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Raman scattering by two-magnon excitation due to the s-d exchange interaction
Volume
44A, number
4
PHYSICS
RAMAN
SCATTERING
18 June
LETTERS
BY TWO-MAGNON
DUE TO THE s-d EXCHANGE
1973
EXCITATION
INTERACTION
A. STASCH * Technical University, Institute of Physics, Poznan, Poland Received
18 May 1973
Theory of the two-magnon Raman scattering is given with particular reference to antiferromagnetic insulators. A two band model for the electrons and holes is used. The extinction coefficient is found to be of the same order as for the first order Raman effect i.e. 10e6 - lo-lo cm-l sr-‘.
The utility of Raman effect for the study of excitations in insulators, metals and semiconductors has been amply demonstrated in recent years in a variety of papers [l-4]. In the present paper we report the calculations of the extinction coefficient for the two-magnon Raman scattering on the basis of the s-d exchange model. The unperturbed Hamiltonian of the model is given by:
Ho = k~Ei‘%,s~kS +k?Eih)h;,,hk,s +CE q 4 (f' lq f lq
+f;qf2q
+ 1) >
(1)
where the first, second and third term describes the energy of the conduction electrons, holes in the valence band and the antiferromagnetic magnons, respectively. The Hamiltonian of the interaction of the electrons, holes and magnons with the radiation field, derived in detail in the papers [l] and [3], is H’ = Hs_d + He_,, H
+6(k-k’-q)f2+4c;_ck,J
uI ’ [P/:k,;‘$-k’&,s
+p-l~k,~h-,-k,sck,slal
,
+ C.C.>
where J(k) k’) denotes the Fourier transform of the s-d exchange integral, V the crystal volume, al is the creation ' is the intraband matrix element operator of the foton with wave vector 1, energy fiiw, and polarization Ul .plFk,k of the momentum operator. Note that the intrabandp-matrix elements are neglected in He_, because as it has been shown by Platzmann and Tzoar [4] these elements are unimportant for the Raman scattering. From (1) and (2), after the standard calculation [S], the extinction coefficient is found to be: 2
c
VN2 4
[Nl(q)+
1]2(Gq)41~,,212
(3)
where M 1,2 =
(4) UI1 ~p~,~w(k--q,412
+U12.P,,;w
+ IJ(k+q,k)12)p~,;:ul,
(Eg+Ao,)2(Eg+Ep+fii,) *Address:
Instytut
Fizyki
Politechniki
Poznanskiej,
- q, 412 +lJ(k+q, k)12,pk’,;;.z$ (Eg-hq)2(Eg+Ep-hw,)
Poznari,
Piotrowo
5, Polska,
1 ’
61 138.
291
Volume
44A, number
4
PHYSICS
LETTERS
IX June
1973
and the magnon energy have been assumed to be smaller than the photon energy fiw,. In (3) the dependence ot the s--d integral on the small wave vectors of the photons is neglected. The parameters I!z’~and 17, are the width of the energy gap and the sum of the width of the valence and conduction band. respectively. Replacement of I!?~;(‘:;, + E(L) by ES + Eb in (4) is stimulated by the experimental results 161, which indicate that the magnons-active in the second order Raman scattering have their wave vectors near the zone edge. On the basis of (3) it is possible to make a numerical estimate of the extinction coefficient. Using typical values for the parameters (E:, -plw, I to 2 eV, E, - 5 cV, G, - 1, pl i -n/u, N,(q) - 0. u = 5 X IO- 8 cm. J(k,k’) - 0.05 0.5 eV) and assuming that the number of the magnons with wave vector q, N, (I/), is equal to /era, we calculate the extinction coefficient to be of the same order as for the one magnon effect, i.e. 10~ 6 10 “’ cm ’ sr I. Note that the so d exchange interaction is inoperative for the first order Raman effect because the generation of the single magnon h!, the operator (2a) is connected with spin-flip scattering of the electron to the conduction hand.
References {I] [ 21 [3] (41 [ 5] 161
292
P.M. Platzmann and N. Tzoar, Phys. Rev. 166 (196X) 514. P.A. Fleury and J.F. Scott, Phy?. Rev. 3B (I 968) 1979. J.A. Izyumov, Fiz. Mett. Mett. 11 (1961) 650. M. Inoue and T. Moriya, J. Phys. SW. Japan 29 (1970) I 17. R. Loudon, Proc. Roy. Sot. 275 ( 1963) 21 X. P.A. Fleury and R. Loudon. Phys. Rev. 166 (I 968) 5 14.
