Pressure-induced Frustration of Magnetic Coupling in Elemental Europium

Pressure-induced Frustration of Magnetic Coupling in Elemental Europium

Shu-Ting Pi Department of Physics, University of California Davis, Davis, California 95616 USA    Sergey Y. Savrasov Department of Physics, University of California Davis, Davis, California 95616 USA    Warren E. Pickett Department of Physics, University of California Davis, Davis, California 95616 USA
July 14, 2019

While elemental lanthanides are known to show strong variations, and often large increases, in their magnetic ordering temperatures under pressure, Eu is unique in that magnetic order vanishes at a critical point (QCP) =82 GPa, giving way to superconductivity up to 3K. We use density functional theory in combination with linear response and the magnetic force theorem to obtain the exchange constants between antiferromagnetic (AFM) sublattices. We confirm that Eu retains divalency with strong () local moments despite structural, Hubbard U, or pressure variations. In the high pressure phase the three sublattice exchange constants decrease strongly with pressure and vanish near , reflecting frustration among interatomic exchange constants. Consideration of the free energy indicates that a combination of magnetic entropy and magnetic energy drives the first order transition into the paramagnetic phase, resulting in a superconducting ground state in the midst of dense disordered but correlated large local moments.

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The behavior of local moments and their ordering as volume varies lies at the root of several paradigmatic phenomena, including the Kondo effect and heavy fermion superconductivity. The shell in lanthanides () has provided a unique platform for the study of this issue, with Ce and Yb showing strong -conduction electron interaction at ambient pressure while reduction in volume is needed to drive other lanthanides into exotic phases. The theory of magnetic coupling in and actinide materials has been advancing from several viewpoints,Pi2014 (); Pi2014b (); Locht2016 (); Tapia2017 () but puzzling behavior remains to be understood. Experimentally, a study by Jackson et al.Jackson2005 () of six metals up to 5-12 GPa indicated a linear decrease in the magnetic ordering temperature T roughly in proportion to the de Gennes factor of the ion. However, higher pressures bring more complex behavior and complications due to structural transitions and band structure changes.

While both Eu and Yb are known to display deviations from trends in behavior, elemental Eu displays distinctive behavior that becomes a challenge to theory. It has been demonstrated that T in varies strongly and often non-monotonically with pressure: for Tb T, varies non-monotonically between 50K and room temperature (RT) in the 15-140 GPa range;Lim2015b () Dy has T above RT at 157 GPa;Lim2015 (); T for Nd increases to 180K in the 40-90 GPa rangeSong2017 () before falling back to 50K. In Eu, however, after non-monotonic behavior in T due to structural transitions,Bi2016 () in the structure that exists in a range around 80 GPa, T falls to 11K at 82 GPa whereupon superconductivity (SC) up to 3K emerges.Debessai2009 ()

Advances in modeling exchange coupling in metalPi2014 (); Pi2014b (); Locht2016 (); Tapia2017 () have dealt with ordering. This apparent first order disappearance of order represents an avoidance of the antiferromagnetic (AFM) quantum critical point that is actively studied in weak AFMs.Knebel2006 () This paramagnetic (PM) phase, with its superconducting (SC) ground state in the midst of disordered spin, may provide a platform for learning more about magnetic interactions, and perhaps more general issues about neighboring phases near a QCP, possibly including a spin liquid or spin gas phase coexisting with SC.

This behavior can be compared with that of Yb. Under pressure, Yb undergoes a valence transition,Syassen1982 (); Fuse2004 (); Dallera2006 (); Ylvisaker2009 () from divalent to somewhere near trivalent through a continuous evolution through intermediate valence and emergence of a local moment, a crossover that has been simulated successfully by dynamical mean field calculationsYlvisaker2009 () up to 40 GPa. Recently Song and Schilling have reportedSong2018 () that Yb, with its local moments, becomes superconducting in the 1.4-4.5 K range at 80 GPa and above. This behavior has parallels with, but distinctions to, that of Eu, to which we return to in the discussion.

The PM phase is unusual having classical spins on an ordered lattice interacting via Heisenberg exchange (the spins are isotropic) yet do not order, a signature of type of frustration that is not apparent and moreover, it only disorders above a critical pressure . Beyond the question of (dis)ordering, there is the perplexing issue of superconductivity in a metal with disordered strong local moments. The simplest scenario is that Eu has been driven through a valence transition to the non-magnetic =0 configuration, in which case there is no magnetic impediment to superconducting pairing. The isovalent rare earth metal Y becomes an impressive superconductor under pressure, with T up to 17 K.Hamlin2006 () We find that Eu, unlike Yb, in not near a change in valence up to 100 GPa or more.

