Inelastic Neutron Scattering Studies of Crystal Field Levels in Pros $ 4

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FIG. 1: (Color online) (a) The cube of 8 Os ions surround- 3+ ing the central Pr ion in PrOs4As12. These give an Oh- symmetric CEF. (b) The corresponding spectrum of Oh- symmetry Pr3+ CEF levels. (black=singlet, blue=doublet, red=triplet). The relative coupling x that gives singlet-triplet FIG. 2: (Color online) (a) Neutron inelastic scattering at T = 1.5 K and T = 200 K with Ei = 32 meV, integrated degeneracy is shown by a dashed vertical. (c) The 12 nearest- ◦ ◦ 3+ over scattering angles from 9 to 30 (low-angle detectors) neighbor As ions surrounding the central Pr in PrOs4As12, ◦ ◦ giving a reduced symmetry (Th) CEF. (d) The corresponding and from 105 to 140 (high-angle detectors). (b) The same 3+ at Ei = 12 meV; (c) Ei = 50 meV. The scattering function As-only Th-symmetry Pr CEF spectrum in PrOs4As12. S(Q,ω) was normalized by comparison to a vanadium stan- dard. confirmed in a large temperature (0.08 K-200 K) and magnetic field (0 T-11 T) range. also observed in the HET data, evidencing the ground state is a magnetic multiplet. III. RESULTS AND DISCUSSION Figures 4(a)-(d) show the temperature dependence of the low-angle scattering for Ei = 32 meV. The CEF peak Figure 2 summarizes the neutron scattering intensity intensities do not change significantly with temperature from PrOs4As12 on HET at temperatures between 1.5 K between 1.5 K and 5 K. At 50 K the intensity in the elas- and 200 K. Since the CEF magnetic scattering decreases tic channel has undergone a substantial decrease, and the with increasing Q whereas the intensity of phonons in- 13 meV peak has shifted to 10 meV. On further increas- creases with Q, a comparison of the neutron intensities ing the temperature to 100 K and 200 K the intensities in the low- and high-angle detectors can distinguish be- at 0 meV, 13 meV and 23 meV continue to decrease, tween magnetic and phonon scattering. Figure 2a shows whereas the scattering at 10 meV increases. the scattering function at T =1.5 K and T = 200 K with The theoretical description of the Pr3+ CEF levels in an incident neutron beam energy of Ei = 32 meV. Com- PrOs4As12 is complicated by the presence of important parison of the low- and high-angle data reveals two clear contributions from two sets of neighboring ions, Os and CEF excitations at 13 meV and 23 meV, with phonons As. The Pr3+ ion in Pr-based FS has a 4f 2 configuration, at 20 meV. Measurements with E = 12 and 50 meV ∼ i which in Russell-Saunders coupling has a ninefold degen- showed no evidence of additional CEF excitations at en- 3 erate H4 ground state. This degeneracy is lifted by the ergy transfers between 2 and 8 meV or above 25 meV CEF interaction, which we assume to be dominated by [Figs. 2(b) and 2(c)]. the 12 nearest neighbor pnictogens (As) and the 8 next To search for CEF excitations at energies below nearest neighbor Os ions; the distances to these ions are 2 meV, we carried out high resolution measurements us- dPr−As =3.23 A˚ and dPr−Os =3.69 A˚ respectively. ing SPINS. At T =0.32 K, energy scans at Q = (1.2, 0, 0) showed a clear peak at 0.4 meV; this mode decreases and A. Single charge model with separate ions becomes broader on warming to 2.5 K and 6 K [Fig. 3(a)]. − Figure 3(b) shows that the energy of the 0.4 meV mode Os ions form a simple cube around the Pr ions and ∼ has weak Q-dependence and decreases in intensity with they alone give an Oh symmetric CEF. The arrangement increasing Q, thus confirming its magnetic nature. Fig- of the 12 pnictogens (As) around Pr3+ forms 3 orthogo- ure 3(d) reveals that the elastic intensity also decreases nally intersected planes where the As-As bonds are shown on warming from 0.08 K to 4 K. This reduction of inten- as solid lines with length L and W is the length of the sity in the elastic channel with increasing temperature is dashed lines in Fig. 1(c) (b = W/L =0.4267 = 1). When 6 3 the 4 pnictogen