44A, number
4
PHYSICS
RAMAN
SCATTERING
18 June
LETTERS
BY TWO-MAGNON
DUE TO THE s-d EXCHANGE
1973
EXCITATION
INTERACTION
A. STASCH * Technical University, Institute of Physics, Poznan, Poland Received
18 May 1973
Theory of the two-magnon Raman scattering is given with particular reference to antiferromagnetic insulators. A two band model for the electrons and holes is used. The extinction coefficient is found to be of the same order as for the first order Raman effect i.e. 10e6 - lo-lo cm-l sr-‘.
The utility of Raman effect for the study of excitations in insulators, metals and semiconductors has been amply demonstrated in recent years in a variety of papers [l-4]. In the present paper we report the calculations of the extinction coefficient for the two-magnon Raman scattering on the basis of the s-d exchange model. The unperturbed Hamiltonian of the model is given by:
Ho = k~Ei‘%,s~kS +k?Eih)h;,,hk,s +CE q 4 (f' lq f lq
+f;qf2q
+ 1) >
(1)
where the first, second and third term describes the energy of the conduction electrons, holes in the valence band and the antiferromagnetic magnons, respectively. The Hamiltonian of the interaction of the electrons, holes and magnons with the radiation field, derived in detail in the papers [l] and [3], is H’ = Hs_d + He_,, H
+6(k-k’-q)f2+4c;_ck,J
uI ’ [P/:k,;‘$-k’&,s
+p-l~k,~h-,-k,sck,slal
,
+ C.C.>
where J(k) k’) denotes the Fourier transform of the s-d exchange integral, V the crystal volume, al is the creation ' is the intraband matrix element operator of the foton with wave vector 1, energy fiiw, and polarization Ul .plFk,k of the momentum operator. Note that the intrabandp-matrix elements are neglected in He_, because as it has been shown by Platzmann and Tzoar [4] these elements are unimportant for the Raman scattering. From (1) and (2), after the standard calculation [S], the extinction coefficient is found to be: 2
c
VN2 4
[Nl(q)+
1]2(Gq)41~,,212
(3)
where M 1,2 =
(4) UI1 ~p~,~w(k--q,412
+U12.P,,;w
+ IJ(k+q,k)12)p~,;:ul,
(Eg+Ao,)2(Eg+Ep+fii,) *Address:
Instytut
Fizyki
Politechniki
Poznanskiej,
- q, 412 +lJ(k+q, k)12,pk’,;;.z$ (Eg-hq)2(Eg+Ep-hw,)
Poznari,
Piotrowo
5, Polska,
1 ’
61 138.
291
Volume
44A, number
4
PHYSICS
LETTERS
IX June
1973
and the magnon energy have been assumed to be smaller than the photon energy fiw,. In (3) the dependence ot the s--d integral on the small wave vectors of the photons is neglected. The parameters I!z’~and 17, are the width of the energy gap and the sum of the width of the valence and conduction band. respectively. Replacement of I!?~;(‘:;, + E(L) by ES + Eb in (4) is stimulated by the experimental results 161, which indicate that the magnons-active in the second order Raman scattering have their wave vectors near the zone edge. On the basis of (3) it is possible to make a numerical estimate of the extinction coefficient. Using typical values for the parameters (E:, -plw, I to 2 eV, E, - 5 cV, G, - 1, pl i -n/u, N,(q) - 0. u = 5 X IO- 8 cm. J(k,k’) - 0.05 0.5 eV) and assuming that the number of the magnons with wave vector q, N, (I/), is equal to /era, we calculate the extinction coefficient to be of the same order as for the one magnon effect, i.e. 10~ 6 10 “’ cm ’ sr I. Note that the so d exchange interaction is inoperative for the first order Raman effect because the generation of the single magnon h!, the operator (2a) is connected with spin-flip scattering of the electron to the conduction hand.
References {I] [ 21 [3] (41 [ 5] 161
292
P.M. Platzmann and N. Tzoar, Phys. Rev. 166 (196X) 514. P.A. Fleury and J.F. Scott, Phy?. Rev. 3B (I 968) 1979. J.A. Izyumov, Fiz. Mett. Mett. 11 (1961) 650. M. Inoue and T. Moriya, J. Phys. SW. Japan 29 (1970) I 17. R. Loudon, Proc. Roy. Sot. 275 ( 1963) 21 X. P.A. Fleury and R. Loudon. Phys. Rev. 166 (I 968) 5 14.