- Email: [email protected]
Indirect exchange via spin-orbit coupled states
Solid State Communications,
Vol. 7, pp. 1813—1818, 1969.
Pergamon Press.
Printed in Great Britain
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES* Peter M. Levy Hammond Laboratory, Yale University, New Haven, Connecticut (Received 6 November 1969 by P.G. de Gennes)
We find new terms are present in the interaction Hamiltonian for indirect exchange when we include the spin—orbit splitting of intermediate states in calculating the exchange. These terms are necessary to properly fit the Curie point data on a series of rareearth intermetallic compounds.
WHEN one explicitly evaluates the indirect exchange between two magnetic ions via nominally non-magnetic states, the spin—orbit splitting of the intermediate states is usually neglected. In this approximation one can immediately sum over the spatial and spin quantum numbers m1 and m8 of these states, and
interaction remains anisotropic. This anisotropy is not directly related to the spin—orbit coupling of the magnetic ions, therefore it can be comparable in magnitude to the isotropic interaction. 3.
The spin part of the Hamiltonian contains
terms which are not bilinear, but linear in only one of the spins. This leads, for example, to a
arrive at an effective interaction Hamiltonian. This Hamiltonjan contains non-scalar as well as scalar products of the orbital variables of the 4 However, wescalar will show, magnetic ions, but contains as only products of their there arespins.~ cases where the spin—orbit splitting of the intermediate states is not negligible.
coupling of the orbit of one ion with the spin of another, i.e., to terms of the form LA S,~and LA X SB, via an exchange interaction. After deriving a Hamiltonian for a specific indirect exchange interaction, we show that the extra terms in this Hamiltonian are necessary to
We have considered several indirect exchange interactions which include the spin—orbit splitting of the intermediate states.5’6 The most important differences between the Hamiltonian we find and other anisotropic indirect exchange Hamiltonians are :~‘~‘~
explain the experimental data on the Curie points of rare-earth intermetallic compounds having a Laves-phase structure.
1. In addition to non-scalar products of operators referring to the orbital variables the Hamiltonian also has products of spin operators which are non-scalar. The anisotropic spin interactions are not small compared to the isotropic interactions,
As a typical example of an interaction described by a Hamiltonian with the above properties, we consider here the indirect exchange between two magnetic ions via the polarization of conduction electrons which are in spin—orbit coupled states. Let us suppose a conduction electron is described by a linear combination of atomic orbitals, ‘I’~,,,..The simplest wave function which is orbitally degenerate (non-zero
2. In the limit of orbital singlets with no spin—orbit coupling of the magnetic ions, e.g., Mn2~,Gd3~,the spin part of the exchange
orbital angular momentum 1) and which is spin—orbit coupled to a total angular momentum J is of the form 1813
1814
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES (r)
111kjm
euI~.r
=
(r)
~
(1)
where ~~sjm (r + R) ~1isjm (~), The translation R is by an integral number of lattice spacings. The electrons on the magnetic ions A and B are described by atomic wave functions of orbital angular momentum L and
derived an explicit form for this Hamiltonian by using wave functions as equation (1) for the conduction and Hamiltonians as equation (2)electrons for the ion-conduction electron interactions.5 We find that the indirect exchange interaction can be represented as a scalar triple product ~ ~ p 2p, k1k2k) x
E
spinS,
1
.s.,, M~ y = A, B By confining ourselves to single pair interchanges we write the spin—dependent interaction =
‘Ly
ML.,,
between a magnetic ion and conduction electron, i.e., the ‘s—d’ or ‘s—f’ exchange interaction, 8’~ as J{(y) = f(L)~,I) (.~a + 2fi ~ s) (2) If the interaction consisted of only direct exchange and if each ion had but one magnetic electron, the parameters a and ~ would simply be one. The lowest order indirect exchange coupling between the magnetic ions comes from the second order perturbation 3’8 of the ground state filled empty energy of the system band, band, = E~ —1 m I .1
kim
Vol.7, No.24
(d”
(LA) k1 k2lc ~ cP2) (LB))~’~x (s~x
~
(0)
(k)J
(4) The orbital operators O~ (L 7) are written as irreducible tensors that transform as spherical harmonics ~ orbital Theirangular ranks momentum p~are onlyl.,~of an limited by the electron in the open shell of the magnetic ion y, e.g., for f electrons I = 3 and p~ 6. The operator S!,,~is a constant for K 1 = 0; for K1 = 1 it is the spin S in spherical tensor notation. The coefficients Q~ are arranged to transform as spherical harmonics. They contain the sums over the conduction electron states, see equation (3), exchange integrals, and the Wigner 3 — 6 — j and 9 j symbols that are used to couple the orbital spin operators into the form of a scalar tripleand product.” The coefficients Q~ are a function of the interionic distance RA — RB I; this has been suppressed in our notation. —
I
k’j’m’
1e~~”~ (RA
— RB)
The primary condition on the ranks of the
~
operators is that their sum p, + p 2 +e~’)~B
RA) 4~AWi’m’t}((A)kI~kim>}.