atoms in each of the 3 orthogonally in- and b = 0.4267 level schemes differ qualitatively, which tersected planes form a square, i.e., b = W/L = 1, the demonstrates the importance of the Th terms in this fourfold rotational symmetry is recovered and the point problem. group symmetry becomes Oh with the simple cubic CEF We find that the Oh singlet and doublet energy eigen- 26 potentials [Fig. 1(c)] . This Oh case is treated by Lea, vectors are unmodified by the Th interaction, consistent Leask, and Wolf (LLW)26 (see their Fig. 9); we have red- with Takegahara et al.25. The singlet eigenvector (in a tot erived their excitation spectrum as shown in Fig. 1(b). J basis) is Ψ1 = 7/12 0 + 5/24( 4 + 4 ) 3+ 3 z | i p | i p | i | − i Both Oh and Th CEF interactions split the Pr H4 and the two doublet states are Ψ = 5/12 0 + ground state into a singlet, a doublet and two triplets25. | 23ai −p | i 7/24( 4 + 4 ) and Ψ = 1/2( 2 + 2 ), con- In the b = 1 T -symmetry case, these multiplets are p 23b p h sistent with| i earlier|− i (numerical)| i results25,26| i .|− Thei singlet referred to6 as Γ (a singlet), Γ (a nonmagnetic dou- 1 23 and doublet energy eigenvalues in our conventions are blet), and Γ(1) and Γ(2) (magnetic triplets). These 4 4 modified by the Th interaction. In terms of the LLW vari- two triplets are linear combinations of the Oh-symmetry 26 25 able x and our parameter b they are E1 = (16/13(1+ triplets, mixed by the new Th CEF interaction ; this b2)2)(91x(1 3b2 + b4) 20(1 x )(2 17b2−+2b4)) and mixing modifies the excitation spectrum and leads to b- − 2 2− − | |2 −4 E23 = (16/13(1+b ) )(13x(1 3b +b )+16(1 x )(2 dependent neutron transition intensities. 17b2 +2−b4)). The corresponding− analytic results−| for| the− The Th-symmetry CEF excitation spectrum has not two Th triplet states for general b are quite complicated, been considered in detail in the literature, and the corre- so we only present numerical results for these states. sponding neutron transition intensities between T CEF h The neutron transition intensities are defined by I = levels have not been considered at all. To aid in the inter- if f J tot i 2, as introduced by Birgeneau27. (There is pretation of our neutron scattering data we carried out z |han| implicit| i| sum over initial and final magnetic quantum these CEF calculations using a point charge model. We numbers.) Our T -symmetry results for these quantities assumed an expansion of the perturbing CEF potential h are shown in Fig. 5(a). The values in the limits x = 1 in spherical harmonics, ± (no ℓ = 6 term, hence Oh symmetry) implicitly check Bir- ℓ geneau’s numerical Oh results; see the off-diagonal entries V (Ω) = X gℓ X MℓmYℓm(Ω), (1) in his Table 1(e). These Oh limits are indicated on the ℓ=4,6 m=−ℓ vertical axis of Fig. 5(a). Next we compare the observed CEF levels and their where the interaction strengths g4 and g6 are treated neutron excitation intensities to the well-known LLW as free parameters. The spherical harmonic moments CEF results for Oh symmetry [Fig. 1(b)] and our calcu- Mℓm are determined by the positions of the 12 As lated CEF predictions for PrOs As under T symmetry {ions, which} we assigned the (scaled) coordinates ~x = 4 12 h [Fig. 1(d) and Fig. 5(a)]. Both Oh and Th CEF spectra ( 1, b, 0), (0, 1, b), ( b, 0, 1). The nonzero inde- have x values that can accommodate a magnetic triplet pendent± ± moments± for± ℓ =± 4, 6 are± M = 21(1 3b2 + 40 − ground state and a nearly degenerate singlet first excited b4)/2√π(1 + b2)2, M =3√13(2 17b2 +2b4)/8√π(1 + 60 − state [vertical lines in Figs. 1(b) and 1(d)]. However, it b2)2 and M = 15√3003b2(1 b2)/16√π(1+b2)3. The 66 − − is evident that the Oh scheme cannot explain the data nonzero M66 for b = 1 (Th symmetry) confirms the pres- (2) 6 6 because the observed 0.4 meV transition Γ4 Γ1 is in- ence of the Bt terms of Takegahara et al. [Eq.