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Optical susceptibilities of polymers: current–current versus dipole–dipole correlation
28 June 1999
Physics Letters A 257 Ž1999. 215–220
Optical susceptibilities of polymers: current–current versus dipole–dipole correlation Minzhong Xu a , Xin Sun
b
a
b
Department of Chemistry, New York UniÕersity, New York, NY 10003, USA Department of Physics, Fudan UniÕersity, and National Laboratory of Infrared Physics, Shanghai 200433, PR China Received 16 February 1999; accepted 27 April 1999 Communicated by A.R. Bishop
Abstract The static current operator leads to definitional zero frequency divergence and unphysical results in studying nonlinear optical susceptibilities of polymers. A well-defined dipole–dipole correlation is superior to the complicated current–current correlation to solve this problem. As illustrative examples, optical susceptibilities under both SSH and TLM models of trans-ŽCH. x are studied. New analytical results are obtained. The reasons of previous improper results are analyzed. q 1999 Elsevier Science B.V. All rights reserved. PACS: 78.66.Qn; 42.65.An; 72.20.Dp; 78.20.Bh Keywords: Zero frequency divergence; Optical susceptibilities; Dipole-dipole correlations
To study the nonlinear optical ŽNLO. properties of polymers, periodic approximate models are necessary to simplify the real systems. Some good approximate models of polymers based on the tight-binding approximation ŽTBA., such as SSH w1,2x and TLM 1 w3x models in weakly correlated systems, Hubbard and Pariser–Parr–Pople models in strongly correlated and electron–electron Ž e-e . interaction systems, etc, have yielded physical insights surpassing the
1 Lagrangian: LsHdx cˆ †Ž x . i" E t q i s 3 Õ F Ex q s 1 D4cˆ Ž x . w3x. The vector potential A is included by changing y i" Ex ™y i" Ex y eA, it will not contain A2 term in the Lagrangian, also see Wu’s work w13x.
complicated non-approximate computations. In considering the optical response of these models, a UŽ1. transformation has been suggested w4–6x to provide the gauge invariance of the TBA Hamiltonian. In linear optical ŽLO. response theory, the Kubo formula based on current–current Ž J-J . correlation w7,8x is widely used because of simplicity. It is commonly held that in discussing the optical susceptibilities of materials, the J-J correlation Ž p P A . will play the same role as that of dipole–dipole Ž D-D . correlation Ž E P r . w9,10x and that the apparent definitional zero frequency divergence ŽZFD. in the NLO susceptibilities definition w11–13x is only a virtual problem, although the proofs only have been shown under certain assumptions w7,14x. However, there exists some discrepancies from the above conclusion
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 8 8 - 1
216
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
under the models. Ži. In the study of conductivity s under TBA models w15,16x, the J-J correlation will have ZFD in ImŽ s . Žrelated to the real part of J-J . although it can give the correct ReŽ s . Žrelated to the imaginary part of J-J .. Žii. In third-harmonic generation ŽTHG. of trans-ŽCH. x , the experimentally observed two-photon absorption peak ŽTP. w17,18x has given rise to theoretical explanations. Based on J-J correlation, the TP was explained w13x by choosing the TLM model. But it has been criticized by others w19–24x with simple parity consideration. Also the spectrum w13x is quite different from that under the D-D correlation w25x. Thus, it casts some doubts in the practical application of J-J correlation. In this letter, we will show that all above controversies are caused by improper application of J-J correlation based only on the static current operator J0ˆ . To recover the correct results, a well-defined D-D correlation is more suitable for studying the optical susceptibilities than the J-J correlation, whose equivalence to the D-D correlation can be satisfied by introducing complicated induced field currents ŽIFCs.. Besides solving the apparent ZFD in the definition of NLO susceptibility, the D-D correlation is to be favored over the J-J correlation due to the lack of the gauge dependence of the vector potential A w10x and simplifying the definitional complexity of the contribution by IFCs in the high order expansions w4,5x. As a deduction, the TP cusp and ZFD w13,15,16x will no longer exist in both SSH and TLM models by the well-defined D-D correlation. However, this seemingly trivial conclusion has not been clearly illustrated by the others although the D-D correlation has already been applied in the NLO response of the real systems w10,19,20,24x. The application of J-J correlation is simplier than that of the D-D correlation in LO response because of the convenient search for the static current J0ˆ w8x compared with that for dipole expression Dˆ under the approximate models. As a typical example, the position operator rˆ is ill-defined in periodic systems w14,26x in real space while J0ˆ is not. Further, the static current J0ˆ still can give the correct results in the LO absorption w8,15,16x because the IFCs only contribute to the real part of J-J correlation. Thus the simplicity of the Kubo formula is still satisfied by applying the static current J0ˆ and ZFD in the real part of J-J correlation could be solved either by the
