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Influence of magnetic order on the crystal field in cubic alloys
Solid State Communications, Vol. 21, pp. 941—943, 1977.
Pergainon Press.
Printed in Great Britain.
INFLUENCE OF MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS R.A.B. Devine* and O.K. Rayt *Laboratoire de Spectrométrie Physique, 53X, 38041 Grenoble Cedex, France, tLaboratoire de Magn~tisme, Centre National de la Recherche Scientifique, 166X, 38042 Grenoble Cedex, France.
Received 14 January 1977
L.F. Bertaut
A simple model is used to study the effect of magnetic ordering on the Eg and T2g band contributions to the fourth order crystal field parameter in rare-earth intermetallic compounds. Reversal of sign of the parameter as found experimentally is predictable from the model.
1
A
4
A4(E ) + A4(T2 ) + A4(latt.) (1) g g We will now study the effect of the exchange splitting of the d bands due to the localised 4f moments. We - use the exchange hamiltonian ~ ~ ~ fd .S
In recent neutron diffraction experiments on paramagnetic ErA12 it has been found that the crystal field parameters deduced differ 2. The diffrom those determined from magnetization meaference in absolute phase magnitude but, surementsis innottheonly ferromagnetic in the case of the fourth order parameter, A 4, a reversal of sign occurs. Independent measuearths rements inon LaA1 paramagnetic dilute alloys of 3. rareSince the significant difference between 2 confirm the reversed sign the two phases is the magnitude of the exchange interaction between the localised 4f and conduction electrons it is clearly of interest to examine the influence of this exchange on the various contributions to the fourth order crystal field parameter. In the following We will use a very simple model to study the effect on the 5d rare-earth bands, The predominant contribution from the conduction electrons to the crystal field is usually4 assumed to come from the d bands. In particular the 5d atomic level associated with the rare-earth ion is degenerated Into bands In the solid and split in a cubic lattice crys— talline electric field Into Eg and T2g bands. For many cases5 there are peaks in the density of states near the Fermi level correlated with the Eg and T2g bands and although the crystalline electric field contains contributions from electrons of all energies up to EF the predominate contribution comes from the Eg and T2g electrons. Thus, as In the problem of band
=
In the molecular field approximation we take 57 ~ 5z (2) so that in the ~ gj paramagnetic fd Z phase <~> is zero and exchange has no effect. In the ferromagnetic phase <3z> will be non-zero, the spin up and spin down bands will then be split by this interaction. For simplicity we assume that the form of the Eg and T2g bands is Lorentzian with width ~. The positions of the centres of the bands will be represented in energy by E(Eg) and E(T2g) and we normalise the total number of electrons possible In the Eg band to four and in the 12g to six. If we represent (gj - 1 !~ci <3~>by y we can write the total number of spin up and spin down electrons in each band up to the Fermi level as f EF n+ = I f(E) O(E + y)dE
I n+
EF F(E) D(E
=
-
y)dE
J+.~ where 0(E)
2
=
E 0
polarisatlon, the problem becomes one essentially Involving electrons near the Fermi energy. For the simple model we will assume that the total fourth order parameter Is a sum of Eg and T2g contributions plus a lattice contribution which Includes all other terms :
I
+
-
E
C
-
~o - E(Eg) or E(T2g)) The Fermi function, f(E), is taken to be unity up to EF in the low temperature approximation. Integration yields E tan
E g
=
-
E
E
±y)/t~)
-
tan
F
{
3
~
+
T2g
tan~ ~ EEg ET2g
941
IA) E
T2g
-
Eg
-
(C
2
T2g
E Eg
-
IA)}
(1)
942
MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS
where the - sign in the bracket refers to n+ and the and the + sign to n+. The total number ofT2g E g electrons+ n’~(T2g). is thus nF(Eg) + n’~(E~) and n+(T2g) The difference ni - ofn+ permits us to calculate the polarization of the band but we will not consider this problem here,
Vol. 21, No. 10
case a negative potential results in the Eg band being lowest in energy. In figure 2, we show a simulation of the result for the CsCl structure 0yZn. The Eg and T2g bands are centred above EF, the T2g approximately 1.8 eV
above the Eg. The lattice contribution is
In figure 1, we show the general results for Eg and T2g bands, we ignore wave vector dependence of the crystal field parameter and assume the contributions :
-
200 A 4(E9)
A (E ) 4 g
(n+[E
=
A 4 CT 2g )
=
g
)
+
(nf(T 2g )
n+(E 3) g +
C1
n4IT2 g )) C 2
A~IatL)=—lgoK
100-
where Ci and ~2 are such that for a totally degenerate E and T2g band A4(E ) A4(T2~) i.e. the d c~arge distribution ~s spherical symmetric. For the general case we have assumed
~ev 2
0
1
3
C
4 _~_.-j
120044(T29)