(7) of → 25 4 6 correctly predicted to have zero intensity due to the Oh Ref. ], in addition to the usual B and B Oh-symmetry c c symmetry. While in the Th scheme the relative neutron terms. (Note that the Th-allowed moment M66 vanishes excitation strengths of the higher levels (at 13 and 23 at the Oh-symmetry point b = 1.) We also confirmed that meV) predicted in Fig. 1(d) seem to be in good agree- the other nonzero moments satisfy the ratios quoted in 25 25 ment with observation at low temperatures, the As CEF Eq.(7) of Ref. . Unlike Takegahara et al. , we do not alone predicts an incorrect spectrum of levels [Fig. 1(d)], introduce a new parameter y for the Th-symmetry terms, (1) with the triplet Γ4 being the highest excitation. The because they are completely determined by g6 and the lattice parameter b in the point charge model. This was calculated neutron transition intensity shown in Fig.5(a) previously noted by Goremychkin et al.18. can not explain the observed intensity at higher temperatures. As temperature increases, the excited states get Diagonalization of this Th CEF interaction within the Pr+3 3H nonet gives our results for the spectrum of CEF populated and the excitations start to decrease in inten- 4 sities. Meanwhile the new excited-state transitions start levels and their associated eigenvectors. These eigen- (1) to increase. If Γ4 , instead of Γ23, is the highest level, vectors depend only on the ratio g6/g4 and the lattice parameter b; the energies in addition have an arbitrary the intensity at 12.6 meV would not increase but that at 22.6 meV would, because the Γ1 to Γ23 transition is overall scale. Our results for the spectrum for b = 1 18 not allowed even in Th symmetry. Goremychkin et al. (Oh symmetry) and b = 0.4267 (PrOs4As12 geometry) are shown in Figs. 1(b) and 1(d), using LLW normal- showed that the highest level in the similar Sb material ization conventions26. (These conventions set our two is the Γ23 doublet. Hamiltonian parameters in Eq.(1) to g4 = (968π/21)x and g = ( 5808π/221)(1 x ).) Note that the b = 1 B. Combined Os As CEF model 6 − − | | − 4

FIG. 5: (a) The theoretical neutron transition intensity for As CEF alone with b=0.4267. (b) Neutron excitation intensities predicted by the combined Os-As CEF model.

Os As g4 and g4 cannot be distinguished because they are summed into a single coefficient of the Oh-symmetry ℓ = 4 interaction. For this reason we introduce com- Os As bined Oh-symmetry Os-As coefficients g4 = g4 + g4 FIG. 3: (Color online) a) Low energy spectrum of CEF ex- Os As and g6 = g + g , which we normalize according to citations observed at T = 0.32, 2.5 and 6.0 K using the 6 6 LLW conventions. As the ℓ = 6 Th-symmetry terms SPINS spectrometer at NCNR. (b) The wave vector depen- As from H(As) in the CEF are proportional to g6 alone, dence of the excitations at Q = (0.8, 0, 0), Q = (1.2, 0, 0), and Os As Q = (1.7, 0, 0). (c) The expected and observed temperature the strengths g6 and g6 can be distinguished. We dependence of the intensity of the 0.4 meV mode. (d) The parametrize these two ℓ = 6 interactions using the to- temperature difference spectrum between 0.08 K and 4 K, tal Oh-symmetry g6 and a Th/Oh relative strength r6, showing clear reduction in magnetic elastic scattering. which is the ratio of the coefficients of Y62 to Y60 in the CEF potential. The energy levels of this Hamilto- nian are E = 6g 8g f, E = 28g 80g , 4(2) − 4 − 6 − 1 4 − 6 E4(1) = 6g4 8g6 + f and E23 = 4g4 + 64g6, where − − 2 2 2 1/2 f = ((20g4 + 12g6) + 960 r6g6) . For r6 = 0 these reduce to the familiar LLW Oh spectrum. In the pure As model, r6 is determined by CEF theory if we assume point As ions, and is given by (11√105/4)b2(1 b2)/(1 + b2)(1 (17/2)b2+b4). For PrOs As we have b−=0.4267, − 4 12 which gives a rather large r6 = 6.901. This drives strong level repulsion between the− two triplets, which explains why the pure As spectrum of Fig.1d differs so greatly from the Oh (pure Os) symmetry spectrum of Fig.1b. Our experimentally observed CEF levels are close to but not exactly consistent with the predictions above of the mixed Os-As model, since the