Quantitative theory of magnetic properties of elemental praseodymium

Abstract

Elemental Pr metal is unique among rare-earth elements in featuring a localized partially filled 4f shell without ordered magnetism. Experimental evidence attributes this absence of magnetism to a singlet crystal-field (CF) ground state of the Pr 4f² configuration. Here, we construct an effective magnetic Hamiltonian for dhcp Pr, by combining density-functional theory with dynamical mean-field theory, in the quasiatomic Hubbard-I approximation. Our calculations fully determine the CF potential and predict singlet CF ground states for both inequivalent Pr sites. The intersite exchange interactions, obtained from the magnetic force theorem, are insufficient to close the CF gap to the magnetic doublets. Hence, ab-initio theory is demonstrated to explain the non-magnetic state of elemental Pr. We also find that the singlet ground state remains robust preventing conventional magnetic orders at the (0001) surface of Pr. Nevertheless, the gap between the ground state and the lowest excited singlet is significantly reduced at the surface, opening the possibility for exotic two-dimensional multipolar orders to emerge within this two-singlet manifold.

Introduction

The interest in the 17 rare-earth (RE) elements as vital components to functional materials is steadily increasing¹. RE-containing modern materials find their applications in devices that convert mechanical energy to electricity, in fuel cells and in batteries^2,3. They are also vital components in light-emitting diodes, in LCD screens and in lasers^4–8. Chemically, in solids, they are often found in a trivalent electronic configuration^9,10 where the outermost valence electrons form itinerant states that contribute to the chemical bonding. The electronic states of the 4f shell of the lanthanides (a subgroup of the rare-earths involving 15 elements) have limited ability to hybridize with other states (except La, Ce and Yb). Instead, these electron states form a localized magnetic moment that has significant spin and orbital contributions (as summarized excellently in ref. ¹¹), that for the most part can be understood from Russell-Saunders (RS) coupling. This involves a treatment of the angular momenta (spin S, orbital L, and total J) of an fⁿ configuration (n-electrons in the f-shell) that is atomic-like. From this, one would expect that any lanthanide element with 14 > n > 0 should form a significant magnetic moment, which is true for the most of the lanthanides and forms an essential background to, e.g., functional magnetic materials like Nd₂Fe₁₄B¹².

Surprisingly, for the solid phase of elemental Pr, that is trivalent with an f² electronic configuration in elemental form, the expected magnetic state (with S = 1, L = 5 and J = 4) is completely missing^13,14. Instead, Pr is a temperature-independent paramagnet. This paramagnetic ground state can be forced to undergo a meta-magnetic transition at extremely high applied magnetic field¹¹. Furthermore, at millikelvin temperatures the nuclear moments can order via nuclear RKKY interaction^11,15, but the 4f shell, with two electrons that in RS coupling are expected to form a stable spin-paired state, shows no experimental evidence of magnetism.

On a theoretical model level, the enigmatic, non-magnetic state of Pr has been somewhat explained by crystal field (CF) theory. In this model, one assumes that Heisenberg exchange of a possible magnetic state results in a total energy that is larger than that of a CF split non-magnetic, singlet level of the J = 4 angular momentum state. A singlet CF state has indeed been detected by inelastic neutron scattering (INS) for the hexagonal site of the dhcp lattice that elemental Pr solidifies in refs. ^16,17, see Fig. 1. Combined with a singlet state that is inferred as a possible state of the cubic site^17,18, the absence of magnetic order of Pr has been proposed.

Fig. 1. The crystal structure of dhcp Pr.

The cubic (hexagonal) sites are depicted with orange (violet) spheres.

Unfortunately, there is no solid experimental evidence of CF level splittings of Pr atoms at the cubic site of the dhcp structure^11,17. A theory that does not rely on experimental data of the CF levels of Pr is also missing, both for the cubic and hexagonal sites of dhcp Pr. This means that the total singlet state of elemental Pr is still undetected, with full evidence missing both from theory and from experiment. In addition, the theory of the interatomic exchange interaction of elemental dhcp Pr, e.g. based on the magnetic force theorem^19,20 (for a review of this method see ref. ²¹), is missing, primarily due to the complexity of the electronic structure of the 4f shell of the rare-earths. There have been some attempts to extract information about the interatomic exchange by fitting experimentally observed magnetic excitations together with susceptibility and magnetization curves data. Long-range RKKY interactions together with a significant two-ion anisotropy needed to be included into the effective magnetic Hamiltonian for quantitative agreement with experimental energies of low-energy excitation branches¹⁷. However, the long-range nature of RKKY interactions, together with the limited number of excitation branches accessible to INS render such fits quite uncertain.

