Thermally activated charge carriers and mid-infrared optical excitations in quarter-filled CDW systems

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by  Ivan Kupcic

Thermally activated charge carriers and mid-infrared optical excitations in
quarter-filled CDW systems
I. Kupˇci´c
Department of Physics, Faculty of Science, University of Zagreb, POB 331, HR-10002 Zagreb, Croatia
The optical properties of the quarter-filled single-band CDW systems have been reexamined in
the model with the electron-phonon coupling related to the variations of electron site energies. It
appears that theindirect, electron-mediated couplingbetween phasephononsand external electro-
magnetic fields vanishes for symmetry reasons, at variance with the infrared selection rules used in
thegenerallyacceptedmicroscopictheory. Itisshownthatthephasephononmodesandtheelectric
8 fieldscoupledirectly,withthecouplingconstantproportionaltothemagnitudeofthecharge-density
0
wave. The single-particle contributions to the optical conductivity tensor are determined for the
0
ordered CDW state and the related weakly doped metallic state by means of the Bethe–Salpeter
2
equations for elementary electron-hole excitations. It turns out that this gauge-invariant approach

n establishes a clear connection between the effective numbers of residual, thermally activated and
a boundcharge carriers. Finally, therelation between these numbersand theactivation energy of dc
J conductivityandtheopticalCDWgapscaleisexplainedinthewayconsistentwiththeconductivity

0 sum rules.
3
PACSnumbers: 71.45.Lr,78.30.-j,78.20.Bh
] Keywords: CDWsystems,optical properties,Bethe-Salpeter equations, single-particleexcitations
l
e
-
r I. INTRODUCTION whichimpliestheindirectelectron-mediated,ratherthan
t
s the direct, photon-phase phonon coupling. The high
t. The electrodynamics of low-dimensional multiband (MIR) frequency analyses [17, 18], on the other hand,
a
were focussed on the precise description of the single-
m systems in which strong correlations responsible for for-
particle excitations, in particular on the rigorous treat-
mation of localized electronic states coexist with cor-
- ment of the square-root singularity in the optical con-
d relations responsible for charge-density-wave ordering
ductivity spectra at ~ω = 2∆. It is shown that the con-
n (CDW)stillattractgreatattention. Thevanadiumchain
o troversies related to the high-frequency optical spectra
compound BaVS is a typical example of such (mag-
c 3 usuallyreflecttheincorrecttreatmentofthediamagnetic
netic) metals with the commensurate CDW instability
[ current in the transverse response approaches,and, con-
[1, 2, 3, 4]. In this system, basic electrodynamic proper-
1 tiesarerelatedtoonewideband,whilemagneticanoma- sequently, can be easily avoided using the longitudinal
v response theory [19].
liesareassociatedwithmoreorlesslocalizedstatesintwo
5
2 narrow bands. The relations between the mid-infrared In this article, we consider a quarter-filledsingle-band
6 (MIR) energy scales, measured at temperatures below CDW system and a related tetramerized metallic sys-
4 the criticalCDWtemperature, the activationenergiesof tem in which all non-retarded short-range interactions
. transport coefficients and the related effective numbers areneglected, andallcoherenceeffects in the correlation
1
0 of conduction electrons are not entirely clear [3]. In or- functionsassociatedwiththeretarded,phonon-mediated
8 der to explain these relations in simple physical terms, interationsaretreatedexactlythroughtheknownanalyt-
0 one needs, in the first place, a precise description of the icalformofthecurrentverticesandtheinverseeffective-
: CDW coherence effects in the quarter-filled wide band. mass tensor. Using this exactly solvable model, we re-
v
i This problem that is still not completely solved for the examine several textbook results [5, 6, 7, 8] for the dc
X single-band case is in the focus of this work. andopticalconductivityofcommensurateCDWsystems.
