Light scattering from magnons in ferromagnetic semiconductors

Solid State Communications, Vol. 18, pp. 785—788, 1976.

Pergamon Press.

Printed in Great Britain

LIGHT SCATTERING FROM MAGNONS IN FERROMAGNETIC SEMICONDUCTORS* S.G. Coutinho Departamento de Fisica, Universidade Federal de Pernambuco, 50 000 Recife, Pe, Brazil and L.C.M. Mirandat Department of Physics, University of Arizona, Tucson, AZ 85721, U.S.A. and Departamento de Fisica,~Universitade de Brasilia, 70000 Brasilia, D.F., Brazil (Received 6 August 1975; in revised form 25 September 1975 by E. Burstein) The effect of the electron—magnon interaction on the Raman scattering by magnons is discussed. It is shown that in addition to Loudon’s spin— orbit coupling mechanism there is another, carrier-mediated, contribution to the Raman line. The frequency dependence of the resonant scattering is discussed and application to CdCr2Se4 is made. LIGHT SCATTERING from ferromagnets has been the 5From subject of much research in the past decade.~ the experimental point of view, however, it is still a relatively unexplored field. Studies of scattering from thermally excited spin waves have only very recently been reported.6 The main light scattering mechanism considered so far has been the spin—orbit mechanism first proposed by Elliott and Loudon.3’4 Although Loudon’s mechanism should always be present, in the case of ferromagnetic semiconductors, there is yet another, electronic, contribution to the light scattering from magnons, namely, the carrier-mediated or indirect process. By carrier-mediated or indirect mechanism we mean the following three step process. In the first place it involves the photoproduction of an electron-hole pair. This is then followed by the scattering of the electron by a magnon, and finally we have the electron—hole recombination. This is similar1’2 to the scattering TheRaman novel point here,by phonons in semiconductors. however, is the use of the electron—magnon interaction to provide another channel for the electron decay into the bottom of the conduction band. Secondly, it also allows us to perform resonant Raman experiments,1 and in this sense it differs considerably from the Loudon’s mechanism.3’4 Thirdly, as we shall see later on, the indirect mechanism is highly dependent on the s—d exchange parameter and therefore provide us with a new tool for its investigation. * Work partially supported by the Brazilian Agencies, CNPq, BNDE, and CAPES. ~ John Simon Guggenheim Fellow 74/75. -~

Apart from recently reported by phonons,7’8 there seems to exist noRaman Ramanscattering studies of magnons in magnetic semiconductors up to date. The reason for this, we believe, is the lack of theoretical estimatives of the size of the effect as well as the experimental difficulties in working at the few microns region of the spectrum. The main motivation for the present calculation is the current rapid growth of Brillouin spectroscopy6 leading to the possibility of studying low-frequency spin waves under high resolution. We assume that a ferromagnetic semiconductor is adequately described by an interacting electron—magnon system.9~Let us describe the electron system by a valence band of energy e~~, ~ and a conduction band of energy ~c, The ferromagnetic system is characterized by ionic moments g~S 1localized at lattice sites R. The main interaction between the carriers and the localized moments considered so has been the description well knownof s—d interaction.911 In far a more detailed this simple model one should also consider the magnetic interactions12’13 between the carriers and the localized spins. These interactions are the spin—orbit interaction due to the coupling of the localized spins with the screened magnetic field of the moving carriers, and the dipolar interaction between the localized spins and the carrier spins. Hence, assuming that the system is below the Curie temperature, the electron—magnon interaction Hamiltonian is given by9’3 ~,

He_rn

=

~.

0 ~

—

~:n

Permanent address.

J

—

npk

785

c~pocnpa

npa

/ a \ 1/2 n) 101 t ~~7I Cnp_ktCnp4Dk \hJV)

+

C.C.

786

LIGHT SCATTERING FROM MAGNONS j(n)

+

P~?1Q3lHe_rnIa)P~

—(c~P+k_k’tcflPt —C~p+k_k~Cnp~)b~’ bk

~ npkk’

M

2N

= ~

Vol. 18, No. 7 + five other terms -

(c~3+Wk —~‘)(w~—c~.~) (5)

(i2 x— k).. k

—Ma, ~ npka

MdiP

~ ~

C~p_kOCflpGb~+

2 k~k+ +~

c.c.

C~p_kaCnpa~+

c.c.

