Theory of Raman scattering in magnetically ordered phases of EuSe and EuTe

Solid State Communications, Vol. 23, pp. 589—592, 1977.

Pergamon Press.

Printed in Great Britain

THEORY OF RAMAN SCA1TERING IN MAGNETICALLY ORDERED PHASES OF EuSe AND EuTe Y. Ousaka, 0. Sakai and M. Tachiki The Research Institute for Iron, Steel and Other Metals, Tohoku University, Katahira,Sendai 980, Japan (Received 26 April 1977 by 7’. Nagamiya) The first order Raman scatteringin various phases of EuSe and EuTe was theoretically studied. The Raman lines which depend on spin structures are caused via the spin-dependent polarizabilities. The phonon dispersion relations of the crystals were calculated on the breathing shell model. The Stokes shifts and intensities of the Raman lines, and their polarization selection rules obtained from the present theory are in good agreement with experiments. RECENTLY Silberstein et a!. [1] have made Raman scattering experiments on EuSe and EuTe and observed the Raman lines in the several magnetic phases occuring under external magnetic fields as shown in Fig. 1. The objective of the present paper is to explain these Rarnan lines. The usual first order phonon Raman scattering is forbidden in Eu-chalcogenides, since their crystal structure is the NaC1 structure. However, the coupling between ordered spins and phonons makes phonons with finite wave numbers Raman-active. This scattering intensity is expressed as a function of the Raman polarizability which depends on both the spins and the

Table 1(a). Values ofparameters used in calculations on the breathing shell model 151 for EuSe and EuTe Parameters A B B” A’ B’

z

Y 6dyncm’) k k 1 (l0 2 (106 dyn cm~)

phonons [3]. Sakai and Tachiki [4] (hereafter referred to as S—T) have shown that the cross effect of the 4! spin—orbit interaction and the exciton—phonon interaction is the most important mechanism for the Raman scattering intensity in the paramagnetic phase of these compounds. On the basis of their theory, we obtain the Stokes shifts, the polarization selection rule, and the scattering intensity using phonon dispersion curves. The phonon dispersion curves are calculated semiempirically by using the breathing shell model [5] and the result is shown in Fig. 2. The parameters introduced by Woods et a!. [6] and by Nusslein and Schröder [5] are listed in Table 1. Details of the calculation will be published in a separate paper [13]. When the polarization directions of the incident and scattered lights are the ~t- and v-directions, respectively, the scattering intensity per solid angle and unit frequency is expressed as follows: 1 I~w) = dt e~o)t
(‘s)

~—J

where a~,indicates the ~ivcomponent of the polarizability and w~the angular frequency of the incident light In (1), (A denotes the thermal average ofA. According to S—T, the polarizability which

EuSe 25.078 —0.517 0.005 1.406 0.278 1.183 6.350 3.782

EuTe 26.266 1.408 0.048 1.153 0.075 1.211 6.140 3.548

—

—

—-

—

--

—

1.598

1.289

Table 1(b). Data used in calculation of T~ble1(a). The frequency ofon thethe TAbasis phonon at the WTA(QL), is calculated ofthe threeL-poznr, body shell model f12];for details see reference [13]

Elastic constants 2) [7] (10~dyncm

C11 C~ 2 C

Dielectric constants [8,9] Infrared frequencies [8,9] (cm~) Lattice constants [10] (A) Ionic 3) polarizabilities [11] (A WTA(QL) (cm’)

I.T0 ~~LO

2r0 ~2

EuSe 11.6 1.2 2 28

EuTe 9.36 0.67 1 63

9.5 4.8 127 182 6.191

8.23 4.18 102.3 141.5 6.5 84

2.4 5.5 82.9

2.4 6.4 76.7

dominantly contributes to the scattering intensity is given by

~‘

589

590

THEORY OF RAMAN SCATTERING IN EuSe AND EuTe 10

i01

a~~— (_2i)f 1

A

x

~

+

~S~(2~k

Vol. 23, No.8

S~k

TI__N ~1L~o

0

0

U

Ui

S~flk~+ ?k

U’

sin k~+ 4 sin k~)

J

‘~

FerrlAF-I~3 AF-fl

.~

sin k 2

—

Xk

S~fl

2k

—

Yk

sin k~)

H:%koe

sin k~)+

{S~(~S~flk~+

+

+ S~(~ksin k~+

4

sin k). )}

+ —,~=-~S~(~k sin k

2

—

I84cm~

Z~sin k~)

+

S~(?k sin k~—Zk

+

cyclic permutation of (x.vz)