r The previous theoretical work [5, 6, 7, 8, 9, 10, 11, The analysis is however limited by the use of several
a
12, 13, 14, 15, 16, 17, 18] on the electrodynamics of approximations. We consider a Holstein-like electron-
the single-bandCDW systems followedtwo basic routes. phonon coupling in which the coupling constant is inde-
First, the collective contributions have been studied to pendentoftheelectronwavevectors. Beinginterestedin
explain the low- and zero-temperature properties of the the optical conductivity spectra obtained by the reflec-
incommensurateCDWsystems,withtheemphasisonthe tivity measurements, which are characterized by a rel-
non-linearconductivityregime[12,13]. Themaincontro- atively poor resolution in the low-frequency part of the
versies characterizing early theoretical investigations are spectra, we approximate the total optical conductivity
resolvedusingaccuratesymmetryanalyses,withapartic- by the sum of the collective and single-particle contri-
ular caredevotedto the localfieldeffects [14]. The prin- butions, with the local field effects included only in the
cipal physical problem in these analyses is that the cou- collective term. In the numerical analysis of the single-
plingbetweeninfrared-activecollectivemodesandexter- particleterm,weconsiderthesimplestcaseinwhichtwo
nalelectromagneticfieldsisshownasasimplefunctionof relaxation rates are assumed to be independent of fre-
thesingle-particle(interband)polarizability[5,7,13,14], quency and temperature. Finally, the commensurability
2
(Umklapp) effects are present in the model through the <u >
A
shift of the phase phononfrequency to a finite value and
throughthe commensuratephoton-phononcoupling (see
Sec. 5). −δε +δε +δε −δε −δε
Thearticleisorganizedasfollows. InSec. 2,theband
dispersionsofthetetramerizedCDWcasearebrieflydis-
FIG. 1: Variations of the electron site energies ( δε, +δε,
cussed. In Secs. 3 and 4.1, we consider the electron-hole −
+δε, δε) induced by the periodic (tetramerization) lattice
excitationsinthetwo-bandversionofthemodelinwhich distor−tion uAq δq,0 uA and φ=π/4 in Eq.(1). Thesolid
the Umklapp effects are neglected. The emphasis is on and dashed arr→ows illuhstriate the Raman-active and infrared-
the clear identification of the effectively free and effec- active q 0 modes. The effective charges of the white and
tively bound charge carriers in the underdamped regime the black≈ions are +q and q, respectively. Here δε = √2∆
−
(the interband damping energy is small in comparison and q=qc/√2.
withtheCDWgap),theirrelationtoboththeactivation
energy of dc conductivity and the MIR scale in the opti-
cal conductivity. The brief discussion of the optical con-
g is the electron-phonon coupling constant. The di-
ductivity spectra in the overdamped regime, tentatively
rect inspection of Fig. 1 shows that the periodic lat-
related to the hybridizations/interactions with localized
tice distortion associated with the replacement u
Aq
electronic states, is given in Sec. 4.2. In Sec. 5, the in- →
δ u +u leads to the charge transfer of the mag-
q,0 A Aq
fraredselectionrulesforthecollectivemodesinthecom- h i
nitude q (determined in Sec. 5) modulated by the wave
c
mensurateCDWsystemsarediscussedinsomedetail. It
vector Q. This lattice distortion makes u and u to
Aq ϕq
is shownthat the (interband) electron-mediatedphoton-
be thelongwavelengthphononmodeswhichare,respec-
phase phonon coupling vanishes for symmetry reasons,
tively, Raman and infrared active. The other phonon
and that the direct coupling is proportional to the mag-
modes are taken into account later in Sec. 3, and are
nitude of the charge-density wave.
treated on the same footing with the static disorder.
