0020-1685/05/4105-0549 © 2005 Pleiades Publishing, Inc. Inorganic Materials, Vol. 41, No. 5, 2005, pp. 549–554. From Neorganicheskie Materialy, Vol. 41, No. 5, 2005, pp. 635–639. Original English Text Copyright © 2005 by Qi Guo, Sien Chi.
1 INTRODUCTION
Exciton–polaritons are also called normal waves of electromagnetic waves in the field of optical physics. The term exciton–polariton is used more frequently in condensed-matter physics, while it is near nothing new but just another term from the optical point of view. Here, we use the terms exciton–polaritons and normal waves interchangeably.
Since the pioneer theoretical works of Pekar [1, 2] and Ginzburg [3], especially the theoretical and exper-imental work by Hopfield and Thomas [4] that con-veyed much of the relevant physics of spatially disper-sive media in a transparently clear way, the exciton– polariton spatial dispersion has been a well-developed and well-documented [5–15] subject. The investigation development of the exciton–polaritons in quantum-con-fined spatially dispersive systems such as GaAs quantum wells [16], as well as in other nanostructures [17], shows the potential for applications of the subject to ultrasmall optoelectronic devices.
Pekar was the first to discuss the phenomenon of optical waves in the region of a particular exciton absorption frequency in media and to shape them with the quantum theory. However, Ginzburg first connected the phenomenon with the spatial dispersion effect and reshaped it with a phenomenological approach. It has been considered [5, 6, 9, 13] for more than 40 years that Pekar’s treatment based on the microscopic quantum theory [2, 9] and Ginzburg’s treatment based on the macroscopic phenomenological approach [3, 10]
1This article was submitted by the authors in English.
would have been different in many of their essential conclusions. Pekar concluded that [6] Ginzburg’s phe-nomenological treatment “leads to consequences (refractive indices) that, generally speaking, are in con-tradiction to the results of the quantum-mechanical investigation and also inconsistent with experiment,” and that (see [9, p. 122]) “the existence of the additional waves cannot be obtained by means of the phenomeno-logical procedure.” Ginzburg defended himself against Pekar’s argument, but, because his defense [11] lacked academic evidence and, therefore, persuasion, even in 1998 there still was a claim that the two theories would have differed substantially [13], and the controversy has not been clarified so far. To clarify this controversy is our purpose in this paper.
PHENOMENOLOGICAL RESULT VS. QUANTUM RESULT
An essential part of Ginzburg’s phenomenological treatment is a phenomenological expansion of the impermeability tensor εij(ω, ), the inverse of the per-mittivity tensor εij(ω, ), in powers of wavevector [3] (see also Section 3.1.2 in [10]),
(1) where all of the coefficients κijl(ω) vanish for nongyro-tropic crystals. The weakness in Ginzburg’s approach, however, is that, although he assumed all of the
coeffi-k
k k
ηij(ω k, ) = ηij( ) iκω + ijl( )klω +βijlm( )klω km,
Does the Phenomenological Approach Contradict
the Quantum Theory of Exciton–Polariton Spatial Dispersion?
1Qi Guo* and Sien Chi**
* Laboratory of Light Transmission Optics, South China Normal University, Guangzhou, Guangdong, China
** Institute of Electro-optical Engineering, National Chiao Tung University, Hsinchu, Taiwan, China e-mail: [email protected]
Received November 22, 2003
Abstract—This paper clarifies the controversial issue over 40 years between the quantum approach by Pekar
and the phenomenological approach by Ginzburg about the exciton–polariton spatial dispersion theory. For an isotropic nongyrotropic medium, the analytical explicit function of the impermeability tensor ηij(ω, ) (the inverse of the permittivity tensor) is obtained from the anisotropic undamped-wave harmonic oscillator model. After expanding ηij(ω, ) with respect to small parameters within it rather than to wavevector , the approxi-mate refractive indices can be determined from the eigenvalue equation. By this treatment, the phenomenolog-ical approach is proved to be the approximation of the quantum approach near the resonance frequency. The condition for the approximation is discussed.
