and 1b, below $B^{\ast}$ $\sim$ 30.8 T the position of In(1) transitions can be calculated (see Supplemental Material) assuming a formal Knight shift $K_{\rm{S}}^{\prime}(1)=$ 7.4(1)% and quadrupolar frequency $\nu_{Q}=$ 6.77(1) MHz.

The subscript $\rm{S}$ stands for the magnetic phase AFS, $B\leq B^{\ast}$, with a small FS and, as introduced later, $\rm{L}$ for AFL, $B>B^{\ast}$, with a large FS. The formal Knight shift bears contributions from both local and itinerant spin susceptibilities. The value of $K_{\rm{S}}^{\prime}(1)$ is consistent with the paramagnetic valueShirer et al. (2012) of $K_{\rm{PM}}^{\|c}(1)\simeq 8.0\%$. The spectrum from In(2) in the AFM phase can be calculated similarly by assuming a periodically oscillating internal field $B_{int,0}^{\|c}(2)=0.27$ T along the $c$-axis,Curro (2006) with nearly the same low-field quadrupolar parametersCurro et al. (2000); Kohori et al. (2000) and a formal Knight shift $K_{\rm{S}}^{\prime}(2)=1.5(1)$%. Taking these parameters into account, we calculate the 115In NMR spectrum that is given by red and green colours for contributions from In(1) and In(2), respectively. The gray solid curve is the simulated (convoluted) overall 115In NMR spectrum from both In signals.

The simulated spectra in Fig. 1b are made on the basis of low-field NMR parametersCurro et al. (2000); Kohori et al. (2000) that account well for spectra in Fig. 1a and agree with experiment for fields up to 30.8 T where some deviation from simulation and experimental results begins just where the new AFL phase sets in. However, above $B^{\ast}\simeq 30.8$ T, the spectra are well simulated by keeping all low-field nuclear quadrupolar parameters but with an abrupt decrease of both In(1) and In(2) formal shifts from $K_{\rm{S}}^{\prime}$(1) = 7.4(1)% to $K_{\rm{L}}^{\prime}$(1) = 5.1(1)% and $K_{\rm{S}}^{\prime}(2)=1.5(1)$% to $K_{\rm{L}}^{\prime}(2)=1.1(2)$%, respectively, indicating absence of a detectable local structural distortion at $B^{\ast}$. The simulation remains comparably good at fields well above $B^{\ast}$ (Fig. 1c). The larger $\Delta K^{\prime}(1)$ compared to $\Delta K^{\prime}(2)$ is consistent with the larger hyperfine coupling constant of In(1),Lin et al. (2015) but the relative decrease of $K^{\prime}$(1) and $K^{\prime}$(2) is similar, implying a decrease in bulk magnetizationCurro (2016) in the high-field state that is reflected in part by a decrease in the slope of the $c$-axis magnetization around $B^{\ast}$.Takeuchi et al. (2001) Opening a density-wave gap in the reconstructed large FS is consistent with the decrease in formal shifts if $K_{c.e.}$, which is proportional to the susceptibility of itinerant quasiparticles, dominates $K_{\rm{L}}^{\prime}$. This is a scenario proposed previously,Jiao et al. (2015); Moll et al. (2015) but, as we have concluded, the nesting wave vector that opens a gap must be similar to the zero-field $\bm{Q}$. A related scenario is that the decrease in formal Knight shifts is due to a decrease in internal field $B_{int}^{\|c}(1)=A_{ord}^{\|c}\mu_{Ce}\cos(\beta/2)$ that arises from a reduction of the ordered moment, $\mu_{Ce}$, and/or a decrease of the hyperfine coupling constant, $A_{ord}^{\|c}$. Both of these depend on the extent to which Ce’s 4$f$ electrons hybridise with band electronsCurro (2016) and, in the limit of stronger hybridisation, would reflect additional $f$ spectral weight being transferred to band states,Chen et al. (2018) with a corresponding increase of the FS. Because a magnetic field tends to weaken Kondo hybridisation as it polarizes spins of both conduction and localized electrons, this scenario superficially seems unlikely but as discussed below is supported by simplified model calculations.

