Spin and orbital moments in Upt3

Solid State Communications, Vol.54,No.5, pp.389-397, 1985. Printed in Great Britain.

0038-1098/85 $3.00 + .00 Pergamon Press Ltd.

SPIN AND ORBITAL MOMENTSIN UPt3 J. S t i c h t and J. KUbler I n s t i t u t f u r Festk~rperphysik, THD, D-6100 Darmstadt, FRG (Received 29.1.1985 by P.H. Dederichs)

The local s p i n - d e n s i t y functional approximation is used to calculate the e l e c t r o n i c structure of UPt3 in assumed non-magnetic, ferromagn e t i c and antiferromagnetic ground states. When s p i n - o r b i t coupling is included i t is found to induce o r b i t a l moments which to a large extend compensate the spin moments of the i n i t i a l l y magnetic ground states.

previous band-structure c a l c u l a t i o n s , we f a i l to get a s u f f i c i e n t l y large DOS at EF, a d i f f e r e n t physical picture possibly emerges f o r the compensation of the magnetic moments: in the ground states which are i n i t i a l l y assumed magnetic, due to the strong s p i n - o r b i t coup l i n g a very large o r b i t a l moment is formed on U which is nearly compensated by the spin moments.

A number of m e t a l l i c compounds having g i g a n t i c s p e c i f i c - h e a t values, y, have recently caused a great deal of i n t e r e s t . They are called heavyfermion materials [ i ] and include CeCu2Si2, CeAI3, UBI3, UPt 3, and a growing number of other systems. Their experimental properties have been reviewed by Stewart [ 2 ] , who also c i t e s a few t h e o r e t i c a l papers. I n t e r e s t i n g l y , many (but not a l l ) heavy-fermion materials (HFM) have a superconducting ground state. This is s u r p r i s i n g because the large s p e c i f i c heat v a l ues, y, and t h e i r high-temperature Curie-Weiss s u s c e p t i b i l i t y would normally be taken to i n d i cate strong magnetic i n t e r a c t i o n s which make a superconducting ground state impossible. Current models [3-5] assume that the conduction electrons compensate the local moments of Ce (or U) at low temperatures by the Kondo e f f e c t , and the system is described as a Kondo l a t t i c e .

The crystal structure of UPt3 is sketched in Fig. 1; two U atoms and s i x Pt atoms are contained in the p r i m i t i v e c e l l . The c a l c u l a t i o n s f o r the assumed non-magnetic ground state are done without s p i n - o r b i t coupling. From the total-energy minimum we obtain the l a t t i c e constant l i s t e d in Table I where the calculated bulk modulus, B, and the heat of forma-

Only l i t t l e work has so f a r been done on the band s t r u c t u r e of these materials. Koelling [6] found that in the case of CeSn3, which has a large y-value (but not large enough to q u a l i f y asaHFM) band-structure c a l c u l a t i o n s do indeed describe the Fermi surface s a t i s f a c t o r i l y . Jarlborg et a l . [7] and KUbler [8] calculated the band s t r u c t u r e of CeCu2Si2 and Strange et a l . [9] that of UPt3. A l l c a l c u l a t i o n s f a i l to give a large density of states (DOS) at the Fermi energy, EF. Preliminary results f o r CeCu2Si2 [10] indicate that s u f f i c i e n t l y large values of y can only be obtained i f the standard band-structure s c a t t e r i n g phase s h i f t of the f - s t a t e s is replaced by a Kondo-resonance phase s h i f t . Here we report r e s u l t s of band-structure c a l culations f o r UPt3 w i t h i n the local spin-dens i t y f u n c t i o n a l approximation [ I I ] . The augmented spherical wave method (ASW) [12] is used to solve the s c a l a r - r e l a t i v i s t i c wave equations [13] and s p i n - o r b i t coupling is i n cluded at the end of the i t e r a t i o n cycles [14]. Motivated by the work of the Amsterdam group [15], who stress the importance of spin f l u c t u a t i o n s in UPt3, we assume in the present c a l c u l a t i o n s , besides a non-magnetic ground state a ferromagnetic and an a n t i f e r romagnetic ground state. Although, as in the

Fig. i

389

MgCd3- or SnNi3 type crystal s t r u c ture of UPt3 [16], c-axis not drawn to scale.