~

sin k3)}J

4

I

(2)

where Xk, etc. are the Fourier amplitudes of the anion displacements and S~,is the Fourier amplitude of the spins. In (2), the lengths are measured in units of the distance between an Eu ion and one of its neighboring chalcogen ions. In the AF—I phase of EuSe (ft~) [21 the averages of the Fourier amplitudes of the spins are given by (S.QL2)

=

(Sq)

=

50

100

~5O

i

Raman shift(cm) Fig. 1. First order phonon Raman spectra of EuSe for T’-~ 4K, obtained by Silberstein, Tekippe, Dresselhaus and Aggarwal [1]. TheSchmutz, inset is the magnetic phase diagram of EuSe in which two antiferromagnetic phases AF—l (1~t~) and AF—I1 (t.L.t.i.), a ferrimagnetic phase (tt4.), and a ferromagnetic phase are included [2].

),

200

200 EuSe

~JN(1±i)(S)/2 0forq±Q~/2

EuTe

_ /

where QL denotes the wave number vector at the L-point in the Briulouin zone of the f.c.c. lattice, (S> a spin in the up-spin sublattice, arid N the number of Eu ions per unit volume. The expression (3) is substituted for the spin operator in (2). Since the magnitude of A in (2) is much larger than those of B, C, and D as discussed in S 1, the xy component of the polarizability is approximately expressed as2)A = ~J[M ~s/~ex(2J~L’ 2wo(QL/2)N]

•

200

~ ‘T>~ 100

g

‘~

—~

6L 2~ + {(aQL26 +a..QL,26)(s + (a_QL/z5 + aQL/,~)(S_QLJ2)}.

0

-~

F

(4)

where 6 denotes the LA and LU phonon branches, w 2) the angular frequency of the phonon with 6 (QLI wave number QL/2, aQL,25 the annihilation operator of the phonon,M 2 the mass of the anion, in (4), e~(2 I?L 12) is the eigenvector of the phonon after the notation introduced by Born and Huang The 2) [14]. are calculated frequencies wLo(QLI2) and ~LA(QL/ to be 174 and 77cm’, respectively. These phonon

L(QL) F L(QL) A axis A axis Fig. EuTe2.inCalculated the [111] dispersion direction. relations for EuSe and

modes can be identified with sharp lines of 176 and 79 cm~,respectively, observed in the AF I. In the ferrimagnetic phase (tt~)[2] the nonvanishing Fourier amplitudes of spins are given by (S±2QL,3) ±‘~/~i)(S)/3 and (Se) = V’N(S)/3. Therefore =../N(l the LO and LA phonon modes at 2QL13 become Raman-active. Since the calculated frequencies

Vol. 23, No.8

THEORY OF RAMAN SCATTERING IN EuSe AND EuTe

591

Table 2. Identification ofthe Raman lines. QL denotes the wave number vector at the L-point in the Brillouin zone. LO expresses the longitudinal optical branch, etc. The vector E1 and E, are the polarization vectors of the incident and scattered lights, respectively. The values in parentheses in the fourth and fifth columns are the experimental

values Magnetic phases

Phonon modes

Stokes shifts (cm~)

Relative intensities

Polarization

174(176) 77(79) 128

1.09(1.20) 1

AF—I

LO(QL/2) LA(QL/2) 2) IO(QL/ LA(QL) TO-phonon

—

E,.LE~ E1.LE, E 1 i.E8 E~II E1 I E9

Ferrimagnetism

LO(2QL/3) LA(2QL13)

167(169) 99 (101)

2.01 (1.96) 1

E,.LE8 E1 .L E8

AF—II

LO(QL)

156

—

E1 I E8

All-phases

LO-phonon

154 (156)

—

E~I E8

AF—Il

LO(QL)

111(112)

—

E1 I E8

EuSe

EuTe

122 (130) 128

(—)

—

—

2)I(S)IA]2 I (a.,o’~’[e~(2I~L~’ M 2~5(QL/2) (6)

~

of the LO and phonons 167and and101 99 cm’, spectively, the LA Raman lines are of 169 cm1 reare

(II) 1O~6~2:~

assigned as corresponding to the LO and LA phonons, respectively. In the AF—Il phase of EuSe and EuTe (f4t.1~)[2] the Fourier amplitude of spins is (SQL> = ~~JN(S>. The amplitudes of the optical phonons with QL are expressed as functions of only the anion displacement. Therefore, only the LO phonon mode at QL becomes Raman-active. The calculated phonon frequencies are 156 cm~for EuSe and Ill cm’ for EuTe. Therefore, the observed line at 112 cm~in EuTe is assigned as corresponding to the LO phonon at QL. In Table 2 the Stokes shifts of the sharp lines are shown along with the calculated values of the corresponding phonon frequencies. The observed extra lines at 169 and 101 cm~in zero external magnetic field are considered to be due to the scattering from the ferrimagnetic spin order [2]. domains coexisting with the AF—I The expression for the intensity of scattered light from a single magnetic domain is obtained by inserting (4) into (1). Thee, the averaged scattering intensity from various magnetic domains is given by