The CDW instability of the quarter-filled Q1D elec-
tronic systems in the case in which the disorder is ab-
II. TETRAMERIZED CDW CASE
sentandtheelectron-phononcouplinggisindependentof
the electron wavevectors is described by the well-known
A. Bloch energies textbookexpressions(seeEqs.(A1)–(A10)inAppendix).
Thecommensurabilityeffectsenterinthetheorythrough
The starting point of the present analysis is the Q1D the φ dependent (Umklapp) contributions in the Bloch
tetramerized model in which the electron-phonon cou- energies E (k). For example, in the 1D limit (t 0),
L b
→
pling is formulated in terms of the displacement vector the four Bloch energies E (k) are (L=C,C,V,V is the
L
band index)
2
u =u +u = eiq·ri u cosφ u sinφ ,
i ϕi Ai rN qX≈0 (cid:2) ϕq i− Aq i(cid:3) EC,C(k) = −p2t2a+2∆2±2t(k),
(1) EV,V(k) = 2t2a+2∆2∓2t(k), (3)
where φ = Q r φ, r is the position of an atom p
and φ isian arb·itriar−y phasie. Two Fourier components t(k) = 2∆2t2a+t4acos22kxa+∆4cos22φ,
p
of interest, uAq and uϕq, are functions of two ordinary withta andtb being the bondenergiesinthe highly con-
components of the wave vectorsq Q, as follows uAq = ducting direction and in the perpendicular direction, re-
±
(1/√2) u +u andu =(i/√2) u u . spectively, and ∆ = g u /√N is the magnitude of the
q+Q q−Q ϕq q+Q q−Q A
− h i
uAi and(cid:2) uϕi correspo(cid:3)nd to two similar l(cid:2)attice deforma(cid:3)- order parameter. When ∆ is small enough, the disper-
tions which phase is shifted by π/2 and which frequency sions of two lowest bands can be approximated by
is expected to be strongly renormalized for low enough
temperaturesandforthebandfillingclosetothequarter E (k) = ε˜(k) ε˜2(k) ε (k)ε (k) U(k). (4)
C,C c c
filling (evidently, u = u† , with λ = A,ϕ being the ∓q − −
λq λ−q
phonon branch index). Q = (0.5π/a,Qy) is the nesting Here εc(k) = ε(k) = − α2tαcoskαaα and εc(k) =
vector of the Fermi surface of the quarter-filled (CDW) ε(k+Q) aretworelevantPbaredispersions,and2ε¯(k)=
case,orthetetramerizationvectorinageneral(metallic) ε|c(k)+εc|(k), ε˜(k) ε¯(k)+∆2/(2ε¯(k))areusefulabbre-
≈
case. The related variations of the electron site energies viations. The Umklapp processes are represented in the
(see Fig. 1), which are assumed to be the main mecha- dispersions (4) through
nism of the electron-phonon coupling here, are
∆2 2∆4 δ4(k) (δ∗(k))4
U(k) α 1+ − − ,(5)
V(r ) = V (r )+V (r ) ≈ (cid:18) 4ε¯2(k)(cid:19) 4ε¯2(k)
i ϕ i A i
2g
= √N qX≈0eiq·ri(cid:2)uϕqsinφi+uAqcosφi(cid:3). (2) Twhitehqδu(kar)t=er-∆filleexdp{αiφ=(k0)}l,imφ(ikt)of=Esqg.n((k4x))wφitahndε˜2α(k=) 1.
≈
3
ε¯2(k)+∆2 correspondsto the usualPeierlsmodelwhich (a)
electrodynamic properties will be studied in more detail
in Secs. 3 and 4.
(b)
B. Optical conductivity in pure CDW systems
(c)
Inthepresentcontext,themaineffectoftheUmklapp
processes is to shift the renormalized frequency ω of
ϕ0
the q = q eˆ = 0 phase phonon mode from zero to a fi-
x x
nite value [5]. Together with other pinning mechanisms, FIG. 2: The collective (a) and single-particle (c) contribu-
this gives the collective contribution to the optical con- tionstotheopticalconductivityinpureregimeofthepresent
ductivity of the form (see Fig. 2(a)) CDWmodel. Thesolidlinesaretheelectronpropagatorsand
thezigzaglinesarethephononpropagators. Theopencircles,
e2n iω theopensquaresandthefullcirclesrepresentthechargever-
σϕ (ω) = , (6) tices, the current vertices and the electron-phonon coupling
xx m∗ ω ω+iγ ω2
ϕq − ϕq constants,respectively. Thedashedlineistheexternalscalar
(cid:0) (cid:1) field Vext(q,ω) and the wavy line is the related electric field
with m∗ being the collective mode effective mass. In the (seeEq.(13)). (b)TheDysonequationforthephononGreen
pure case, where the scattering of electrons on both the functions. The phonon self-energy is given by the expression
disorderandthephononmodesisneglected,ω andγ (A10) in Appendix.
ϕq ϕq
inEq.(6)arethesolutionsoftheusualfunctionalintegral
approach [10], or the solutions of the Dyson equation
showninFig.2(b)[5,6,7]. Thestructureofm∗ andω
ϕq A. Relaxation processes on the (quasi)static
is discussed in more detail in Sec. 5 and Appendix. disorder
The related single-particle contribution to the optical
conductivity (Fig. 2(c)) is given, for example, by the
Having in mind the known structure of the current
gauge-invariantexpression
vertices related to the bands (8) [21], or, alternatively,
calculating the long wavelength charge vertices (29), we
iω
σsp (ω) = χ (q ,ω)= iωα(q ,ω) (7) can make the generalization of the optical conductiv-
αα qα2 1,1 α − α ity (7) to the case in which the (quasi)static disorder
is present. This can be easily done by using the longitu-
obtained by means of the longitudinal response theory dinal equation-of-motion approach [20, 22], because this
[20] (the left-hand side of Fig. 2(c), with q = qαeˆα). approach treats the (intraband) charge continuity equa-
χ1,1(qα,ω)isthesumoftheintra-andinterbandcharge- tion in a natural way (the calculation is more compli-
charge correlation functions defined in Appendix, and cated in the transverse response theory [23]). The result
α(qα,ω) is the related polarizability. (Eqs.(22)and(25))isequaltoEq.(7)withtheadiabatic
termηinthecharge-chargecorrelationfunctionsreplaced
by two phenomenological damping energies ~Γintra and
α
~Γinter.