Historically, a valence transition in Eu has been anticipated well below 80 GPa, and some interpretations of xray and spectroscopic data have been interpreted to indicate that the formal valence is reduced significantly from =3 for pressures in the few tens of GPa range.Rohler () However, the most recent x-ray absorption data indicates that Eu retains its () moment even in the SC phase above 82 GPa.Bi2012 (); Song2018 () If that is the case, the SC phase appears to be of an exotic type in which pairing occurs within a dense array of large but disordered moments. These questions have led us to perform systematic calculations of the electronic structure and magnetic coupling of Eu at pressures up to the 100 GPa range.

Figure 1: (color online) The ratio as a function of lattice constant (a) bcc (b) hcp (c) . The lattice structures were obtained from experiments. Red dash lines indicate the ratio of pressures where the structures are confirmed stable in experiments.

Like all metals, Eu displays structural transformations with increasing pressure. The bcc phase extends from 0-12 GPa. The common bcc-to-hcp transition for metals occurs at 12 GPa, with single phase hcp in the 12-18 GPa range. The 18-66 GPa regime is more complex, with mixed-phases reported.Husband2012 () Above 66 GPa, a single phase regime is reached, which covers the pressure range of interest here. We will simplify and consider the structures as three stages: bcc, 0-12 GPa; hcp, 12-40 GPa; , 40 GPa. In Table I the experimental lattice structures at specific pressures are listed. The volume decrease ratio ( is the ambient volume) regions of stability of the three phases are shown with the structures in Fig. 1.

Although not mapped in detail, the evidence is that Eu displays antiferromagnetic (AFM) order from ambient to GPa. At , magnetic order vanishes and superconductivity emerges with critical temperature T=1.7K, increasing with pressure up to 2.8K at 142 GPa.Debessai2009 () As recently reportedBi2016 () and as we confirm from calculation, the large moment on Eu persists (the moment in Gd is calculatedYin2006 () to persist to 500 GPa), making the interplay between large but disordered moments and SC, and its dependence on pressure, outstanding issues.

P = 4 GPa 14 GPa 75 GPa
Lattice a=4.1961 a=3.3501 a=4.977
constants b=3.3501 b=4.264
() c=5.2962 c=2.944
x 0 1/3 0.325
y 0 2/3 1/4
z 0 1/4 0.029
Table 1: Crystal structures of Eu metal in various pressures. are the Wyckoff positions, and there are 1, 2 and 4 atoms per primitive cell, respectively. Only in the bcc case is the unit cell doubled.

Our density functional theory (DFT) calculations employ the full potential linearized-muffin-tin-orbital method (LMTO).Savrasov1992 () The local spin density approximation (LSDA) with Hubbard U correction (LSDA+U) on the localized shell orbitals of Eu. A reasonable value is = 6-7 eV at ambient pressure; at high pressure we investigate smaller values of . Note that we use LSDA+U rather than LDA+U because the spin-density mediated intra-atomic coupling that polarizes the conduction electrons is important to include and assess.

Rather than attempt to calculate conduction electron mediated RKKY coupling out to several neighbors, we focus on the AFM sublattice exchange constants, which are linear combinations of interatomic exchange constants out to arbitrary distance. An effective and efficient method is to use linear response theory and the magnetic force theorem.AIL () Consider the electronic Kohn-Sham Hamiltonian , where is kinetic energy, is the spin-independent potential, and is the spin-dependent (effective Zeeman) potential including the contribution from . We write


where is the vector of Pauli matrices. B appears as an effective magnetic field arising from the spin-dependent exchange-correlation potential. If one rotates the moment on AFM sublattices , in unit cells R, R by infinitesimal angles , respectively, the second order energy difference is related to the exchange constants by


The linear response expression in momentum space is


where are band indexes, are Cartesian coordinates, are wave vectors, is the Fermi function, and are the LSDA+U energies and eigenstates. This method has been confirmed to work well in several transition metal oxides and rare earth compounds.A version extended to systems with strong spin-orbit coupling and multipolar exchange interactions was also proposed and applied successfully.Pi2014 (); Pi2014b ()