k
550
INORGANIC MATERIALS Vol. 41 No. 5 2005 QI GUO, SIEN CHI
cients in expansion (1) to be dependent on frequency ω, he did not know how this dependence arose; therefore, he dealt with all of the coefficients except ηij(ω) as con-stants [3] (see also Chapter 4 in [10]), which is unrea-sonable from both the physical and mathematical points of view. Although, as we will see in this paper, βijlm(ω) can at last be proved to be approximately inde-pendent of frequency near the resonance frequency, the treatment of them as constants in the beginning is vul-nerable to be criticized, as done by Pekar [6]. To over-come this shortcoming, we should first find a way to obtain the analytical explicit function of εij(ω, ), and a start point for this work is the anisotropic undamped-wave harmonic oscillator model to formulate a linear relation between the polarization and the electric field , suggested first by Agranovich and Kaganov [7], which reads (see also Section 4.5.1 in [10])
(2)
where the tensors ρij, υij, γijl, and others are practically independent of ω near the individual exciton dipole absorption frequency, and they are determined by the type of crystal symmetry. The last two terms on the left-hand side of Eq. (2) take into account spatial dispersion, which originates from the direct interaction between the oscillators permitting energy transmission that is not electromagnetic in origin [4, 18], and the first two terms, as well as the term on the right-hand side, have the same physical meaning with the counterparts in the isotropic harmonic oscillator model in the absence of spatial dispersion [19–21]. The media are considered ideal, so that the dissipation term σij∂Pj/∂t is not included in Eq. (2). It is clear that γijl = 0 for nongyro-tropic media. Considering the harmonic solution of
E(t, ) = E(ω, )exp[i( –ωt)]/2 + c.c. and P(t, ) =
P(ω, )exp[i( – ωt)]/2 + c.c., where = k0n , n is
the refractive index, k0= ω/c, and is the unit vector of , from Eq. (2) we find the explicit form of the permit-tivity tensor εij(ω, ), taking into consideration spatial dispersion (3) k P E ρij∂ 2 Pj(t r, ) ∂t2 ---+υijPj(t r, ) +γijl∂Pj(t r, ) ∂xl --- µijlm∂ 2 Pj(t r, ) ∂xl∂xm ---+ = Λij( )0Ej(t r, ), r k k r⋅ r k k r⋅ k kˆ kˆ k k
εij(ω k, ) ( )ijεr 4π∆il 1 – ω k, ( )Λlj( )0 , + =
where (εr)ij is the background permittivity, and
(4) Now, let us take into account isotropic nongyrotro-pic media, the simplest crystals with the highest sym-metric property. For isotropic media, we have
(5)
where ρµlo and ρµtr are two independent components of
µijlm for isotropic media (see Section 4.5.2 in [10]). Introducing Eq. (5) into Eqs. (4) and (3), we obtain the analytical expression for the permittivity tensor
(6) where εtr and εlo are the transverse and longitudinal
parts of εij(ω, ), respectively,
(7)
and ωtr is the particular resonance frequency (the
trans-verse frequency). Furthermore, the intrans-verse of εij(ω, ) can be got
(8)
where Ωtr = k2µtr/(ω2 – ), Ωlo = k2µlo/(ω2 – ), and
ωlo = ( + /εr)1/2 is the longitudinal frequency. It is
obvious that an isotropic medium must, in general, be characterized by a tensorial rather than a scalar permit-tivity as a consequence of spatial dispersion. For the specific configuration that is directed along the z axis of the coordinate system, Eq. (6) degenerates as
(9) ∆ij(ω k, ) ρijω2
– +υij+iγijlkl–µijlmklkm. =
εr
( )ij εrδij, ρij ρδij, υij ρωtr 2 δij, = = = Λij( )0 ρω0 2 4π ---δij, γijk 0, = = µijlmklkm ρµtrk 2 δij+ρ µ( lo–µtr)kikj, =
εij(ω k, ) = εtrδij+(εlo–εtr)kikj,
) ) k εtr εr ω0 2 ω2 k2µtr–ωtr2 + ---, – = εlo εr ω0 2 ω2 k2µlo–ωtr2 + ---, – = k ηij(ω k, ) δijε r ---= + ω0 2 εr 2 ω2 ωlo 2 – ( ) --- δij–kikj 1+Ωtr --- kikj 1+Ωlo ---+ ) ) ) ) , ωlo 2 ω lo 2 ωtr 2 ω 0 2 k εxx = εyy = εtr, εzz = εlo, εij = 0 i( ≠j).
INORGANIC MATERIALS Vol. 41 No. 5 2005
DOES THE PHENOMENOLOGICAL APPROACH CONTRADICT THE QUANTUM THEORY 551 On the other hand, Hopfield and Thomas [4] have
obtained a semiphenomenological model of the scalar permittivity in the specific configuration that the polar-ization and electric field are parallel and both perpen-dicular to . As a matter of fact, their model is only εxx or εyy, the transverse part of εij(ω, ). Comparing εtr in
Eq. (7) with their model (see Eq. (9) in [4]) and the model without spatial dispersion [19–21], we have
(10) where Np is the number of molecules per unit volume with Z electrons per molecule; fp is the number of elec-trons per molecule with binding frequency ωtr, the
oscillator strength, satisfying the sum rule Σfp = Z; e is the charge of the electron; m is the rest mass of the elec-tron; and is the effective mass of the exciton.