<~~ kimIH(B)Ic~k’i’m’>< The vector
(3)
+
K1
+
K2
must beFor even. If the twoinversion magnetic ions are and even. lattices with identical, the sum of the ranks p +symmetry K must be bands which have at least a reflection plane
R
gives the position of the magnetic ion y with respect to an arbitrary point in the crystal. We have neglected the energy difference between the states and ç~,,,therefore we are restricted in the states to which can apply the effective Hamiltonian. Thewe indirect exchange Hamiltonian is defined so that when acting on the states of the magnetic ions its matrix elements are the energy E (a) equation (3), 7
~
,
E(2)
<~A ~Bi
Rex
I c~,çS,~>. 1 we have
With the aid of Racah’s algebra,’
containing the Z axis in k space, time invariance of the interaction further requires that the rank A is even. The interaction Hamiltonian (4) differs from 3’ ~71t other anisotropic exchange Hamiltonians. contains non-scalar (p, K ~ 0) as well as scalar (p = K = 0) products of orbital and spin operators. Conventional Hamiltonians contain only scalar products of spin operators. It is not necessary for the rank K, to equal K 2, therefore it is possible to have interactions in which K, = 0
Vol.7, No.24
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
and K2 = 1, and vice versa. For example for p, = K2 = 1 and p2 = K, = 0 the Hamiltonian (4) contains the terms [Q(A)(101, 011) x (L~ x SA~~ )~]~for A = 0, 1, 2. This gives a contribution offor the 5B for A = 0, 0, ‘LAX 5B form Q0 L~ A = 1, and (Q 5B 3Q~LA S 2: LA 8) for A = 2. For orbital singlets the ranks p,, p2 and concomitantly sum p, + p p are zero. The condition that the 2 + K, + K2 is even requires K1 to equal K2, are onlyto 0 or Nonetheless the rank KasisK,, notK2 restricted be 1. zero, and we find anisotropic spin—spin interactions besides the magnetic dipole interaction exist between S-state ions. In our calculation of indirect exchange the spin—orbit coupling of the intermediate states has not beer 1 considered to be small compared to other interactions. Therefore, in contradistinction to previous treatments of 7 the anisotropic spin—spin interactions, magnitudes of the anisotropic interactions, A ~ 0, do not have to be small compared to the isotropic interactions, A = 0. —
1815
We have found it necessary to use the above form of indirect exchange to fit the data on the Curie points of a series of rare-earth intermetallic compounds having a Laves-phase structure, RA12, R1r2 and The Ruderman—Kittel interaction2 has R0s2. been rather successful in
‘
If we neglect the dependence of the energy of the conduction electrons Ekjm on the spatial quantum number m, we can sum over m and m’~ This simplification does not alter the form of the Hamiltonian (4), or any of the conditions on the ranks of the operators. When we further neglect the spin—orbit splitting of the conduction electrons, or consider a band with no orbital degeneracy, i.e., 1 = 0, the energy is independent of the quantum numbers j and m, E&im = Ek. By summing over j and I’, as well as m and m’, we find the coefficients Q(~’.)are simplified Q(f~)(p,p
explaining the variation of the paramagnetic Curie point with the8’ number of 4fone electrons theable rare13 however has notfor been earth metals, using this interaction to fit the ferromagnetic4 Curie points1. ofThe thefits rare-earth see Table are not intermetallics,’ appreciably improved by taking into account either the observed variation of the s—f exchange integral across the rare-earth series,15 or the higher degree contributions to the bilinear isotropic exchange from the conventional anisotropic exchange interaction.~ The Curie temperature T~is estimated by considering only bilinear scalar exchange’7 and by neglecting crystal field splittings. In the molecular field approximation T~is proportional to the sum F2 of the scalar interactions between the magnetic moments of the rare-earth ions kT~ = (1/3) F2 J(J where F2
—
KlKl0)~K1K2t5KO~
Therefore, in the limit of negligible spin—orbit splitting of the conduction bands we retrieve the conventional 3”2form for an anisotropic exchange interaction.