III. BETHE–SALPETER EQUATIONS α
The longitudinal equation-of-motion approach is one
example of the calculations which describe the elemen-
Wetakethecollectivemodecontribution(6)asideand tary electron-hole excitations in the effective single-
continue with the examination of the (single-particle) particle multiband models in the gauge-invariant way.
electrodynamic properties of the α = 0 two-band model The Bethe–Salpeter approach, described below, is an-
of Eqs. (4),(5). The Bloch energies are otherexample. Inbothcases,the elementaryexcitations
arerepresentedbythe Matsubaraelectron-holepropaga-
E (k) = ε¯(k) (1/4)ε2 (k)+∆2, (8) tor
C,C ∓q cc LL′(k,k ,k′ ,k′,τ)= LL′(k,k ,k′ ,k′,τ),
D + + D(2n) + +
with εcc(k) = εc(k) εc(k). To make the analysis more Xn
−
general, we consider two regimes of the two-band model
where
close to the quarter filling in which an important role is
playedbythermallyactivatedelectronsinbothtransport LL′(k,k ,k′ ,k′,τ)= (−1)n+1 β~dτ ... β~dτ
and optical properties. The first regime corresponds to D(n) + + ~nn! Z 1 Z n
0 0
tohnedorredgeirmede iCsDaWssosctiaatteedcownistihdetrheedmabeotavlel,icwshtialetet,hwehseerce- ×hTτ H1′(τ1)...H1′(τn)c†Lkσ(τ)cL′k+qσ(τ)
the doubled Fermi wave vector is 2kF 6=Qx =0.5π/a. ×c(cid:2)†L′k′+qσ(0)cLk′σ(0) i (9)
(cid:3)
4
form (see Fig. 3(a))
(a)
i~νn+ELL′(k,k+)+ΣL(k,iωn) ΣL′(k+,iωn+)
−
(cid:2) ×ΦLL′(k;iωn)= ~1 GL(k,iωn)−GL′(k+,iωn+(cid:3))
(cid:2) (cid:3)
(b) ×(cid:26)δk,k′ − β1~ (cid:12)GλN(q′)(cid:12)2Dλ(q′,iνm)
λqX′iνm (cid:12) (cid:12)
(c) ΦLL′(k+q′;iωn+iνm) , (11)
× (cid:27)
in obvious notation (ELL′(k,k+) = EL(k) EL′(k+) is
−
the electron-hole-pair energy). In the intraband chan-
FIG. 3: (a) The Bethe–Salpeter equations. (b) The related
nel, these equations transform into two coupled equa-
self-consistentequationsfortheinduceddensitiesδnLL′(k,ω). tions, one is the charge continuity equation and the
Thediamondrepresentstheelectron-holeself-energy. (c)The
other is the transport equation. These two equations
generalization of Fig. 2(c).