Having the analytical explicit expression for ηij(ω, ), we can further expand it with small parameters within it. First, it is reasonable to assume that µtr and µlo
have the same order; then, Ωtr and Ωlo also have the
same order. With the condition
(11)
(now |Ωlo| 1 also, vice versa), Eq. (8) can be
expanded with respect to Ωtr and Ωlo as
(12)
where
Although here ηij(ω, ) is expanded with respect to the small parameters Ωtr and Ωlo, rather than with
respect to about = 0, it can be found, as a matter of fact, that the result will be the same with expansion (12) provided that Eq. (8) is expanded with respect to , fol-lowing Ginzburg’s idea [3, 10]. Then, comparing Eq. (12) with the concrete form of expansion (1) for
k k ω0 2 4πe 2 Npfp m ---, µtr ωtr me* ---, – = = me* k Ωtr k02µtrn2 ω2 ω lo 2 – ---= 1 ηij(ω k, ) = η0δij +βtr kikj k 2δij – ( ) β– lokikj+o(Ωtr) η0 ω2 ω tr 2 – εr ω 2 ωlo2 – ( ) ---, βlo µloω0 2 /εr 2 ω2 ωlo 2 – ( )2 ---, = = βtr µtrω0 2 /εr 2 ω2 ω lo 2 – ( )2 ---. = k k k k
isotropic media (see Section 3.2.2 in [10]), we can obtain
(13)
where β1 = βxxxx and β2 = βxxzz are two independent
com-ponents of βijlm for isotropic media. Of course, κijl = 0 as mentioned above. Clearly, β1 and β2 are functions of ω,
rather than constants, but near the resonance we have ω2 – ≈ – = – /ε
r; β2 is then reduced to a
constant β, i.e., is independent of frequency ω,
(14) which is just what was obtained by Ginzburg (Eq. (4.2.9b) in [10]).
Now, directly quoting the result for refractive indi-ces obtained by Ginzburg [3] (see also Eq. (4.2.6) in [10]) in the coordinate system whose z axis coincides with , we have
(15)
The results above can be extended to the cases of cubic nongyrotropic media belonging to the two classes and m3m, the cubic crystals with highest property in the cubic system, as long as the wavevector is directed along [0, 0, 1] (or [1, 0, 0], [0, 1, 0]) (see [10, p. 169]). Using Eq. (10) and ω ≈ ωtr, we can obtain near the resonance
(16) where
(17) However, Pekar got the following result on the refrac-tive index in cubic crystals more than 40 years ago:
(18) Here, µq and bq have the same expressions with µp and
bp in Eq. (16) near the resonance frequency [2]. Using ηij( )ω = η0( )δijω , β1( )ω = –βlo( ), βω 2( )ω = –βtr( ),ω ωlo 2 ωtr 2 ωlo 2 ω0 2 β β2( )ωtr ωtr ω0 2 me* ---, = = k n±2 η0 2k02β --- 1 4 --- η0 k02β --- 2 1 k02β ---+ . ± – = 43m k µp –η0 k02β --- Cp 1 ωtr ω ---– , bp 1 k02β --- 8πme*c 2 ap 2ω tr 3 ---≈ , = ≈ = Cp 2me*c 2 ωtr ---, ap e2Npfp 2m ---. = = n±2 1 2 ---(µq+εr) 1 4 ---(µq–εr)2+bq. ± =
INORGANIC MATERIALS Vol. 41 No. 5 2005 the typical concrete data presented by Pekar [2] (see
also Section 12 in [9]), we have bp ≈ 5.8 × 104, Cp ≈
5.1 × 105, and ε
r = 2. Therefore, although the results on
the refractive index from the phenomenological and quantum approaches appear different, when Cp and bp are very much larger than εr the two results on the
refractive index are identical near the resonance. A CRITERION:
MATERIAL CHARACTERISTIC PARAMETER αc
It goes without saying that the error between the quantum result (18) and the phenomenological one (15) will occur both if ω is detuned away from the resonance and if Cp and bp are not larger enough than εr even in the resonance. To discuss this issue quantitatively, we use condition (11) to find the frequency region where expansion (12) holds. Because the n– (upper) exciton– polariton branch cannot propagate throughout the bulk crystal near the resonance [8, p. 83], only the case for
n+ (lower) branch is discussed here. In order to discuss
the issue, condition (11) is rewritten as its equivalent form
(19) where α is a constant. When |α| 1, Eq. (19) is just the expansion condition (11). Supposing αmax (0 < αmax < 1)
is the upper limit of the expansion condition (11) and introducing n+ in Eq. (18) into Eq. (19), we have the
corresponding ωmax for the case that > 0 (µtr < 0)
(only positive effect mass has been treated because it is the usual case for exciton resonances in direct band gap semiconductors),
(20) where k0 ≈ ωtr/c,
(21)
and ωLT = ωlo – ωtr, given by ωLT≈ /(2εrωtr), is the
longitudinal–transverse splitting. As will be seen, αc in
