Magnetic interaction in europium chalcogenides

0038-1098/83 $3.00 + .00 Pergamon Press Ltd.

Solid Sta/,e Communications, Vol. 48, No. 9, pp. 795-797, 1983. Printed in Great Britain.

MAGNETIC INTERACTION IN EUROPIUM CHALCOGENIDES Ven-Chung Lee Department of Electronics Engineering, National Taiway Institute of Technology, Taipei, Taiwan, Republic of China* and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60201, U.S.A. and L. Liu Department of Physics and Astronomy, Northwestern University, Evanston, IL 60201, U.S.A.

(Received 23 June 1983 by F. Bassani) The magnetic interaction in EuO, EuS and EuSe is characterized by a ferromagnetic nn interaction and an antiferromagnetic nnn interaction. We propose a mechanism for this interaction; it is an indirect exchange mediated by electrons occupying chalcogen valence bands. The LCAO energy bands are used in a perturbative calculation of the indirect exchange between two Eu ions. The calculated results agree with experiments both in sign and in magnitude for all three compounds considered. EUROPIUM OXIDE and europium chalcogenides are well-known magnetic semiconductors, which crystallize in simple rock salt structure. It has been determined experimentally that while the nearest neighbor (nn) Eu ions interact ferromagnetically, the next nearest neighbor (nnn) ions have an antiferromagnetic interaction. Despite several attempts [ 1] to find the underlying mechanism for this magnetic interaction, the results so far are far from being satisfactory. One of us [2] (LL) has recently interpreted this interaction as an indirect exchange mediated by chalcogen p valence band electrons which are magnetically polarized by the localized f electrons. Since the above mentioned behaviour of the magnetic interaction in europium compounds is very similar to what has been found recently [3] for the indirect exchange interaction in semiconductors with s-p bands, we decided to pursue this interpretation in a more quantitative way. The energy bands for EuO, EuS, and EuSe are first obtained by the linear combination of atomic orbital method (LCAO), and then used in a perturbative calculation of the indirect exchange. The calculated results agree with the measured nn and nnn magnetic interaction not only in sign but also in magnitude for all three substances considered. We assume that the well localized felectrons play no other role than forming a local spin Si which magnetically polarizes other valence band electrons. The non-magnetic Bloch electrons are represented by LCAO wave functions

*Permanent Address.

~nk : ~. Cn(k) Z eik'Ri~Pkv(r --Ri),

(1)

i

where ekv denotes the atomic orbital with index v and quantization axis along k. The band electron mediated indirect exchange interaction between two local spins as obtained by second-order perturbation theory is of the Heisenberg form [4] / , ~ = - 2/(ai~)si. sj.

(2)

The interband coupling constant 1is a function of interspin distance Rij and may be calculated once the band structure is known according to: 1 lgl2 e~(k'-k) • R/j /(Rij) = ~ n ~ ' _ E,,(k') - e , ( k ) + c.c.,

(3)

where E~,(k') and'E~(k) are the conduction and the valence band energy, respectively. The wave vector summation is over the entire band. The interband exchange matrix element is given by M = ~

' C~~'(k)C[, (k)O~v(g2)Juv ,

(4)

/2, V

where J~v is the exchange integral involving the localized ]'orbital and the s, p, or d atomic orbital, all centered in the same unit cell. From symmetry, we only have to consider three such exchange integrals, ]ss, Jpp and ,lad, associated with s-f, p - f , and d - f exchange, respectively. The function 0 arises from the overlap between the angular part of an atomic orbital oriented along the valence band k-axis and that quantized along the conduction band k'-axis, and hence depends on the angle ~2 between the two wave vectors. 795

796

MAGNETIC INTERACTION IN EUROPIUM CHALCOGENIDES

Table 1. Angular factor 0(~ ) associated with Jpp and Jdd in the interband exchange matrix elements

Vo1.,48, No. 9

( e"