From the high-field data and spectra simulation, we can conclude that the magnetic structure does not change qualitatively through $B^{\ast}$. One possibility is that the magnetic structure adopts the commensurate order with $\bm{Q}=(0.5,0.5,0.25)$ observed for CeRhIn5 when $B^{\bot{c}}\gtrsim 2$ TRaymond et al. (2007); Fobes et al. (2018) that is not so different from the low-field incommensurate $\bm{Q}=(0.5,0.5,0.297)$. For a commensurate $\bm{Q}$, the internal field $B_{int,0}^{\|c}(2)$ at In(2) will take only distinct values, but an incommensurate $\bm{Q}$ creates an oscillating $B_{int,0}^{\|c}(2)$ that produces a characteristic ”double horn” spectral distribution pattern. Such a distribution is, indeed, revealed by the NMR data and simulation presented in Figs. 1b and 1c. We conclude that the magnetic structure of CeRhIn5 remains incommensurate with a similar, if not identical, $\bm{Q}=(0.5,0.5,0.297)$ above $B^{\ast}$.

At high fields, the In spectrum, acquired in a hybrid 45 T magnet, broadens as shown in Figs. 1b and 1c. This broadening is more evident for the equidistant In(1) transitions where the linewidth increases from $\Delta L\simeq$ 0.020(5) T in the low-field limit to $\Delta L\simeq$ 0.10(1) T in the high-field limit. We consider possibilities for this broadening. Though not dramatically, the linewidth increases with increasing fields from 26 to 42 T, which likely is due to the crystal experiencing a slight field gradient in the hybrid magnet. From the magnet’s known (in)homogeneity, we estimate that the linewidth would increase by at most 9 % in this field range. Field-induced electronic anisotropy from the proposed $XY$ nematicityRonning et al. (2017) in principle should contribute to NMR line-broadening. Such a nematic electronic texture would induce anisotropy in the in-plane hyperfine field component at the In(1) site (Fig. 1d), resulting in line-broadening or even splitting each In(1) transition, and by breaking local tetragonal symmetry of the In(1) site, would produce non-equidistant In(1) transitions due to a modified electric field gradient (EFG). Within the accuracy of our measurements, however, the separation between In(1) transitions remains constant for fields spanning $B^{\ast}$, and there is no clear evidence for splitting of In(1) transitions. Though the pronounced in-plane symmetry breaking of magnetotransport appears at $B^{\ast}$, weak magnetoresistive anisotropy begins to developRonning et al. (2017) already near 17 T where specific heat and de Haas-van Alphen measurements with field along the $c$-axis also find the onset of enhanced Sommerfeld coefficient and quasiparticle mass.Jiao et al. (2019) Whether these effects are precursors to proposed nematicity above $B^{\ast}$ is unknown but, whatever their origin, conceivably could manifest in larger linewidths shown in Figs.1b and 1c. Nevertheless, In(1) lineshapes remain symmetric and do not broaden noticeably as field is swept through $B^{\ast}$. The absence of a change in crystal and magnetic structures as a function of field and particularly the abrupt decrease in formal Knight shift at $B^{\ast}$ (Fig. 2) are primary conclusions that come directly from our NMR measurements.

The ground states of CeRhIn5 and its isostructural family members, CeCoIn5 and CeIrIn5, depend on the orbital character of their 4$f$ wavefunctions that determines the extent of $f$ hybridization with In electronic states.Willers et al. (2015) In a tetragonal environment, the CEF splits the $J$ = 5/2 manifold of CeRhIn5’s 4$f^{1}$ state into three doublets whose energy separation and wave-functions (see Supplemental Material) have been determined by linear-polarised soft-X-ray absorption and inelastic neutron scattering experiments in zero magnetic field.Christianson et al. (2002); Willers et al. (2010) Fields of order $B^{\ast}\simeq 30$ T ($\Delta_{\rm{CEF}}\simeq 7$ meV $\simeq$ 81 K) are sufficient to induce mixing of the wave-functions of the $\Gamma_{7}^{2}$ doublet ground state with the first excited doublet state $\Gamma_{7}^{1}$. This level mixing manifests as a bending of the field-dependent CEF energy levels (see Supplemental Material).

