Optical phonons of Rare-Earth halides in a magnetic field

Z. Physik B 26, 323 - 328 (1977) Zeitschrift for Physik B © by Springer-Verlag 1977

Optical Phonons of Rare-Earth Halides in a Magnetic Field

Peter Thalmeier and Peter Fulde

Max-Planck-Institut ftir Festk6rperforschung, Stuttgart, Germany

Received December 22, 1976

A theory is developed which explains the observed magnetic field splitting of doubly degenerate E-phonons in Rare Earth Chlorides. It is based on first and second order magnetoelastic interactions. Quantitative calculations are done for CeC13 for which the field dependence of the splitting is derived and estimates of the coupling constants are given. Furthermore we propose a mechanism for the observed reduction of the phonon- linewidth in a magnetic field.

I. Introduction

Recently it was shown by G. Schaack I-1, 2] in a series of Raman-scattering experiments that an applied magnetic field can have pronounced effects on some of the optical phonons of paramagnetic Rare Earth (RE)-salts. The systems which were in- vestigated are the RE-trifluorides and trichlorides, respectively, and the measurements were done at a temperature of approximately 2K. Due to the uniaxial symmetry of these compounds (D4a for RE-F 3 and CZh for RE-C13) they have doubly degenerate E- phonon branches. It was found by Schaack that some of the Elg and Ezg-modes show a splitting in the presence of an applied magnetic field. In fields parallel to the symmetry axis of the crystal the splitting increases linearly for small fields and saturates for higher fields. In fields perpendicular to the symmetry axis the splitting is quadratic in the field and shows no saturation in fields up to 6 T. Furthermore it was found that many of the measured k = 0 phonons show a pronounced reduction in the line width with increasing magnetic field strength. It was also demonstrated by Schaack that the above effects are absent in the isomorphous LaC13 and LaF 3 compounds. This suggests strongly that the origin of the observed mode splitting and reduction in line width lies in the coupling of the 4f-electrons to the optical phonons. The aim of the present in- vestigation is to provide a theoretical description of

those interactions and of the experimental observations. In order to be specific we will restrict our- selves to the case of CeC13 since the crystaline- electric field (CEF) states of Ce 3 + are known in detail in this compound.

II. Crystal Field States and Magnetoelastic Interactions in CeCl 3

The CeC13 unit cell contains 2 Ce and 6 C1 atoms [3]. The point symmetry of the Ce3+-sites and consequently the symmetry of the CEF-potential is C3h. The groundstate configuration of the free Ce 3 + ion is 2F5/2. If one does not take into account the mixing with higher ( j=7 ) states, the CEF-potential can be written in terms of Stevens-Operators [-4].

H c F = B 2 0 o 02 + B 4 04. (1)

This potential splits the six degenerate J=3-states of the Ce 3+ Kramers-ion into three Kramers-doublets [ -+ 3) ] -+ ½) I - 3) whose energies are E + ~ = 0 cm- 1, E+¢=47 cm -1, E+~= 116 cm -1 [2]. One should note however, that the assumption of vanishing mixing with higher J-multiplets is not quite correct. This would imply a vanishing transversal g-coefficient g± = g5/2 (5/21Jx ] - 5/2) with gs/2 = 6/7 of the groundstate doublet. In distinction to this the experimental value is g±--0.23 [5]. The large value is attributed to

324 P. Thalmeier and P. Fulde: Optical Phonons of Rare-Earth Halides

a strong sixth-order contribution to the CEF- potential which is mixing J = 7 / 2 states into the groundstate doublet. However in the following it is not necessary to consider these J = 7 admixtures explicitly if we remember that g± ~ 0. Phonons can in- teract with the CEF-states by deforming the neigh- bourhood of the RE-sites and thereby changing the CEF-potential. The form of the resulting coupling can be derived by group theoretical methods [6, 7]. For this purpose we have to perform the reduction of the C6h-representations, which classify the phonon modes with respect to C3h which is the site-symmetry of the Ce 3 +-ions. For the doubly degenerate E-modes we obtain [3] the results:

C6h C3h

Elg E" E2g E '