-1.0
-100
-
lowest centred at 0.8 eV above the
900 Fermi level. The T2g band is centred at 2.8 eV above the Fermi level.
Figure 200 2
Example of the case of the Eg band 0
-200
0
0 3.0
I 2
1
-
3
0.5/
3ev
-
-400
-
-600
ev Figure 1
: A4(E 3 and A4(T2g) In °K as a func~ion of exchange parameter and position of the centre of the band with respect to the Fermi level. The bandwidth is taken to be 0.5 eV and the Fermi level at 7.0 eV.
that C1 is equal to 300 K per Eg electron and -200 K per T2g electron, the T2g contrib~tion 8g . In order to use the general result given in being sign to that of the the Eg figure opposite 1. it is in sufficient to locate and T2g bands relative to the Fermi level and
for an exchange value given ly) add A4CEg) and A4CT2g) to the known lattice contribution, A 4(latt.). We will consider two examples, one coincident with a negative lattice potential and one with a positive potential. In the former
400 ~
-800
600
I 0
1
~‘ev
2
3
Figure 3 : Example of the case of a Tlg band lowest for various positions of the E bend. The centre of the T2g band
i~taken at 0.5 eV below the Fermi level. Left hand scale, A4, righ hand scale A4CEg) + A4(T2g). both in units of °K.
Vol. 21, No. 10
MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS 943 6. As can be seen a reversal so that for a relatively close lying Eg band taken bethe -190 K of signto of crystal field parameter might (eg the case of Eg at the Fermi level) a be anticipated but only for large exchange reversal of sign of the crystal field parameter value. In reality the value of ~ is found7 between paramagnetlc and ferromagnetic phases to be 0.04 eV So that (g 3z> Is is possible. typically a maximum of 0.15 In this case One may conclude that based on the simple 3 - eV. I) Jfd < no slgnifican change in crystal field parameter model the two ingredients for sign reversal, with exchange is anticipated, or indeed or indeed strong parameter variation11, with observed experimentally, state of order may be a large exchange parameter For the second example the T2g band i8 and preferably a small splitting between the lowest - this we take to be representative of T2g and Eg bands. Both of these conditions the case of rare-earth A12 although the band might be satisfied for the cases of rare-earth structure calculations8 for LaAl 2 do not clearly Al2, Fe2, Co2 and Ni2 compounds. The latter indicate which band would be lowest. Because of condition might also be satisfied for rare-earth the large value of co-ordination number for Al3. Measurements of the crystal field parameters rare-earth ions in the cubic Laves phase in both paramagnetIc and ordered phases of these compounds the potential tends to be nearly compounds would therefore be desirable. spherical and consequently the splitting between the Eg and T2g bands is expected to be small. The results are given in figure 3. In this case we assume a point charge lattice sum Acknowledgements 9 based on rare-earth neighbours For with We Drs.gratefully B. Barbara and P. Morin. Mr. 0. of +230 K from a simple point charge only. calculaacknowledge discussions the tion Al 2 compounds 10 thanthe the exchange DyZn caseis and found of the to be Schmitt of results Is his thanked APW band for communicating calculations for to usOyZn the orderlarger of 0.2 eV. y may thus be equal to 0.75 eV prior to publication. much
I
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11.