The initial questions to address are the occupation and the position of the levels with respect to the Fermi energy E. In Fig. S1 of the Supplemental Material (SM), the occupations (all majority spin) and the total moment inside the atomic sphere are displayed for bcc, hcp, and phases across their pressure range of stability. In each case the full contribution is present, with a conduction band () contribution of 0.1-0.2 , each decreasing somewhat with reduced volume as the overlap of neighboring orbital increases. Thus the configuration persists and polarizes somewhat the conduction states. Figure S2 of SM shows the band structures, for each phase at pressures near the bottom and top pressures of their stability range. In each case the bands are centered near -5 eV, and the main change with pressure is that the band “width” increases, primarily a crystal field increase rather than a hopping amplitude increase. The configuration is stable and inert, remaining at -5 eV for pressure of interest here. The configuration of Gd has been calculated to remain stable to 500 GPa and above.Yin2006 ()

To indicate the magnitude of the exchange constants and provide connection with future experiment, the spin wave spectrum for the ambient pressure bcc phase is provided in Fig. S3 in the SM. The spinwave velocity is in the meV range with maximum energy =12-16 meV, depending on the value of the Hubbard repulsion . Experimental measurements would help in assessing our accuracy and pinning down the effective value in Eu metal under pressure.

Figure 2: (Color online) The exchange constants for various pressures, as labeled. Panels (a) and (b) show bcc and hcp, respectively. The others are for the phase at higher pressure: (c) U=6eV, (d) U=5eV, (e) U=4eV. Panel (f) focuses more closely on the sign change region in (e), showing that the zero crossings lie close to 80 GPa. Red dashed lines represent experimental pressures at the displayed lattice constants.

Magnetic coupling of the metals in general and Eu in particular, with their non-overlapping local moments within an itinerant electron sea, is due to the conduction electron mediated RKKY exchange mechanism of Eq. 3. Throughout the pressure range studied, and in particular in the regime where magnetic order vanishes, the Fermi surface is large and multisheeted, as pictured in Fig. 3. Large sheets are separating in the vicinity of , but this change in Fermi surface topology does not lead to significant van Hove singularities in the density of states nor to identifiable structure in versus pressure. While spiral magnetic order is commonly identified with nesting of sheets of Fermi surface, stable AFM order while the Fermi surface evolves argues against any nesting origin of ordering.

Figure 3: (color) The Fermi surfaces of Eu at the relative volumes, with respect to that at 75 GPa) of +6%, 0, -4%, and -10%. At all volumes the surfaces are large and multisheeted, varying through changes in topology with the only effect being the decrease in exchange constants and hence the ordering temperature, which vanishes around 82 GPa.

Equation 3 was used to calculate the exchange constants between spin-up and spin-down sublattices, based on the AFM ordered state. Since ions are spherical and the electronic structure is three dimensional, the exchange constant tensor is effectively diagonal in Cartesian coordinates, simplifying the coupling. The RKKY expression includes momentum-conserving virtual excitations, with those near the Fermi level having larger weight. In the limit, inter-sublattice exchange constants contain distinct intraband and interband terms. The former is the zone sum of a Fermi level density of states factor modulated by matrix elements – the RKKY result. This contribution makes exchange coupling sensitive to the Fermi surface, and several examples of incommensurate (usually spiral) order in lanthanides have been traced back to Fermi surface calipers. The interband contribution may still be important but, being unrelated to the FS it will be continuous and slowly varying compared to the intraband contribution.

The sublattice exchange couplings versus pressure are shown in Fig. 2 for =5, 6, 7 eV. For bcc and hcp Eu, Fig. 2(a),(b) respectively, the single sublattice coupling is AFM in sign and increases monotonically over the ranges of interest. Larger enhances the coupling; AFM order is very stable in these two phases.

The behavior in the high pressure structure is different. has four atoms per primitive cell, hence four magnetic sublattices. Only three of the six intersite coupling constants are independent. In Fig. 4(c)-(f) the exchange constants , and at =0 are shown, with increasing pressure and for =4, 5, 6 eV. affects primarily the magnitude, not changing the behavior as the volume is reduced. The trend with increasing pressure is for all three sublattice couplings to decrease in magnitude and pass through zero almost simultaneously, signaling a collapse of the spinwave spectrum and AFM order. This trend is independent of the value of ; the collapse of coupling – the incipient QCP – corresponds to the experimental observation of loss of order best with 4.5 eV. The collapse occurs at lower pressure as is decreased. is expected to decrease under pressure from the eV value that is realistic at ambient pressure.