(1)

0k

script 0isdenotes the electronic and the ~ and j3 Here i~ the dielectric constantground of the state, medium, subrun over complete sets of intermediate states. p~ denotes the momentum the longmatrix wavelength element limit in of thethe direction component i, andof n 1,

np

n2 and ~k are the number of incident photons, scattered

where 2irieh f2g/i~Mo’\1~~2 mc v ,,

and MthP

photons and magnons present, respectively. Since we are interested in the spontaneous scattering we shall take the number n2 of photons w2 in the initial state to be

\

=

v

41

=

)

PB (2~~~Mc~

(2) Here A ±= A~± L43, M0 = g/.t~NS/vis the saturation magnetization, N ais crystal the number of magnetic atoms(s~.&~ of = ionic g-valueg in of volume V, X~= wa/c 4ire2n 0/me is the plasma is the screening 1B is thefrequency) Bohr magneton. .J~is the exwave vector, and Ichange integral between the localized spins and the carriers of band n. cn~(c~a) is the usual annihilation (creation) operator for an electron in band n with wave vector p and spin projection a. Here a = + 1 for up carrier moments, and a = 1 for down moments. Finally, bk(b~)is the magnon annihilation (creation) ,

—

operator. One notices that the contribution from the first term in equation (1) produces a spin splitting of the bands, i.e. —

2

a,

fore assume two have parabolic bands modelmass in which the electron and ahole reduced effective p, wavevector p and forbidden energy gap hc~g.The intermediate state energy is then (6) hw~= hwg + 2ji 2 As w 1 is increased to approach the forbidded gap frequency some of the denominators in equation (5) tend to zero. The divergence is strongest’ in the first term on the right-hand side of equation (5). Assuming further ~

—

—

that the matrix elements p~4are independent of p we may approximate M by where we have neglected the band splitting due to the v 2 ~a d3p Me—rn external magnetic field (usually much smaller than J<’~S). M = ~~3Ip~~I Next we proceed to evaluate the transition probability for the indirect Raman process. The procedure is < ~2\ -1 =

e,~,

zero.The frequency dependence of the scatteringamplitude M in the resonance situation can be further calculated if one specifies the band structure for the system. The band structure of ferromagnetic semiconductors has 14 Nevertheless, not yet been solved to the reliable extent. there is evidence of the usual parabolic band. We there-

(3)

(

similar to that of phonon scattering as described in reference (1). We assume that the crystal is in its electronic ground state before each scatteringeven with the valence band full and the conduction band empty. In all the intermediate states there is an electron in the conduction band and a hole in the valence band. Hence, using third-order perturbation theory, the transition probability per unit time that one incident photon ~ has been destroyed, 1’2 and a photon w2 and magnon Wk have been created is W(T) T

—

—

8~3e4

2m4h2

n 1(n2 + l)(nk +

k

v~n

2

)

~

2~(wi Wk), x IMI where the scattering amplitude M is given by —

—

(4)

$

+

Wk

W 1

—.

aw~+

—

~

~

—

—

aw8 +

(7)

~

2p

2p/

J

where hw8 = (J(c) J(v)\,~/2and Me_rn stands for the matrix element of the appropriate electron—magnon interaction. Let us now show that the only effective electron— magnon interaction mechanism in the indirect Raman process is the dipolar one. This can be seen as follows. The interband transition arising from the optical absorption conserves the spin projection. Hence the electron— —

hole recombination after the creation of one magnon in the conduction band is only possible if the electron— magnon scattering does not change the conduction-band spin state. To be more specific consider the interband transition v-lj c.4~according to the s—d interaction -÷

Vol. 18, No. 7

LIGHT SCATTERING FROM MAGNONS

[second term in equation (1)], after the creation of one magnon the electron will be in the ct state. Now the electron in the ct state cannot drop into the valence band since vt is completely occupied; the vacancy in the valence band is in the vJ., state. Hence as far as one magnon processes are concerned one is left only with the other two electron—magnon mechanisms, namely, the spin—orbit andspin the dipolar Both processes allow do notfor involve carrier flip andones. would, in principle, the electron—hole recombination. However, due to the p-dependence of its matrix element [i.e. (p x k).] the spin—orbit contribution to equation (7) vanishes, Therefore the only non-vanishing contribution for the one-magnon scattering is the dipolar one as given by the last term of equation (1). In this context, it should be emphasized that the present mechanism yields a nonzero contribution even in the case of zero spin-orbit scattering gives zeroLoudon intensity. coupling for which s theory for the one-magnon Hence, substituting Me_rn in equation (7) by the dipolar vertex and performing the integration over p one obtains ,

M

— —

v (2p\312 Ip~,,jM~~.~,F(kik h— lTWk\

2 4 4

2 (11) ma4h3c 3 Vw2n 1/2 ~IS(w)I In arriving at equation (11) we have taken the magnon dispersion to be parabolic, and approximated ~ by Wa and p by m. Here a is the lattice spacing, and n is the 4g e /iBN~l(n + 1)

W(I) T

=

Wo,

—

2)S(wk. k1

/

-

k2)

(8)

where F”~’— k k ‘k2 + /c2 ~. 1 — z +1 1

‘- -‘

and S(w)