Based on the discussion above, one must conclude that an accurate theoretical calculation or experimental estimate of the energetics relevant to determining the energy balance and the competition between CF splitting of a non-magnetic spin-singlet state and a magnetically ordered state of dhcp Pr is missing. This paper attempts to provide such an analysis for bulk dhcp Pr, based on ab-initio electronic structure theory combined with dynamical mean field theory. In addition, we address a possible magnetic state of Pr for both surface terminations with a cubic and hexagonal site.

Results

Theoretical model

A minimal model considered here is an effective low-energy Hamiltonian comprising CF and intersite Heisenberg exchange, that act within the ³H₄ ground-state multiplet of the Pr³⁺ ion:

$$H_{\mathrm{eff}}=\sum _i{H}_i^{\mathrm{CF}}-\sum _{ij}{I}_{ij}{\mathbf{J}}_i\cdot {\mathbf{J}}_j,$$ 1

where $H_i^{\mathrm{CF}}$ is the single-site CF term acting on the site i, I_ij is the intersite exchange coupling between the total angular momentum operators $J_{i\left(j\right)}^{\alpha }$ (α = x, y, z) at the corresponding sites. J_i(j) is the Cartesian vector of these operators (in contrast to a classical treatment of the angular momentum that considers J_i(j) to be an ordinary Cartesian vector). In Eq. (1) we neglect the anisotropic exchange coupling and magnetoelastic effects that were shown to be important for a detailed description of field-induced magnetic response and the excitation spectra of Pr^11,17. These smaller terms are unlikely to strongly affect the overall energetics and the competition between a magnetic ordered state and a possible singlet CF state.

In the standard Stevens formalism, the CF term reads

$$H^{\mathrm{CF}}=\sum _{kq}{B}_k^q{O}_k^q,$$ 2

where $O_k^q$ are the Stevens operator (ref. ²², see, e.g. refs. ^23,24 for a review) of the rank k and projection q acting within the ³H₄ multiplet, $B_k^q$ are the corresponding CF parameters (CFPs). Due to the point group symmetry of the hexagonal and cubic sites in the dhcp lattice, the CFPs can be non-zero for the following kq combinations only:

$$\mathrm{hex}:kq=20,40,60,66$$ 3

$$\mathrm{cub}:kq=20,40,43,60,63,66.$$ 4

In this work, we use first-principles approaches to calculate all parameters in Eq. (1). We detail approach for calculating the CFPs $B_k^q$ and the exchange interactions I_ij in the Methods.

Crystal field and intersite exchange in bulk Pr

The calculated CFPs for the two dhcp sites are listed in Table 1, where we also show the CFPs values of ref. ¹⁷ estimated from neutron scattering and magnetization measurements. For the hexagonal site, our values overall agree with the experimental estimates within the uncertainties of the latter.

Table 1.

Calculated CF parameters for bulk and (0001) relaxed surface of dhcp Pr (in meV)

	$B_2^0\times 10^2$	$B_4^0\times 10^4$	$B_4^3\times 10^4$	$B_6^0\times 10^4$	$B_6^3\times 10^4$	$B_6^6\times 10^4$
Bulk
Hex. site	14.0	−4.17	–	0.82	–	10.3
Hex. site (EE)	19 ± 4	−5.7 ± 5	–	1.0 ± 0.1	–	9.6
Cub. site	3.05	11.6	−462	0.9	10.0	11.2
Cub. site (EE)		29	−820	0.8	10	8

(0001) surface
Hexagonal termination
surf. l. (h)	−2.26	−6.17	−15.07	0.97	3.06	4.20
subsurf. l. (c)	−3.93	8.06	−182.09	0.81	13.45	8.81
Cubic termination
surf. l. (c)	−5.76	2.48	81.1	1.1	5.89	3.92
subsurf. l. (h)	4.88	−3.96	141	0.86	6.68	7.61

The bulk values are compared to the corresponding experimental estimates of Houmann et al.¹⁷. Only the $B_4^0$ and $B_6^0$ CFPs were fitted in ref. ¹⁷ for the cubic site. That work employed the ideal-cubic approximation for $B_k^q$ with q ≠ 0 resulting in the values shown in italic. We use the coordinate frame with y∣∣b, z∣∣c. ‘EE’ stands for ‘experimental estimate’, ‘h’ means ‘hexagonal’, ‘c’ refers to ’cubic, ‘l.’ stands for ‘layer’.