Kramers–Kronig ŽKK. relation w8x or by subtracting ²w j, j x:Ž v s 0. w15x. It is not a big surprise to see why the correct results still can be preserved for the LO response through J0ˆ . However, the advantage of J-J correlation application in LO response is no longer available for the NLO response because of the IFCs w5,11,12x, thus the definition of n-th order J-J correlation based only on the static current w13x will lead to the incorrect results. Besides the difficulty in obtaining the correct IFCs in approximate models, J-J correlation will make the definition of NLO susceptibilities unable to be written in a general form as defined in Wu’ work w13x and will make the computations very tedious w5x, although it may give the correct result. In periodic systems, the imposed boundary condition requires Ži. Bloch wave functions to describe the extended electronic states and Žii. all operators including the position operator rˆ to have at least the same periodicity w27x. A band index n and crystal momentum k are used to label the Bloch states: < n, k : s u n, k Ž r . e i kP r , where u n, k Ž r . is the periodic function under the translation of lattice vector. If the homogeneous electric field is applied, the energy level EnŽ k . should be changed to EnŽ k . w6x and the static current operator Jˆ0 Ž k . should be replaced by a new current operator JˆŽ k ., where k s k y eAr" w11,12x. By Taylor expansion of JˆŽ k . based on powers of the vector potential A, we will obtain the IFCs related to all orders of A besides the static current J0ˆ . There is no doubt that the IFCs will contribute to the optical response. Complexities in handling IFCs Žusually related to the intra-band currents w11,12x. in NLO studies could be imagined. Polymer models w1–3,13x usually share the same properties as periodic systems. The direct consequence of dropping the IFCs will result in ZFD and unphysical results w13,15,16x. Although the unphysical results in NLO response have been recurrently questioned by the others w19–24x, the direct reason has never been revealed and correct analytical results have never been obtained. The chief difficulty is that direct ways to include IFCs in many approximate models are forbidden w1–3,13,15x and the static current could easily be improperly used w13,16x. By a well-defined D-D correlation, we will show that all difficulties caused by the improper use of J-J correlation could be solved. To give an intuitive picture,
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
we will illustrate those correlations by studying the simple one-dimensional Ž1-d . TBA electron–lattice models of trans-ŽCH. x ŽSSH and TLM. whose optical properties have been widely studied by others w13,16–24,28x. To avoid the ill-definition and to provide the periodicity of the position operator r, ˆ we express rˆ under < n, k : as w26x:
s. In the continuum limitation, the above SSH model will give the TLM model w3,13x. In momentum space, the above Hamiltonian with electron-photon Ž e-A . interaction could be found as follows: H Ž k ,t . s Ý ´ Ž k . cˆ k†, s Ž t . s 3 cˆ k , s Ž t . y Dˆ P Ee i v t , k,s
rn k , nX kX s i dn , nX= k d Ž k y kX . q V n , nX Ž k . d Ž k y kX . , Ž 1. ) Ž . X Ž . where V n, nX Ž k . s Õi HÕ u n, k r = k u n , k r dr and Õ is unit cell volume.
1. Susceptibilities definition by D-D correlation Without considering retardation effect w10x, optical susceptibilities are usually defined by expanding the optical polarization in powers of the transverse electric field w9x. Under that approximation, the general nth-order susceptibility is a purely material quantity defined as w10x:
x Ž n. Ž V ; v 1 , . . . , v n . 1
i
s n! "
n
Hdr
1
PPP drn dt1 PPP dt n
H
ˆ ˆ Ž r ,t . = drdt eyi kP rqi V t ²TD
H
=Dˆ Ž r 1 ,t 1 . PPP Dˆ Ž rn ,t n . : ,
Ž 2.
where V ' yÝ nis1 v i , Tˆ is the time-ordering operator and Dˆ is dipole operator. Based on the periodic TBA, the SSH Hamiltonian w1,2x is given by: HSSH sy
Ý l, s
t 0 q Ž y1. l
D 2
† ˆ ˆ† ˆ Ž Cˆlq1 , s C l , s qC l , s C lq1 , s . ,
where t 0 is the transfer integral between the nearestneighbor sites, D is the gap parameter and Cˆl,† s Ž Cˆl, s . createsŽannihilates. an p electron at site l with spin
217
Ž 3.
(
2 2 where ´ Ž k . s 2 t 0 cos Ž ka . q Dsin Ž ka . and † †c † Õ cˆ k, s Ž t . s Ž aˆ k, s Ž t ., aˆ k, s Ž t .. is the two-component spinor describing excitations of electrons in the conduction band and valence band. The long wave approximation w8x is applied in electromagnetic field E with frequency v . The dipole operator Dˆ could be obtained by the Eq. Ž1.:
Dˆ s e Ý b Ž k . cˆ k†, s s 2 cˆ k , s q i k,s
ž
E Ek
cˆ k†, s cˆ k , s ,
/
Ž 4.
where b Ž k . s yD t 0 ar´ 2 Ž k ., is the coefficient related to the interband transition between the conduction and valence bands in a unit cell 2 a and the second term is related to the intraband transition w14x, e is the electric charge and s are the Pauli matrixes. We neglect the relative distortion h Ž' 2 ura. in the dipole operator because it is relatively small in the optical contribution w5,25x. Due to the fact that p electrons in the SSH model are non-localized w28x, the dipole approximation w10,24x is no longer valid in the extended states and will lead to wrong results as pointed out by some authors w5x. The Fourier transform of Eq. Ž4. to coordinate space shows that the transition dipole is related to the electron hopping to all the other sites besides the nearest neighbor sites. Thus the dipole approximation fails for the extended states in periodic systems. Because of the failure of dipole apˆ proximation through the polarization operator P, 2 computations show a magnitude difference of 10 in x Ž1. and 10 4 in x Ž3. w25x and quite different shape in spectrum compared with the results through the dipole operator Dˆ and the experimental values w17,18,28x in trans-ŽCH. x , although the position of some resonant peaks may be correctly obtained.
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
218
2. LO response by D-D Ž1. Ž The LO susceptibility x SSH V , v 1 . can be obtained from Eqs. Ž2. and Ž4.: Ž1. x SSH Ž yv 1 , v 1 .
i
s2
e2 Ý
"
k
`
Hy`Tr
E
½
= i
E GŽ k ,v . i
Ek
Ek
GŽ k ,v y v1 .
E qb Ž k . s 2 G Ž k , v . i
Ek
GŽ k ,v y v1 .
E qi
b Ž k . G Ž k , v . s2G Ž k , v y v 1 .
