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Analysis of optical conductivities of cuprates via doped Hubbard model
PHYSICA ELqEVIER
Physica C 282-287 (1997) 1743-1744
Analysis of optical conductivities of cuprates via doped Hubbard model Jongbae Hong* ~ and H.-Y. Kee b ~Department of Physics Education, Seoul National University, Seoul 151-742, Korea bDepartment of Physics, Rutgers State University, Piscataway, NJ 08854, USA Analysis of optical conductivities measured for cuprates has been done via the doped Hubbard model in infinite dimensions using the analytical variant of the dynmlfic Lanczos method. We examine the frequency behavior in the low frequency region at comparatively large doping, the midinfrared band, and the isobestic point in the optical conductivity of cuprates.
'File normal state property of high Tc materials having strong correlation is important to test whether a microscopic model is suitable or not. One of interesting normal state behaviors is seen in the optical conductivity of cuprates. The interesting unusual features of the optical conductivity are (1) ~ l/a; behavior in a low frequency region at large doping (x _> 0.2), (2) the so called midinfrared(MIR) band in the charge transfer gap upon small hole or electron doping, (3) there is a crossing (isobestic) point at a special frequency at which optical conductivities of different doping concentrations coincide.Ill. These interesting behaviors have not been fully studied. The proper model to describe normal state of cuprates is the well-known doped Hubbard model with nearest neighbor hopping. We study this model on the Bethe lattice with infinite connectivity which captures the mean-field nature of the real system appropriately. We believe that the description in infinite dimensions[3] may reveal the characteristics of normal states of cuprates to some extent. The optical conductivity in infinite dimensions is obtained using the following formula[2].
=f where Ao(e) = ~ for the Bethe lattice, ~,(a;t) = f(u.c)-f(cd+¢o) f(a;) is t h e F e r m i d i s oJ ~N tribution function, a0 = " t2~v
where a , N , V
*This work has been supported by SNU-CTP and Basic Science Research Institute Program, Ministry of Education. 0921-4534/97/$17.00 © ElsevierScienceB.V. All rights reserved. PII S0921-4534(97)00995-7
are lattice constant, number of lattice sites, volume, respectively, and A(e, a;) = - l I m G ( k , a;) = - ~ I m I a ; ÷ i 0 - £ - E ( a ; ) ] -1. Use of the momentumindependence of the self-energy in infinite dimensions has been made. The self-energy E(a;) gives A(e, a;) using the relation E(a;) = a ; - ~ 1 which is valid for a paramagnetic state on Bethe lattice.J4,5] The Green's function G(a;) can be written as
iG(a;)
=:
(~ol{Cjo(r),C}o}l~o)e wT-'Tdr
=
(2)
where
det(zI - M)oo ao(z) = d e t ( z I - M) '
(3)
where numerator is the cofactor of the element z - Moo of the determinant in denominator, I is the unit matrix, and M is the matrix whose elements are M ~ = ~ , where L is the LiouviIle operator and the inner product is defined by (fv,f~,)=: (~ol(fv,f~}l~O). Fig. 1 shows the optical conductivity (thinsolid line) obtained by using Eq. (1) for U := 6t. using ~? = 0.6t. in units of a0. One can see MIR bands in x = 0.077 and x -- 0.1. As for the anomalous frequency behavior at the low frequency regime, our result can only show a trend approaching ~ 1/w decay (dashed line) as x increases. In the figure we divide the optical conductivities into two parts to analyze the MIR band and the so called isobestic point. T h e thicksolid line is obtained by the transition to gap
3 [long, H.-Y Kee/Physica C 282-287 (1997) 1743-1744
1744
states and the dotte.d line is to upper llut)bard band fi'om stales below I.'crnd level. The fi)rmer may be relat(,d to the enhanced spin fluctuations due to increased holes and the latter to charge excitations. Thin-solid lines show the hill contribution of electron excitations fl'om occupied to unoccupied states. It is interesting to note the effect of phenonmnological parameter r/which gives rise to a dispersed Drude peak in optical conductivity and hides MIR band at x = 0.125 and x = 0.2 in which comparatively large free carrier contributions occur. Th(;refore, the phenomenological parameter 7/plays a crucial role in simulating exp(;riment. Our theoretical lines ch,arly show a crossing l)oint, callc(l isolmstic poitlt around u~ 7z 3/.. Our analysis shows that the combination of the contribution from charge excitation and that from spin fluctuation is exactly the ~une at this l)articular energy excitation for various dopings. We add the optical conductivity for the half-filled case (x = 0) in Fig. 1. This has been obtained analytically for the paramaguctic state with 77 = 0 in our previous workl(i,dJ. We have a good compari~)n with experiment on La2_,.5'r,Cu04[l], i.e., the chargetransfer gap is 2eV an(l the isobestic point rises
at about 1.5eV. And 7/-= 0.3eV which is the same as that used ill other work[7]. Therefore, we predict that the effective on-site Coulomb repulsion U in La2_~SrxCu04 is about 3eV. In conclusion, the optical conductivity of the doped Hubbard model with large r/describes the anomalous behaviors of normal state of cuprates. Enhanced and weakened free-carrier strength at large and small doping, respectively, may explain the change of slope of the optical conductivity in the low frequency regime. Spin fluctuations are mainly responsible for MIR band, though charge excitation contribute a little at small doping. The isobestic point is one at which the sum of spin fluctuation and charge excitation is the same fi)r different (loping concentrations. We also stress to study the origin of finite 7/taking real effects into account, because that makes theory simulate experiment very well. As a final comment, we mention that all of the above unusual features of cuprates obtained for the doped Hubbard model in infinite dimensions have also been studied to some extent, but not fully, by numerical work in two dimensions[8]. There are also some works in infinite dimensions via numerical method.J3,9] These including present work imply that the phenomena treated in this work are generic propertics of the strongly correlated system.