390

SPIN AND ORBITAL MOMENTS IN UPt 3

Table I

Vol. 54, No. 5

Calculated ground-state properties of UPt3 assuming a non-magnetic (unpo].), ferromagnetic and antiferromagnetic ground states, a: l a t t i c e constant, B: bulk modulus, AH: heat of formation per UPt3.

a/X (a)

B/Mbar (c)

AH/eV

unpol.

5.786 (a)

2.6

- 3.03

ferrom.

5.786 (a)

2.6

- 3.058

antiferrom.

5.786 (a)

2.5

- 3.061

(a)

throughout c/a = 0.85 [16]

(b)

experimental value is a = 5.764 X [16]

(c)

experimental value at T = 300 K is B = 2.1Mbar [17]

tion, AH, as well as some experimental values are listed, too. The heat of formation is cert a i n l y an overestimate because t h e t o t a l energy of the constituent metals is underestimated since U was calculated assuming an hcp crystal structure, which is not the observed one. The partial d-DOS of Pt is graphed in Fig. 2(a);

is a factor of 24 too small when i t is compared with the specific-heat y-value [15]. S t i l l , compared with 3d-transition-metal compounds, the DOS at EF due to the U atom of about 5.7 eV- I per U is rather large. So, i f we assume a Stoner exchange parameter of I = 0.5 eV, which, in fact, we verify below ' UPt 3(unpOl.)'

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Fig. 2 Densities of states (DOS)for assumed non-magmetic UPt3. (a) partial d-DOS of Pt, (b) partial d-DOS of U, (c) partial f-DOS of U, and (d) total DOS. the peak at E = 0.4 eV above EF results from hybridization with the f-DOS of U shown in Fig. 2(c). Figur 2(b) shows the partial d-DOS of U and Fig. 2(d) the total per c e l l . Rounded, Pt has 1.5 s and p electrons and 8.5 d electrons, whereas U has 1.1 s and p electrons, 2 d electrons (not 1) and 2.6 f electrons, the remaining fraction going into higher angular-momentumstates. As in the case of CeSn3 [6] the f electrons are accomodated in the long t a i l of the DOS [Fig. 2(c)] stretching down to 6 eV below EF. The total DOS at EF is N(EF) = 7.3 eV- I per UPt3 which

by the spin polarized calculations, then the Stoner product, I.N(EF), is larger than 1. The non-magneticgroundstate, most l i k e l y , is thus unstable. The results shown in Fig. 2 are in qualitative agreementwith those of Strange et al. [9], who, however, obtain some more fine structure. We now turn to the spin polarized calculations, f i r s t without spin-orbit coupling. Here we do not redetermine the l a t t i c e constant but use the one given in Table I ; one sees that the bulk modulus does not change

'

Vol. 54, No. 5

The DOS at EF is highest here: N(EF) =9.6 eV- I per UPt3, which s t i l l implies an enhancement f a c t o r of 18.6 when compared with the s p e c i f i c heat. S p i n - o r b i t and exchange s p l i t t i n g and the magnetic moments (Table I I ) are only l i t t l e changed compared with the ferromagnetic ground state; the t o t a l moments, mtot = mspin + morb, are i d e n t i c a l . For both ground states studied spin and o r b i t a l cont r i b u t i o n s are of opposite sign since the low energy, occupied part of the f-band consists of j = c - I/2 = 5/2 states. The ov e r a ll features of the t o t a l DOS [Fig. 4(c)] are a broad double peak below EF which is dominated by Pt d-states and above EF a more narrow double peak up to 2 eV dominated by U f states and another s l i g h t l y structured peak between 2 and ~ 4 eV due predominantly to U

but the total energy is lower by 28 meV per UPt3 compared with the non-magnetic ground state. For the antiferromagnetic calculations we assume alternating ferromagnetic planes normal to the c-axis, i . e . in Fig. 1 the spins are aligned up in the bottom plane and down in the center plane; thus the size of the unit c e l l does not change. The t o t a l energy (Table I) of the antiferromagnetic state is only lower by about 3 meV per UPt3 compared with the ferromagnetic state. I t is thus l i k e l y that the interatomic exchange coupling is very weak, but, as was shown in previous work on Heusler a l l o y s [18], one needs more than one antiferromagnetic c o n fi g u r a ti o n to estimate exchange constants. In Table I I we give the spin moments, m~pin, of this set of calculations. Table I I