Ek
qb Ž k . s 2 G Ž k , v . b Ž k . s 2 G Ž k , v y v 1 . dv
=
, 2p where the Green function GŽ k, v . s
5 Ž 5.
vq v k s3
with
v 2 y v k2 q i e
q
v k ' ´ Ž k .r" and e ' 0 . Ž1. Ž . Ž1. Ž By Eq. Ž5., we have x SSH v ' x SSH yv , v .: Ž1. x SSH Ž v.
e2 Ž 2 t0 a.
s
2pD
2
1r d
H1
dx
ž
Ž1yd 2 x2 .
1 2
2
= Ž x y 1.
y1 2
2
2
x Ž x yz .
/
,
where x ' " v krD, z ' " vrŽ2 D . and d ' DrŽ2 t 0 .. If the continuum limitation is applied, that is, d ™ 0 and 2 t 0 a ™ "Õ F , the above integral gives the Ž1. Ž . LO susceptibility x TLM v under the TLM model w3,13x as follows: Ž1. x TLM Ž v . sy
e 2 "Õ F 2 2
2pD z
Ž1yf Ž z. . ,
Ž 6.
where
°
arcsin Ž z .
~
Ž z 2 -1 . ,
'
z 1y z 2
Ž 7.
f Ž z.'
¢
coshy1 Ž z .
y
ip q
'
z z 2 y1
'
2 z z 2 y1
Ž z 2 )1 . .
The conductivity s Ž v . given by yi v P x Ž1., is exactly the same as based on J-J correlation w16x. However, we should point out that the direct computation based on the static current from J-J correlation under TLM model w3,13,16x shows no first term in Eq. Ž6. w25x, ZFD in real part of J-J correlation is obvious although the correct imaginary part still can be given. These difficulties have never been clearly addressed previously, provided the reason that the static current J0ˆ is still valid for obtaining the correct imaginary part in the LO absorption. To include the IFCs by changing k ™ k y eAr" in the static current operator J0ˆ Ž k ., ZFD could be solved to give the same result as that under D-D correlation w29x. But the complicated way to include the IFCs compared with simple D-D correlation already makes J-J correlation impractical even for the LO response. Fortunately, the IFCs only contribute for the real part of J-J correlation and have no influence on the absorption. Attempts to obtain IFCs directly from TLM Hamiltonian fail because it is forbidden by the model to include A2 term w3,13x. In the discrete SSH Hamiltonian, we have the chance to include the IFCs through Peierls substitution w5x, but a straightforward computation easily shows that they are not correct IFCs to cancel the ZFD w29x. It gives us an impression of the difficulty to obtain the correct IFCs besides the complexity of handling their contributions to the optical response even in well-defined 1-d periodic models. From the above examples, the feasibility of J-J correlation in more general models will be questioned and the application of D-D correlation is more reasonable under approximate models to obtain the correct results.
3. NLO susceptibilities of trans-(CH)x chain There are many elegant works discussing NLO susceptibilities of polymer chains w13,17–24x. The TP w13x obtained from J-J correlation was doubted in the literature w17–24x since it is forbidden by momentum conservation and parity consideration in both TLM and SSH models. Based on the D-D correlation, our analytical results of THG show explicitly that TP no longer exists in both the SSH and the TLM models. This unphysical TP is caused by the same reason – the improper use of the static
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
219
assumed to be oriented. x, z and d are defined the Ž1. Ž . same as in x SSH v . Eq. Ž8. is an elliptical integration and can be numerically integrated if one changes x ™ x q i e considering the life-time of the state w19–21x. For polyacetylene, by choosing t 0 s 2.5 eV, D s 0.9 eV, ˚ n 0 s 3.2 = 10 14 cmy2 , a s 1.22 Aand e ; 0.03 w21x, we have d s 0.18 and x 0Ž3. f 1.0 = 10y1 0 esu. The THG Ž . absolute value of x SSH v is plotted in Fig. 1. It shows good agreement with the experimental value w17,18x around z s 1r3. Let d ™ 0q and e ™ 0q in Eq. Ž8., we obtain the analytical result of THG under TLM model as follows:
TH G Ž .< Fig. 1. < x SSH v for e s 0.03 with z ' " v rŽ2 D ..
THG x TLM Ž v.
current Jˆ0 and omission of the IFCs in periodic chain. Although the reason for J-J correlation is very clear, we only give the computational results based on D-D correlation. It is expected that if the correct IFCs are considered, J-J correlation will give the same results as D-D correlation although much more complicated computations are inevitable. After a similar definition as Eq. Ž2. and tedious derivations, the new result of THG per unit length under SSH model for infinite chains is recovered as: THG x SSH Ž v.
s x 0Ž3.
½
45
= y
q
dx
1r d
H 128 1
2
Ž 1 y d x . Ž x 2 y 1.
128
½
14 y 3z
Ž 37 y 24 z 2 . 8 z8
8
4 y
15 z 4
f Ž z. q
Ž1y8 z2 . 24 z 8
f Ž3 z .
5 Ž 9.
x 0Ž3.
where f Ž z . and are defined in Eqs. Ž7. and Ž8.. The comparison between our result Ž D-D . and Wu’s THG resultŽ J0-J0 . w13x on the absolute value of x TLM is plotted in Fig. 2. The TP disappears in our analytical results, which is more reasonable for the physical situation and consistent with the previous numeral computations w19–24x. It also implies that the e-e interactions
8 x8Ž x2 yz2 .
3 Ž 1 y d 2 x 2 . Ž x 2 y 1. 2
9 47 y 48 Ž 1 q d 2 . x 2 q 48 d 2 x 4 8 x8 Ž x2 y Ž3 z .
q
1r2
q
45
47 y 48 Ž 1 q d 2 . x 2 q 48 d 2 x 4
x6 Ž x2 yz2 . q
2
s x 0Ž3.