REFERENCES
0.03 ~
.
"'''.
0.02
0
0
0.5
" "" I
"'"
1.6
2
2.5
~(eV')
Figure 1. Optical conductivities using 77 = 0.6t. for various doping concentrations. But 7 / : 0 for x : 0. l~'equcncy has been rcscaled in units of t. = 0.5eV.
1. S. Uchida et al., Phys. Rev. B 43, 7942 (1991). 2. Th. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 (1993). 3. A. Georges, G. Kotliar, W. Krauth, and M. .l. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 4. .I. llong and I]-Y. Kee, Europhys. Let(. 33, 453 (1996). 5. A. Gcorges and W. Krauth, Phys. Rev. B 48, 7167 (1993). 6. J. tlong and H-Y. Kee, Phys. Rev. B 52, 2415 (1995). 7. E. Dagotto, Rev. Mod. Phys. OO, 763 (1994). 8. E. Dagotto et al., Phys. Rev. B 45, 10107 (1992). 9. M. Jarrell, J. K. Freericks, and Th. Pruschke, Phys. Rev. B 51, 11704 (1995).
Physica C 282-287 (1997) 1743-1744
Analysis of optical conductivities of cuprates via doped Hubbard model Jongbae Hong* ~ and H.-Y. Kee b ~Department of Physics Education, Seoul National University, Seoul 151-742, Korea bDepartment of Physics, Rutgers State University, Piscataway, NJ 08854, USA Analysis of optical conductivities measured for cuprates has been done via the doped Hubbard model in infinite dimensions using the analytical variant of the dynmlfic Lanczos method. We examine the frequency behavior in the low frequency region at comparatively large doping, the midinfrared band, and the isobestic point in the optical conductivity of cuprates.
'File normal state property of high Tc materials having strong correlation is important to test whether a microscopic model is suitable or not. One of interesting normal state behaviors is seen in the optical conductivity of cuprates. The interesting unusual features of the optical conductivity are (1) ~ l/a; behavior in a low frequency region at large doping (x _> 0.2), (2) the so called midinfrared(MIR) band in the charge transfer gap upon small hole or electron doping, (3) there is a crossing (isobestic) point at a special frequency at which optical conductivities of different doping concentrations coincide.Ill. These interesting behaviors have not been fully studied. The proper model to describe normal state of cuprates is the well-known doped Hubbard model with nearest neighbor hopping. We study this model on the Bethe lattice with infinite connectivity which captures the mean-field nature of the real system appropriately. We believe that the description in infinite dimensions[3] may reveal the characteristics of normal states of cuprates to some extent. The optical conductivity in infinite dimensions is obtained using the following formula[2].
