Z. Physik B 32, 1- 3 (1978) Zeitschri f t f~r Physik B © by Springer-Verlag 1978

Magnetic-Field Dependence of Polaritons in Rare-Earth Systems

Peter Thalmeier and Peter Fulde

Max-Planck-Institut for Festk6rperforschung, Stuttgart, Germany

Received September 4, 1978

A theory is given for the magnetic field dependence of infrared active optical phonons in Rare-Earth systems. It results from a coupling of those phonons to the different crystal- field states of the incomplete 4f-shell. As a consequence magnetic field dependent polaritons should exist which should be detectible in a transmittance or reflectance experiment. The theory is applied in particular to CeC13.

In a series of Raman experiments Schaack [1,2] and Ahrens and Schaack [3] have shown that some of the doubly degenerate optical Eg-phonons in Rare Earth- trichlorides (space group CZh) and -trifluorides (space

D3a ) show a distinct behaviour in an applied group 4 magnetic field. If the field is parallel to the crystal- symmetry axis 2~ the experiments show that at k = 0 the two degenerate phonon branches split up and are of circular polarization. The splitting is linear for small fields and saturates for fields pBH~>kT. For fields H perpendicular to the symmetry axis the split- ting is proportional to H 2 and does not show satu- ration. In addition the two modes are linearly polar- ized (e.g. along the x- and y-axis if the field is along the x- or y-axis). A replacement of the Rare-Earth (RE) ions by La ions with an empty 4f-shell reduces the splitting in proportion to the La concentration. This proves that the splitting in an applied field is due to the magnetoelastic coupling of phonons to the crystalline-electric field (CEF) split 4f-states o f the RE-ions. A theory for this phenomenon has been developed in [4]. In addition to the Eg-phonons which are even under the inversion symmetry oper- ation the trichlorides and trifluorides have also E u- phonons, which are odd under inversion symmetry. They cannot be seen in a Raman scattering experi- ment but some of them are infrared active. In prin- ciple the E,-phonon should also show a splitting in a magnetic field induced by similar magnetoelastic cou- pling mechanism as in the case of Eg-phonons.

In the following we will calculate this splitting by restricting ourselves to the case HpI~ and we shall mention only briefly the results for the transverse case. As in the case of the Eg-phonons the first order magnetoelastic coupling for the E. phonons can be written as

Hme = - ~ g(k, p) [O~( - k, p) opt(k) + Oh(-- k, p) (pb(k)] kp (1)

(p~,% denote the phonon operators for the two de- nenerate branches E~ 'b. Oa, Ob are the corresponding quadrupolar operator which have transition-matrix elements between the CEF-states Ira) with the cor- responding energies Era. (Oa,Ob) form a two dimen- sional representation of the RE-point group. For example in the case of E1,-phonons in CeCI 3 we have in terms of the total angular momentum operator J of the incomplete 4f-shell:

Oa=JZ-J~

Ob=J~Jy+JyJ x. (2)

g(k, p) are the magnetoelastic coupling constants with p denoting the RE-site within a unit cell. If the sites p and p' are connected by an inversion operation we have g(k, p)= -g(k , p'). This is contrary to the situa- tion of Eg phonons where the coupling constants are equal for p and p'. Since the coupling constants enter quadratically in second order perturbation theory the

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2 P. Thalmeier and P. Fulde: Magnetic-Field Dependence of Polaritons

calculation of the renormalized phonon frequencies due to the Hamiltonian (1) proceeds in exactly the same way as for Eg-phonons. The details of this calculation can be found in [4]. For H[I~ the E, ,- phonons split according to

a - - ~ + - ~ =ao tanh - 2kBT

with

(3)

ao=~,2(2o{(f22-A2)2 +4ff, ZY2oA}i; ~ 2 = S M 2 g 2.

Qo is the bare E, , -phonon frequency in the absence of magnetoelastic interactions, d and M are the en- ergy difference and the quadrupolar matrix element for the transition between the I_+~> and l+½> CEF- Kramers doublets of Ce 3+ (d =47cm-1) . Furthermore cS(H)=glI#~H where gll is the parallel ground state g-factor. O + , £ 2 denote the frequencies of the magnetic field split right and left handed circularly polarized El , components. CeC13 has two optical E~,-modes with frequencies of 138 cm -* and 210cm -1 which could develop such a splitting provided that the coupling constants are not too small. For example the coupling constants for the Eg-phonons are such, that some of them lead to a corresponding mode splitting of the order of 10- 20 cm- ~ for fields of 6 T. For other Eg-modes the coupling constants are too small in order to observe a mode splitting at such field strength. Thus there is no a priori prediction one can make for the size of the splitting of the optical E~,-modes. In the following we will discuss two methods by which such a splitting would be observable. Both rely on the fact that an external electromagnetic field couples to the infrared active E~,,-modes resulting in polariton excitations. Their dispersion relation is gi- ven by [53

C 2k2 - - ± (~Zv--gt)~ (4)

where Otv are the frequencies of the doubly de- generate transverse E,,-branches and (2~, are the frequencies of the corresponding longitudinal branch-

± is the transverse high-frequency dielectric es. goo constant, c is the speed of light. Equation (4) leads to the usual polariton modes and also gives the rest- strahlen bands £2t~<co<Q,~. A splitting of the £2t~ branches in an external field according to (3) implies a splitting of the polariton branches which is de- scribed by (4). The latter should be detectable experi- mentally by a transmittance or reflectance measure- ment.

