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Raman scattering in a locally anharmonic model with strong electron correlations
PHYSlCA® ELSEVIER
Physica C 341-348 (2000) 171 - i 72
www.elsevier,nl/Iocate/physc
Raman Scattering in a Locally Anharmonic Model with Strong Electron Correlations I. V. Stasyuk and T. S. Mysakovych Institute for Condensed Matter Physics NASU, 1 Svientsitskii str., UA-79011 Lviv, Ukraine Anharmonic phonon (pseudospin) contributions to the Raman light scattering in the HTSC crystals described by the pseudospin- electron model with strong electron correlations are investigated. The frequency dependence of the scattering spectrum components connected with excitations of pseudospin wave type as well as with the reconstruction of electron spectrum at the pseudospin reorientation is analyzed.
This work is devoted to the study of some features of Raman light scattering in high-T~ superconducting (HTSC) crystals caused by the strong local anharmonicity of the lattice. It is one of factors that can influence the electron or magnon scattering process and be a reason why the observed Raman line profiles in crystals of the Y B a 2 C u 3 0 7 - 6 group deviate appreciably from the predicted ones by the magnon mechanism of scattering [1]. The pseudospin-electron model (PEM) [2] is used in this work for investigation of the Raman scattering on the local anharmonic vibrations (pseudospin reorientations) modified by the interaction with conducting electrons. The pseudospin contributions to the polarizability operator and the Raman scattering tensor are separated. The frequency dependences of the scattering intensity at the certain values of the electron concentration n and temperature T are analyzed. In our calculation we base on the expression for the Raman scattering tensor +oo
g
;~---k2,kl( 0 ) 1 , 0 J 2 )
----
X
x { {M~(#, t ) l M ~ (L s)}},
where wl,w2 are incident and scattered light frequencies;_, kl, k2 are corresponding wave vectors; MC'(k) is a Fourier transform of a dipole momentum of a crystal unit cell; the symbol {{/~/~(k3, t)l_~ra(f¢, s)}} stands for "unaveraged" Green's function defined as {{A(t)lB(t')} } = - i O ( t - t ' ) [ A ( t ) , B(¢)] with operators being written in Heisenberg representation. The Hamiltonian of the PEM has the form [2]:
H : Z ~i + ~ ti,j~L~., i
where the single-site term
Hi = Uni~nij~ - #(nit + ni~)+ +9(ni, + ni+)S~ - 12S~ - hS~
+o~
-o¢ 0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V PI[ $ 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 0 4 3 2 - 9
(4)
describes the pseudospin-electron interaction (gterm); the tunneling splitting of the vibrational mode (fl-term), the asymmetry of local potential (h- field) and the Hubbard electron correlation (U-term). After reducing to the diagonal form
(1)
/5 is the polarizability operator
(3)
i,j,a
Hi = Z
× ( ~kH (_0)~ t ) Y ~ , (0)~,0)), 2_kl
(2)
)'rX~'r'
(5)
r
with A,7 = -/~n~ + U(62,~ + 52,;-)+ 1
nl = 0 ; n 2 = 2 ; n u - - n 4 = l . All rights reserved.