—
RB!),
—
From our Hamiltonian (4), the sum F2 is given as 2(n)
1p2A,
~RAB’I~’ (IRA
=
and the coefficient ç~?is defined by the interaction Hamiltonian = ~2 (IR RB! ) JA’ JB
1(p 2p,K,K2K)= Q”
1)
+
2
F
=
a~Q
010, + b~Q,0~0-,-c~Q2,2, + 2a~b~Q,00, +2a~c~Q0,2, + 2b~c~Q,02, (5)
where K
K
=
~
~
+j1 2
+P + I
(2p
+ 1)3~’2
~Ilp22
The form of the Hamiltonian (4) is not unique to the indirect exchange via conduction electrons. We have found an identical Hamiltonian for the indirect exchange between magnetic ions via the closed (non-magnetic) outer 5 shells on the ions or on intermediate ions.
W (p1 K, 1/K2 p2
~
~
(p, p2 p, K,K2 p R,~
AB
and n is the number of 4f electrons on the rareearth ion. The coefficients Q~ are defined in equation (4), and W is Racah coefficient or
1816
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
Vol.7, No.24
Table 1. Fits to the Curie points of the rare-earth intermetallic compounds R0s2. Element
Experirnent*
Nd SmH Gd Tb Dy Ho Er
Conventional two parameter fit §
New two parameter fit**
Thre~parameter fits
(°K)
(°K)
(°K)
(°K)
(°K)
23 34 66 34 15
6.5 16 55 37 25
7.1 16 55 37 25
17 31 70 36 16
23 35 66 33 16
9 3
16 9.0
16 10
5.7 0.5
7.6 3.4
*
See Bozorth et a!., Ref. 14.
~
Q010,
=
One parameter fitt
6.4 x i0~eV.
3 eV.
Q0101== 8.1 6.3 xx i0~eV, Q,010= 0.35 x id i0~eV, Q, 00, = 3.4 x 10 ~ eV. ~ Q0101= 7.6 x i0~eV, Q,0,0= 2.6 x i0~eV, ~
**§
~
3.8 x i0~eV. Samarium was not used in finding the fits. The Curie points in columns 2—5 are those predicted on the basis of a J = 5/2 ground state by using the parameters QPKP’K’ that were found to fit the other data.
6 I symbol.* The coefficient a~is proportional to (gj 1) where g,, is the Lande g-factor, b~is proportional to (2 ga), and c~is proportional to the ratio of the reduced matrix elements (J7II[0~ (L)x S~’~ ~ and (J7 IIJ(1) From an analysis of our fits of the Curie —
—
—
14)
114).
points of the rare-earth intermetallics using the parameters Q in equation (5), we find that the quality of the fits is relatively insensitive to the values of the parameters Q~,21,Qc,~2, and
*
This is primarily due to the coefficients c,~,a,.~c,3 and b~c~being small compared to the others.
~ This coupling is distinct from and much stronger than the magnetic spin—other—orbit coupling between two ions. Its strength is proportional to the ‘s—f’ exchange integrals contained in f(L7, I), see equation (2).
Q,02, .* Setting these parameters equal to zero hardly affects the fits. The remaining three parametersm provide a fit to the Curie temperatures that is far superior to the one obtained using only the one parameter Q010, present in the 3 The Ruderman—Kittel—deGennes interaction.’ data and fit on R0s 2 given in Table 1 are also typical of the other intermetallics RA12 and R1r2. Of the two parameter fits we find that by using Q,00, which does not appear in the conventional Hamiltonian for indirect exchange, we obtain a considerable improvement over the one parameter fit. If we use the conventional Hamiltonian parameter Q,010 as the second parameter the fit is only marginally improved. We conclude from our analysis that the term Q(O) (101, 01i;R118) LA SB must be in the Hamiltonian for the indirect exchange. ,
.
between rare-earths in the intermetallics RA12, R1r2 and R0s2.t This term enters only when we *
The Racah coefficients enter because we have recoupled the orbital and spin operators in equation (4) such that the operators referring to the same ion are coupled t” together as a first rank tensor (vector) .J
Vol.7, No.24
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
consider the spin—orbit splitting of the conduction electron states. Therefore we further conclude that this splitting is not negligible in the rare-earth intermetallics. To summarize, we have found additional terms are present in Hamiltonians describing indirect exchange interactions when there is appreciable spin—orbit coupling of the intermediate (non-magnetic) states. These terms contribute both to the isotropic and anisotropic parts of the exchange interaction. The additional isotropic contributions are needed to properly
1817
describe the rare-earth exchange interaction in intermetallic compounds having a Laves-phase structure. The most striking feature of the anisotropic contributions is that they exist even for ions in orbital singlet states; therefore they provide a new mechanism for large anisotropic spin—spin interactions between S-state ions.