For the cubic site, only two CFPs were experimentally estimated from fitting magnetic susceptibility and the lowest magnetic excitation by Houmann et al.¹⁷. A perfect cubic symmetry for that site was assumed in their analysis. In this approximation $B_2^0$ becomes zero, while $B_k^q$ for k = 4, 6 and q > 0 are related to the corresponding $B_k^0$ by well-known relations²³. However, the “cubic” site of the dhcp-structure possesses the actual cubic symmetry only in the case of an ideal c/a = 3.266 ratio (c/a = 3.222 in dhcp Pr). Our values for the cubic site CFPs differ significantly from the estimates of Houmann et al., with, in particular, the calculated value of $B_4^0$ being almost three times smaller. Moreover, $B_4^3$ and $B_6^6$ are significantly (about one-third) larger compared to what one would obtain from $B_4^0$ and $B_6^0$ using the relations for the ideal cubic case. We note that the Stevens operators $O_k^q$ are not normalized to unity. Their norm (which can be defined as, e.g., $\sqrt{\mathrm{Tr}\left[O_k^q\cdot O_k^q\right]}$) scales approximately as J^k. Hence, the formally small $B_k^q$ CFPs for k = 4 and 6 in Table 1 lead to significant contributions to the CF potential.

In Fig. 2a, b, we display the calculated CF level schemes for the two sites, expressing the CF wavefunctions as superpositions of the J_z eigenstates $∣J=4;M_J⟩$ of the ³H₄ atomic configuration. The CF ground state is a singlet in both cases, with the gap to the lowest excited doublet $∣\pm 1⟩$ for the hexagonal and cubic sites is 2.8 and 6.8 meV, respectively, as compared to the experimental estimates of 3.5 and 8.4 meV¹⁷. The theoretical values are thus underestimated by about 20%. We note that the singlet $\left(∣-3⟩-∣+3⟩\right)/\sqrt{2}$ is lower than the $∣\pm 1⟩$ by 0.3 meV (Fig. 2b), in contrast to the experimental CF scheme for the hexagonal site (Fig. 2c), though the excitations from the GS singlet to this excited one will not be directly detectable by inelastic neutron scattering experiments. Otherwise, the order of theoretical CF levels and the composition of corresponding wavefunctions agree with the experimental picture, apart from a systematic underestimation by about 20%. In particular, the experimentally measured energy for the first excited state at the cubic site is well reproduced using our full set of CFPs values that differs significantly from previously available experimental estimates¹⁷.

Fig. 2. Crystal field levels of bulk Pr.

Calculated crystal-field splitting of the Pr ³H₄ configuration for the cubic (a) and hexagonal (b) site in bulk dhcp Pr. The CF wavefunctions are written in the $∣M⟩\equiv ∣J=4;M_J⟩$ basis and are defined in the same coordination frame as the CFPs in Table 1. In panel (c), we reproduce the experimentally inferred CF level scheme of ref. ¹⁷ for the hexagonal site.

The exchange interactions (see Eqs. (7) and (8) for their definition) were calculated both for the dhcp and a hypothetical hcp structure, similar to what was done in ref. ²⁵. In Fig. 3, we show the exchange interactions of dhcp Pr and compare them to those of dhcp Nd, reported in ref. ²⁵. The intersite exchange of hcp Pr is also added for comparison. We can see that ${\tilde{I}}_{ij}$ s in dhcp Pr are similar to that of dhcp Nd²⁵, as at short range the interactions are weak and antiferromagnetic. Pr, in general, has weaker exchange interactions compared to Nd. Similarly to Nd, we find that Pr has stronger exchange interactions in the hcp structure. hcp ${\tilde{I}}_{ij}$ s are ferromagnetic for short-range coupling for both materials, additional details can be found in Supplementary Section 1.

Fig. 3.

Exchange interactions ${\tilde{I}}_{ij}$ for the dhcp crystal structures of Pr (current work) and Nd²⁵, as well as for the hcp Pr (current work).

Crystal field and intersite exchange on the (0001) surface

In order to investigate the difference in exchange interactions between the bulk and surface layers of the dhcp structure of Pr, we performed slab calculations for the two possible surface terminations (with a cubic or a hexagonal layer as the surface layer). An ab initio optimization of the slab geometry was first performed, for details see the Methods section. As was similarly shown in ref. ²⁵ for dhcp Nd, the bulk-like magnetism is restored within just a few atomic layers. We found that for the hexagonal termination, the surface layer relaxes 2.4% inward, while the subsurface layer moves by 4.8% outward. For the cubic termination, these values are 1.34% and 4.4%.

Magnetic moments were then calculated with RSPt with similar results for the surface layers of both terminations and the bulk. The 1^st (surface) layer it is 1.77 μ_B and 1.78 μ_B for the cubic and hexagonal terminations, respectively. For the 2^nd (subsurface) layer, the values are 1.72 μ_B. In the case of the bulk, we obtained 1.72 μ_B and 1.71 μ_B for the cubic and hexagonal sites. For the surfaces, the exchange parameters were calculated with the RSPt code in a manner similar to the bulk case using the relaxed slab structure. The values of ${\tilde{I}}_{ij}$ for the surface and subsurface layers deviate considerably from the bulk data, as is outlined in detail in the Supplementary Section 2. As an example, ${\tilde{I}}_{1,2}$ is FM (0.14 meV for the cubic termination and 0.02 meV for the hexagonal termination), where bulk values are AFM, as we see in Fig. 3. At the 3^rd layer, the values of ${\tilde{I}}_{ij}$ s are almost restored to the bulk values.