(6)
I.V. Stasyuk, T.£ Mysakovych/Physica C 341-348 (2000) 171-172
172
Here the Hamiltonian is written in terms of the operators X [ ' = li,r > < i,s I. We use the equations of motion procedure for the quantities {{A(t)lB(t')} } to construct the P operator in the form of operator series in powers of the electron transfer constant tij. In zero order with respect to the parameter tij
{{MkIM'}} = Y~
sin4¢~Sk, td`2 (X/7 - X[%)+
4Wl
r
5k tds 2
+
4( 1
Err)(sin4OrX;r-
r
- s i n e 2 ¢ r ( Z ; r - X;~) ) 5k Ids 2 r
7"*
small ones is overlapped by the above mentioned band. The first (incoherent) of these components is connected with interband transitions from the occupied states of the e~l subband to the unoccupied ones of the e ~ subband and reflects the reconstruction of electron states at the reorientation of pseudospin. The second (coherent) component has a pure pseudospin origin. It expresses the collective (pseudospin wave-type) dynamics that is formed by the effective interaction between pseudospins via conducting electrons. At the temperature T ~ 0 the band in the scattering spectrum becomes broader due to the smearing of the Fermi distribution and can cover now the coherent component of the spectrum creating the sufficiently narrow peak on the band background (Fig.l). At the small values of n the intensity cur-
- s i n 2 2 ¢ r ( X ; ~ - X/r)), where cos2¢r =
4-
nrg - h
(b)
(8)
~/(nrg -- h) 2 + ~2
I O0
v
(3 2"
Retaining only the pseudospin contributions to the polarizability operator, we consider Raman scattering on the local anharmonic vibrations that is modified by the effective retarded interaction between pseudospins and by the electron interband transitions. The case 0 < n < 1; U, g >> h, 9t >> W (W = ~ j tij ) is considered. The electron subbands e~a = ~1, 4e~x= e~, where
epq(k) = Ap - Aq + Apq(X pp + xqq}tk,
(9)
and Agi- = cos2¢x, Aga = sin2¢x, taking part in the transitions with the pseudospin reorientations, are considered. The pseudospin correlation functions (XPq(t)X re) (pq, rs = 11, 11, 11, 11) are calculated with the use of the generalized random phase approximation (GRPA) at the basic allowance for the short-range electron correlations. The pseudospin "boson" and loop-like electron contributions are taken into account [3]. As a result, the two main contributions to the Stokes component of the Raman scattering intensity I(w) ... Hk2,kx (W) can be separated at T = 0: a) the relatively broad band, the width of which Aw = 4(1-n)W changes with th change of n • 2--n ' b) the narrow peak that at large electron concentrations is in the shape of the 6-function while at
-0.02
50
0.o1 co o.d4-~ ~
• " 0.50
co
0.56
Figure 1. Raman scattering intensity for h = 0.4, fl = 0.5, W = 0.1,n = 0.9, a)T = 0;b)T = 0.005;w = (w2 - Wl + v / ~ - + fl2)/10W. ves possess presumably asymmetric shape with the wing on the large frequency side. On the base of obtained results the study of the pseudospin dynamics influence on the magnon scattering and the consistent analysis of experimental data can be performed. REFERENCES 1. M. Pressl, M. Mayer, P. Knoll, S. Lo, U. Hohenster, J. Raman Spectr., 27 (1996) 343. 2. K.A. Mfiller, Z. Phys. B, 80 (1990) 193. 3. I.V. Stasyuk and T.S. Mysakovych, J. Phys. Studies, 3 (1999) 344.
Physica C 341-348 (2000) 171 - i 72
www.elsevier,nl/Iocate/physc
Raman Scattering in a Locally Anharmonic Model with Strong Electron Correlations I. V. Stasyuk and T. S. Mysakovych Institute for Condensed Matter Physics NASU, 1 Svientsitskii str., UA-79011 Lviv, Ukraine Anharmonic phonon (pseudospin) contributions to the Raman light scattering in the HTSC crystals described by the pseudospin- electron model with strong electron correlations are investigated. The frequency dependence of the scattering spectrum components connected with excitations of pseudospin wave type as well as with the reconstruction of electron spectrum at the pseudospin reorientation is analyzed.
This work is devoted to the study of some features of Raman light scattering in high-T~ superconducting (HTSC) crystals caused by the strong local anharmonicity of the lattice. It is one of factors that can influence the electron or magnon scattering process and be a reason why the observed Raman line profiles in crystals of the Y B a 2 C u 3 0 7 - 6 group deviate appreciably from the predicted ones by the magnon mechanism of scattering [1]. The pseudospin-electron model (PEM) [2] is used in this work for investigation of the Raman scattering on the local anharmonic vibrations (pseudospin reorientations) modified by the interaction with conducting electrons. The pseudospin contributions to the polarizability operator and the Raman scattering tensor are separated. The frequency dependences of the scattering intensity at the certain values of the electron concentration n and temperature T are analyzed. In our calculation we base on the expression for the Raman scattering tensor +oo
g
;~---k2,kl( 0 ) 1 , 0 J 2 )
----
X
x { {M~(#, t ) l M ~ (L s)}},
where wl,w2 are incident and scattered light frequencies;_, kl, k2 are corresponding wave vectors; MC'(k) is a Fourier transform of a dipole momentum of a crystal unit cell; the symbol {{/~/~(k3, t)l_~ra(f¢, s)}} stands for "unaveraged" Green's function defined as {{A(t)lB(t')} } = - i O ( t - t ' ) [ A ( t ) , B(¢)] with operators being written in Heisenberg representation. The Hamiltonian of the PEM has the form [2]:
H : Z ~i + ~ ti,j~L~., i
where the single-site term
Hi = Uni~nij~ - #(nit + ni~)+ +9(ni, + ni+)S~ - 12S~ - hS~
+o~
-o¢ 0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V PI[ $ 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 0 4 3 2 - 9
(4)
describes the pseudospin-electron interaction (gterm); the tunneling splitting of the vibrational mode (fl-term), the asymmetry of local potential (h- field) and the Hubbard electron correlation (U-term). After reducing to the diagonal form
(1)
/5 is the polarizability operator
(3)
i,j,a
Hi = Z
× ( ~kH (_0)~ t ) Y ~ , (0)~,0)), 2_kl
(2)
)'rX~'r'
(5)
r
with A,7 = -/~n~ + U(62,~ + 52,;-)+ 1
nl = 0 ; n 2 = 2 ; n u - - n 4 = l . All rights reserved.