Acknowledgements This work was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant no. 1258—67. —
REFERENCES 1.
ANDERSON P.W., Phys. Rev., 79, 350 (1950).
2.
RUDERMMJ M.A. and KITTEL C., Phys. Rev. 96, 99 (1954).
3.
KAPLAN T.A. and LYONS D.H., Phys. Rev. 129, 5072 (1963); KASUYA T. and LYONS D.H., J. Phys. Soc., Japan 21, 287 (1965). ELLIOTT R.J. and THORPE M.F., J. appi. Phys. 39, 802 (1968).
4. 5. 6.
7.
LEVY P.M., (to be published). COQBLIN B. and SCHRIEFFER J.R., Phys. Rev., (in press) have taken into account the spinorbit coupling of the conduction electrons in alloys containing cerium impurities. Their analysis is restricted because they specifically considered the hybrid exchange interaction, which is large only in cerium and ytterbium. We consider a general class of spin-dependent interactions which have much wider applicability. VAN VLECK J.H., Phys. Rev. 52, 1178 (1937); STEVENS K.W.H., Rev, mod. Phys. 25, 166 (1953); DZIALOSHINSKY I., J. Phys. Chem. Solids, 4, 241 (1958), and MORIYA T., Phys. Rev. Lett. 4, 228 (1960), Phys. Rev. 120, 91 (1960).
8.
De GENNES P.G., J. Phys radium, 23, 510 (1962).
9.
WATSON R.E., KOIDE S., PETER M. and FREEMAN A.J., Phys. Rev. 139, A167 (1965).
10. ii. 12. 13.
LEVY P.M., Chem. Phys. Lett. 3, 556 (1969), and COPLAND G.M. and LEVY P.M., Phys. Rev. (in press). FANO U. and RACAH G., irreducible Tensorial Sets (Academic Press, New York, 1959). IRKHIN YU.P., Soviet Phys. —JETP 23, 253 (1966); DRUZHININ V.V. and IRKHIN YU.P., Ibid. 24, 1250 (1967); IRKHIN YU.P., DRUZHININ V.V. and KAZAKOV A.A., ibid. 27, 633 (1968). De GENNES P.G., Compt. rend., 247, 1836 (1958).
14.
BOZORTH R.M., MATTHIAS ~T., SUHL H., CORENZWIT E. and DAVIS D.D., Phys. Rev. 115, 1595 (1959); WILLIAMS H.J., WERNICK J.H., NESBITT E.A. and SHERWOOD R.C., J. Phys. Soc. Japan, 17, Supp. B—i, 91(1962).
15.
JACCARINO V., MATTHIAS B.T., PETER M., SUHL H. and WERNICK J.H., Phys. Rev. Lett. 5, 251 (1960).
16.
KANEMATSU K., J. Phys. Soc. Japan 25, 628 (1968) has considered the effect of 4f orbital currents in polarizing the conduction electrons. This effect is not sufficient to explain the observed data on the Curie points of RAI2.
1818 17.
18.
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
Vol.7, No.24
The symmetry at the rare-earth sites in the cubic Laves compounds, RA12 and R1r2, is Td, see reference 15, and WERNICK J.H. and GELLER S., Trans. Met. Soc. AIME 218, 866 (1960). For the hexagonal Laves compounds ROs2, the symmetry is nearly tetrahedral (Td), see FRIAUF J.B., Phys. Rev. 29, 34 (1927). For this symmetry and with the molecular field approximation, anisotropic terms of the third and higher degrees only can enter the total interaction Hamiltonian for an ion and its surroundings. As we obtain a good fit to T~without these and other higher degree scalar terms we do not consider them in this analysis. The relative magnitudes of the higher degree parameters Q,~,0and Q,~,compared to Q010, are reasonable. See references 3, 4 and SPECHT F., Phys. Rev. 162, 389 (1967).
Quand nous tenons compte du couplage spin-orbite des états intermédiaires, nous trouvons qu’il existe de nouveaux termes dans l’hamiltoriien d’é~hangeindirect. Nous montrons que ces termes sont nécessaires pour expliquer la variation de la temperature de Curie d’une série de composes intermdtalliques de terres rares.