=f where Ao(e) = ~ for the Bethe lattice, ~,(a;t) = f(u.c)-f(cd+¢o) f(a;) is t h e F e r m i d i s oJ ~N tribution function, a0 = " t2~v
where a , N , V
*This work has been supported by SNU-CTP and Basic Science Research Institute Program, Ministry of Education. 0921-4534/97/$17.00 © ElsevierScienceB.V. All rights reserved. PII S0921-4534(97)00995-7
are lattice constant, number of lattice sites, volume, respectively, and A(e, a;) = - l I m G ( k , a;) = - ~ I m I a ; ÷ i 0 - £ - E ( a ; ) ] -1. Use of the momentumindependence of the self-energy in infinite dimensions has been made. The self-energy E(a;) gives A(e, a;) using the relation E(a;) = a ; - ~ 1 which is valid for a paramagnetic state on Bethe lattice.J4,5] The Green's function G(a;) can be written as
iG(a;)
=:
(~ol{Cjo(r),C}o}l~o)e wT-'Tdr
=
(2)
where
det(zI - M)oo ao(z) = d e t ( z I - M) '
(3)
where numerator is the cofactor of the element z - Moo of the determinant in denominator, I is the unit matrix, and M is the matrix whose elements are M ~ = ~ , where L is the LiouviIle operator and the inner product is defined by (fv,f~,)=: (~ol(fv,f~}l~O). Fig. 1 shows the optical conductivity (thinsolid line) obtained by using Eq. (1) for U := 6t. using ~? = 0.6t. in units of a0. One can see MIR bands in x = 0.077 and x -- 0.1. As for the anomalous frequency behavior at the low frequency regime, our result can only show a trend approaching ~ 1/w decay (dashed line) as x increases. In the figure we divide the optical conductivities into two parts to analyze the MIR band and the so called isobestic point. T h e thicksolid line is obtained by the transition to gap
3 [long, H.-Y Kee/Physica C 282-287 (1997) 1743-1744
1744
states and the dotte.d line is to upper llut)bard band fi'om stales below I.'crnd level. The fi)rmer may be relat(,d to the enhanced spin fluctuations due to increased holes and the latter to charge excitations. Thin-solid lines show the hill contribution of electron excitations fl'om occupied to unoccupied states. It is interesting to note the effect of phenonmnological parameter r/which gives rise to a dispersed Drude peak in optical conductivity and hides MIR band at x = 0.125 and x = 0.2 in which comparatively large free carrier contributions occur. Th(;refore, the phenomenological parameter 7/plays a crucial role in simulating exp(;riment. Our theoretical lines ch,arly show a crossing l)oint, callc(l isolmstic poitlt around u~ 7z 3/.. Our analysis shows that the combination of the contribution from charge excitation and that from spin fluctuation is exactly the ~une at this l)articular energy excitation for various dopings. We add the optical conductivity for the half-filled case (x = 0) in Fig. 1. This has been obtained analytically for the paramaguctic state with 77 = 0 in our previous workl(i,dJ. We have a good compari~)n with experiment on La2_,.5'r,Cu04[l], i.e., the chargetransfer gap is 2eV an(l the isobestic point rises
at about 1.5eV. And 7/-= 0.3eV which is the same as that used ill other work[7]. Therefore, we predict that the effective on-site Coulomb repulsion U in La2_~SrxCu04 is about 3eV. In conclusion, the optical conductivity of the doped Hubbard model with large r/describes the anomalous behaviors of normal state of cuprates. Enhanced and weakened free-carrier strength at large and small doping, respectively, may explain the change of slope of the optical conductivity in the low frequency regime. Spin fluctuations are mainly responsible for MIR band, though charge excitation contribute a little at small doping. The isobestic point is one at which the sum of spin fluctuation and charge excitation is the same fi)r different (loping concentrations. We also stress to study the origin of finite 7/taking real effects into account, because that makes theory simulate experiment very well. As a final comment, we mention that all of the above unusual features of cuprates obtained for the doped Hubbard model in infinite dimensions have also been studied to some extent, but not fully, by numerical work in two dimensions[8]. There are also some works in infinite dimensions via numerical method.J3,9] These including present work imply that the phenomena treated in this work are generic propertics of the strongly correlated system.
REFERENCES
0.03 ~
.
"'''.
0.02
0
0
0.5
" "" I
"'"
1.6
2
2.5
~(eV')
Figure 1. Optical conductivities using 77 = 0.6t. for various doping concentrations. But 7 / : 0 for x : 0. l~'equcncy has been rcscaled in units of t. = 0.5eV.
1. S. Uchida et al., Phys. Rev. B 43, 7942 (1991). 2. Th. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 (1993). 3. A. Georges, G. Kotliar, W. Krauth, and M. .l. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 4. .I. llong and I]-Y. Kee, Europhys. Let(. 33, 453 (1996). 5. A. Gcorges and W. Krauth, Phys. Rev. B 48, 7167 (1993). 6. J. tlong and H-Y. Kee, Phys. Rev. B 52, 2415 (1995). 7. E. Dagotto, Rev. Mod. Phys. OO, 763 (1994). 8. E. Dagotto et al., Phys. Rev. B 45, 10107 (1992). 9. M. Jarrell, J. K. Freericks, and Th. Pruschke, Phys. Rev. B 51, 11704 (1995).