(6)
I.V. Stasyuk, T.£ Mysakovych/Physica C 341-348 (2000) 171-172
172
Here the Hamiltonian is written in terms of the operators X [ ' = li,r > < i,s I. We use the equations of motion procedure for the quantities {{A(t)lB(t')} } to construct the P operator in the form of operator series in powers of the electron transfer constant tij. In zero order with respect to the parameter tij
{{MkIM'}} = Y~
sin4¢~Sk, td`2 (X/7 - X[%)+
4Wl
r
5k tds 2
+
4( 1
Err)(sin4OrX;r-
r
- s i n e 2 ¢ r ( Z ; r - X;~) ) 5k Ids 2 r
7"*
small ones is overlapped by the above mentioned band. The first (incoherent) of these components is connected with interband transitions from the occupied states of the e~l subband to the unoccupied ones of the e ~ subband and reflects the reconstruction of electron states at the reorientation of pseudospin. The second (coherent) component has a pure pseudospin origin. It expresses the collective (pseudospin wave-type) dynamics that is formed by the effective interaction between pseudospins via conducting electrons. At the temperature T ~ 0 the band in the scattering spectrum becomes broader due to the smearing of the Fermi distribution and can cover now the coherent component of the spectrum creating the sufficiently narrow peak on the band background (Fig.l). At the small values of n the intensity cur-
- s i n 2 2 ¢ r ( X ; ~ - X/r)), where cos2¢r =
4-
nrg - h
(b)
(8)
~/(nrg -- h) 2 + ~2
I O0
v
(3 2"
Retaining only the pseudospin contributions to the polarizability operator, we consider Raman scattering on the local anharmonic vibrations that is modified by the effective retarded interaction between pseudospins and by the electron interband transitions. The case 0 < n < 1; U, g >> h, 9t >> W (W = ~ j tij ) is considered. The electron subbands e~a = ~1, 4e~x= e~, where
epq(k) = Ap - Aq + Apq(X pp + xqq}tk,
(9)
and Agi- = cos2¢x, Aga = sin2¢x, taking part in the transitions with the pseudospin reorientations, are considered. The pseudospin correlation functions (XPq(t)X re) (pq, rs = 11, 11, 11, 11) are calculated with the use of the generalized random phase approximation (GRPA) at the basic allowance for the short-range electron correlations. The pseudospin "boson" and loop-like electron contributions are taken into account [3]. As a result, the two main contributions to the Stokes component of the Raman scattering intensity I(w) ... Hk2,kx (W) can be separated at T = 0: a) the relatively broad band, the width of which Aw = 4(1-n)W changes with th change of n • 2--n ' b) the narrow peak that at large electron concentrations is in the shape of the 6-function while at
-0.02
50
0.o1 co o.d4-~ ~
• " 0.50
co
0.56
Figure 1. Raman scattering intensity for h = 0.4, fl = 0.5, W = 0.1,n = 0.9, a)T = 0;b)T = 0.005;w = (w2 - Wl + v / ~ - + fl2)/10W. ves possess presumably asymmetric shape with the wing on the large frequency side. On the base of obtained results the study of the pseudospin dynamics influence on the magnon scattering and the consistent analysis of experimental data can be performed. REFERENCES 1. M. Pressl, M. Mayer, P. Knoll, S. Lo, U. Hohenster, J. Raman Spectr., 27 (1996) 343. 2. K.A. Mfiller, Z. Phys. B, 80 (1990) 193. 3. I.V. Stasyuk and T.S. Mysakovych, J. Phys. Studies, 3 (1999) 344.