Vol. 7, pp. 1813—1818, 1969.
Pergamon Press.
Printed in Great Britain
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES* Peter M. Levy Hammond Laboratory, Yale University, New Haven, Connecticut (Received 6 November 1969 by P.G. de Gennes)
We find new terms are present in the interaction Hamiltonian for indirect exchange when we include the spin—orbit splitting of intermediate states in calculating the exchange. These terms are necessary to properly fit the Curie point data on a series of rareearth intermetallic compounds.
WHEN one explicitly evaluates the indirect exchange between two magnetic ions via nominally non-magnetic states, the spin—orbit splitting of the intermediate states is usually neglected. In this approximation one can immediately sum over the spatial and spin quantum numbers m1 and m8 of these states, and
interaction remains anisotropic. This anisotropy is not directly related to the spin—orbit coupling of the magnetic ions, therefore it can be comparable in magnitude to the isotropic interaction. 3.
The spin part of the Hamiltonian contains
terms which are not bilinear, but linear in only one of the spins. This leads, for example, to a
arrive at an effective interaction Hamiltonian. This Hamiltonjan contains non-scalar as well as scalar products of the orbital variables of the 4 However, wescalar will show, magnetic ions, but contains as only products of their there arespins.~ cases where the spin—orbit splitting of the intermediate states is not negligible.
coupling of the orbit of one ion with the spin of another, i.e., to terms of the form LA S,~and LA X SB, via an exchange interaction. After deriving a Hamiltonian for a specific indirect exchange interaction, we show that the extra terms in this Hamiltonian are necessary to
We have considered several indirect exchange interactions which include the spin—orbit splitting of the intermediate states.5’6 The most important differences between the Hamiltonian we find and other anisotropic indirect exchange Hamiltonians are :~‘~‘~
explain the experimental data on the Curie points of rare-earth intermetallic compounds having a Laves-phase structure.
1. In addition to non-scalar products of operators referring to the orbital variables the Hamiltonian also has products of spin operators which are non-scalar. The anisotropic spin interactions are not small compared to the isotropic interactions,
As a typical example of an interaction described by a Hamiltonian with the above properties, we consider here the indirect exchange between two magnetic ions via the polarization of conduction electrons which are in spin—orbit coupled states. Let us suppose a conduction electron is described by a linear combination of atomic orbitals, ‘I’~,,,..The simplest wave function which is orbitally degenerate (non-zero
2. In the limit of orbital singlets with no spin—orbit coupling of the magnetic ions, e.g., Mn2~,Gd3~,the spin part of the exchange
orbital angular momentum 1) and which is spin—orbit coupled to a total angular momentum J is of the form 1813
1814
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES (r)
111kjm
euI~.r
=
(r)
~
(1)
where ~~sjm (r + R) ~1isjm (~), The translation R is by an integral number of lattice spacings. The electrons on the magnetic ions A and B are described by atomic wave functions of orbital angular momentum L and
derived an explicit form for this Hamiltonian by using wave functions as equation (1) for the conduction and Hamiltonians as equation (2)electrons for the ion-conduction electron interactions.5 We find that the indirect exchange interaction can be represented as a scalar triple product ~ ~ p 2p, k1k2k) x
E
spinS,
1
.s.,, M~ y = A, B By confining ourselves to single pair interchanges we write the spin—dependent interaction =
‘Ly
ML.,,
between a magnetic ion and conduction electron, i.e., the ‘s—d’ or ‘s—f’ exchange interaction, 8’~ as J{(y) = f(L)~,I) (.~a + 2fi ~ s) (2) If the interaction consisted of only direct exchange and if each ion had but one magnetic electron, the parameters a and ~ would simply be one. The lowest order indirect exchange coupling between the magnetic ions comes from the second order perturbation 3’8 of the ground state filled empty energy of the system band, band, = E~ —1 m I .1
kim
Vol.7, No.24
(d”
(LA) k1 k2lc ~ cP2) (LB))~’~x (s~x
~
(0)
(k)J
(4) The orbital operators O~ (L 7) are written as irreducible tensors that transform as spherical harmonics ~ orbital Theirangular ranks momentum p~are onlyl.,~of an limited by the electron in the open shell of the magnetic ion y, e.g., for f electrons I = 3 and p~ 6. The operator S!,,~is a constant for K 1 = 0; for K1 = 1 it is the spin S in spherical tensor notation. The coefficients Q~ are arranged to transform as spherical harmonics. They contain the sums over the conduction electron states, see equation (3), exchange integrals, and the Wigner 3 — 6 — j and 9 j symbols that are used to couple the orbital spin operators into the form of a scalar tripleand product.” The coefficients Q~ are a function of the interionic distance RA — RB I; this has been suppressed in our notation. —
I
k’j’m’
1e~~”~ (RA
— RB)
The primary condition on the ranks of the
~
operators is that their sum p, + p 2 +e~’)~B
RA) 4~AWi’m’t}((A)kI~kim>}.