gap ratio (E4(1) E )/(E E ) 0.57 is slightly below the theo-− 4(2) 23 − 4(2) ≈ FIG. 4: (Color online) The temperature dependence of the retical lower bound of 7/12. The parameters we esti- excitations observed on HET with Ei = 32 meV at (a) T = mate from the measured gaps are g 0.24 meV and 4 ≈ 1.5 K, 5 K; (b) 50 K; (c) 100 K, and (d) 200 K. The lines are g6 0.20 meV. The value of r6 is not determined by the theoretical results for neutron excitation intensities, from the measured≈ energies due to the inconsistency mentioned combined Os-As CEF model, with an arbitrary overall scale above, although r6 < 0.5 appears plausible. A more sen- factor. sitive determination∼ of r6 is possible through the mea- surement of the inelastic neutron excitation intensities we discuss below. The twin constraints of having the Γ23 level at the top (2) The neutron excitation intensities in this combined Os- of the spectrum and having a large Γ Γ neutron 4 1 As Hamiltonian depend only on a single parameter θ, excitation strength requires both Os and As↔ terms in the which is the mixing angle of the triplet energy eigenvec- CEF interaction. We therefore introduce a combined Os- tors when expanded in an O -symmetry 3 , 3′ basis, As Hamiltonian, h | i | i ′ H = H(Os)+ H(As). (2) 4(1) = + sin(θ) 3 + cos(θ) 3 | i | i | ′i 4(2) = + cos(θ) 3 sin(θ) 3 . (3) Although this model nominally has four parameters | i | i− | i Os Os As As (g4 ,g6 ,g4 and g6 ), only three are independent; This mixing angle is related to the Hamiltonian param- 5 eters by tan(2θ) =2√15 r6/(5(g4/g6)+3). The singlet and doublet Oh energy eigenvectors are unchanged. The nonzero neutron excitation intensities in terms of (2) 2 s = sin(θ) and c = cos(θ) are Γ4 Γ1 = (20/3)s , (2) (1) 2 2 (2) ↔ 2 Γ4 Γ4 = 7/2+8c s , Γ4 Γ23 = 4+(16/3)s , ↔ (1) 2 (1) ↔ 2 Γ1 Γ4 = (20/3)c , Γ4 Γ23 = 28/3 (16/3)s , (2)↔ (2) ↔ 2 2 (1)− (1) Γ4 Γ4 = (25/2)(1 (4/5)s ) , and Γ4 Γ4 = (1/2)(1+4↔ s2)2. The calculated− neutron scattering↔ intensity of different transitions as a function of θ is shown in Fig. 5(b). We recover the Oh-symmetry results of Birgeneau (Table 1(e) of Ref.27) for s =0,c = 1. We carried out a least-squares fit of our neutron excitation data at 1.5 K, 50 K, 100 K and 200 K (Fig. 4) to the theoretical intensities given above, which gives an estimate of the triplet mixing angle θ in PrOs4As12,

◦ θ 22.5 . (4) FIG. 6: (Color online) (a) The magnetic field dependence ≈ of the low-lying CEF excitations at Q = (1.2, 0, 0) and When combined with the values of g and g from the T = 0.08 K. The inset shows the field dependence of the first 4 6 excited state. (b) The difference spectrum between 0 T and spectrum, this θ corresponds to r 1.2. In this fit 6 ≈ 11 T at Q = (1.2, 0, 0) and T = 0.08 K. The effect of an ap- the relatively isolated Γ(2) Γ(1) peak at 23 meV was 4 → 4 plied field is to suppress intensity at ¯hω = 0 meV, and to split used to infer the background, which was taken to be con- the spin-triplet ground state; the latter results in the field stant plus linear. The assumed lineshapes were Loren- dependence of the 0.4 meV peak in (a), which may involve zians with a common linewidth, fixed by the 23 meV an intra-triplet transition. The elastic intensity suppression peak. The calculated intensities of the individual transi- effect essentially disappears for fields above 4 T. (c) The field- ≈ tions (dotted lines) and the total intensity (solid lines) for induced CEF excitation at 1.1 meV is weakly wave vector each temperature are shown in Figs. 4(a)-(d). We note dependent, and shows essentially no temperature dependence that the intensity reduction at 0.4 meV on warming from between T = 0.08 K and 4 K. 0.32 K to 2.5 K is larger than that expected from the CEF model [Fig. 3(c)], thus suggesting Pr-Pr interactions below TN (=2.3 K) are important. On the other hand, the large difference between the calculated and expected intensity around 8 meV in the 200 K data is presumably due to thermally populated phonons (Fig. 2a).