Using the basis functions of E', E" we can now construct the relevant magnetoelastic interactions up to second order in the phonon coordinates. For this purpose we first express these interactions in a C3h- invariant way i.e. in terms of the local distortion amplitudes and 4f-shell operators transforming like C3h-representations. By expanding the local distor- tions into phonon coordinates which correspond to C6h-representations at k = 0, we obtain:

HIme = - E gI( k, P) [ Oa( -- k, p) q~.(k) + Ob( -- k, p) g0b(k)] k, p

I I _ _ H.,~- - ~ g~2)(kk',p)[0(+2)(-k-k',p) kk', p

• (9.(k) goo(k') - gOb(k ) %(k')) (2)

+ 0(2)(- k - k', p) go.(k) (Pb(k')].

Here p = 1, 2 denotes the different RE-sites in the unit-cell and O(k, p) = (N)- ~/2 ~ O(i, p) e-~k B, (i = unit cell index). g~, gH are the first and second order magnetoelastic coupling constants. ~o,, % are the conventional phonon operators for the E"' b-components of the doubly degenerate E-modes. Finally,

Oo(E,g) = O?)(E2.) = Jx J~ + L L O~(EIg) = O~)(E2.) = J, J~ + 4 J,

- - ( 2 ) - - 2 2 Oo(E~) - 0 + ( E ~ ) - J; - J;

Ob(E2g) = O(2-)(Elg) = Sx Jy -~- Jy Sx.

(3)

In addition there are also contributions to Hme that contain operators of fourth order in d~, dy,J~; for

example:

j£(j; 2 2 j z ) j 2 O?)(Elg)= 2 2 j ; ) + ( j ; _ (4)

O~)(Elg) = J~2 (Jx J, + J, Jx) + ('Ix Jy + Yy J~,) j2

Such terms can originate from the sixth-order contribution to Hcv, and the corresponding coupling constant g~) can be comparable to that of the second order terms, gl 2). In the following we will calculate the effect of (2) on the optical E-modes when a magnetic field H is applied. The Zeeman Hamiltonian takes thereby the form

Hz = g5/2 #B ~ HJ(i , p) (5) i,'p

where #B is Bohr's magneton.

III. Calculation of the Elg , E2g-splitting in an External Field A possible splitting of the E-modes in an applied field can be understood from symmetry considerations alone. The two components of a twofold degenerate E-mode correspond to complex conjugate representations of C6h and are necessarily degenerate only if time reversal invariance holds. This is not the case if a magnetic field is applied and therefore a splitting of the modes can be expected. This was pointed out in detail by Anastassakis et al. [8]. However these symmetry considerations cannot explain the explicit field dependence of the splitting. Microscopic calculations as presented here are therefore desirable. We shall consider in the following the complete CEF- level scheme of Ce 3 + as it turns out to be necessary. The mode splitting is calculated by solving Dyson's equation for the Greensfunction of the E-phonons at k= 0. It is of the form

-- 1((0) ~- ~ ( 0 ) - 1((0) -- oCf((0) ~a((0) = 5OI((0) ~_ ~galI((0). (6)

Since we are dealing with twofold degenerate phonon modes the Greensfunction N((0), N(o)((0) as well as the self-energy 5°((0) resulting from the magnetoelastic interaction have to be described by 2 x 2 matrices. The second line of Equation (6) shows the decom- position of the selfenergy due to Hie and H~e (see Eq. (2)). The Greensfunction in the absence of magnetoelastic interactions D(°)((0) is of the form

D~°~)((0) = 2 (006~/( (02 - (02) (7)

where (00 is the unrenormalized E-phonon frequency and e, fl=a,b denote the two independent components of the E-mode.