Eq. (21) is a critical parameter determined uniquely by the characteristic parameters of the material. The fre-quency region where the expansion (12) holds for the condition |Ωtr| < αmax is the interval (–∞, ωmax). From
Eq. (20), it can be derived that
(22) k02µtrn2 ω2 ωlo 2 – --- = α, me* ωmax 2 ωtr 2 – k0 2µ trεr αmax --- 1 αmax 2 αc ---– , = αc εr 2 bp --- εrωtr 2 2ωLTc2me* ---, = = ω0 2 ωmax–ωtr ωLT --- αmax 1 αc αmax 2 ---– α max( ).<1 < =
The inequality (22) is obtained because αc > 0 and gives
another inequality ωmax – ωtr < ωlo – ωtr, which tells us
that, no matter what materials are, ωmax will always be
less than ωlo; that is, expansion (12) exists only at the
lower end of the longitudinal frequency ωlo for the n+
branch. ωlo is a singularity of Eq. (15), and n deter-mined by Eq. (15) is completely unreliable at the higher end of ωlo. From Eq. (22), taking αmax = 0.1 (a general
case), we also conclude that, when αc > 0.01(= ), then ωmax < ωtr and ωmax is located at the lower end of
the resonance frequency; otherwise, ωmax > ωtr and ωmax
is at the higher end of the resonance frequency. For the case that αc > 0.01 (ωmax < ωtr), the truncated error in
expansion (12) cannot be neglected as ω gets close to ωtr. The approximate n+ from Eq. (15) will introduce
neglected-less error even in the interval (–∞, ωmax),
because β is the value of β2(ω) at ω = ωtr. In contrast,
when αc < 0.01, ωtr ∈ (–∞, ωmax), and Eq. (15) is a good approximation of Eq. (18) in the interval. There-fore, the critical parameter αc of a material is a
crite-rion to judge whether Eq. (15) can approximate Eq. (18) well or not for the material. The bigger the material’s αc, the larger the error between the two results. To illuminate the conclusion, we give the com-parison of the two results for three concrete materials with different αc, as shown in Fig. 1. For the fictitious
material given by Pekar [2] (see also Section 12 in [9]), its αc is about 3 × 10–5; therefore, a much better
approximation can be obtained.
A remark is given at the end of the paper. The micro-scopic quantum theory of spatial dispersion is the most rigorous but obscure [4]. The rudiments of the theory with less rigorous but without loss of physical essence are needed from the applicable and technical point of view. Such rudimentary models available now are the phenomenological one suggested by Ginzburg [3] and semiphenomenological one by Hopfield and Thomas [4]. The Hopfield–Thomas model allows one to obtain the same results with the quantum model, but their scalar permittivity can be used only in a very specific config-uration (the condition of the model can be found in the paragraph above Eq. (4) in [4] and the first two para-graphs of Section 3 in the same paper). However, the expansion of ηij(ω, ) has the advantage of making it possible to obtain an approximate analytical solution for an arbitrary configuration. As a matter of fact, the phenomenological approach was used to prove that exciton–polaritons propagate along the [1, 1, 0] and [1, 1, 1] directions in cubic crystals (see Section 4.3.1 in [10]). Therefore, in our point of view, the phenome-nological description has its potential merit, especially in applicable and technical problems (when the spatial
αmax 2
INORGANIC MATERIALS Vol. 41 No. 5 2005
DOES THE PHENOMENOLOGICAL APPROACH CONTRADICT THE QUANTUM THEORY 553
dispersion theory is put into practice), although it is only an approximation of the “exact” solution along the principal axes.
CONCLUSIONS
We clarified the controversy on the exciton–polari-ton spatial dispersion theory between the quantum point of view and the phenomenological point of view over 40 years. The result from the phenomenological approach does not contradict that from the quantum approach but is an approximation of it. The approxi-mate extent is determined by the approxi-material’s characteris-tic parameter αc defined by Eq. (21). The greater is the
material’s αc, the larger is the error between the phe-nomenological result and the quantum result.
ACKNOWLEDGMENTS
This research was supported by the National Natural Science Foundation of China (grant no. 10474023) and the Natural Science Foundation of Guangdong prov-ince, China (grant nos. 031516 and 04105804).
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