391

SPIN AND ORBITAL MOMENTS IN UPt 3

Calculated magnetic moments of assumed ferromagnetic and antiferromagnetic ground states, per UPt3. m~oin from c a l c u l a t i o n s without s p i n - o r b i t coup l i n g , other entries include s p i n - o r b i t coupling. o

mspi n/UB

mspi n/UB

morb/UB

mtot/U B

ferrom.

1.15

1.23

- 1.66

- 0.43

antiferrom.

1.06

1.12

1.55

- 0.43

Spin-orbital coupling is next included as a perturbation [14] using the formulation of MacDonald et al. [19]; this last step of the calculation is not self-consistent even though the Fermi energy was redetermined which changes the spin moments, msDin, s l i g h t ly, as is apparent from Table I I . The orbital contribution to the magnetic moment is obtained from EF morb = uB z Z m S N~ms(E) dE , ~s m=-~

d-states. We believe that the structure below EF is in agreement with recent photoemission studies [21]. I t w i l l be i n t e r e s t i n g to see i f bremsstrahlung isochromat studies w i l l y i e l d the rather pronounced structures between EF and ~ 4 eV above EF, Both the assumed ground states give rather s i m i l a r results except f o r the Fermi surfaces; f e r r o magnetic order leads to very anisotropic surfaces whereas antiferromagnetic order, s u r p r i s i n g l y , gives rise to much more spherical features.

where uR is the Bohr magneton and Ncms(E) a partial-DOS to angular momentum c, magnetic quantum number m, and spin s [20]. Figure 3 shows our results for the ferromagnetic ground state. The f-electron DOS, especially Fig. 3(c), clearly shows a spin-orbit s p l i t ting of approximately 0.7 eV. The exchange s p l i t t i n g (conveniently obtained from the Hankel-function energies inherent in the ASW-method [12]) is only ~ = 0.57 eV. Thus with the calculated value of m~pin, one obtains f o r the Stoner exchange parameter, I = A/m~pin = 0.5 eV. The occupancies of the various ~ubbands is approximately the same as in the unpolarized c a l c u l a t i o n s when maj o r i t y - and m i n o r i t y - s p i n contributions are added. The calculated magnetic moments are l i s t e d in Table I I . Both spin and o r b i t a l co n t rib u t i o n s are due to the U f - e l e c t r o n s , the Pt d-electrons giving corrections of the order of 1-2% only. Figure 4 shows our state densities f o r an assumed antiferromagnetic ground state including s p i n - o r b i t coupling.

The foregoing discussion is not meant to imply that our band-structure results can, in d e t a i l , describe the e x c i t a t i o n properties of a highly correlated system as UPt3. As discussed l u c i d l y by von Barth [22], the differences between the l o c a l - d e n s i t y eigenvalues and the true e x c i t a t i o n energies can be enourmous when the e f f e c t i v e electron mass is much larger than the bare mass. But, as in the case of, f o r instance, the 3d t r a n s i t i o n metals where s i m i l a r reservations are in order, our results should at least be a zeroth order picture of r e a l i t y . That the band approach is not an unreasonable s t a r t ing point is demonstrated by Fig. 5, where we compare an f - with a d-radial wave function obtained f o r the ferromagnetic state. The calculated ground-state properties should be more r e l i a b l e , provided we f i n d the correct ground state to study. The t o t a l energies indicate that a proper s t a r t i n g point is an i n i t i a l l y magnetic ground state which subsequently can change d r a s t i c a l l y

392

Vol. 54, No. 5

SPIN A N D O R B I T A L M O M E N T S IN UPt 3 _

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Fi@. 3 Density of states (DOS) for assumed ferromagnetic UPtR including spinorbit coupling. (a) partial f-DOS of U, majority-spin electrons (b) partial d-DOS of U, majority-spin electrons, (c) and (d) same as (a) and (b) but for minority-spin electrons, (e) majority-total per c e l l , (f) minority-total per cell.