to/~ 1 / / 1"~ H) B_/9~ / /

10

~2+I~1 i (2)

(~t-Wo}/~t _[1_} . . . . . . . . . . . . ~ ~ . . . ~ . _ _ _ _ ~

0.5

qLk./ko Iik_/k o I I ! I t I *

0 0.5 1.0 1.5 2.0 k / k o

Fig. 1. Dispersion curves for left and right handed circularly polarized polaritons in an external magnetic field. A ratio ~]f2~ =0.84 has been assumed, corresponding to the transverse E1,(138c m t) and longitudinal A,(165cm -1) phonons of CeC13, A typical maximal splitting of cro(E1,)=15 cm -1 has been used, Reflectance bandwidths are denoted by B+, B .

ko=QlV/(~)u/c where #~El,(138cm -~)

a) Transmittance Experiment

Suppose the Qt, phonon (# refers to one of the Elu- modes) splits up into right and left circularly polar- ized branches £2+t,, O2u, respectively. Furthermore assume that the frequency co of the incident, linearly polarized infrared radiation is slightly less than Ot,, say co=f2,u-co o with COo>½ao(f2tu) where ao(f2,u ) is the maximal splitting of the ~2~u-transverse phonon. The reason for this choice of co is apparent from Figure 1. One is in that case in a regime in which the polariton dispersion is strongly changing with k. Within the sample the linearly polarized incident wave can be visualized as a superposition of right and le•handed polaritons with different wave num- bers k+, k which are given by

c2k+_2 2 2 ~--~2#_(.02 , ~ ± f21v-f2t, o .= - v..I1 - + = - o , ( s )

Here the phonon splittings in the nonresonant part of (5) have been neglected. Consequently, the polariza- tion plane of the incident radiation will be Faraday- rotated by an amount

=½(kf -kf). (6)

Using (5) the Faraday rotation per unit length q0, is approximately given by

(e~)~ co 2 £2 2, - £2,2. . /2 qo.= c2k. 4£2c" co2_(o_/2) 2. (7)

P. Thalmeier and P. Fulde: Magnetic-Field Dependence of Polaritons 3

ku=½(k; +k~) is approximately the zero-field wave- k number corresponding to frequency co. (e~), is defined

by

L __ ± ~''tv--~'~t~O 2 2

( g co)# - - g aO vll=r/~ ~ t v ¢D2 - - 0 2 # ' (8)

cr(f2tu ) is the field and temperature dependent splitting of the f2,,-phonon. From (7) one notices that in the case a(f2t,)/2-~co0 or f2~ ~ co the angle of rotation diverges. Using the definition (6) this can be im- mediately seen in Figure l. In reality this singularity will be suppressed due to a finite phonon linewidth and the resulting strong ir-absorption below Ot~. In any case on expects a strongly temperature and field dependent Faraday rotation in the frequency range just below ~2t, due to the temperature and field dependence of a(g2t,,,H, T). Inverting (7) gives the phonon splitting in terms of the observed Faraday rotation angle.

b) Reflectance Experiment- Field and Temperature Dependent Reststrahlen Bands

In the absence of a magnetic field no polariton mode can propagate in the frequency range f2tu<co<(2~ ,. Therefore all of the incident intensity is reflected for frequencies within the reststrahlen band. Without field there are two identical bands for right and left circularly polarized radiation. By applying an exter- nal field the bandwidths B,~ for right and left handed circularly polarized incident radiation are no longer equal; their difference is given by

a(u) B; - B ; = ao(f2t,, A)tanh 2kB~. (9)

In this way by measuring the field- and/or tempera- ture dependence of the different reststrahlen bands one should be able to determine the phonon splitting. The same method should be applicable for a trans-

verse field HIlL In this case the reststrahlen bands for incident radiation polarized parallel and perpendicu- lar to the field will have different widths, according to

x y 2 Bu - B~ = a~ H . (1 O)

o- a is the transverse splitting factor which can be calculated to second order in the magnetoelastic in- teraction [4]. Its experimental measurement would enable one to determine the unknown magnetoelastic coupling constant. As in the transmittance experi- ment the situation is somewhat complicated by the finite phonon linewidth which itself is magnetic field dependent [1-3]. One can include this effect on the reflectance bands by extending (4) such as to include a phenomenological phonon-linewidth. Although our discussion referred explicitely to CeC13 it is in principle also applicable to CeF 3. The only difference is a more complicated expression for the splitting factor ~r 0 in (3). This is due to the lower RE- site symmetry in CeF 3 which implies that all CEF- transitions contribute to the phonon splitting. In fact CeF 3 should be more favourable for finding the described effects since it has a larger number of doubly degenerate phonon branches because the number of atoms per unit cell is higher.

References

1. Schaack, G.: Solid State Communications 17, 505 (1975) 2. Schaack, G.: Z. Physik B 26, 49 (1977) 3. Ahrens, K., Schaack, G.: Proceedings of the International Con-

ference on Lattice Dynamics Paris, 1977, ed. by M. Balkanski 4. Thalmeier, P., Fulde, P.: Z. Physik B 26, 323 (1977) 5. Kurosawa, T.: Journ. Phys. Soc. Japan 16, 1298 (1961)

Peter Thalmeier Peter Fulde Max-Planck-Institut ftir Festk6rperforschung Heisenberg Str. 1 D-7000 Stuttgart 80 Federal Republic of Germany