P.Thalmeier and P. Fulde: Optical Phonons of Rare-Ear th Halides 325

To lowest order in perturbation theory the selfenergies are given by

Sift((/)) = SIc~fl(co) ' -].- i SI~(co) ''

sl,~(co)'=g~,,~,Z CO~_(~M_~M,):

• ( O ~ ' O ~ ' ~ ~ - r ~ ' _ ~. ~a~'~,

SI ' /~ (co ) " -= igZco N,M'E C O 2 _ ( / ~ M _ / ~ M , ) 2

• (o~M' o f M - o y ' o y ' ~ )

(8)

Here M, EM, ffM denote the different CEF-states, their energies and occupation numbers respectively in an external magnetic field. Furthermore

l I _ _ &o - - s II = - (g~)<o(+ 2~) + gl, ~ < o ~ ) )

SII __ VII - - __ [ a ( 2 ) / 0 ( 2 ) \ _~_ or(4-)(0(_4))) ab- - ~ b a - - \&II X -- / &ll

(9)

where the thermal averages are defined by (0) =~_ ffMO~M. Fourth order terms mentioned pre-

M viously have been included in (9). One notices that the selfenergy resulting from Hnme is frequency independent• With the help of Equations (8), (9) we de- termine the splitting of the E-modes.

a) H il Let us first consider the E~g-mode. Since the CEF- states are also eigenstates of the Zeeman Hamiltonian as long as H II ~ they are not mixed. Only their energies are changed according to

EM = EM + gs/2 #B MH. (10)

In this case all thermal averages in (9) vanish as can be seen by inspecting (3), (4). Therefore 5Pu=0 i.e. only 5pI(co) can lead to a splitting of the Elg-modes. In order to calculate 5~I(co) we need the matrix elements of Oo(E~g), Ob(E~g) between the crystal field states:

(±~100[ -t-28>= ± 2 1 ~ ( + ~ 1 0 b I + ~ > = 2 i 1 ~ 5

(+½1 0~1 + ~ > = ±21/~ ( + ½ l O b l + ~ > = 2 i l / ~ (11)

As a consequence, sloo(CO)=SIb(CO)=, SI(CO); S~(CO)* =Slob(co).Furthermore i , ,_ S d(CO ) - - O, Slab((2)) ' = O.

The Dyson equation now reads:

1 l(co)=2coo

[co2 _ cog _ 2 coo S](CO)' D

2i coo SIob(co) '' -- 2i coo Slob( co)''

coe - coo ~ - 2coo s](co) ! (12)

with

S~(co)' = 41<~10o 1~}12 gi2/(co 2 - A 2)

SIob(CO) '' = 4K231 Oo 15}12 g2 CO(g_ ~ _ ~0/(CO2 _ A 2). (13)

Here the following approximations have been made• Transitions 1_+~),--~1+½) which are allowed by the selection rules (11) have been neglected due to the thermal depopulation of these states at T = 2 K. Furthermore Zeeman energies were neglected against CEF-energy differences i.e.

ff~+ 3 / 2 - - ff~+_5/2~- E+_3/2 - E+_s / z = A

Equation (12) demonstrates that a splitting of the two modes can result only if ~ " Sob(CO ) is different from zero. Physically, these contributions result from virtual transitions I+~)~1 3 +~), 1-~)+--~1--~) and corresponding virtual absorption of an E{g-phonon and emission of an E]g-phonon. For H = 0 , the two contributions have different signs and cancel each other

a b so that (12) is diagonal i.e. the E,'g-modes are degenerate. But if H ~ 0 , the Ce3+-doublets split and their components obtain different occupation numbers i.e. ff5/24=ff_5/2 etc. As a consequence, the two contributions to the nondiagonal selfenergy have now different magnitude and don't cancel any more. Therefore I ,, I tt Sob(CO ) =Sbo(CO ) has a finite value for H4=0 which is clearly seen in (13). The new normal modes and their frequencies can be calculated by diagonalizing ~-~(co). This implies that the secular equation detN-1(o))=0 has to be solved. This equation can be transformed into

2 2 (~(H) (CO2 - - ~ 2 ) ( C O 2 - - ~ _ ) ~- gi coo co tanh 2 ~ = 0 (14)

where

2 1 t . 2 + A 2 ) + 1 2 2 2 _+=>% _[a (coo-a ) +~2cooa] '2

~i ~ = 8 I<~1 oo I~>12 g~

5(H) =E~ - E _ ~ = g l l ~B H (gll =4.037).