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Fi 9 . 4 Density of states (DOS) for assumed antiferromagnetic UPtR including spin-orbit couping. (a) partial ~ublattice f-DOS of U, majority-spin electrons (b) partial sublattice d-DOS of U, majority-spin electrons, (c) and (d) same as (a) and (b) but for minorityspin electrons, (e) total DOS for both spin directions per UPt3 .

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12

Vol. 54, No. 5

SPIN AND ORBITAL MOMENTS IN UPt 3

0.5 9-

0

r-

-0.5

UPt 3 ( F )

r (au} Fig. 5 Radial wave functions of a 5f spinup electron of U for an energy equal to EF, and of a 6d spin-up electron for an energy 3 eV above EF; ferromagnetic order assumed. when spin-orbit (SO) coupling is "turned on". That the l a t t e r is an important effect

Rws

we know from previous calculations on UN [20]. Now, our treatment of SO coupling as a perturbation, which is not f u l l y self-consistent, is dictated by necessity and is not unproblematic. Because, unlike the case of Gd [23] where the exchange s p l i t t i n g is very much larger than SO s p l i t t i n g , we here deal with a situation where both effects are of comparable magnitude. So the numbers appearing in Table I I may only be q u a l i t a t i v e l y correct, but they indicate that the orbital moment more or less compensates the spin moment. I t is interesting to note that coupling the orbital and spin moments parallel results in a total moment of 2.7 - 2.9 uB which is the effective moment measured in the high-temperature Curie-Weiss susceptibility [15].

Acknowledgement: We are indebted to Drs. Franse, Frings, and de Visser for their i n i t i a l help in characterizing UPt3. Part of this work was supported by Sonderforschungsbereich 65, Darmstadt-Frankfurt. References

[i]

Steglich, F., J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, and J. Sch~fer, Phys. Rev. Lett. 43, 1892 (1979)

[2]

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[3]

Grewe, N., Z. Phys. B 53, 271 (1983) and Z. Phys. B 56, ITI-TI984 )

[4]

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[5]

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[6]

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[7]

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[8]

KUbler, J., (unpublished)

[91

Strange, P., B.L. Gyorffy, Proc. of the Conference on Electronic Structure of Properties of Rare Earth and Actinide Intermetallics, to be published

[10]

d'Ambrumenil, N. and J. Sticht, to be published

[11]

Hedin, L. and B.I. Lundqvist, J. Phys. C 4, 2064 (1971); U. von Barth and L. l~6-arin, J. Phys. C 5, 1629 (1972); Janak, J.F., Sol. ~ t e Com. 2_55, 53 (1975)

[12]

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[13]

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393

[14]

Andersen, O.K., Phys. Rev. B 12, 3060 (1975)

[15]

Frings, P.H., J.J.M. Franse, F.R. de Boer, and M. Menowsky, J.M.M.M. 31-34, (1983); Palstra, T.T.M., P.H. K e ~ A. Mydosh, A. de Visser, J.J.M. Franse, and A. Menowsky, Phys. Rev. B 30, 2986 (1984); A. de Visser, J . J . M . ~ n s e , A. Menowsky, and T.T.M. Palstra, J. Phys. F 14, L191 (1984)

[16]

Heal, T.J., G.I. Williams, Acta Cryst. 8, 494 (1955); Lam, D.J., J.B. Darby, ~nd M.V. Nevitt, Electronic structure and related properties, A.J. Freeman, J.B. Darby eds., (Academic Press, New York, 1974) Vol. II

[17]

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[18]

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[19]

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[20]

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[21]

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[22]

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[23]

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