have to be solved self-consistently. For this purpose,
we retain in Eqs. (11) only the most singular scatter-
ing processes by putting iωn+ EL′(k+)/~ and iωn
and k = k+q. Here H′ describes the scattering of EL(k)/~, in two single-particle→self-energies, ΣL(k,iω→n)
conduc+tionelectronsonthe1(quasi)staticdisorder. Using and ΣL′(k+,iωn+), and in the related vertex corrections
on the right-hand side of the equations. In this way, the
this approach, it is possible to treat the relaxation pro-
expressionin the brackets on the left-hand side does not
cesses on impurities, on lattice imperfections, and even
depend on iω , and the equations can be rewritten in
on soft phonons. The hybridizationwith other (uncorre- n
terms of the intraband (L=L′) and interband (L=L′)
lated) bands can also be studied in this way. induced densities δnLL′(k,ω) defined by 6
δnLL′(k,ω) = 1 LL′(k,k ,k′ ,k′,ω)
B. Relaxation processes on phonons ~D + +
Xk′
eqL′L(k ,k)Vext(q,ω). (12)
In the presence of inelastic electron scattering pro- × +
cesses on phonon modes, or other boson modes in gen- Here
eral,onecanuseasimilarlongitudinaldiagrammaticap-
proachbasedontheself-consistentBethe–Salpeterequa- eqL′L(k ,k)Vext(q,ω) ~JαL′L(k) iE (ω), (13)
tions for + ≡ EL′L(k+,k) α
ΦLL′(k;iω ) ΦLL′(k,k ,k′ ,k′,iω ,iω ) with Eα(ω) andVext(q,ω)=(i/qα)Eα(ω) being, respec-
n ≡ + + n n+ tively, the external electric field and the related scalar
potential (notice that for L=L′ Eq. (13) reduces to the
(iωn+ =iωn+iνn). In this case H1′ in Eq.(9) represents expression eVext(q,ω) = e(i/q )E (ω) that is indepen-
α α
the electron-phonon Hamiltonian (A4) with the phonon dent of k). Furthermore, qL′L(k ,k) is the charge ver-
wavevectorq′ runningovertheentireBrillouinzoneand tex, given by Eqs. (A8) and (29),+JL′L(k) is the current
the phonon branch index λ running over all (elastic and α
inelastic) scattering channels. ΦLL′(k;iωn) is defined by verItnexEqan.d(1q2)=, qαLeˆLα′(.k,k ,k′ ,k′,ω) is obtained by the
the Fourier transform of DLL′(k,k+,k′+,k′,τ), analyticalcontiDnuationof+DL+L′(k,k+,k′+,k′,iνn),iνn →
ω+iη. The summation of Eqs. (11) over iω makes the
LL′(k,k ,k′ ,k′,iν ) (10) n
D + + n single-particle self-energy contributions and the related
=(~/β) ΦLL′(k,k ,k′ ,k′,iω ,iω ). vertex corrections to appear on the same footing, result-
+ + n n+
Xiωn ing in the electron-hole self-energy of the form
~ΠLL′(k,ω) (14)
Fortemperaturesanddopingsnottooclosetothecrit- α
ical values characterizing the CDW metal-to-insulator Gλ(q′) 2 1 J˜αLL′(k+q′)
phasetransition,theinterbandelectron-phononcoupling ≈− (cid:12) N (cid:12) (cid:26) − J˜LL′(k) (cid:27)
in H′ can be neglected (notice that beyond this ap- Xλq′ s=X+1,−1(cid:12) (cid:12) α
1
proximationone needs four band indices in the electron- fλb(q′)+f(sEL′(k+q′))
hole propagator (9)). Now, for GLλL(k + q′,k) ≈ ×(cid:20)~ω+iη+ELL′(k,k+q′)+s~ωλq′
pG~λ/(q(2′)M2ω,λtqh′)e×BgegtλLhLe–(kSa+lpqe′t,ekr)eaqnudat(cid:12)(cid:12)GioLλnLs([k2+4]qt′a,kke)(cid:12)(cid:12)2th≈e +~ω+fiηλb(+q′E)L+Lf′((ksE+Lq(′k,k+)q′)s)~ωλq′(cid:21).
−
(cid:12) (cid:12)
(cid:12) (cid:12)
5
The corresponding generalization of ΠLL′(k,ω) to the plicitly for the (single-band) case in which ∆ = 0 and
α
case with the interband electron-phonon coupling is the relaxation rate Γ is presumably small but finite
straightforward; however, it is beyond the scope of the (Im Π0(k,ω) Γ). These expressions will make the
present work. fb(q′) and f(E (k)) f (k) in Eq. (14) com{paαrison o}f t≈he present approach with the common
λ L ≡ L
are,respectively,theBose–EinsteinandFermi–Diracdis- CDW approaches in Sec. 5 straightforward.
tribution functions. Finally, notice that the electron For ∆=0 andq=q eˆ , the electron-hole-pairenergy
x x
scattering on the static disorder, described by the po- is ε(k ) ε(k) ~ω (q) = ~q v0(k), with ω2(q) being
tential V (q′), is included here through the replacement indepe+nd−ent of k≈for0k k .xExq. (15) can b0e written
1 F
ωλq′ → 0 and (1+2fλb(q′))|Gλ(q′)|2/N → |V1(q′)|2 in now in the textbook form≈[22]
the λ=0 boson branch [20].