63 Ž 1 y d 2 x 2 . Ž x 2 y 1 . x6 Ž x2 y Ž3 z .
where x 0Ž3. '
4
3 8 e n 0 Ž2 t 0 a .
45
p
D6
2 2
.
2
.
5
Ž 8.
and n 0 is the number of
chains in unit cross area; the polymer chains are
Fig. 2. Computed D-D value Žsolid line. versus J0 -J0 value TH G Ž .< Ždashed line. of < x TL Ž . M v with z ' " v r 2 D .
220
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
w23,24x, disorders, quantum fluctuations or finite chain size effects w21x should be taken into account to explain this experimentally observed TP w17,18x. Fig. 1 shows another new resonant peak z s 1 at a ratio of 1r10 of z s 1r3. This new peak hasn’t been reported by the experiments w17,18x because of the experimental scanning range. The cancellation of ZFD by J0-J0 correlation w13x is actually a coincidence under the TLM model, because in the SSH Ž3. model, we find ZFD in x SSH through J0-J0 correlation w25x. Keldysh formalism w13x is not necessary to apply in this equilibrium system w19,20x. From THG of trans-ŽCH. x , D-D correlation is superior to J-J correlation to obtained NLO susceptibilities. As a conclusion, the principle about the equivalence of D-D versus J-J correlation is still correct under the approximate models while practicality favors D-D over J-J correlation. Although our discussion is chiefly based on 1-d polymer chains, the main results in this letter can be generalized for 2-d or 3-d periodic systems.
Acknowledgements This work was supported by the Chemistry Department, New York University, Project 863 and the National Natural Science Foundation of China Ž59790050, 19874014.. Very helpful discussions with Z. Bacic, ˇ ´ J.L. Birman, H.L. Cui, A.J. Epstein, G. Gumbs, N. Horing, Y.R. Shen, Z.G. Shuai, Z.G. ¨ Soos, M.E. Tuckerman, Z. Vardeny, C.Q. Wu, Z.G. Yu and J.Z.H. Zhang are acknowledged.
References w1x W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Lett. 42 Ž1979. 1698. w2x W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22 Ž1980. 2099.
w3x H. Takayama, Y.R. Lin-Liu, K. Maki, Phys. Rev. B 21 Ž1980. 2388. w4x E. Fradkin, Field Theories of Condensed Matter Systems Addison-Wesley, 1991, p. 9. w5x F. Gebhard, K. Bott, M. Scheidler, P. Thomas, S.W. Koch, Phil. Mag. B 75 Ž1997. 1. w6x D.K. Ferry, C. Jacoboni ŽEds.., Quantum Transport in Semiconductors, Plenum, New York, 1992, p. 57. w7x P. Martin, J. Schwinger, Phys. Rev. 115 Ž1959. 1342. w8x G.D. Mahan, Many-Particle Physics, Plenum, New York, 1981, ŽChapter 3.. w9x Y.R. Shen, The Principles of Nonlinear Optics, Wiley, 1984. w10x S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford, New York, 1995, and references therein. w11x P.N. Butcher, T.P. McLean, Proc. Phys. Soc. ŽLondon. 81 Ž1963. 219. w12x P.N. Butcher, T.P. McLean, 83 Ž1964. 579. w13x Weikang Wu, Phys. Rev. Lett. 61 Ž1988. 1119. w14x D.E. Aspnes, Phys. Rev. B 6 Ž1972. 4648. w15x I. Batistic, A.R. Bishop, Phys. Rev. B 45 Ž1992. 5282. w16x K. Maki, M. Nakahawa, Phys. Rev. B 23 Ž1981. 5005. w17x W.S. Fann, S. Benson, J.M.J. Madey, S. Etemad, G.L. Baker, F. Kajzar, Phys. Rev. Lett. 62 Ž1989. 1492; w18x A.J. Heeger, D. Moses, M. Sinclair, Synth. Met. 17 Ž1987. 343. w19x J. Yu, B. Friedman, P.R. Baldwin, W.P. Su, Phys. Rev. B 39 Ž1989. 12814. w20x J. Yu, W.P. Su, Phys. Rev. B 44 Ž1991. 13315. w21x C.Q. Wu, X. Sun, Phys. Rev. B 42 Ž1990. R9736. w22x Z. Shuai, J.L. Bredas, Phys. Rev. B 44 Ž1991. R5962. ´ w23x F. Guo, D. Guo, S. Mazumdar, Phys. Rev. B 49 Ž1994. 10102. w24x F.C. Spano, Z.G. Soos, J. Chem. Phys. 99 Ž1993. 9265. w25x Minzhong Xu, Xin Sun, to be submitted. w26x J. Callaway, Quantum Theory of the Solid State, 2nd ed., Academic Press, 1991, p. 483. w27x G. Weinreich, Solids: Elementary Theory for Advanced Students, Wiley, New York, 1965, p. 136. w28x A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 Ž1988. 781, and references therein. w29x In the TLM model, using the static current operator J0 s Ý k AŽ k . c k† s 1 c k q B Ž k . c k† s 3 c k , where AŽ k . sy eÕ F D r ´ Ž k . and B Ž k . s eÕ F2 "k r ´ Ž k .. By k ™ k y eA r ", we have the IFC related to x Ž1. : JA sÝ k C Ž k . c k† s 3 c k A, where C Ž k . syŽ eÕ F D . 2 r ´ 3 Ž k .. The computation of JA gives the first term in Eq. Ž6.. But JA cannot be directly obtained from the field currents in the previous works w3,5,13x, thus it shows the difficulty to obtain IFCs in general models.