Angular dependence Transition Parallel coupling AI--A 1 As-As Perpendicular coupling AI-A s AI-A2 As-A2 As-A2

Jaa

Jpp

~(3 cos2~2 -- 1) cos 2 ~ cos ~2

cos ~2 [~os ~2

½sin 2 ~ sin2~ sin ½sin 2£Z

sin~2

"

1

EuSe

As has been found previously [3], the angular factor 0 in the interband exchange matrix element plays an important role in determining the sign of the contribution to the indirect exchange. We give the form of this angular function in Table 1. Two types of angular dependence are distinguished. For two bands having p - d symmetry mixing with each other, the exchange coupling is the strongest when the two wave vectors are parallel to each other. On the other hand, a valence band state which does not mix with the conduction band favors a perpendicular coupling with the latter. While the parallel coupling favors ferromagnetic exchange, the perpendicular coupling contributes an antiferromagnetic term. The sign of the indirect exchange depends on the relative magnitude of the two competing contributions. Although the existence of competing components can be understood from qualitative consideration [3], in order to obtain quantitative results we still have to know the realistic band structure. We have calculated the band structure for EuO, EuS, and EuSe by an empirical LCAO method along the I ' - X axis. Ten basic orbitals were taken; they are the 5d and 6s orbitals of the europium and the valence s and p orbitals of the anion. The E u - E u and anion-anion overlap integrals are assumed to be small, but the overlap between Eu 5d and anion s and p orbitals is taken into account. The energy and the overlap integrals are adjusted to fit several measured energy gaps and also the width of the p band. Our calculated band structures as shown in Fig. 1 agree with the APW calculations of Cho [5] in their over-all features. The most significant feature of the band structure relevant to our present calculation is the upward hump of the A~ (stemming from F12) conduction band and the downward hump of the Al (stemming from F~s) valence band. They signify a strong p - d mixing between these two bands. In addition, p - d mixing also occurs between the As (stemming from P2s,)

I

F

A

X F

A

X F

A

X

Fig. 1. Calculated energy band structure for EuO, EuS and EuSe along the P - X axis. The numerical numbers label the symmetry of the band. The magnetic f bands have not been included in our band structure calculation. conduction band and the As (stemming from P~s) valence band. Since the s-like A~ conduction band is free-electron like, it does not mix with other bands. But there is s - d mixing between the lowest A1 valence band and the highest A~ conduction band in Fig. 1. For the purpose of calculating the indirect exchange, we assume that the band structure is isotropic and represented by that along the F - X axis throughout a spherical Brillouin zone of radius 2~/Ro where R0 is the lattice constant. All interband transitions listed in Table 1 have been taken into account. The calculated results based on equation (3) are presented in the following form:

I(Rij ) = A(Rij)J~p + B(Rij)JppJdct + C(Rij)J~ct +

+ D(gi/Mdd + E ( R i / J .

(5)

Since we have made an assumption of isotropic bands, the indirect exchange only depends on the magnitude of Rii. In Table 2 we give our results at Rii = 0 and Rii = Ro (i.e., nnn magnetic ion distance). As can be seen from Table 2, the most important term which determines the sign of the magnetic interaction at Rij = Ro is B(Ro). This term comes from parallel coupling A1-A~ and As-As between bands with p - d mixing. The process is such that if the electron-hole excitation (A~ -+ A~) is via the p component, the

Vol. 48, No. 9

MAGNETIC INTERACTION IN EUROPIUM CHALCOGENIDES

797

Table 2. Calculated values for A, B, C, D and E defined in equation (5) R 0.