Finally, the selection rules governing the scattering can easily be derived from equation (5), (8) and (9). It can be shown for example that exciting light linearly polar. ized along the z-axis gives rise to scattered light circularly polarized in the xy-plane. However, when the incident and scattered polarization vectors are parallel, the Raman scattering vanishes only if k1 and k2 are parallel, 4 contrary to Loudon’s selection To proceed further with therules. evaluation of the transition probability per unit time we suppose that the mcident radiation is directed along the z-axis (magnetization axis) and polarized along the x-axis, and that the scattered radiation is observed in a small solid angle ~7in the direction of the y-axis and polarized along the z-axis. Substituting equations (8)—(l0) into equation (4) and performing the integration over k 2 one gets

magnon population at frequency c~= wo + hk~/M

2

~-~-

787

where w0 is the Zeemen frequency and M is the magnon effective mass. 2 as the ratio of the Defining the Raman efficiency” number N 2 of scattered photons unit cross-sectional area Raman of the crystal perproduced unit timeper to the number N1 of incident photons crossing unit area in

=

~

a[(w~+ w

w1

—

—

unit time one gets 2e4/4LSk~T(N/V) IS(w)12

aw8)i/2 (10)

a=+1,-1 —

(Wg

—

Wi

—

aw

ma c Ii w

2]. 9)~

Equation (8) is similar to Loudon’s well k~.own result for the Raman scattering by phonons’2 The main difference, however, lies on the fact that the intermediate state summation now involves the spin pro-

c~

(12)

4g

=

w

where we have replaced (n + 1) by kBT/hw. L is the crystal thickness. Equation (12) is very sensitive to the actual value of S(w)12. In fact, at the resonance (w = arbitrarily w8), if one 2 one gets1an high approximates the Sletbyusw” value for e. Hence rewrite 5(w) in the following —

jection explicitly. In other words, the conduction is effectively splitted into upand down-spin bandsband differing in energy by 2hw 8, and the magnon creation can occur in both bands. The total scattering amplitude is then the sumThe of both band contributions. effectupof and this down-conduction band splitting could alternately be seen as a renormalization of the forbidden gap, namely, = Wg aw 3.

form 2+ (r + s)~’2 (r + s + —

—.j~= (r +

1)1~’2—

l)b~2,

r’~’

(13) where the dimensionless parameters r and s are defined

—

This means that the actual gap for the f-transitions is Wg W~whereas for the 4~-transitionsthe gap is wg + w8. Therefore, in keeping with the assumption that w1 and w2 are small for the photon to be able to induce real transitions it follows that w1 should be smaller than the smallest electron—hole frequency, namely, Wg W8. The resonance now means w1 approaching Wg w~.

by —

w~ w1 —

=

rw

and

s

=

2w~/w.

(14)

—

—

—

r measures the photon matching of the resonance whereas s describes the conduction band splitting. One notices that for s = 0 (i.e. J~ = j(v)) the Raman efficiency vanishes, This is a consequence of the fact that the total probability amplitude for the indirect

788

LIGHT SCATTERING FROM MAGNONS

one-magnon process is the sum of the contributions from both spin bands, each one multiplied by a. Also, at the- resonance (r = 0), S(w)/w~2is approximately equal to one. However, for typical values of the parameters involved, r is equal to zero only if the resonance is achieved with an error, say, in the fifth decimal place. This may, in fact, not be the case of the actual experimental situation. To get a more realistic estimate of the Raman efficiency we consider a typical ferromagnetic semiconductor such as CdCr 2Se4 with thew~ following values for the physical ~ = 2.00 x lO’~sec~,m = 1028g, J = J(~)_J~’)= 0.1 eV, N/V = 1023 cm3. From the absorption data15 it follows that CdCr 2Se4 is transparent for wavelengths greater than 1 pm. Hence assuming a crystal of 1 cm thick with a detector receiving the photons from unit solid angle one estimates9at the Raman efficiencythe to be T= lOO°Kusing X of the order of 0.5 x lO 1 = 1O1SOAYAG line (w1 = 1.88 x 1015 sec~).This value of about l0~0 for the Raman efficiency is within the same of 4 order for insumagnitude as that of Loudon’s mechanism lators. For r = 0, however, the Raman efficiency is of the

Vol. 18, No. 7

order of 10-6. Furthermore, takingf = 0.01 eV and keeping the other parameters fixed the Raman efficiency becomes of the order of 0.2 x 10-il. Finally, in order that the Raman signal be detected the scattered photon flux must be strong enough to lie above the noise level of the appropriate detector. In the region of 1—5 pm one may use PbS which responds to a photon flux greater than l0~ photons/sec. This in turn, entails that, for e 10~0,the incident power should be 2 which is well within the exof about 10—20 perimental reach.mW/cm In conclusion, it should be emphasized that the considered model contains a number of simplified assumptions. Nevertheless, some essential conclusions can be drawn therefrom. Amongst them, the present mechanism suggests to us the use of the one-magnon scattering as a new tool for the investigation of the s—d exchange parameter. Acknowledgement One ofJr., us (L.C.M.M.) is very grateful to Professor W.E. Lamb, for the hospitality and for making possible his visit to the University of Arizona. —

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