We calculated the CFPs for both surface terminations of the (0001) dhcp surface using the relaxed DFT geometries described above. The relaxed interlayer distances were employed for the first four surface layers; the distances for deeper layers were fixed at the bulk value. The calculated CFPs values for both terminations are listed in Table 1. For the hexagonal sites, the on-site inversion symmetry is lifted by the surface, with their point group symmetry $\overline{6}m2$ correspondingly reduced to 3m. The CFPs $B_4^3$ and $B_6^3$ thus become non-zero in hexagonal layers. With our choice of the supercells, the global inversion symmetry with respect to the middle-layer cubic sites is preserved. Therefore, all sites have the same set (4) of real non-zero CFPs.

For both terminations, one observes CFPs to strongly deviate from the corresponding bulk values that are also displayed in Table 1. In particular, the values of $B_2^0$ for hexagonal surface and subsurface layers are significantly reduced compared to the bulk, while $B_4^3$ takes rather large values. One also observes a significant overall reduction of high-rank CFPs for the cubic site. As was shown in ref. ²⁶, high-rank CFPs are enhanced by hybridization effects; one can expect those effects to be reduced on the surface due to the reduced coordination number. In Fig. 4 we show the evolution of the $B_2^0$ CFP for the hexagonal sites and the $B_4^3$ CFP for the cubic ones vs. layer’s depth with respect to the surface. The value of $B_4^3$ is seen to increase quite monotonously and in a similar way for both terminations; the bulk value is virtually reached at the fifth layer. The behavior of $B_2^0$ is broadly similar, though displaying some oscillations vs. the depth.

Fig. 4. Layer-resolved crystal field parameters.

Evolution of the $B_2^0$ CFP on hexagonal sites (a) and the $B_4^3$ CFP on cubic sites (b) vs the layer depth with respect to the surface. The surface layer is the first one; the corresponding values for bulk are shown on the right-hand side. Since the sign of $B_4^3$ in the global coordination frame is flipped between the two dhcp cubic sites related by the inversion symmetry, we display the value $B_4^3$ in its local frame (i.e., we flip the sign of $B_4^3$ for every second cubic layer starting from the bulk one).

The calculated CF level schemes for surface and subsurface layers are shown in Fig. 5. As could be anticipated from the reduced CFPs values, we find correspondingly reduced CF splittings at the surface. The lifting of inversion symmetry on the hexagonal sites leads to mixing of the bulk GS ($∣0⟩$) with the excited singlet $\left(∣-3⟩-∣+3⟩\right)/\sqrt{2}$ by $B_k^3$ CF terms. For the hexagonal surface layer, a strong reduction of $B_2^0$ diminishes the splitting between those levels to about 1 meV. Similarly, one finds the splitting between the singlet GS and excited levels to decrease significantly for the cubic termination surface layer due to the decay of all high-rank CFPs $B_k^q$ for q > 0 (Table 1). For the subsurface (hexagonal) layer of the same termination, we find the CF gap between the GS singlet and the excited one to reduce to 0.7 meV only. This can be explained by a significantly enlarged interlayer distance between the subsurface and second surface layer predicted by our DFT calculations, which results in an especially strong deviation from the hexagonal symmetry reflected by a large value of $B_4^3$ (Table 1). Indeed, setting this CFP to zero, which mimics approximately restoring hexagonal symmetry, increases the gap back to above 2 meV.

Fig. 5. Surface crystal field levels of Pr.

Calculated crystal-field level splitting of the Pr ³H₄ multiplet at the (0001) dhcp surfaces with hexagonal (a) and cubic (b) termination. The CF wavefunction representation and coordination frame are the same as in Fig. 2. For both cases, we show the levels for the surface and subsurface site. In the subsurface layer the site symmetries are reversed with respect to the surface one, becoming cubic in (a) and hexagonal in (b), respectively.

Magnetic vs nonmagnetic ground state

Having calculated the CFPs and intersite exchange interactions ${\tilde{I}}_{ij}$, we obtain the full effective Hamiltonian, $H_{\mathrm{eff}}$, in Eq. (1) for dhcp Pr. This allows for a full determination of the magnetic ground state. We solved the Hamiltonian, using the single site quantum mean field approach of the McPhase package²⁷ together with an in-house module that implements the J_α and $O_k^q$ operators. We converted the calculated inter-site exchange coupling of the classical model, Eq. (7), to the J = 4 quantum model of Eq. (1) as $I_{ij}={\tilde{I}}_{ij}/20$, taking into account the length of quantum angular momentum as $\sqrt{J\left(J+1\right)}$.