Physics Letters A 257 Ž1999. 215–220
Optical susceptibilities of polymers: current–current versus dipole–dipole correlation Minzhong Xu a , Xin Sun
b
a
b
Department of Chemistry, New York UniÕersity, New York, NY 10003, USA Department of Physics, Fudan UniÕersity, and National Laboratory of Infrared Physics, Shanghai 200433, PR China Received 16 February 1999; accepted 27 April 1999 Communicated by A.R. Bishop
Abstract The static current operator leads to definitional zero frequency divergence and unphysical results in studying nonlinear optical susceptibilities of polymers. A well-defined dipole–dipole correlation is superior to the complicated current–current correlation to solve this problem. As illustrative examples, optical susceptibilities under both SSH and TLM models of trans-ŽCH. x are studied. New analytical results are obtained. The reasons of previous improper results are analyzed. q 1999 Elsevier Science B.V. All rights reserved. PACS: 78.66.Qn; 42.65.An; 72.20.Dp; 78.20.Bh Keywords: Zero frequency divergence; Optical susceptibilities; Dipole-dipole correlations
To study the nonlinear optical ŽNLO. properties of polymers, periodic approximate models are necessary to simplify the real systems. Some good approximate models of polymers based on the tight-binding approximation ŽTBA., such as SSH w1,2x and TLM 1 w3x models in weakly correlated systems, Hubbard and Pariser–Parr–Pople models in strongly correlated and electron–electron Ž e-e . interaction systems, etc, have yielded physical insights surpassing the
1 Lagrangian: LsHdx cˆ †Ž x . i" E t q i s 3 Õ F Ex q s 1 D4cˆ Ž x . w3x. The vector potential A is included by changing y i" Ex ™y i" Ex y eA, it will not contain A2 term in the Lagrangian, also see Wu’s work w13x.
complicated non-approximate computations. In considering the optical response of these models, a UŽ1. transformation has been suggested w4–6x to provide the gauge invariance of the TBA Hamiltonian. In linear optical ŽLO. response theory, the Kubo formula based on current–current Ž J-J . correlation w7,8x is widely used because of simplicity. It is commonly held that in discussing the optical susceptibilities of materials, the J-J correlation Ž p P A . will play the same role as that of dipole–dipole Ž D-D . correlation Ž E P r . w9,10x and that the apparent definitional zero frequency divergence ŽZFD. in the NLO susceptibilities definition w11–13x is only a virtual problem, although the proofs only have been shown under certain assumptions w7,14x. However, there exists some discrepancies from the above conclusion
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 8 8 - 1
216
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
under the models. Ži. In the study of conductivity s under TBA models w15,16x, the J-J correlation will have ZFD in ImŽ s . Žrelated to the real part of J-J . although it can give the correct ReŽ s . Žrelated to the imaginary part of J-J .. Žii. In third-harmonic generation ŽTHG. of trans-ŽCH. x , the experimentally observed two-photon absorption peak ŽTP. w17,18x has given rise to theoretical explanations. Based on J-J correlation, the TP was explained w13x by choosing the TLM model. But it has been criticized by others w19–24x with simple parity consideration. Also the spectrum w13x is quite different from that under the D-D correlation w25x. Thus, it casts some doubts in the practical application of J-J correlation. In this letter, we will show that all above controversies are caused by improper application of J-J correlation based only on the static current operator J0ˆ . To recover the correct results, a well-defined D-D correlation is more suitable for studying the optical susceptibilities than the J-J correlation, whose equivalence to the D-D correlation can be satisfied by introducing complicated induced field currents ŽIFCs.. Besides solving the apparent ZFD in the definition of NLO susceptibility, the D-D correlation is to be favored over the J-J correlation due to the lack of the gauge dependence of the vector potential A w10x and simplifying the definitional complexity of the contribution by IFCs in the high order expansions w4,5x. As a deduction, the TP cusp and ZFD w13,15,16x will no longer exist in both SSH and TLM models by the well-defined D-D correlation. However, this seemingly trivial conclusion has not been clearly illustrated by the others although the D-D correlation has already been applied in the NLO response of the real systems w10,19,20,24x. The application of J-J correlation is simplier than that of the D-D correlation in LO response because of the convenient search for the static current J0ˆ w8x compared with that for dipole expression Dˆ under the approximate models. As a typical example, the position operator rˆ is ill-defined in periodic systems w14,26x in real space while J0ˆ is not. Further, the static current J0ˆ still can give the correct results in the LO absorption w8,15,16x because the IFCs only contribute to the real part of J-J correlation. Thus the simplicity of the Kubo formula is still satisfied by applying the static current J0ˆ and ZFD in the real part of J-J correlation could be solved either by the