A (eV -1)

B (eV -1 )

C (eV -1 )

D (eV -1 )

E (eV -1)

0 Ro

6.5 5.9

8.2 --1.5

X 10 -4

2.4 x 10 -a --1.6 x 10 -s

1.I x 10 -s --1.6 x 10 -s

2.6 x 10 -a 2.4 × 10 -s

EuS

0 Ro

7.7 x 10 -3 9.5 x 10 -6

1.0 x 10 -s --2.0 x 10 -4

3.2 x 10 -3 --2.1 x 10 -s

1.3 x 10 -s --1.8 x 10 -s

2.5 x 10 -a 2.4 x 10 -s

EuSe

0 Ro

8.7 x 10 -3 1.0 × 10 -s

1.2 x 10 -s --2.3 × 1 0 - 4

3.9 x 10 -3 --2.6 x 10 -s

1.3 x 10 -s --1.8 x 10 -s

2.4 x 10 -3 2.3 x 10 -s

EuO

× 1 0 -3 X 10 -6

× 10 -6

Table 3. Comparison o f calculated indirect exchange with experimental values. The coupling constant I 1 and I2 (in units o f degree Kelvin) denote nn and nnn Eu ion interaction, respectively. Calculated values are based on the assumption that Jss = Jpp = Jdd = 0.2 eV. The value for I 1 is obtained by assuming an interpolation formula I(Rij ) = a ( R o -- Rti)t~ + I2for 0 ~< Rij ~< R0. The values o f fl used are also given in the table

Ii(exp.[6l°K) ~ (exp. [7] °K) It (calc. °K) ~3 I 2 ( e x p . [6] °K) ~ ( e x p . [7] °K) /2 (calc. °K)

EuO

EuS

EuSe

0.58~0.67 0.55 0.60 1.7 --0.07 0.15 --0.071

0.2 "0.21 0.20 2.5 --0.06"~--0.14 --0.11 --0.096

0.11 0.12 2.8 ~/d~iI2L --0.09 --0.11

II, l~llz[

e l e c t r o n - h o l e annihilation (A1 <- A1) involves the d component. There are several factors which affect the sign of B: the wave function orthogonality between the interacting bands, the angular factor associated with the excitation and annihilation process, and the phase factor e i ( k ' - k ) ' R i j in equation (3). At R ~ / = Ro, the signs of B for all three compounds are negative, if we assume that Jss, Jpp and Jaa are approximately equal to each other, the net sign of the coupling constant in equation (5) is negative, in other words, the indirect exchange at nnn magnetic ion distance is antiferromagnetic. When we use the atomic exchange integral value of 0.2 eV for Jss ~ Jpp ~ Jaa, the calculated nnn interaction strength agrees with the experimental value. The comparison is made in Table 3. As our scheme only allows us determine the

interaction at inter-spin distance equal to an integer multiple of the lattice constant, we can not calculate the nn magnetic interaction precisely. However, we can make a reasonable estimate using the interaction values at R o = 0 and R a = R0 given in Table 2. The estimated interactions at R t / = Ro/x/2 are also in good agreement with the measured values. We note that while the ferromagnetic nn interaction decreases in strength from EuO to EuS and then EuSe, the antiferromagnetic nnn interaction increases in magnitude. This experimentally determined trend is reproduced in our calculation. From Fig. 1 one sees that the conduction bands get closer to the valence band in going from left to right. It is this variation in band structure which produces the above mentioned trend. Details of the present calculation will be reported elsewhere.

Acknowledgements - One of us (VCL) would like to thank Professor T.Y. Wu for his assistance in obtaining a government fellowship for him to come abroad and accomplish the present piece of research. REFERENCES 1. 2. 3. 4. 5. 6. 7.

See, for example, T. Kasuya, CRC Critical Reviews in Solid State Sciences, 3 , 1 3 1 (1972). L. Liu, SolidState Commun. 46, 83 (1983). V.C. Lee & L. Liu, Solid State Commun. 48, 341 (1983). L. Liu, Phys. Rev. B26,975 (1982). S.J. Cho, Phys. Rev. B 1 , 4 5 8 9 (1970). S. Methfessel & D.C. Mattis, Encyclopedia of Physics, Vol. 18/1. p. 389. Springer Verlag, Berlin (1968). W. Zinn, JMMM 3, 23 (1976).