HAPPEL, H., v. BLANCKENHAGEN, P.. KNORR, K. and MURANI, A.P., Proc. of the 2nd Intern. Conf. on Crystal Fields In Metals and Alloys, Zurich, 1976 (in press). PURWINS, H.G., WALKER, E., BARBARA, B., ROSSIGNOL, M.F. and FURRER, A., J. Phys. C9, 1025 (1976). HOENIG, H.E.. HAPPEL, H., NJOO, H.K. and SEIM, H., Proc. of the 1st Conf. on Crystal Fields In Metals and Alloys, Montreal 1974. DAS, K.C. and RAY. O.K., Solid Stat. Commun. 8, 2025 (1970). BELAKHOVSKY, M., PIERRE, 3. and RAY, O.K., 3. Phys. F 12, 2274 (1975). TANNOUS, C., RAY, O.K. and BELAKHOVSKY, M., 3. Phys. F6, 2091 (1976). A.P.W. calculations for the paramagnetic phase for OyZn and OyRh give A4(Eg) “~ 300 K per electron - there is evidence here for wavevector dependence CO. Schmitt, to be published). We Ignore wavevector dependence within the framework of our very simple model. MORIN, P., Ph.D. Thesis, C.N.R.S. N° A.0. 9323. SWITENOICK, A.C., Proc. of the 10th Rare Earth Res. Conf. (1973, unpublIshed). BLEANEY, 5., Proc. Roy. Soc. A276, 28 (1963). BERTHIER, Y. • BARBARA, 8. and DEVINE, R.A.B., to be published. An Irregular variation of the fourth order crystal field parameter in RRh compounds has been observed by Chamard-Bois (Thesis, University of Grenoble. 1974, C.N.R.S. N° A.0. 9748).
Pergainon Press.
Printed in Great Britain.
INFLUENCE OF MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS R.A.B. Devine* and O.K. Rayt *Laboratoire de Spectrométrie Physique, 53X, 38041 Grenoble Cedex, France, tLaboratoire de Magn~tisme, Centre National de la Recherche Scientifique, 166X, 38042 Grenoble Cedex, France.
Received 14 January 1977
L.F. Bertaut
A simple model is used to study the effect of magnetic ordering on the Eg and T2g band contributions to the fourth order crystal field parameter in rare-earth intermetallic compounds. Reversal of sign of the parameter as found experimentally is predictable from the model.
1
A
4
A4(E ) + A4(T2 ) + A4(latt.) (1) g g We will now study the effect of the exchange splitting of the d bands due to the localised 4f moments. We - use the exchange hamiltonian ~ ~ ~ fd .S
In recent neutron diffraction experiments on paramagnetic ErA12 it has been found that the crystal field parameters deduced differ 2. The diffrom those determined from magnetization meaference in absolute phase magnitude but, surementsis innottheonly ferromagnetic in the case of the fourth order parameter, A 4, a reversal of sign occurs. Independent measuearths rements inon LaA1 paramagnetic dilute alloys of 3. rareSince the significant difference between 2 confirm the reversed sign the two phases is the magnitude of the exchange interaction between the localised 4f and conduction electrons it is clearly of interest to examine the influence of this exchange on the various contributions to the fourth order crystal field parameter. In the following We will use a very simple model to study the effect on the 5d rare-earth bands, The predominant contribution from the conduction electrons to the crystal field is usually4 assumed to come from the d bands. In particular the 5d atomic level associated with the rare-earth ion is degenerated Into bands In the solid and split in a cubic lattice crys— talline electric field Into Eg and T2g bands. For many cases5 there are peaks in the density of states near the Fermi level correlated with the Eg and T2g bands and although the crystalline electric field contains contributions from electrons of all energies up to EF the predominate contribution comes from the Eg and T2g electrons. Thus, as In the problem of band
=