ωδn (k,ω) ω (q)δn (k,ω)=0,
0 0 1
−
ω+iΓ δn (k,ω) ω (q)δn (k,ω)
1 0 0
C. Transport equations and optical conductivity −
(cid:0) (cid:1) ∂f(k)
=( ) iev0(k)E (ω). (17)
− ∂ε(k) x x
The resulting self-consistent equation for the densities
δnLL′(k,ω)representsthegeneralizationofthetransport
Introducingthebosonfieldψ(q,ω),whichtimederivative
equation [20, 22],
is proportional to the current induced by the external
~ω+ELL′(k,k+) δnLL′(k,ω)+~ΠαLL′(k,ω)δn˜LL′(k,ω) electric field,
(cid:2) = fL(k)−(cid:3) fL′(k+)i~JL′L(k)E (ω). (15) jind(ω)= 1 ev0(k)δn (k,ω) eψ˙(q,ω), (18)
EL′L(k+,k) α α x V Xkσ x 1 ≡ π
In this equation δn˜LL′(k,ω) is the contribution to
one obtains the equation of motion
δnLL′(k,ω)proportionaltoJ˜L′L(k)(theeffectivecurrent
α
verticesinEqs.(14)and(15)areJ˜LL(k)=JLL(k)inthe m ω ω+iΓ ω2(q) ψ(q,ω) = eπneff,0E (ω).(19)
intraband channel and J˜αLL(k) =α~ωJαLL(kα)/ELL(k,k) (cid:2) (cid:0) (cid:1)− 0 (cid:3) xx x
in the interband channel). Equation (15) is solved con- The resulting optical conductivity is given by the inter-
sistently with the charge continuity equation and com- band term in Eq. (16), i.e., by the expressions
bined with the definition ofthe opticalconductivity ten-
e2neff,0 iω
sor. Theresultistheopticalconductivityoftheeffective σ (ω) = xx
single-particle multiband models [19, 20, 21, 25] xx m ω ω+iΓ ω2(q)
− 0
σαspα(ω)= ωi V1 LXL′kσ(cid:18)EL′L~(kω+,k)(cid:19)nLL′(cid:12)JαLL′(k)(cid:12)2 (16) ≡ i~ω(cid:18)πe(cid:19)2V(cid:0)Dψ0(q,(cid:1)ω). (20)
(cid:12) (cid:12)
fL(k) fL′(k+) Here 0(q,ω)isthebarepropagatorofthefieldψ(q,ω),
− , Dψ
×~ω+~ΠαLL′(k,ω)+ELL′(k,k)− EL2′L′~(ωk,k+) nxeffx,0 is the effective number of conduction electrons,
with nLL = 1 in the intraband channel and nLL = 2 neff,0 = m v0(k) 2∂f(k) = 2maavx0(kF), (21)
in the interband channel. In the dynamical limit, the xx −V Xkσ (cid:0) x (cid:1) ∂ε(k) π ~
last term in the denominator can be safely neglected.
At this point it should be recalled that the current ver- v0(k) = (1/~)∂ε(k)/∂k is the bare electron group ve-
tices JLL′(k) can be shown in simple physical terms loxcity and m =2~2/(txa2) is the mass parameter.
