Softening of the LO—phonon frequencies in SmS

Solid State Communications, Vol. 35, pp. 175—177. Pergamon Press Ltd. 1980. Printed in Great Britain. SOFTENING OF THE LO—PHONON FREQUENCIES IN SmS K. Baba7 M. Kobayashi, H. Kaga and I. Yokota Department of Physics, Niigata University, Niigata 950-21, Japan (Received lOMarch 1980 byM. Cai~iona) The zone-boundary LO—phonon frequencies of SmS show a “softening” in the semiconducting as well as in the valence-fluctuating metallic phase. This has been interpreted as a renormalization of the phonon frequencies due to the phonon-induced on-site f—d hybridization interaction. Renormalized phonon frequencies, which are calculated as a function of the f—d energy gap, show a larger “softening” in the semiconducting phase than in the metallic phase, being in qualitative agreement with experiment.

VALENCE FLUCTUATION and its associated phenomena in the rare-earth (RE) mixed-valence compounds are the subject of current interest [1, 2]. Recently, Guntherodt et al. [3] made Raman scattering studies on the optical phonons of various rare-earth divalent (RE2~S)and trivalent (RE34S) sulfides including the mixed-valence compound SmS. They found a sizeable reduction in the longitudinal optical (LO) phonon frequencies of SmS in the semiconducting divalent phase as well as in the metallic mixed-valent state. Neutron inelastic scattering on Sm 0•75Y0~25S by Mook eta!. [4] also show a similar result. This softenmg of the zone-boundary phonons rn the Sm compounds appears to relate to the large vanation of Sm ionic radius due to a promotion of the 4f electron to the 5d conduction band. The idea of LO—phonon frequency “softening” due to phonon-induced on-site f—d hybridization in such systems has been considered by different authors [3, 5—71. In this letter we calculate the renormalization of the LO—phonon frequencies by the phonon-induced on-site hybridization as a function of the f—df—d energy gap [8] ininteraction both the semiconducting and metallic phases of SmS. l’his calculation may serve to understand why only SmS compounds2~S show andsuch RE3~S, “softening” compared other REaverage energy because theseaslatter have to much larger gaps, though a more direct and quantitative comparison must await future experimental and theoretical studies. We use the following effective Hamiltonian: H

=

~

I

p

._

I

5d

5d

//

0 -

G

()

(a) Semiconductin~rhase

//

4f

(b) Metallic phase

Fig. 1. The energy scheme of the 4f level and Sd conduction band used in the numerical calculation of Fig. 2. V.

~-.

+

4

g(ftd~cb~ + h.c.) +

~,

+

hll(bqbq + ~)

(1)

with = b + b~ = q -q q~ ~ -q~ where d~.f~,and b~are the creation operators of Sd 6k~ 4f localized conduction electron with energy electron at site i withkenergy E 0, and phonon with wave-vector q, respectively. V is the off-site f—d hybridization energy, and g is the phonon-induced on-site f—d interaction constant. We are interested only in the particular zoneboundary LO—phonons which are softened by the breathing motion of surrounding sulfur atoms around each valence-fluctuating Sm. We denote the bare phonon frequency by q-independent ~1.The fourth term, the ,

ekd~dk+ ~ E~f~f

+ 1 ~ V(f~d,~ + h.c.) k

i

I. k

______________

*

W

Present address: Department of Physics, Nagoya University, Nagoya 464, Japan.

phonon-inducedf--d interband transition, has been suggested by Entel eta!. [5] to be responsible for a discontinuous change in the occupancy of 4f-electrons for some values of the above parameters. The 5d conduction band is assumed to have a simple square-band of constant density of states p and width W. 175

176

SOFTENING OF THE LO—PHONON FREQUENCIES IN SmS

3S

280 ~—Sm

2BOL(~~SmS ~~Sm~S

Sn~’S(mixed valent) —

260 ,,.~

.

.

semiconducting

~.