Solving $H_{\mathrm{eff}}$, Eq. (1), for the bulk dhcp phase we correctly obtained a nonmagnetic state, with both crystallographic sites having the same singlet ground state as shown in Fig. 2. It agrees with experimental observations and illustrates the competition between interatomic exchange, which favors a magnetically ordered state, and crystal field effects, which for Pr favor a non-magnetic, singlet state. Following the experimental observations, the singlet state has the lowest energy. It means that the energy gain that would come from a magnetically ordered state, as quantified by the second term of Eq. (1), is smaller than the gain of the singlet crystal field effect that arises due to the Coulombic interaction of the J = 4 state of Pr in the dhcp crystal structure. In the Supplementary Section 3, we analyze the magnetic contribution to the specific heat, and a Schottky anomaly that occurs due to the excited CF levels of Pr.

We also show the calculated magnetization curves for bulk Pr with a magnetic field applied along the [100] (a-axis) and [001] (c-axis) directions. This was obtained by supplementing the effective Hamiltonian, Eq. (1), with a Zeeman term. The resulting curves are shown together with the corresponding experimental data¹¹ in Fig. 6. The easy-plane anisotropy is reproduced by the theory. However, theory on this level of approximation does not obtain a quantitative agreement with the measurements, with the anisotropy underestimated and the observed first-order metamagnetic transition at 30 Tesla absent. We notice that the suggested Hamiltonian^11,17 based on experimental data, includes significant two-site anisotropy as well as magnetoeslastic terms, which are neglected in the present theory. Those terms seem to be necessary to quantitatively reproduce the magnetization curves of dhcp Pr.

Fig. 6. Calculated magnetization curves in dhcp Pr (in red) compared to experiment (in blue).

The experimental data are from Fig. 7.13 of ref. ¹¹.

Similarly to the Hamiltonian for bulk Pr, we calculated the magnetic state of the dhcp Pr (0001) surface, for both terminations (cubic and hexagonal) using the layer-resolved CFPs and I_ij’s described above. The mean-field simulation supercell employed 15 and 17 layers for the hexagonal and cubic terminations, respectively. In both cases, we found the singlet paramagnetic state with no magnetic phases appearing. The decrease of CF splitting at the surface (Fig. 5) is thus not sufficient for the magnetic order to prevail. We note that even when increasing the values of the surface I_ij’s by a factor of two does not affect this result, showing that the non-magnetic GS state is sufficiently robust with respect to small changes in the effective Hamiltonian (1).

In Fig. 7 we display the magnetization of the hexagonal surface layer vs. external field applied along the in-plane (a) and out-of-plane (c) directions compared to the corresponding bulk data. The field in these calculations was applied to all layers of the supercell to simulate an experimental uniform field. One observes a drastic reduction of the anisotropy on the surface, with the magnetization curves for both directions almost coinciding, in a sharp contrast to the bulk case. The origin of this anisotropy collapse can be understood from the CF level scheme for the hexagonal surface layer (Fig. 5a). The singlet $\left(∣-3⟩+∣+3⟩\right)/\sqrt{2}$ is shifted down by about 5 meV as compared to the bulk (Fig. 2b) and occurs right above the magnetic doublet. Moreover, due to the $B_k^3$ CFPs being non-zero on the surface (Table 1), the singlet GS also acquires a $∣\pm 3⟩$ admixture. In the bulk, only the in-plane moment operators M_x(y) = g_JJ^x(y) couple the singlet GS $∣0⟩$ to other CF levels, namely, to the low-lying CF doublet $∣\pm 1⟩$. In the surface case, one may easily show that the matrix elements of the out-of-plane magnetic moment operator M_z between the GS singlet and the lowest excited singlet are also non-zero. As a result, M_α for all three directions couple the GS with excited CF levels located at about the same energy, leading to an isotropic behavior in the applied field.

Fig. 7.

Magnetic moment of the hexagonal site in the bulk and in the surface layer vs. applied field.

A strongly reduced CF gap between the GS and excited singlet levels could be naively expected to lead to a magnetic ordering on the (0001) surface. However, both these levels are superpositions of the $∣0⟩$ and $\left(∣-3⟩-∣+3⟩\right)/\sqrt{2}$ states. All matrix elements of the total angular momentum operators J^α can be easily shown to be zero within the space spanned by those two states. Hence, conventional magnetic orders cannot be hosted by them. The splitting between the singlet GS and the lowest magnetic doublet remains about 2 meV.