<~~ kimIH(B)Ic~k’i’m’>< The vector
(3)
+
K1
+
K2
must beFor even. If the twoinversion magnetic ions are and even. lattices with identical, the sum of the ranks p +symmetry K must be bands which have at least a reflection plane
R
gives the position of the magnetic ion y with respect to an arbitrary point in the crystal. We have neglected the energy difference between the states and ç~,,,therefore we are restricted in the states to which can apply the effective Hamiltonian. Thewe indirect exchange Hamiltonian is defined so that when acting on the states of the magnetic ions its matrix elements are the energy E (a) equation (3), 7
~
,
E(2)
<~A ~Bi
Rex
I c~,çS,~>. 1 we have
With the aid of Racah’s algebra,’
containing the Z axis in k space, time invariance of the interaction further requires that the rank A is even. The interaction Hamiltonian (4) differs from 3’ ~71t other anisotropic exchange Hamiltonians. contains non-scalar (p, K ~ 0) as well as scalar (p = K = 0) products of orbital and spin operators. Conventional Hamiltonians contain only scalar products of spin operators. It is not necessary for the rank K, to equal K 2, therefore it is possible to have interactions in which K, = 0
Vol.7, No.24
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
and K2 = 1, and vice versa. For example for p, = K2 = 1 and p2 = K, = 0 the Hamiltonian (4) contains the terms [Q(A)(101, 011) x (L~ x SA~~ )~]~for A = 0, 1, 2. This gives a contribution offor the 5B for A = 0, 0, ‘LAX 5B form Q0 L~ A = 1, and (Q 5B 3Q~LA S 2: LA 8) for A = 2. For orbital singlets the ranks p,, p2 and concomitantly sum p, + p p are zero. The condition that the 2 + K, + K2 is even requires K1 to equal K2, are onlyto 0 or Nonetheless the rank KasisK,, notK2 restricted be 1. zero, and we find anisotropic spin—spin interactions besides the magnetic dipole interaction exist between S-state ions. In our calculation of indirect exchange the spin—orbit coupling of the intermediate states has not beer 1 considered to be small compared to other interactions. Therefore, in contradistinction to previous treatments of 7 the anisotropic spin—spin interactions, magnitudes of the anisotropic interactions, A ~ 0, do not have to be small compared to the isotropic interactions, A = 0. —
1815
We have found it necessary to use the above form of indirect exchange to fit the data on the Curie points of a series of rare-earth intermetallic compounds having a Laves-phase structure, RA12, R1r2 and The Ruderman—Kittel interaction2 has R0s2. been rather successful in
‘
If we neglect the dependence of the energy of the conduction electrons Ekjm on the spatial quantum number m, we can sum over m and m’~ This simplification does not alter the form of the Hamiltonian (4), or any of the conditions on the ranks of the operators. When we further neglect the spin—orbit splitting of the conduction electrons, or consider a band with no orbital degeneracy, i.e., 1 = 0, the energy is independent of the quantum numbers j and m, E&im = Ek. By summing over j and I’, as well as m and m’, we find the coefficients Q(~’.)are simplified Q(f~)(p,p
explaining the variation of the paramagnetic Curie point with the8’ number of 4fone electrons theable rare13 however has notfor been earth metals, using this interaction to fit the ferromagnetic4 Curie points1. ofThe thefits rare-earth see Table are not intermetallics,’ appreciably improved by taking into account either the observed variation of the s—f exchange integral across the rare-earth series,15 or the higher degree contributions to the bilinear isotropic exchange from the conventional anisotropic exchange interaction.~ The Curie temperature T~is estimated by considering only bilinear scalar exchange’7 and by neglecting crystal field splittings. In the molecular field approximation T~is proportional to the sum F2 of the scalar interactions between the magnetic moments of the rare-earth ions kT~ = (1/3) F2 J(J where F2
—
KlKl0)~K1K2t5KO~
Therefore, in the limit of negligible spin—orbit splitting of the conduction bands we retrieve the conventional 3”2form for an anisotropic exchange interaction.