-. -~

3 240

—~

— ~-

~

~

‘J~~—

220

~

260

~~N)

220 g=aoI9ev (3J

l4

(UV”OO2eV

x

200

(2,g Q0l5#i/. (3;g~aog9ev

~ di o~ d~04~~L3

sernwnckicting—.

240

V=QOIeV X ~2t~

(mixed valent)

I

(I)~:QOlW

200

______

Vol. 35, No.2

(2)V QOIeV (3)V0007#I

-Q5~ 0 01 02 03 OJ 1.3 14

(5

G (eV) (a)

15

G (eV) (b)

Fig. 2. Calculated renormalized LO—phonon frequencies ~ are shown as a function of the energy gap G. which is taken to be positive in the semiconducting and negative in the metallic phase, as defined in the text. (a) is the effect of the on-site hybridization g, and (b) the effect of the2 off-site ~S and fictitious hybridization Sm3 ~s) V. The Thebare crosses frequency are the ~2is experimental chosen so as to reproduce the points marked byand solid circles (Eu points of the semiconducting Sm2~S the mixed-valent Sm2~S,which are plotted for the estimated gaps. The4f-levelisseparatedbyagapGfromthebottomof theconductionband[8].TheFermienergyeFandthe number of 4f-electrons, fl~ ~(f~f,>/N (N the number of atoms), are related to each other due to the conservation of total electron number and the local screening of 4f-electrons by conduction electrons. = I

flf+PCF

(T=0).

—

F

(2)

—

for C=0 fl~rW F = G(G + H’) and for C <0

tan

—

—

—

(5b)

,

G(G +

EF)

fG_+_w\

.~

~~

4f2g2p n

=

~2

—

—

h

-~

~ (G + eF)2 C G2—C —

2

in

(G + ~2 G2 j~2

F ~ (G + Hr)2 C G2—C

tan’

(~))~ (Sc)

—

‘

~ = V + g(Ø~,), ‘(3 + 6F ~ + = IT tan~\ irpV / 2 —

2g ~

—

I

-~--,

(T = 0).

(6)

(ftd~+ h.c.) + hfZ(cb~> = 0,

(7)

‘

(4) where forC>0

—

In the above linearization procedure of the equations of motion the following two relations are obtained:

—

—

—

—

/ G

~

+~ (tan-i (G~~) and where 2 2 ~2 —

(5a)

_______ —

The operators bq~b~,~f —

-~

—

F =

1 (~f~), d~d~ ~Zdk) (3) and f~dk (ftdk) ±h.c., which appear successively for the calculation of 4~, (h>, also constitute the equations of motion. These equations are linearized with respect to the participating operators corresponding to the mean-field approximation. The self-consistent phonon solution of the frequency: equation of motion gives the following LO—phonon

______

—

~ G + CF VC’ + ~ G G + CF + .,/?~ G + ...J?~ j

-

= cz(bqb~).

[~q(øq),H1/h

(

1 G+W~~..JZr 2\/~ nr\l1) G+w+~J~ Iii G+~J~ ,—

—

In order to obtain the phonon frequency the equation of motion is constructed for the fluctuating part, cb~ (~),of the phonon coordinates çb~from the Hamiltonian in equation (1). =

—

(ek~EO)(f~dk+h.c.)

~ Thus, from equations (7) and (8)

= 0.

(8)

Vol. 35, No.2

SOFTENING OF THE LO—PHONON FREQUENCIES IN SmS

(

—

=

G + CF 4gVp ~ln G I G + CF

h~2+ 8~p In

~ G+W G ) (3 + w \

—

G

—

h~ G

177

metallic phase is not sensitive enough to enable to distinguish between the values of these parameters. (9)