Discussion

Using a Hamiltonian (1) that consists of CF splittings and Heisenberg inter-site exchange terms, with all parameters obtained from ab initio, electronic structure theory, we demonstrate the absence of magnetic order in bulk, dhcp Pr. This result is in perfect agreement with experimental observations. It is important to note that the level of electronic structure theory used here goes beyond standard formulations based on density functional theory and the common approximations of the exchange and correlation functional. Instead, we used the combination of density functional with dynamical mean field theory in the Hubbard-I approximation for the calculation of crystal field parameters, since it is difficult to see how theories that treat electron correlations less precisely, could have the needed accuracy to reproduce the singlet state of dhcp Pr. Hence, electronic structure theory, as outlined here, is capable of reproducing the complex magnetic state of bulk Pr and this is a central result obtained in this investigation.

The theoretical results of the Pr (0001) surface are in some ways similar to those of bulk Pr. As shown in Fig. 5, the splitting between the GS singlet and lowest magnetic doublet on the hexagonal sites is only moderately reduced at the surface or subsurface hexagonal layers. In contrast, our calculations predict the splitting between the GS and lowest excited singlet to reduce for surface and subsurface hexagonal layers down to 1 meV and below (Fig. 5). The space of the two singlets cannot host conventional dipolar magnetic moments. However, higher-order multipolar operators can have non-zero matrix elements in this space. Multipolar orders and fluctuations arising from two closely spaced non-magnetic singlets have been discussed, e.g., for the prototypical hidden-order system URu₂Si₂²⁸ and the unconventional superconductor UTe₂²⁹.

Evaluating inter-site multipolar exchange in Pr is beyond the scope of the present work. However, we have explored possible multipolar orders on the basis of the compositions of the two singlets. To that end, we evaluated matrix elements of the multipolar Stevens operators $O_k^q$ (see, e.g., ref. ³⁰ for the explicit form of odd-k Stevens operators) within the space of the two singlets finding non-zero matrix elements only for the even-rank (charge) multipoles with q = 0, 3 and the odd-rank (magnetic) multipoles with q = −3. Since the surface CF Hamiltonian, Eq. (2) and Table 1, already contains all even-rank terms with q = 0, 3, no even-rank order parameter is possible within the two-singlet space. Among odd-k multipoles $O_k^{-3}$, we find the largest matrix elements for the octupolar moment operator, $O_3^{-3}=\frac{i}{2}\left[{\left(J^{-}\right)}^3-{\left(J^{+}\right)}^3\right]$. A sufficiently strong exchange coupling between these octupoles can lead to an unconventional surface octupolar order on the Pr (0001) surface.

Direct detection of high-rank magnetic multipolar orders represents a significant challenge even when they occur in the bulk, see, e.g., refs. ^31–34. Moreover, the relevant spectroscopical x-ray or neutron scattering probes are typically bulk sensitive. However, octupolar orders can manifest themselves indirectly by coupling to the strain^35,36. Namely, an appropriate strain applied to an octupole-ordered system can generate conventional dipole moments that are easily detectable. In the context of surface science, strains can be generated by depositing adatoms or by the inclusion of surface impurities. We simulated the effect of strain in the presence of an octupolar $O_3^{-3}$ order by supplementing the single-site CF Hamiltonian with the corresponding exchange and strain-induced terms

$$H^{\mathrm{CF}}+I_{\mathrm{o}}{O}_3^{-3}+K{O}_2^q,$$ 5

where I_o is the exchange mean-field due to the $O_3^{-3}$ order; the last term is induced by strain coupled to the quadrupole operator $O_2^q$ of matching symmetry. By diagonalizing (5) using the CFPs for hexagonal surface and subsurface layers (Table 1) we obtained the GS expectation value for dipole magnetic moments induced in the octupolar phase by various strains. We find the largest effect in the case of x² − y² orthogonal strain. This strain, which couples to the $O_2^2$ quadrupole, corresponds to the hexagonal layer being compressed along the a lattice parameter and extended in the orthogonal direction. The strain is found to induce an in-plane dipole magnetic moment along a. Therefore, the conjectured octupole order could indeed be detectable by inducing local strains, e.g., by depositing adatoms. The dipole moments are then expected to appear in the vicinity of the lattice perturbation below the octupolar transition temperature.

Methods

Calculation of crystal-field parameters

In order to calculate the CFPs, we use the approach of ref. ³⁷ that is based on density-functional theory (DFT)+dynamical mean-field theory³⁸ (DMFT) framework^39–41 in the Hubbard-I (HI) approximation^40,42. Our charge-density self-consistent implementation^43,44 of DFT+HI is based on the Wien2k linearized augmented-plane-wave (LAPW) full-potential code⁴⁵ and “TRIQS" library for implementing DMFT^46,47. In these calculations, we used LDA exchange correlations and the LAPW basis cutoff $R_{\mathrm{mt}}{K}_{\mathrm{MAX}}$=8. The on-site Coulomb repulsion was specified by the parameters F⁰ = U = 6 eV and Hund’s rule coupling J_H = 0.7 eV, which are in the commonly accepted range of Coulomb interaction parameters for Pr^3+48,49. Moreover, the CFPs calculated by DFT+HI have been shown to be rather insensitive to varying U and J_H³⁷. The double-counting correction was calculated in the fully localized limit using the nominal occupancy f² of Pr, as has been shown to be appropriate for the DFT+HI framework⁵⁰. We employ projective Wannier functions⁴³ to represent the 4f orbitals, see further details below. The experimental dhcp Pr lattice parameters a = 3.672 Å and c = 11.833 Ź¹ are used in bulk calculations.