Kramers–Kronig ŽKK. relation w8x or by subtracting ²w j, j x:Ž v s 0. w15x. It is not a big surprise to see why the correct results still can be preserved for the LO response through J0ˆ . However, the advantage of J-J correlation application in LO response is no longer available for the NLO response because of the IFCs w5,11,12x, thus the definition of n-th order J-J correlation based only on the static current w13x will lead to the incorrect results. Besides the difficulty in obtaining the correct IFCs in approximate models, J-J correlation will make the definition of NLO susceptibilities unable to be written in a general form as defined in Wu’ work w13x and will make the computations very tedious w5x, although it may give the correct result. In periodic systems, the imposed boundary condition requires Ži. Bloch wave functions to describe the extended electronic states and Žii. all operators including the position operator rˆ to have at least the same periodicity w27x. A band index n and crystal momentum k are used to label the Bloch states: < n, k : s u n, k Ž r . e i kP r , where u n, k Ž r . is the periodic function under the translation of lattice vector. If the homogeneous electric field is applied, the energy level EnŽ k . should be changed to EnŽ k . w6x and the static current operator Jˆ0 Ž k . should be replaced by a new current operator JˆŽ k ., where k s k y eAr" w11,12x. By Taylor expansion of JˆŽ k . based on powers of the vector potential A, we will obtain the IFCs related to all orders of A besides the static current J0ˆ . There is no doubt that the IFCs will contribute to the optical response. Complexities in handling IFCs Žusually related to the intra-band currents w11,12x. in NLO studies could be imagined. Polymer models w1–3,13x usually share the same properties as periodic systems. The direct consequence of dropping the IFCs will result in ZFD and unphysical results w13,15,16x. Although the unphysical results in NLO response have been recurrently questioned by the others w19–24x, the direct reason has never been revealed and correct analytical results have never been obtained. The chief difficulty is that direct ways to include IFCs in many approximate models are forbidden w1–3,13,15x and the static current could easily be improperly used w13,16x. By a well-defined D-D correlation, we will show that all difficulties caused by the improper use of J-J correlation could be solved. To give an intuitive picture,
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
we will illustrate those correlations by studying the simple one-dimensional Ž1-d . TBA electron–lattice models of trans-ŽCH. x ŽSSH and TLM. whose optical properties have been widely studied by others w13,16–24,28x. To avoid the ill-definition and to provide the periodicity of the position operator r, ˆ we express rˆ under < n, k : as w26x:
s. In the continuum limitation, the above SSH model will give the TLM model w3,13x. In momentum space, the above Hamiltonian with electron-photon Ž e-A . interaction could be found as follows: H Ž k ,t . s Ý ´ Ž k . cˆ k†, s Ž t . s 3 cˆ k , s Ž t . y Dˆ P Ee i v t , k,s
rn k , nX kX s i dn , nX= k d Ž k y kX . q V n , nX Ž k . d Ž k y kX . , Ž 1. ) Ž . X Ž . where V n, nX Ž k . s Õi HÕ u n, k r = k u n , k r dr and Õ is unit cell volume.
1. Susceptibilities definition by D-D correlation Without considering retardation effect w10x, optical susceptibilities are usually defined by expanding the optical polarization in powers of the transverse electric field w9x. Under that approximation, the general nth-order susceptibility is a purely material quantity defined as w10x:
x Ž n. Ž V ; v 1 , . . . , v n . 1
i
s n! "
n
Hdr
1
PPP drn dt1 PPP dt n
H
ˆ ˆ Ž r ,t . = drdt eyi kP rqi V t ²TD
H
=Dˆ Ž r 1 ,t 1 . PPP Dˆ Ž rn ,t n . : ,
Ž 2.
where V ' yÝ nis1 v i , Tˆ is the time-ordering operator and Dˆ is dipole operator. Based on the periodic TBA, the SSH Hamiltonian w1,2x is given by: HSSH sy
Ý l, s
t 0 q Ž y1. l
D 2
† ˆ ˆ† ˆ Ž Cˆlq1 , s C l , s qC l , s C lq1 , s . ,
where t 0 is the transfer integral between the nearestneighbor sites, D is the gap parameter and Cˆl,† s Ž Cˆl, s . createsŽannihilates. an p electron at site l with spin
217
Ž 3.
(
2 2 where ´ Ž k . s 2 t 0 cos Ž ka . q Dsin Ž ka . and † †c † Õ cˆ k, s Ž t . s Ž aˆ k, s Ž t ., aˆ k, s Ž t .. is the two-component spinor describing excitations of electrons in the conduction band and valence band. The long wave approximation w8x is applied in electromagnetic field E with frequency v . The dipole operator Dˆ could be obtained by the Eq. Ž1.:
Dˆ s e Ý b Ž k . cˆ k†, s s 2 cˆ k , s q i k,s
ž
E Ek
cˆ k†, s cˆ k , s ,
/
Ž 4.
where b Ž k . s yD t 0 ar´ 2 Ž k ., is the coefficient related to the interband transition between the conduction and valence bands in a unit cell 2 a and the second term is related to the intraband transition w14x, e is the electric charge and s are the Pauli matrixes. We neglect the relative distortion h Ž' 2 ura. in the dipole operator because it is relatively small in the optical contribution w5,25x. Due to the fact that p electrons in the SSH model are non-localized w28x, the dipole approximation w10,24x is no longer valid in the extended states and will lead to wrong results as pointed out by some authors w5x. The Fourier transform of Eq. Ž4. to coordinate space shows that the transition dipole is related to the electron hopping to all the other sites besides the nearest neighbor sites. Thus the dipole approximation fails for the extended states in periodic systems. Because of the failure of dipole apˆ proximation through the polarization operator P, 2 computations show a magnitude difference of 10 in x Ž1. and 10 4 in x Ž3. w25x and quite different shape in spectrum compared with the results through the dipole operator Dˆ and the experimental values w17,18,28x in trans-ŽCH. x , although the position of some resonant peaks may be correctly obtained.
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
218
2. LO response by D-D Ž1. Ž The LO susceptibility x SSH V , v 1 . can be obtained from Eqs. Ž2. and Ž4.: Ž1. x SSH Ž yv 1 , v 1 .
i
s2
e2 Ý
"
k
`
Hy`Tr
E
½
= i
E GŽ k ,v . i
Ek
Ek
GŽ k ,v y v1 .
E qb Ž k . s 2 G Ž k , v . i
Ek
GŽ k ,v y v1 .
E qi
b Ž k . G Ž k , v . s2G Ž k , v y v 1 .