This simultaneous vanishing of sublattice exchange couplings at indicates a coherent cancellation of positive and negative couplings from various Eu-Eu neighbors, that is, frustration to (AFM) magnetic order. All RKKY couplings are included in at each . It is not feasible to evaluate these on a fine enough to invert Eq. 3 and obtain small, long range exchange constants, but they might not provide further understanding anyway. Eu thus provides a strong contrast to the Fe-pnictides where impact of magnetic interactions on the phase diagram has been actively studied. Our methods applied to Fe pnictides led to (1) effective short-range coupling, and (2) AFM order that vanishes due to first neighbor () and second neighbor () coupling as approaches unity.Yin2008 (); Han2009 (). Such a model near frustration, with spins damped by conduction electrons, was found by Wu, Si, and AbrahamsSi2008 (); Si2016 () to account for the quantum critical point versus isoelectronic AsP doping in BaFeAs. Pressure tuning of Eu is of course isoelectronic, but the mechanism of frustration is quite different.

Discussion. Both experiment and theory concur that Eu retains its strong local moment to the pressures discussed here, without valence change.Bi2016 (). The calculated sublattice exchange couplings have been obtained, relative to an underlying antiferromagnetic alignment that is observed across the pressure range. Unlike for the configuration itself which is impervious to the environment, the crystal structure and accompanying electronic structure plays a critical role by inducing a AFM-PM transition, at the volume seen in experiment. That the three independent couplings vanish together at 82 GPa suggests that the Kondo coupling between spin and conduction electrons dominates RKKY coupling and has decreased dramatically with pressure. We have calculated the hybridization functionHan2008 () and determined that this is not the case.

Figure 4: (color online) Schematic depiction of the experimental phase diagram of Eu under pressure, showing the first order transition at P=82 GPa. Phases are: PM, paramagnetic; AFM: antiferromagnetic; SC, superconducting.

Another item concerns the character of the transition from AFM to PM at . A schematic phase diagram based on experimental data is presented in Fig. 4. Magnetic order decreasing to 11K vanishes at , and a SC electronic phase emerges within the PM magnetic phase whose driving force should be considered. This ground state represents a new phase: a superconducting condensate in the midst of large disordered moments below T=1.7K. In the free energy


the two DFT terms, available from DFT calculations, are expected to be linear in P and T-independent at low T since (1) no structural change is detected, and (2) the magnetic moments remain, only the order vanishes. The electronic entropy , where the bandwidth is eV, is orders of magnitude smaller than the magnetic terms and has not been displayed.

The focus becomes the magnetic contributions, and from spinwaves in the AFM phase, or spin disorder in the PM phase, as providing the free energy change. The difference in entropy between ordered and uncorrelated moments at high temperature is S = for . A rough (factor of two) estimate of the entropy of the ordered phase can be taken from the linear spin-wave expression ST; is P-dependent because it depends on the exchange couplings . The entropy just above T is roughly /2, a common value for AFMs. Equating these at T, one obtains the change across the transition as temperature is lowered for


which is smooth and small across the magnetic transition but becomes sizable at lower temperature. However, supposing uncorrelated moments for , the change in entropy across has the same form: the increase in entropy contributes to the loss of magnetic order above , with a finite jump for =11K but vanishing at , giving no driving force for a first order transition at this point in the phase diagram.

The magnetic energy of thermally excited spinwaves, in terms of the spinwave density of states and the Bose occupation factor , vanishes above and is replaced with nothing if the spins are totally uncorrelated in the PM phase. Total lack of correlation is unrealistic, in fact considerable short-range correlation may survive while leaving a finite jump in the free energy at . The magnetic subsystem may exhibit behavior characteristic of a spin liquid or spin gas.Savary2017 ()

To conclude, we comment on the unconventional electronic state in the SC phase. The scenario that has emerged is that of superconducting pairs co-existing with a spin gas or spin liquid. It is a classical given the large value of the moments. with negligible quantum fluctuation and the temperature being low compared to other scales, one has pairing in the midst of effectively static spins. Our calculated exchange splitting of the Eu bands for ferromagnetic alignment indicates a local on-site Hund’s exchange strength of =0.75 eV. This substantial coupling suggest comparable spin-disorder broadening of the conduction bands, hence washing out of the Fermi surface. Spin-disorder is normally destructive of pairing, unless the mechanism actually proceeds through, and depends on, the dynamic spin system. Such pairing, if it is responsible, lies in a different regime in Eu than for the cuprates and Fe-based superconductors, where magnetic fluctuations of small moments are intimately mixed into the conduction states. Yb at high pressure, as discussed in the introduction, presents a SC phase that may possess similarities to that of Eu.

We thank Wei-Ting Chiu, R. R. P. Singh, and J. S. Schilling for helpful communications. W.E.P and S.-T.P. were supported by DOE NNSA grant DE-NA0002908. For computational facilities, the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by DOE under Contract No. DE-AC02-05CH11231, as well as an in-house cluster at UC Davis, are gratefully acknowledged.


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