In order to evaluate the CF splitting at the (0001) surface, we carried out DFT+HI calculations using 17(15)-layer slabs in the case of cubic (hexagonal) terminations, respectively, with about 15 Å of vacuum spacer. We always kept the in-plane lattice parameter at the bulk value, while the interlayer spacings were fixed at the relaxed DFT values obtained as detailed below.

The HI approximation (HIA) was chosen as it provides a pragmatic balance between accuracy and computational efficiency. Studying the electronic structure of f-electron systems with more exact numerical approaches, like the exact diagonalization (ED) or continuous time quantum Monte Carlo (CT-QMC), is in general feasible, see, e.g., refs. ^51–54. However, these methods are hardly applicable to the particular problem of a singlet non-magnetic state in dhcp Pr that is caused by the crystal field. First, off-diagonal elements in the one-electron level positions or hybridization function worsen the CT-QMC sign problem, so they have to be neglected in practical calculations. Therefore, the off-diagonal CF contributions (those with q ≠ 0, i.e., $B_4^3$, $B_6^3$, and $B_6^6$ in our case) need to be neglected. Also, one needs to study the temperature range below the CF energy of the excited magnetic doublet, about 2 meV, which would be a challenge for CT-QMC.

The number of bath sites in ED should also be increased to include those off-diagonal elements, or the same approximation suppressing off-diagonal elements has to be applied. Finally, the CFPs, which are given by the Kohn-Sham potential, are strongly modified due to the charge density self-consistency in DFT+DMFT (see ref. ³⁷ for discussion of this point). Carrying out full self-consistent DFT+DMFT calculations that require a lot of DMFT loops with ED or CT-QMC is not feasible.

The HIA avoids the complexity associated with handling large hybridization functions and reduces computational resources by orders of magnitude. As previously demonstrated⁴⁹, it accurately describes occupied and unoccupied states and magnetic moments in rare-earth elemental metals. While hybridization effects are not explicitly included in the HI framework, their impact on CFPs can be effectively included by a judicious choice of the correlated orbital basis representing 4f states^26,37. Namely, we employ spatially extended Wanniers to represent 4f correlated orbitals that are formed from a narrow energy window, including essentially only hybridized 4f bands. The hybridization affects the Wannier’s shape leading to the impact of hybridization on the CFPs being effectively included into the on-site level position as shown by previous works, in particular, ref. ²⁶.

In rare-earth elemental metals like dhcp Pr, the 4f bands are crossed by 5d ones, and one cannot completely unambiguously define the 4f band range. However, the 4f contribution rapidly diminishes for Kohn–Sham bands away from the Fermi level. The optimal window can be chosen by requiring the narrowest window enclosing all bands with more than 50% of 4f character. In our calculations, we employ the projective window [−h: h] around the centerweight of the Kohn–Sham Pr-4f band with h = 1.1 eV (which corresponds to [−0.86:1.34] eV with respect to the Kohn–Sham Fermi level) that satisfies this condition (see Supplementary Fig. 4). This prescription is not fully precise, changing h, e. g., by ±0.1 will approximately satisfy the same condition; one could also “optimize” the window by making it slightly not symmetric with respect to the 4f centerweight. These small changes of the window size have no impact on the overall CF level structure, e.g., changing h by ±0.1 eV alters the CF splitting magnitudes by about 15%.

In the surface calculation we employ, with respect to the KS Fermi level, the same projective window.

As shown in refs. ^37,55, the non-spherical 4f-electron charge density induces a DFT self-interaction contribution to the CFPs. In order to suppress this unphysical contribution, we average the Boltzmann weights of the nine states within the ³H₄ ground-state multiplet during self-consistent DFT+HI iterations, following the prescription of ref. ³⁷.

Having converged the DFT+HI calculations, we extract the CFPs for each crystallographically inequivalent Pr site a from the 4f one-electron level positions that read

$$H_a=E_a^0+H_a^{\mathrm{SO}}+H_a^{\mathrm{CF}},$$ 6

where the terms on the right-hand side are the uniform shift, spin-orbit interaction and crystal-field term. We find inter-multiplet mixing in dhcp Pr to be totally negligible. Therefore, we calculated the matrix elements of H_a in the basis of J_z eigenstates $∣J;M⟩\equiv ∣M⟩$ of the Pr f² ground state multiplet ³H₄ and extracted the CFPs by fitting the resulting matrix $⟨M∣H_a∣M^{\prime }⟩$ to the form (2).