—
RB!),
—
From our Hamiltonian (4), the sum F2 is given as 2(n)
1p2A,
~RAB’I~’ (IRA
=
and the coefficient ç~?is defined by the interaction Hamiltonian = ~2 (IR RB! ) JA’ JB
1(p 2p,K,K2K)= Q”
1)
+
2
F
=
a~Q
010, + b~Q,0~0-,-c~Q2,2, + 2a~b~Q,00, +2a~c~Q0,2, + 2b~c~Q,02, (5)
where K
K
=
~
~
+j1 2
+P + I
(2p
+ 1)3~’2
~Ilp22
The form of the Hamiltonian (4) is not unique to the indirect exchange via conduction electrons. We have found an identical Hamiltonian for the indirect exchange between magnetic ions via the closed (non-magnetic) outer 5 shells on the ions or on intermediate ions.
W (p1 K, 1/K2 p2
~
~
(p, p2 p, K,K2 p R,~
AB
and n is the number of 4f electrons on the rareearth ion. The coefficients Q~ are defined in equation (4), and W is Racah coefficient or
1816
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
Vol.7, No.24
Table 1. Fits to the Curie points of the rare-earth intermetallic compounds R0s2. Element
Experirnent*
Nd SmH Gd Tb Dy Ho Er
Conventional two parameter fit §
New two parameter fit**
Thre~parameter fits
(°K)
(°K)
(°K)
(°K)
(°K)
23 34 66 34 15
6.5 16 55 37 25
7.1 16 55 37 25
17 31 70 36 16
23 35 66 33 16
9 3
16 9.0
16 10
5.7 0.5
7.6 3.4
*
See Bozorth et a!., Ref. 14.
~
Q010,
=
One parameter fitt
6.4 x i0~eV.
3 eV.
Q0101== 8.1 6.3 xx i0~eV, Q,010= 0.35 x id i0~eV, Q, 00, = 3.4 x 10 ~ eV. ~ Q0101= 7.6 x i0~eV, Q,0,0= 2.6 x i0~eV, ~
**§
~
3.8 x i0~eV. Samarium was not used in finding the fits. The Curie points in columns 2—5 are those predicted on the basis of a J = 5/2 ground state by using the parameters QPKP’K’ that were found to fit the other data.
6 I symbol.* The coefficient a~is proportional to (gj 1) where g,, is the Lande g-factor, b~is proportional to (2 ga), and c~is proportional to the ratio of the reduced matrix elements (J7II[0~ (L)x S~’~ ~ and (J7 IIJ(1) From an analysis of our fits of the Curie —
—
—
14)
114).
points of the rare-earth intermetallics using the parameters Q in equation (5), we find that the quality of the fits is relatively insensitive to the values of the parameters Q~,21,Qc,~2, and
*
This is primarily due to the coefficients c,~,a,.~c,3 and b~c~being small compared to the others.
~ This coupling is distinct from and much stronger than the magnetic spin—other—orbit coupling between two ions. Its strength is proportional to the ‘s—f’ exchange integrals contained in f(L7, I), see equation (2).
Q,02, .* Setting these parameters equal to zero hardly affects the fits. The remaining three parametersm provide a fit to the Curie temperatures that is far superior to the one obtained using only the one parameter Q010, present in the 3 The Ruderman—Kittel—deGennes interaction.’ data and fit on R0s 2 given in Table 1 are also typical of the other intermetallics RA12 and R1r2. Of the two parameter fits we find that by using Q,00, which does not appear in the conventional Hamiltonian for indirect exchange, we obtain a considerable improvement over the one parameter fit. If we use the conventional Hamiltonian parameter Q,010 as the second parameter the fit is only marginally improved. We conclude from our analysis that the term Q(O) (101, 01i;R118) LA SB must be in the Hamiltonian for the indirect exchange. ,
.
between rare-earths in the intermetallics RA12, R1r2 and R0s2.t This term enters only when we *
The Racah coefficients enter because we have recoupled the orbital and spin operators in equation (4) such that the operators referring to the same ion are coupled t” together as a first rank tensor (vector) .J
Vol.7, No.24
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
consider the spin—orbit splitting of the conduction electron states. Therefore we further conclude that this splitting is not negligible in the rare-earth intermetallics. To summarize, we have found additional terms are present in Hamiltonians describing indirect exchange interactions when there is appreciable spin—orbit coupling of the intermediate (non-magnetic) states. These terms contribute both to the isotropic and anisotropic parts of the exchange interaction. The additional isotropic contributions are needed to properly
1817
describe the rare-earth exchange interaction in intermetallic compounds having a Laves-phase structure. The most striking feature of the anisotropic contributions is that they exist even for ions in orbital singlet states; therefore they provide a new mechanism for large anisotropic spin—spin interactions between S-state ions.