Ek
qb Ž k . s 2 G Ž k , v . b Ž k . s 2 G Ž k , v y v 1 . dv
=
, 2p where the Green function GŽ k, v . s
5 Ž 5.
vq v k s3
with
v 2 y v k2 q i e
q
v k ' ´ Ž k .r" and e ' 0 . Ž1. Ž . Ž1. Ž By Eq. Ž5., we have x SSH v ' x SSH yv , v .: Ž1. x SSH Ž v.
e2 Ž 2 t0 a.
s
2pD
2
1r d
H1
dx
ž
Ž1yd 2 x2 .
1 2
2
= Ž x y 1.
y1 2
2
2
x Ž x yz .
/
,
where x ' " v krD, z ' " vrŽ2 D . and d ' DrŽ2 t 0 .. If the continuum limitation is applied, that is, d ™ 0 and 2 t 0 a ™ "Õ F , the above integral gives the Ž1. Ž . LO susceptibility x TLM v under the TLM model w3,13x as follows: Ž1. x TLM Ž v . sy
e 2 "Õ F 2 2
2pD z
Ž1yf Ž z. . ,
Ž 6.
where
°
arcsin Ž z .
~
Ž z 2 -1 . ,
'
z 1y z 2
Ž 7.
f Ž z.'
¢
coshy1 Ž z .
y
ip q
'
z z 2 y1
'
2 z z 2 y1
Ž z 2 )1 . .
The conductivity s Ž v . given by yi v P x Ž1., is exactly the same as based on J-J correlation w16x. However, we should point out that the direct computation based on the static current from J-J correlation under TLM model w3,13,16x shows no first term in Eq. Ž6. w25x, ZFD in real part of J-J correlation is obvious although the correct imaginary part still can be given. These difficulties have never been clearly addressed previously, provided the reason that the static current J0ˆ is still valid for obtaining the correct imaginary part in the LO absorption. To include the IFCs by changing k ™ k y eAr" in the static current operator J0ˆ Ž k ., ZFD could be solved to give the same result as that under D-D correlation w29x. But the complicated way to include the IFCs compared with simple D-D correlation already makes J-J correlation impractical even for the LO response. Fortunately, the IFCs only contribute for the real part of J-J correlation and have no influence on the absorption. Attempts to obtain IFCs directly from TLM Hamiltonian fail because it is forbidden by the model to include A2 term w3,13x. In the discrete SSH Hamiltonian, we have the chance to include the IFCs through Peierls substitution w5x, but a straightforward computation easily shows that they are not correct IFCs to cancel the ZFD w29x. It gives us an impression of the difficulty to obtain the correct IFCs besides the complexity of handling their contributions to the optical response even in well-defined 1-d periodic models. From the above examples, the feasibility of J-J correlation in more general models will be questioned and the application of D-D correlation is more reasonable under approximate models to obtain the correct results.
3. NLO susceptibilities of trans-(CH)x chain There are many elegant works discussing NLO susceptibilities of polymer chains w13,17–24x. The TP w13x obtained from J-J correlation was doubted in the literature w17–24x since it is forbidden by momentum conservation and parity consideration in both TLM and SSH models. Based on the D-D correlation, our analytical results of THG show explicitly that TP no longer exists in both the SSH and the TLM models. This unphysical TP is caused by the same reason – the improper use of the static
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
219
assumed to be oriented. x, z and d are defined the Ž1. Ž . same as in x SSH v . Eq. Ž8. is an elliptical integration and can be numerically integrated if one changes x ™ x q i e considering the life-time of the state w19–21x. For polyacetylene, by choosing t 0 s 2.5 eV, D s 0.9 eV, ˚ n 0 s 3.2 = 10 14 cmy2 , a s 1.22 Aand e ; 0.03 w21x, we have d s 0.18 and x 0Ž3. f 1.0 = 10y1 0 esu. The THG Ž . absolute value of x SSH v is plotted in Fig. 1. It shows good agreement with the experimental value w17,18x around z s 1r3. Let d ™ 0q and e ™ 0q in Eq. Ž8., we obtain the analytical result of THG under TLM model as follows:
TH G Ž .< Fig. 1. < x SSH v for e s 0.03 with z ' " v rŽ2 D ..
THG x TLM Ž v.
current Jˆ0 and omission of the IFCs in periodic chain. Although the reason for J-J correlation is very clear, we only give the computational results based on D-D correlation. It is expected that if the correct IFCs are considered, J-J correlation will give the same results as D-D correlation although much more complicated computations are inevitable. After a similar definition as Eq. Ž2. and tedious derivations, the new result of THG per unit length under SSH model for infinite chains is recovered as: THG x SSH Ž v.
s x 0Ž3.
½
45
= y
q
dx
1r d
H 128 1
2
Ž 1 y d x . Ž x 2 y 1.
128
½
14 y 3z
Ž 37 y 24 z 2 . 8 z8
8
4 y
15 z 4
f Ž z. q
Ž1y8 z2 . 24 z 8
f Ž3 z .
5 Ž 9.
x 0Ž3.
where f Ž z . and are defined in Eqs. Ž7. and Ž8.. The comparison between our result Ž D-D . and Wu’s THG resultŽ J0-J0 . w13x on the absolute value of x TLM is plotted in Fig. 2. The TP disappears in our analytical results, which is more reasonable for the physical situation and consistent with the previous numeral computations w19–24x. It also implies that the e-e interactions
8 x8Ž x2 yz2 .
3 Ž 1 y d 2 x 2 . Ž x 2 y 1. 2
9 47 y 48 Ž 1 q d 2 . x 2 q 48 d 2 x 4 8 x8 Ž x2 y Ž3 z .
q
1r2
q
45
47 y 48 Ž 1 q d 2 . x 2 q 48 d 2 x 4
x6 Ž x2 yz2 . q
2
s x 0Ž3.