/

Quantities n,., 6F and (4~)are determined self-consistently from equations (2), (6) and (9). When these quantities are substituted into equation (4), the renormalized phonon frequency ~ is obtained as a function of the energy gap G alone. In the semiconducting phase of Sm2~Sthe 4f.level separation G from the bottom of the conduction band happens to be(Gvery1small eV) as compared the other RE2~S eV). (~0.1 The effective energy gaptocan be further reduced (to < 0.1 eV [9]) by the f—d Coulomb screening effect (Falicov—Kimbal term, which is not considered here) concurrent with the on-site hybridization interaction. In the metallic phase, however, the negative energy gap G (Fig. 1) taken as such is not changed by this effect (because both f-to-d and d-to-f hybridizations are balanced in the equilibrium mixed-valence state), and is roughly estimated as —~0.4 eV from the conduction electron number. Taking V and g as parameters we calculate the renormalized LO—phonon frequencies w from equation (4) as a function of the energy gap IGI in both the semiconducting RE2~Sand the metallic RE3~Sphases. The bare phonon frequencies ~2are chosen so as to reproduce the observed LO—phonon frequency (240 cm~)of Eu2~Sin the semiconducting phase and the trivalent Sm3~Sphonon frequency (‘-~275 cm~) with a fictitious lattice constant interpolated from the other RE3~Sin the metallic phases. ~2is therefore the LO—phonon frequency for the large gap I GI in each of the two phases. The d-band parameters p = 2 eV1 and W = S eV are used. In Fig. 2 we show the effects of the phonon-induced on-site f—d hybridization g [Fig. 2(a)] and of the off-site hybridization V [Fig. 2(b)] on the “softening” of the renormalized phonon-frequencies w for three typical values of each parameter. The experimentally observed LO—phonon frequencies of 2+ 2.8+ the divalent Sm S and the mixed-valent Sm S are indicated by crosses in these figures, whose gaps are estimated as in the previous paragraph. From Figs. 2(a) and (b) we see that the effect of the on-site hybridization is larger than that of the offsite interaction for the same values of the interaction parameters. The experimental points can be explained well if we use the optimum parameters g = 0.019 eV and V= 0.007eV; the IGI-dependence of w in the

Hereafter we assume this parameter set. These results indicate the relative importance of the on-site hybridization g as compared to the off-site one V. The observed larger “softening” of the semiconducting Sm2~Sin comparison with the metallic mixed-valent Sm28~Scan be explained as due to an enhanced on-site hybridization effect for the smaller energy gap, G <0.1 eV. The chemically collapsed metallic Sm 0~75Y0~25 S shows much larger “softening” (15%) than the 2~8~S (‘~2%) for almost the metallic same Smmixed-valent valence + 2.7Sm [3,4]. The inclusion of substitutional Y impurity atoms may lead to a narrower Sm Sd-conduction band, thus resulting in a smaller average negative gap I GI. If this is the case, then a sizable phonon-frequency renormalization is expected also in the metallic phase (I GI small) from Fig. 2. The gap G as defmed here is the bare f—d band gap, but should be replaced by an effective energy gap for comparison with experiment, for the nonperiodicity of the mixed-valence phase,f—d Coulomb screening effect and other possible effects are not considered here. In conclusion, we have shown that the phononinduced on-site f—d hybridization interaction plays a major role in the effect of the observed “softenings” of the zone-boundary LO—phonon frequencies. This effect is particularly enhanced in the semiconducting divalent Sm2~Swith the small f—d band gap G. We hope that more experimental study is conducted for the energy gap dependence of LO—phonon frequencies of SmS and other (RE)S either by applying hydrostatic pressure or by collapsing chemically.

1. 2. 3. 4. ~ 6. 7. 8. 9.

REFERENCES C.M. Varma, Rev. Mod. Phys. 48, 219 (1976). J.M. Robinson, Physics Reports 51, 1 (1979). G. Guntherodt, R. Merlin, A. Frey & M. Cardona, Solid State Commun. 27, 551(1978). H.A. Mook & R.M. Nicklow,Phys. Rev. B18, 2925 (1978) i. ~ H.J: Leder & N. Grewe, Z. Phys. B30, 277 (1978). N. Grewe, P. Entel & H.J. Leder, Z. Phys. B30, 393 (1978). N. Grewe & P. Entel,Z. Phys. B33, 331 (1979). The average energy gap G is taken as negative for the metallic phase as shown in Fig. 1(b). E. Kaldis & P. Wachter, Solid State Commun. 11, 907 (1972).