From this formula, the ratio of the scattering intensity of the LO phonon to that of the LA phonon is calculated to be 1.09, which is compared with the experimental value of 1.20. The corresponding intensity ratio in the ferrimagnetic phase, 1.96, is in very good agreement with the calculated value 2.01. The polarization of the scattered light is expected to be perpendicular to that of the incident light. This polarization agrees with-the observed polarization of the Raman line in the AF—II phase of EuTe. For the broad peak at 130 cm~in the AF—l phase, the following three possible mechanisms are considered. One is the scattering from the TO phonon via the second, third and fourth terms in the square 2brackets which isof (2). The of the isTO phonon to at QL/ active in frequency the AF—l phase calculated be 128 cm~. The second mechanism is the scattering via the polarizability associated with a nearest neighbor pair of Eu ions. In this process the LA phonon at QL whose frequency is calculated to be 122 cm~is Raman-active.

1

1

—

—

~

5~(/,)>~2

(5)

where Q runs over the four equivalent wave numbers in the directions of (111 } and ~‘ over the equivalent three easy directions of spins in the (111) plane. When the thermal excitation of phonons is neglected, the intensity in the AF—l phase is given by

=

The polarization of the scattered light is expected to be parallel to that of the incident light in this scattering [4]. The third mechanism is the scattering assisted by the excitation of spin waves. The scatteringdue to this process has two peaks at 128 and lS4cm’ which originate respectively from the spectral densities of TO and LO phonons [13]. The peak at 128 cm’ may be assigned as corresponding to the observed line at 130 cm~.Another broad peak at 156 cm~observed

592

THEORY OF RAMAN SCATTERING IN EuSe AND EuTe

in all phases may correspond to the calculated peak at 154 cm~.A relatively sharp resonant line observed at 184 cm’ in the ferromagnetic phase of EuSe remains unexplained, Recently Safran et a!. [15] have considered the exchange mechanism for the Raman scattering in these compounds on the basis of symmetry argument. Their assignment of the Raman lines is different from ours, because the exchange mechanism was used there. The

Vol. 23, No.8

phonon dispersion curves inferred from their assignment are in discrepancy with the dispersion curves calculated by using the breathing shell model. Further more, the Raman intensities estimated from the exchange mechanism are in disagreement with the experimental ones. Acknowledgements — We would like to acknowledge Drs. M. Kataoka and S. Maekawa for helpful discussions.

REFERENCES 1. 2. 3.

SILBERSTEIN R.P., SCHMUTZ L.E., TEKIPPE V.J., DRESSELHAUS M.S. & AGGARWAL R.L., Solid State Commun. 18,1173(1976). GRIESSEN R., LANDOLT M. & OTT H.R.,Solid State Commun, 9,2219(1971). SUZUKI N. &KAMIMURA H.,J. Phys. Soc. Japan 35,985 (1973).

4. 5.

SAKAI 0. & TACHIKI M. (submitted to J. Phys. Chem. Solids). NUSSLEIN V. & SCHRODER,Phys. Status Solidi 21,309 (1967).

6.

WOODS A.D.B., COCHRAN W. & BROCKHOUSE B.N.,Phys. Rev. 119,980(1960).

7. 8.

SHAPIRA Y. & REED T.B.,AIPConf Proc. 5,837 (1972). AXE J.D.,J. Phys. Chem. Solids 30, 1403 (1969).

9. 10.

HOLAH G.D., WEBB J.S., DENNIS R.B. & PIDGEON C.R., Solid State Commun. 13,209 (1973). WYCKOFF R.W.G., Crystal Structures, Vol. 1. p. 87. Wiley, New York (1963).

11.

KASUYA T.,J. App!. Phys. 41, 1090 (1970).

12.

VERMA M.P. & SINGH R.K., Phys. Status Solidi 33, 769 (1969); VERMA M.P. (private communication).

13.

OUSAKA Y., SAKAI 0. & TACHIKI M. (in preparation).

14.

BORI’J M. & HUANG H., Dynamical Theory ofCrystal Lattices, p. 297. Clarendon Press, Oxford and London

15.

SAFRAN N.A., LAX B. & DRESSELHAUS C., Solid State Commun 19, 1217 (1976).

(1954).