Acknowledgements This work was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant no. 1258—67. —
REFERENCES 1.
ANDERSON P.W., Phys. Rev., 79, 350 (1950).
2.
RUDERMMJ M.A. and KITTEL C., Phys. Rev. 96, 99 (1954).
3.
KAPLAN T.A. and LYONS D.H., Phys. Rev. 129, 5072 (1963); KASUYA T. and LYONS D.H., J. Phys. Soc., Japan 21, 287 (1965). ELLIOTT R.J. and THORPE M.F., J. appi. Phys. 39, 802 (1968).
4. 5. 6.
7.
LEVY P.M., (to be published). COQBLIN B. and SCHRIEFFER J.R., Phys. Rev., (in press) have taken into account the spinorbit coupling of the conduction electrons in alloys containing cerium impurities. Their analysis is restricted because they specifically considered the hybrid exchange interaction, which is large only in cerium and ytterbium. We consider a general class of spin-dependent interactions which have much wider applicability. VAN VLECK J.H., Phys. Rev. 52, 1178 (1937); STEVENS K.W.H., Rev, mod. Phys. 25, 166 (1953); DZIALOSHINSKY I., J. Phys. Chem. Solids, 4, 241 (1958), and MORIYA T., Phys. Rev. Lett. 4, 228 (1960), Phys. Rev. 120, 91 (1960).
8.
De GENNES P.G., J. Phys radium, 23, 510 (1962).
9.
WATSON R.E., KOIDE S., PETER M. and FREEMAN A.J., Phys. Rev. 139, A167 (1965).
10. ii. 12. 13.
LEVY P.M., Chem. Phys. Lett. 3, 556 (1969), and COPLAND G.M. and LEVY P.M., Phys. Rev. (in press). FANO U. and RACAH G., irreducible Tensorial Sets (Academic Press, New York, 1959). IRKHIN YU.P., Soviet Phys. —JETP 23, 253 (1966); DRUZHININ V.V. and IRKHIN YU.P., Ibid. 24, 1250 (1967); IRKHIN YU.P., DRUZHININ V.V. and KAZAKOV A.A., ibid. 27, 633 (1968). De GENNES P.G., Compt. rend., 247, 1836 (1958).
14.
BOZORTH R.M., MATTHIAS ~T., SUHL H., CORENZWIT E. and DAVIS D.D., Phys. Rev. 115, 1595 (1959); WILLIAMS H.J., WERNICK J.H., NESBITT E.A. and SHERWOOD R.C., J. Phys. Soc. Japan, 17, Supp. B—i, 91(1962).
15.
JACCARINO V., MATTHIAS B.T., PETER M., SUHL H. and WERNICK J.H., Phys. Rev. Lett. 5, 251 (1960).
16.
KANEMATSU K., J. Phys. Soc. Japan 25, 628 (1968) has considered the effect of 4f orbital currents in polarizing the conduction electrons. This effect is not sufficient to explain the observed data on the Curie points of RAI2.
1818 17.
18.
INDIRECT EXCHANGE VIA SPIN—ORBIT COUPLED STATES
Vol.7, No.24
The symmetry at the rare-earth sites in the cubic Laves compounds, RA12 and R1r2, is Td, see reference 15, and WERNICK J.H. and GELLER S., Trans. Met. Soc. AIME 218, 866 (1960). For the hexagonal Laves compounds ROs2, the symmetry is nearly tetrahedral (Td), see FRIAUF J.B., Phys. Rev. 29, 34 (1927). For this symmetry and with the molecular field approximation, anisotropic terms of the third and higher degrees only can enter the total interaction Hamiltonian for an ion and its surroundings. As we obtain a good fit to T~without these and other higher degree scalar terms we do not consider them in this analysis. The relative magnitudes of the higher degree parameters Q,~,0and Q,~,compared to Q010, are reasonable. See references 3, 4 and SPECHT F., Phys. Rev. 162, 389 (1967).
Quand nous tenons compte du couplage spin-orbite des états intermédiaires, nous trouvons qu’il existe de nouveaux termes dans l’hamiltoriien d’é~hangeindirect. Nous montrons que ces termes sont nécessaires pour expliquer la variation de la temperature de Curie d’une série de composes intermdtalliques de terres rares.