For ~i=0 one finds f2+~co o, Fa_~A i.e. ~2+ denote the frequencies of the coupled Elg phonon and CEF- excitation with energy A if no magnetic field is present. Equation (14) cannot be solved exactly, but using a linear expansion around f2+, the Elg-splitting can be calculated explicitly• One finds for the energy differ- ence

co+ - co_ _ ~i 2 tanh 6 (H) coo ]/(co~- A2)2 +4~2 coo A 2kT" (15)

326 P. Thalmeier and P. Fulde: Optical Phonons of Rare-Earth Halides

The normal coordinates of the split modes E-~g, E~g are given by

1 i ~P+ =~(q~- i q~b) q)- = ~ (q°, + i q)b)" (16)

Therefore a magnetic field parallel to the crystal axis leads to a circular polarization of the spiit modes. This result was found experimentally in the case of CeF 3 [1] which is similar to CeC13. Equation (15) exhibits the experimentally observed saturation of the splitting for fields such that 6(H)>>k B T. The saturation splitting o- depends on g~ and the relative position of the phonon frequency coo and the corresponding CEF-energy separation A. One finds

o-=(gx/coo) z A<coo, A>coo

° = ~gi/coo) A ~coo. (17)

If the phonon is in "resonance" with the corresponding CEF-transition, the splitting is expected to be largest. In CeC13 one is with co o (Elg) = 1.69 A some- what between the two extremes described by (17). Figure 1 shows the experimentally determined Elg- splitting fitted by (15) with a coupling constant of g~ = 4.2 cm-1, determined from the saturation splitting. In a similar way one can explain the splitting of the E/g-phonons. Using the correct operators for these modes (see (3)) the splitting is obtained from (15) by making the substitutions ~i(Elg)~ffi(Ezg), coo(Elg)--->coo(E2g), A ----~A' =E+ 1/2--E+ 5/2. CEC13 has three types of Ezg-modes, namely E(21~ (109 cm-1),

~-(1) a E(22~ (182 cm -1) and ~2~(3)g (218 cm- 1). Only for ~2g splitting was observed. Probably the magnetoelastic coupling constants of ~-(2) ~,(3) are too small to yield ~2g~ *-~2g a splitting within experimental resolution. But even if they would be of the same size as that of

.-(lq 3 ~2g]~'(1) (g//52g= cm-1) the corresponding splitting would be smaller. This is due to the fact that the high frequencies of the ~(2) ~(3) modes bring them further ~2g~ ~2g "off resonance" with the CEF-excitation energy of A' = 47 cm-1. As a consequence one would have

(1) . a(E~22g)): o'(E~) = 1 : 0.3 : 0.2. o(Ezg).

b) H ± ~

In Reference 8 it was shown by symmetry arguments that in a uniaxial crystal a transverse field cannot produce a splitting of the E-modes linear in the field. This was confirmed by Schaack [1, 2] but at the same time he found an appreciable splitting which increased quadratically with the field. In order to calculate the Elg-splitting in this case we need the CEF-states in the transverse field. We choose H = H. ~. In the limit H-+0 the correct eigenfunctions

I O(x)o ~ - , ~ ' - ~ : : .,

0.5 / H IT]

I I I I I I I

0 1 2 3 ~ 5 6

Fig. l. Normalized splitting of the Elg-mode in the case H II z. ( + = experimental values; - = theoretical curve)

of Hcv + H z are given by the following linear com- binations

[g -+ ) = (2)- a/2 (15/2) _+] - 5/2))

Id + > = (2)-1/2(11/2) +I- 1/2)) (18)

le ___ ) = (2)- 1/2 (13/2) + ] - 3/2)).

One can check by inspection that (+ lJx l - )=0 where 1+), I - ) refers to the above Iv+), Iv2) states. Thus a transverse field does not mix I + ) and I - ) states in any order of perturbation theory. Furthermore ( + ] O, I + ) = ( - I O, [ - ) = 0 and ( + [Ob [ - ) = 0. Consequently, 5el(co) has no off- diagonal contributions and the mechanism which led to a phonon splitting in the case HII~ does not apply here. Thus a different physical effect must be re- sponsible. Detailed calculations show that due to the above selection rules for 0 , , O b the diagonal terms Slaa(co) and S~b(co ) are no longer equal. This is due to the third order Zeeman splitting of the le+) doublet. But the resulting Elg-splitting is proportional to H 4 and therefore cannot account for the experiment. The correct explanation of the experiment findings is obtained by studying 5 ~n, the selfenergy due to Hne. Since the transverse field defines a preferred direction within the xy-plane one expects that /r~(2)\ (0(+4)) \ v *_+ /~

are nonzero. Such a finite value results from the second order Zeeman mixing of Id_+} states into the ground state doublet Ig_+). The perturbed states are given by