α aa a
as: JLL(k) = evL(k) in the intraband channel and
α α
JLL(k) = (i/~)PLL(k)E (k,k) in the interband chan-
α α LL
nel; vL(k) is the electron group velocity and PLL(k) is IV. DISCUSSION
α α
therelatedinterbanddipolevertex. Itshouldalsobeno-
A. Underdamped regime
ticedthatthecoherencefactorinthetextbookexpression
for the interband contributions [5, 8] is slightly different
fromthat in Eq.(16), as wellas that the phenomenolog- For comparison with the textbook expressions for the
icalextensionofthetextbookexpressionisfoundtolead single-particle contributions to the optical conductivity,
tothenon-physicalin-gapspectra(seeFig.1inRef.[17]). itisconvenienttoturnbacktothesimplestcasediscussed
inSec.3.1. Theelectron-holeself-energyisapproximated
by its imaginarypartandthe imaginarypartis assumed
D. ∆=0 case to be small when compared to the interband threshold
energy; i.e., ΠLL′(k,ω) iΓLL′(k,ω) and ΓLL′(k,ω)
Before turning to the detailed numerical analysis of ΓLL′ (ΓLL Γαintra and≈ΓLLα Γinter, hereaftαer). In th≈e
the expression (16), let us write Eqs. (15) and (16) ex- unαderdaαmp≡ed rαegime (Γintαer ≡ 2∆α), this approximation
α ≪
6
0.5
00..65 V0 n 1.21 effV n0xx00..43
eff(m/m) Vnaa0 xx000...243 γ (k)xxx01000.....224681 blaorweb eabrna dnd upbpaenrd effeff (T)n / nxx0xx000...468 t(m/m)b aa=0 01..20150 1m0e mVe0VT.5/T0 1
0.1 00 0.2 20a.4kx 0/ .π6 0.8 1 0.2 1 meV0 meV
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1
δ = 2ak / π T / T
F 0
FIG. 4: Main figure: The normalized effective number of FIG. 5: Main figure: The temperature dependence of the
conductionelectrons(maa/m)V0nexffxasafunctionoftheband normalized effective number of conduction electrons in the
filling for 2ta = 0.5 eV, 2tb =0, T → 0 and ∆0 = 0 (dashed (imperfectly nested Qy = 0) CDW state for 2ta = 0.5 eV,
line) and 10 meV (solid line). The diamonds represent the 2t =0(solidline),1meV(dashedline),10meV(dot-dashed
b
dimensionless electron group velocity [2/(taπ2)]vx0(kF). In- line) and 15 meV (dot-dot-dashed line), 2kF =Qx =0.5π/a,
set: The inverse effective mass tensors γxCxC(k) and γxCxC(k) T0 =66.7 K and ∆0 =10 meV. The diamonds represent the
for Qx = 0.5π/a and ∆0 = 10 meV (solid and dotted lines, function exp{−∆(T)/(kBT)}.
respectively). The bare-band vertex (∆ = 0, dashed line) is
given for comparison.
For example, the intraband conductivity along the
highly conducting direction is proportional to the effec-
tive number of conduction electrons neff, as seen from
xx
doesnotaffectthespectruminacriticalmanner,neither
in the metallic regime (with 2kF not too close to Qx), σintra(ω) ie2neαffα , (22)
nor in the CDW state at temperatures well below the αα ≈ m(ω+iΓintra)
α
critical temperature. In the metallic case, the effective
numbers of conduction electrons in the resistivity, or in forα=x. neff includes the contributions ofbothbands,
αα
theDrudepartoftheopticalconductivity,consistoftwo anditcanbeshownintwoalternativeways,asasurface
parts, the residual one, which remains finite at T 0, integral in the k space, or the related volume integral,
→
and the thermally activated part. In the t 0 Peierls
casethereareonlythermallyactivatedelectbro→ns. Finally, neff = neff(L) = m vL(k) 2∂fL(k)
in the Peierls case with tb/∆ not too small there is a αα XL αα −V LXkσ(cid:0) α (cid:1) ∂EL(k)
portion of the Fermi surface on which the CDW gap is 1
not developed, leading again to a finite residual number = V γαLαL(k)fL(k). (23)
of conduction electrons. LXkσ
To compare briefly the correct estimations of these Inthefirstrepresentation,itisgivenintermsoftheelec-
numbers with the oversimplified “semiconducting” ex- trongroupvelocity,while inthe secondrepresentationit
pressions encountered in the literature [26], we apply depends on the dimensionless inverse effective-mass ten-
Eq. (16) now to the simplest metallic situation corre- sorγLL(k)=(m/~)∂vL(k)/∂k . Thedependenceofneff
αα α α xx
sponding to Q = (0.5π/a,0) and t 0. In this re- onthe electron doping δ =2ak /π is shownin Fig.4 for
b F
→
spect, it should be noticed that the present analysis is two choices of the distortion potential, ∆ = 0 and 10
0
notfocussedonthesingle-particlepropertiesoftheusual meV, and T 0. The electron doping δ =nV (n is the
0
→
strictly 1D model, the physics of which is dominated by concentrationofelectronsandV istheprimitivecellvol-
0
the singular nature of the forward scattering processes umeofthe∆ =0lattice)andthedimensionlesselectron
0
in the single-electron self-energy. Instead, we consider group velocity are also given for comparison. The latter
the electron-hole excitations in the typical Q1D model represents the analytical solution of the t 0 model.