63 Ž 1 y d 2 x 2 . Ž x 2 y 1 . x6 Ž x2 y Ž3 z .
where x 0Ž3. '
4
3 8 e n 0 Ž2 t 0 a .
45
p
D6
2 2
.
2
.
5
Ž 8.
and n 0 is the number of
chains in unit cross area; the polymer chains are
Fig. 2. Computed D-D value Žsolid line. versus J0 -J0 value TH G Ž .< Ždashed line. of < x TL Ž . M v with z ' " v r 2 D .
220
M. Xu, X. Sun r Physics Letters A 257 (1999) 215–220
w23,24x, disorders, quantum fluctuations or finite chain size effects w21x should be taken into account to explain this experimentally observed TP w17,18x. Fig. 1 shows another new resonant peak z s 1 at a ratio of 1r10 of z s 1r3. This new peak hasn’t been reported by the experiments w17,18x because of the experimental scanning range. The cancellation of ZFD by J0-J0 correlation w13x is actually a coincidence under the TLM model, because in the SSH Ž3. model, we find ZFD in x SSH through J0-J0 correlation w25x. Keldysh formalism w13x is not necessary to apply in this equilibrium system w19,20x. From THG of trans-ŽCH. x , D-D correlation is superior to J-J correlation to obtained NLO susceptibilities. As a conclusion, the principle about the equivalence of D-D versus J-J correlation is still correct under the approximate models while practicality favors D-D over J-J correlation. Although our discussion is chiefly based on 1-d polymer chains, the main results in this letter can be generalized for 2-d or 3-d periodic systems.
Acknowledgements This work was supported by the Chemistry Department, New York University, Project 863 and the National Natural Science Foundation of China Ž59790050, 19874014.. Very helpful discussions with Z. Bacic, ˇ ´ J.L. Birman, H.L. Cui, A.J. Epstein, G. Gumbs, N. Horing, Y.R. Shen, Z.G. Shuai, Z.G. ¨ Soos, M.E. Tuckerman, Z. Vardeny, C.Q. Wu, Z.G. Yu and J.Z.H. Zhang are acknowledged.
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w3x H. Takayama, Y.R. Lin-Liu, K. Maki, Phys. Rev. B 21 Ž1980. 2388. w4x E. Fradkin, Field Theories of Condensed Matter Systems Addison-Wesley, 1991, p. 9. w5x F. Gebhard, K. Bott, M. Scheidler, P. Thomas, S.W. Koch, Phil. Mag. B 75 Ž1997. 1. w6x D.K. Ferry, C. Jacoboni ŽEds.., Quantum Transport in Semiconductors, Plenum, New York, 1992, p. 57. w7x P. Martin, J. Schwinger, Phys. Rev. 115 Ž1959. 1342. w8x G.D. Mahan, Many-Particle Physics, Plenum, New York, 1981, ŽChapter 3.. w9x Y.R. Shen, The Principles of Nonlinear Optics, Wiley, 1984. w10x S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford, New York, 1995, and references therein. w11x P.N. Butcher, T.P. McLean, Proc. Phys. Soc. ŽLondon. 81 Ž1963. 219. w12x P.N. Butcher, T.P. McLean, 83 Ž1964. 579. w13x Weikang Wu, Phys. Rev. Lett. 61 Ž1988. 1119. w14x D.E. Aspnes, Phys. Rev. B 6 Ž1972. 4648. w15x I. Batistic, A.R. Bishop, Phys. Rev. B 45 Ž1992. 5282. w16x K. Maki, M. Nakahawa, Phys. Rev. B 23 Ž1981. 5005. w17x W.S. Fann, S. Benson, J.M.J. Madey, S. Etemad, G.L. Baker, F. Kajzar, Phys. Rev. Lett. 62 Ž1989. 1492; w18x A.J. Heeger, D. Moses, M. Sinclair, Synth. Met. 17 Ž1987. 343. w19x J. Yu, B. Friedman, P.R. Baldwin, W.P. Su, Phys. Rev. B 39 Ž1989. 12814. w20x J. Yu, W.P. Su, Phys. Rev. B 44 Ž1991. 13315. w21x C.Q. Wu, X. Sun, Phys. Rev. B 42 Ž1990. R9736. w22x Z. Shuai, J.L. Bredas, Phys. Rev. B 44 Ž1991. R5962. ´ w23x F. Guo, D. Guo, S. Mazumdar, Phys. Rev. B 49 Ž1994. 10102. w24x F.C. Spano, Z.G. Soos, J. Chem. Phys. 99 Ž1993. 9265. w25x Minzhong Xu, Xin Sun, to be submitted. w26x J. Callaway, Quantum Theory of the Solid State, 2nd ed., Academic Press, 1991, p. 483. w27x G. Weinreich, Solids: Elementary Theory for Advanced Students, Wiley, New York, 1965, p. 136. w28x A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 Ž1988. 781, and references therein. w29x In the TLM model, using the static current operator J0 s Ý k AŽ k . c k† s 1 c k q B Ž k . c k† s 3 c k , where AŽ k . sy eÕ F D r ´ Ž k . and B Ž k . s eÕ F2 "k r ´ Ž k .. By k ™ k y eA r ", we have the IFC related to x Ž1. : JA sÝ k C Ž k . c k† s 3 c k A, where C Ž k . syŽ eÕ F D . 2 r ´ 3 Ž k .. The computation of JA gives the first term in Eq. Ž6.. But JA cannot be directly obtained from the field currents in the previous works w3,5,13x, thus it shows the difficulty to obtain IFCs in general models.