Ig'+ ) = Ig + ) + (]/lO/2A A')(g} #B H) 2 Id_+ ). (19)

From this one obtains

(O~)) =]fliO (g_~ #B H) 2 Re(g+ ]O~ ) Id+)IAA'. (20)

Here Re denotes the real part which is nonzero only for the O~)-matrix elements. Equation (20) implies that E]g and E~g are renormalized differently according to (9). Their respective selfenergies have different signs. This leads to a mode splitting which shows the correct field dependence. Let us assume for simplicity that gl? )-c'(4)-c~- 6 I I - - SII' Then one finds for the mode

P. Thalmeier and P. Fulde: Optical Phonons of Rare-Ear th Halides 327

splitting energy

('0b - - (Da gII (g~ #B H)2/AA' (21) (D O (D O

where

glI : ] ~ Re (g + I o~ ) + o~)[d + } glp

By using the experimental Elg splitting energy in a field of 6T we obtain gn(Elg)= 21 cm -1. In distinction to the case H 1[ ~ the splitting does not show any saturation for/@ H > k B T and the polarizations of the split modes are unchanged.

IV. Magnetic Field Dependence of the Phonon Linewidth

It was demonstrated by Schaack [2] that some of the Raman-active phonons such as E~g show an appreciable reduction in line width with increasing field strength (H 11 z). The experimental observations can be described phenomenologically by

Fm(H)/Fm(O) = 1 - tanh 2 6(H) (22) 2kT

where F~(H)=F(H)-F(~) is the field dependent part of the total phonon line width F(H). A similar field dependence was observed for the line width of the CEF excitations. We want to demonstrate that an equation of the form of (22) can be obtained by considering the fluctuating magnetic fields at the RE- sites and their influence on the phonon line width. There is a certain analogy between the present problem and the EPR-line width problem [4]. The fluctuating fields can be caused by interionic interactions of the form

H D = - ½ ~ 1,~,,,(i,j) J,,(i) J,,,(j) (23) i j

Ga'

where i,j denote different RE-sites and a, a' =x, y, z. The origin of I~o, may be of magnetic dipolar or exchange nature• We suggest that via the second order magnetoelastic interactions H~e the interaction given by (23) is re- sponsible for the field dependent linewidth. In first order perturbation theory H~e leads to the following linewidth of a phonon of symmetry type ~ and energy (D0

fl MM"

Here

qS~M' = (M[ O~a [M') - (O~p} 5~u,

(24)

and N~((D) is the density of states of a phonon of symmetry type ft. As before g~t ~ denote the coupling constants• It is important to notice that without including the interaction given by (23) the line width (24) is nearly field independent. This is so since

(MI O~ a[M) = ( - M [ O~al-M) ( -MIO~a[M)=O

which is a general property of Kramers states [4]. As a consequence there are no contributions to (24) coming from scattering processes within the ground state doublet (M, M'= +_ 5/2). The remaining contributions to (24) are due to inelastic scattering processes and have only a weak field dependence via Np((D). However if the interaction (23) is included then its non-diagonal parts (~Ixz, Irz) mix [___3/2) contributions into the ground state doublet. Although the thermal average of the mixing contributions is zero the squared matrix elements for scattering within the ground state configuration is nonzero and is given by

(I A-+~, -+~l 2 ) =41( ---31L I --+~>1 e I( -t-310~,el+~)[ z

• ~ K(j, k) { (J~(j) J~(k)) - (Jz)2}. (25) jk

Here only matrix elements with equal signs for both states are implied. Furthermore

K(j, k) = ~, Ix~(i,j) I=(i, k)/A 2 i

= Z l,,(i,j) l,z(i , k)/A 2. i

By including in (25) only terms with j = k one obtains

+~]e)=q52fl . , 2c5(H)] ( 1 ¢ ~ : ' - t a n n ~ ; (26)