We now consider the consequences of magnetic degrees of freedom. Although a general solution of a theory of a strongly interacting Kondo lattice like CeRhIn5 has not been solved, we incorporate the magnetic Rudderman-Kittel-Kasuya-Yosida (RKKY) interaction into the above electronic framework. This magnetic interaction is represented by an effective spin-spin interaction term, $j_{k}\mathbf{J_{m}}\cdot\mathbf{J_{n}}$. Specifically, we consider a simplified mean-field model with intra- and inter-layer nearest-neighbor (nn) exchange couplings ($j_{x}=j_{y}\equiv j_{0}$ and $j_{z}\equiv j_{1}$, Fig. 1d) to play the role of an effective RKKY interaction combined with the appropriate CEF hamiltonian term (see Supplemental Material).

Our model does not explicitly include the Kondo interaction but considers it to renormalise the bare spin-spin exchange, so that $j_{0}$ and $j_{1}$ are effective exchange coupling constants. With this simple mean-field model we calculate the specific heat thermal dependence (see Supplemental Material) constraining the value of calculated constants to give the zero-field Neel temperature $T_{\rm{N}}$ = 3.8 K and keeping the experimentally determined ratio $j_{0}/j_{1}\simeq 8$.Das et al. (2014) For $B$ = 0, we find effective $j_{0}$ = 0.72 K and $j_{1}$ = 0.088 K, which are an order of magnitude smaller than those derived from a model that gives the zero-field magnetic structure.Das et al. (2014) This is consistent to the fact that thermal fluctuations tend to suppress the mean field ordering temperature for a quasi 2D system like CeRhIn5 ($j_{0}/j_{1}\simeq 8$).

Following the same approach, we estimate the field dependence of $j_{0}$ and $j_{1}$, shown in the inset to Fig. 2, that is required to reproduce the $T_{\text{N}}(B)$ phase boundary. As seen, $j_{0}$ and $j_{1}$ decrease linearly up to 30 T before increasing above $B^{\ast}$. From the Shrieffer-Wolff transformation, the Kondo exchange is proportional to the square of the $f-c.e.$ hybridisation matrix element.Schrieffer and Wolff (1966) A reasonable interpretation of the increase in exchange constants above $B^{\ast}$, then, is that this reflects an enhanced hybrisiation in the high-field state due to field-induced change in the orbital character of the $4f$ wave function. Obviously, a more realistic theoretical framework that explicitly takes into account the Kondo interaction as well as a frustrating inter-layer next nn exchange and orbital degrees of freedom is desirable to substantiate our interpretation.

Our NMR measurements and model calculations thus provide a microscopic basis for the origin of the unusual electronic state that emerges at high fields in the Kondo-lattice CeRhIn5: field-driven mixing of the orbital character of the 4$f$ wave function enhances Kondo hybridisation that induces a large FS above $B^{\ast}$ = 30 T where it experiences a density-wave instability due to nesting at a $\bm{Q}$ close to, if not the same as, that characterising magnetic order in the zero-field antiferromagnetic state. There is no detectable change in local structure at fields to 42 T. Except for the field scale $B^{\ast}$, which is specific to the Kondo interaction and crystal-field wave functions of CeRhIn5, similar high-field states should be generic to Kondo-lattice materials. With the essential role of the orbital nature of wave functions and its consequences for Kondo coupling, $B^{\ast}$ could be considered in the zero-temperature limit to reflect an orbitally selective type of Kondo-breakdown quantum-critical pointPépin (2007); Si et al. (2001) within the ordered state. This is an interpretation suggested initially by Jiao et al.Jiao et al. (2015) and now we provide a microscopic rationale for it.