C. Field eﬀect on the CEF gap

(2) If the ground state of PrOs4As12 is indeed the Γ4 triplet, application of a magnetic field should Zeeman split it, resulting in a field dependent energy gap. There should also be a reduction in the intensity of the zero- (2) (2) energy Γ4 Γ4 magnetic scattering. Figure 6 shows that these expectations→ are indeed qualitatively satisfied. The first excited state at 0.4 meV shifts toward higher energies as the applied field increases. The field-dependent FIG. 7: (Color online) The boundary between singlet and transition energy is linear only at higher fields (between (2) triplet ground states in skutterudites (E1 = E4 ) as a func- 6 T and 11 T). The drop of intensity in the elastic chan- tion of r6 and g4/g6, and the observed PrOs4As12 spectrum. nel is almost constant for all applied fields, as shown in the inset of Fig. 6(b). Figure 6(c) shows that the wave vector dependence is also present with applied mag- neutron scattering intensities that are also modified by netic field (H =9 T ). Normally the field-splitting of the (2) their field-induced Γ4 -Γ1 mixing. ground state multiplet would be a very clear test of our Our determination of the CEF levels in PrOs As (2) 4 12 Γ4 triplet assignment for the ground state. However, reveals the reasons for the wide range of behaviours in PrOs4As12 is complicated by the near degeneracy of the different FS. The spectrum of CEF levels is largely de- (2) Γ4 and Γ1 levels, which mix strongly under an applied termined by the Oh symmetry field of the eight nearest field. This results in a more complicated spectrum of neighbor ions, and for Os the near equality of the ℓ = 4 low-lying states, with several low-field level crossings and and ℓ = 6 strengths g4 and g6 implies nearly degener- 6 ate low-lying singlet (insulator) and triplet (AF) levels. must incorporate the As ions’ contribution to the CEF The low temperature magnetic properties are determined Hamiltonian24,25, in addition to the usual Os cubic field by which of these phases happens to be the true ground terms. A comparison of our CEF calculations using this state. In the CEF model, this is specified by the two more general Hamiltonian with our experimental results 3+ parameters g4/g6 and r6 (Fig. 7); in PrOs4As12, which shows that the Pr CEF level scheme in PrOs4As12 con- (2) has a triplet ground state, we estimate g4/g6 1.15 and sists of a Γ magnetic triplet ground state, a nearly ≈ 4 r6 1.2. The Th symmetry pnictogen CEF (propor- (1) ≈ degenerate Γ1 singlet excitation, and higher Γ4 mag- tional to r6) acts to stabilize the triplet state, and can netic triplet and Γ23 nonmagnetic doublet excited states. itself lead to a triplet ground state if r6 is sufficiently We find that contributions in the CEF Hamiltonian due large to cross the phase boundary shown in Fig. 7. to As are important in determining the neutron exci- In principle, one can extend our approach to calculate tation intensities in PrOs4As12; our results differ qual- the ground states of other Pr-FS by determining its crys- itatively from the predictions of the conventional CEF tal structure and g4/g6 ratio. The necessity of using the Hamiltonian26,27, and therefore provide a microscopic Th-symmetry of As rather than the Oh-symmetry pure understanding for its AF ground state. Os form to explain the observed excitations shows that the detailed pnictogen geometry is important in determining the CEF levels. Indeed, the AF-ordered ground ACKNOWLEDGEMENT (2) state in PrOs4As12 can arise from a Γ4 triplet magnetic ground state, while the superconducting PrOs4Sb12 We thank R. J. Birgeneau, B. C. Sales and D. Schultz has nonmagnetic Γ1 singlet ground state. The nearly for helpful discussion. The work at UT/ORNL was degenerate first excited state Γ1 at 0.4 meV ( 4 K), and its temperature and field dependence (Figs.∼ 4 and supported by the U.S. DOE under grant DE-FG02- 6), may explain the presence of multiple transitions in 05ER46202, ORNL is managed by UT-Battelle, LLC, the specific heat (in C(T )/T versus T ) and its field for the U.S. DOE under contract DE-AC05-00OR22725. dependence14,15,22. Work at UCSD was supported by the U.S. DOE under grant DE-FG02-04ER46105 and by the NSF under grant II. SUMMARY DMR-0335173. Work on SPINS was supported in part by the National Science Foundation under Agreement No. To understand the observed Pr3+ CEF levels, one DMR-0454672.

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