L

with

I+~)1 •Y lxzO, l) /A . q s ~ = 4 1 ( + 3 1 J x l + ~ ) 1 2 1 ( _ + 2 3 1 G e 5 2 . . 2 2 J

By using (24) we can write the field dependent part of the line width in the final form

F~((Do) = 2 n(g~1) z N~ ((D0) ~bg {1 - tanh 2kTJ6(H)'~" (27)

This expression has the observed field dependence. Since the scattering is elastic, only terms with 7=f i were considered in (24). g~1 is an effective coupling constant which is obtained by averaging g~(0, k) over the surface co~=(D o. Physically the reduction of F~((Do) with increasing external field H is due to the freezing out of the fluctuating fields at a RE-site originating from H D. This in turn reduces the scattering strength of the ground state configuration (see (26)). An estimate of the absolute value of F~((Do) is difficult because of the

328 P.Thalmeier and P. Fulde: Optical Phonons of Rare-Earth Halides

unknown parameters in (27). Considering the Elg- mode we make the optimistic estimate that gn--10 c m - 1. Fur thermore for N(coo) -- lIB (B = dispersion of the Elg-phonon ) we assume B~-10 cm-1 . In addit ion

we set (~lZz(i,j))-l/2~-2 cm -1, This number is ob= J

tained by assuming that the non-diagonal parts of Io~, are as big as the diagonal ones. The latter can be estimated from the observed widths of the electronic R a m a n lines [2]. With these values we obtain for the zero field width F~, ~ (H = 0 ) - 2.5 c m - 1 which is of the same order of magni tude as the observed value [2]. Experimentally it is found that the magni tude of the line width reduct ion is different for the two components of the Elg-mode. This feature is not con- tained in the present theoretical model.

V. Conclusions

We have demonst ra ted that the magnetic field in- duced splitting of E -phonon modes in CeC13 as well as their line width reduction can be explained on the basis of magnetoelast ic interactions. The absolute value of the line width could be only estimated qualitatively. We want to stress that it is not possible to explain the experiments by considering the CEF- ground state doublet of Ce 3+ only. Instead one has to include the excited states. Lately Schaack [9] has extended his experiments to the i somorphous (Pr, Nd, Eu, Gd) C13 compounds . The Pr and Nd compounds showed similar be- haviour as CeCI 3. In particular the same E-modes showed a splitting which indicates that only these

modes have sufficiently large magnetoelastic coupling constants. In the case of Eu (J = 0) and Gd (L = 0) no splitting was observed in agreement with the asser- t ion that the magnetoelast ic interactions are re- sponsible for the observed phenomena.

We would like to thank Professor G. Schaack for making his experimental findings available to us before publication and for many stimulating discussions. Furthermore we would like to thank Dr. I. Peschel and Dr. V. Dohm for useful discussions and remarks.

References

1. Schaack, G.: Solid State Communications 17, 505 (1975) 2. Schaack, G.: Z. Physik B 26, 49 (1977) 3. Murphy, J., Caspers, H.H., Buchanan, R.A.: The Journal of

Chemical Physics, 40, 743 (1964) 4. Abragam, A., Bleaney, B.: Electron Paramagnetic Resonance of

Transition Ions. Oxford: Clarendon Press 1970 5. Hellwege, K.H., Orlich, E., Schaack, G.: Phys. Kondens. Materie

4, 196 (1965) 6. Elliott, R.J., Harley, R.T., Hayes, W., Smith, S.R.P.: Proc. R.

Soc. Lond. A328, 217 (1972) 7. Gehring, G.A., Gehring, K.A.: Rep. Prog. Phys. 38,, 1 (1975) 8. Anastassakis, E., Burstein, E., Maradudin, A.A., Minnick, R.: J.

Phys. Chem. Solids 33, 519 (1972); ibid. 33, 1091 (1972) 9. Schaack, G.: to be published

Peter Thalmeier Peter Fulde Max-Planck-Institut fiir Festk6rperforschung Biisnauer Straf3e 171 D-7000 Stuttgart (Btisnau) Federal Republic of Germany

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Optical phonons of Rare-Earth halides in a magnetic field