In the molecular field approximation we take 57 ~ 5z (2) so that in the ~ gj paramagnetic fd Z phase <~> is zero and exchange has no effect. In the ferromagnetic phase <3z> will be non-zero, the spin up and spin down bands will then be split by this interaction. For simplicity we assume that the form of the Eg and T2g bands is Lorentzian with width ~. The positions of the centres of the bands will be represented in energy by E(Eg) and E(T2g) and we normalise the total number of electrons possible In the Eg band to four and in the 12g to six. If we represent (gj - 1 !~ci <3~>by y we can write the total number of spin up and spin down electrons in each band up to the Fermi level as f EF n+ = I f(E) O(E + y)dE
I n+
EF F(E) D(E
=
-
y)dE
J+.~ where 0(E)
2
=
E 0
polarisatlon, the problem becomes one essentially Involving electrons near the Fermi energy. For the simple model we will assume that the total fourth order parameter Is a sum of Eg and T2g contributions plus a lattice contribution which Includes all other terms :
I
+
-
E
C
-
~o - E(Eg) or E(T2g)) The Fermi function, f(E), is taken to be unity up to EF in the low temperature approximation. Integration yields E tan
E g
=
-
E
E
±y)/t~)
-
tan
F
{
3
~
+
T2g
tan~ ~ EEg ET2g
941
IA) E
T2g
-
Eg
-
(C
2
T2g
E Eg
-
IA)}
(1)
942
MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS
where the - sign in the bracket refers to n+ and the and the + sign to n+. The total number ofT2g E g electrons+ n’~(T2g). is thus nF(Eg) + n’~(E~) and n+(T2g) The difference ni - ofn+ permits us to calculate the polarization of the band but we will not consider this problem here,
Vol. 21, No. 10
case a negative potential results in the Eg band being lowest in energy. In figure 2, we show a simulation of the result for the CsCl structure 0yZn. The Eg and T2g bands are centred above EF, the T2g approximately 1.8 eV
above the Eg. The lattice contribution is
In figure 1, we show the general results for Eg and T2g bands, we ignore wave vector dependence of the crystal field parameter and assume the contributions :
-
200 A 4(E9)
A (E ) 4 g
(n+[E
=
A 4 CT 2g )
=
g
)
+
(nf(T 2g )
n+(E 3) g +
C1
n4IT2 g )) C 2
A~IatL)=—lgoK
100-
where Ci and ~2 are such that for a totally degenerate E and T2g band A4(E ) A4(T2~) i.e. the d c~arge distribution ~s spherical symmetric. For the general case we have assumed
~ev 2
0
1
3
C
4 _~_.-j
120044(T29)
-1.0
-100
-
lowest centred at 0.8 eV above the
900 Fermi level. The T2g band is centred at 2.8 eV above the Fermi level.
Figure 200 2
Example of the case of the Eg band 0
-200
0
0 3.0
I 2
1
-
3
0.5/
3ev
-
-400
-
-600
ev Figure 1
: A4(E 3 and A4(T2g) In °K as a func~ion of exchange parameter and position of the centre of the band with respect to the Fermi level. The bandwidth is taken to be 0.5 eV and the Fermi level at 7.0 eV.
that C1 is equal to 300 K per Eg electron and -200 K per T2g electron, the T2g contrib~tion 8g . In order to use the general result given in being sign to that of the the Eg figure opposite 1. it is in sufficient to locate and T2g bands relative to the Fermi level and
for an exchange value given ly) add A4CEg) and A4CT2g) to the known lattice contribution, A 4(latt.). We will consider two examples, one coincident with a negative lattice potential and one with a positive potential. In the former
400 ~
-800
600
I 0
1
~‘ev
2
3
Figure 3 : Example of the case of a Tlg band lowest for various positions of the E bend. The centre of the T2g band
i~taken at 0.5 eV below the Fermi level. Left hand scale, A4, righ hand scale A4CEg) + A4(T2g). both in units of °K.