Calculation of Heisenberg exchange

The calculations of the Heisenberg exchange were performed according to refs. ^19–21. We considered the spin-exchange only to the Heisenberg interaction and therefore made the substitution

$$\sum _{ij}{I}_{ij}{\mathbf{J}}_i\cdot {\mathbf{J}}_j\to \sum _{ij}{I}_{ij}^{\prime }{\mathbf{S}}_i\cdot {\mathbf{S}}_j,$$ 7

where according to the Russell-Saunders coupling S = J(g_J − 1), where g_J is the Landé g-factor. Moreover, according to refs. ^19–21, we included the value of the spin angular momentum in the exchange parameter (see Eq. (1.3) in ref. ¹⁴), which means that for the Heisenberg exchange of Eq. (1) the following substitution was considered:

$$\sum _{ij}{I}_{ij}^{\prime }{\mathbf{S}}_i\cdot {\mathbf{S}}_j\to \sum _{ij}{\tilde{I}}_{ij}{\mathbf{e}}_i\cdot {\mathbf{e}}_j.$$ 8

Note that these equations represent a system where exchange interaction is dominated by the spin moment, and not the orbital magnetic moment. Since it is primarily the itinerant valence electrons of the rare-earths that mediate the exchange for which the orbital moment is essentially quenched, the approach presented here captures the dominating contributions to the magnetic couplings.

For the calculations of ${\tilde{I}}_{ij}$, the full-potential linear muffin-tin orbital method (FP-LMTO) as implemented in the RSPt code^56,57 was used. The PBE functional⁵⁸ for exchange and correlation was employed. Experimental unit cell parameters were used for the calculations. 4f electrons were treated as localized and unhybridized particles with a magnetic moment according to Russell-Saunders coupling, according to the standard model of the lanthanides. The calculations were performed using 8000 k-points for bulk dhcp Pr.

To obtain the relaxed unit cell of the hcp structure, the Vienna Ab Initio Simulation Package (VASP)^59,60 was used within the Projector Augmented Wave (PAW) method⁶¹. PAW-PBE Pr_3 pseudopotential was employed, the plane wave basis energy cut-off was 500 eV with the k-point grid of 11 × 11 × 11. The optimization of the slab geometries was performed for 11 Pr layers with a vacuum region of 15 Å, where the three central layers were fixed to their bulk positions. For the surfaces, the exchange parameters were calculated with the RSPt code in a manner similar to the bulk case using the relaxed slab structure. The simulations were done using 5600 k-points of the full BZ.

Though we use two different band structure codes, Wien2k and RSPt, to calculate the terms in Eq. (1), we believe that combining those two frameworks does not affect our conclusions on the robustness of the non-magnetic state. A comprehensive comparison by Lejaeghere et al.⁶² demonstrated that electronic structure calculations performed by modern codes, including Wien2k and RSPt, yield nearly indistinguishable results, with deviations comparable to differences between high-precision experimental measurements. With respect to the use of 4f-in-core approximation in our calculations of the exchange interactions ${\tilde{I}}_{ij}$, we do not believe that hybridization effects impact ${\tilde{I}}_{ij}$ as significantly as CFPs. Intersite exchange in RE metals is induced by the intra-atomic Hund’s rule coupling between localized 4f and itinerant d or s states. This coupling should be only weakly affected by hybridization of quasiatomic 4f orbitals since the resulting 4f delocalization is expected to be small. To verify these expectations, we evaluated the overlap between the “small-window" spatially extended 4f Wannier set used in our calculations and a set of well-localized “large-window" Wannier orbitals (defined by the window [-9:9] eV). The latter orbitals are, in practice, indistinguishable from the local orbitals used by RSPt. We find the average overlap of 98.5%, confirming a very small fraction of 4f charge that delocalizes due to hybridization. This small fraction is very important for CFPs, since the bulk of 4f electron density, which is localized and spherically symmetric, does not contribute to the CF splitting. In contrast, the whole 4f local moment is involved into Hund’s rule coupling, so the contribution of the small delocalized part into RKKY should be indeed insignificant.

Author contributions

M.I.K. proposed the project. L.V.P. carried out the ab initio crystal field and mean-field calculations. A.M. carried out the DFT total energy and intersite exchange calculations. O.E. coordinated the project. L.V.P., O.E., and A.M. drafted the initial manuscript and supplementary. All authors contributed to analyzing the results and editing the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41524-025-01803-2.