Vol. 21, No. 10
MAGNETIC ORDER ON THE CRYSTAL FIELD IN CUBIC ALLOYS 943 6. As can be seen a reversal so that for a relatively close lying Eg band taken bethe -190 K of signto of crystal field parameter might (eg the case of Eg at the Fermi level) a be anticipated but only for large exchange reversal of sign of the crystal field parameter value. In reality the value of ~ is found7 between paramagnetlc and ferromagnetic phases to be 0.04 eV So that (g 3z> Is is possible. typically a maximum of 0.15 In this case One may conclude that based on the simple 3 - eV. I) Jfd < no slgnifican change in crystal field parameter model the two ingredients for sign reversal, with exchange is anticipated, or indeed or indeed strong parameter variation11, with observed experimentally, state of order may be a large exchange parameter For the second example the T2g band i8 and preferably a small splitting between the lowest - this we take to be representative of T2g and Eg bands. Both of these conditions the case of rare-earth A12 although the band might be satisfied for the cases of rare-earth structure calculations8 for LaAl 2 do not clearly Al2, Fe2, Co2 and Ni2 compounds. The latter indicate which band would be lowest. Because of condition might also be satisfied for rare-earth the large value of co-ordination number for Al3. Measurements of the crystal field parameters rare-earth ions in the cubic Laves phase in both paramagnetIc and ordered phases of these compounds the potential tends to be nearly compounds would therefore be desirable. spherical and consequently the splitting between the Eg and T2g bands is expected to be small. The results are given in figure 3. In this case we assume a point charge lattice sum Acknowledgements 9 based on rare-earth neighbours For with We Drs.gratefully B. Barbara and P. Morin. Mr. 0. of +230 K from a simple point charge only. calculaacknowledge discussions the tion Al 2 compounds 10 thanthe the exchange DyZn caseis and found of the to be Schmitt of results Is his thanked APW band for communicating calculations for to usOyZn the orderlarger of 0.2 eV. y may thus be equal to 0.75 eV prior to publication. much
I
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11.
HAPPEL, H., v. BLANCKENHAGEN, P.. KNORR, K. and MURANI, A.P., Proc. of the 2nd Intern. Conf. on Crystal Fields In Metals and Alloys, Zurich, 1976 (in press). PURWINS, H.G., WALKER, E., BARBARA, B., ROSSIGNOL, M.F. and FURRER, A., J. Phys. C9, 1025 (1976). HOENIG, H.E.. HAPPEL, H., NJOO, H.K. and SEIM, H., Proc. of the 1st Conf. on Crystal Fields In Metals and Alloys, Montreal 1974. DAS, K.C. and RAY. O.K., Solid Stat. Commun. 8, 2025 (1970). BELAKHOVSKY, M., PIERRE, 3. and RAY, O.K., 3. Phys. F 12, 2274 (1975). TANNOUS, C., RAY, O.K. and BELAKHOVSKY, M., 3. Phys. F6, 2091 (1976). A.P.W. calculations for the paramagnetic phase for OyZn and OyRh give A4(Eg) “~ 300 K per electron - there is evidence here for wavevector dependence CO. Schmitt, to be published). We Ignore wavevector dependence within the framework of our very simple model. MORIN, P., Ph.D. Thesis, C.N.R.S. N° A.0. 9323. SWITENOICK, A.C., Proc. of the 10th Rare Earth Res. Conf. (1973, unpublIshed). BLEANEY, 5., Proc. Roy. Soc. A276, 28 (1963). BERTHIER, Y. • BARBARA, 8. and DEVINE, R.A.B., to be published. An Irregular variation of the fourth order crystal field parameter in RRh compounds has been observed by Chamard-Bois (Thesis, University of Grenoble. 1974, C.N.R.S. N° A.0. 9748).