b
→
in which the Q1D characteris incorporatedin the afore- The metallic regime of the primary experimental in-
mentioned form of the electron-hole self-energy, where terest corresponds to the case of a low electron/hole
the forward scattering contributions are removed by the doping with respect to the quarter-filled band. In this
Ward identity cancellation of the q 0 self-energy and case, the residual part in neff (shown in Fig. 4) is small
≈ xx
vertex corrections in Eq. (14). The t 0 limit serves when compared to its ∆ = 0 value (dashed line in the
b
→
here onlyto replacemostof the integrationsoverthe 2D figure). Even for relatively low temperatures, the ther-
Brillouin zone by the 1D integrations. The results ob- mally activated part in neff becomes important, leading
xx
tained in this way can be comparedterm by term to the to pronounced thermal effects in neff. The thermal ef-
xx
well-known analytical expressions. fects in the Drude part of the optical conductivity, or in
7
1
Anotherimportantobservationisthattheopticalcon-
T = 67 K ductivitymodel(16)hasaquitegeneralformandisthus
64
0.8 56 applicable to variety of low-dimensionalsystems. In this
T)0 4224 respect one should notice that the values of the effec-
DC( 0.6 5 tive numbers neff are generally a direct consequence of
σ}/xx the coherent scαaαttering of conduction electrons on the
σxx0.4 crystal potential or on the periodically distorted lattice,
{
e irrespectiveof the band filling. These numbers represent
R
0.2 the numbers of effectively free charge carriers, with the
thermally activated part being usually very sensitive to
0 temperature, providing the estimation of the activation
0 0.01 0.02 0.03 0.04 0.05
energy (eV) energy ∆, or the charge-transfergapin the generalcase.
The corresponding numbers of bound charges, on the
FIG. 6: The development of the normalized total optical other hand, are the numbers of excited electrons across
conductivity with temperature in the underdamped regime: the CDW gap [8] or across the charge-transfergap. The
2ta = 0.5 eV, 2tb = 1 meV, T0 = 66.7 K, ∆0 =10 meV and latter numbers depend on temperature as well, as seen
~Γiantra=~Γiantra=5 meV. fromtheinterbandpartofEq.(16),whichcanberewrit-
ten in the form
σinter(ω) iω 1 PLL′(k) 2
theresistivity,arefurtherillustratedinFig.5fortheim- αα ≈ V α
perfectly nested CDW case (corresponding to the elec- L6=XL′kσ(cid:12)(cid:12) (cid:12)(cid:12)
tron doping δ = 0.5 in Fig. 4). The temperature de- fL(k) fL′(k)
− . (25)
pendence of the activation energy ∆ is parametrized as ×~ω+i~Γiαnter+ELL′(k,k)
∆(T)=∆ (1 T/T )β, with β =1/2. neff is shown for
0 − 0 xx
four different values of t and comparedto the empirical The temperature dependence of the total optical con-
b
expression neff(T) = exp ∆(T)/(k T) neff(T ) (the ductivity σsp(ω)=σintra(ω)+σinter(ω) is calculated for
xx {− B } xx 0 xx xx xx
residualpartinneff correspondsto thezero-temperature several underdamped regimes. The damping energies
xx
interception value). The dashed and dot-dashed lines in ~Γintra and ~Γinter are assumed to be temperature in-
a a
the inset of figure represent the partial contributions of dependent, for simplicity, and small in comparison with
the lowerandupper bandfor t =0,neff(C) andneff(C) the scale2∆. The results for the imperfectly nested case
b xx xx
in Eq. (23), respectively. The latter two have to be con- are shown in Fig. 6 and are essentially the same as the
trasted to the numbers resultsofthe analogousnearlyhalf-filleddimerizedband
[19, 25]. Although it is physically clear that at tempera-
1 1 turesT wellbelowT ΓinterismuchlargerthanΓintra,the
n = [1 f (k)], n = f (k) (24) c a a
h V − C e V C spectra shown in Fig. 6 are calculated for Γintra =Γinter,
Xkσ Xkσ a a
for clarity. The shift of the MIR maximum in the spec-
invokedfrom the theory of the ordinary semiconductors. tra (~ωMIR) with temperature follows roughly the en-
The numbers (24) are often taken as a good approxima- ergy scale 2∆, giving an alternative method of estimat-
tion for neff(C) and neff(C). However, the disagreement ing ∆ [5]. The spectra are also characterized by the in-
xx xx
between the expressions (24) and experimental observa- gap optical excitations, mainly in the energy range from
tionistypicallyoftwoordersofmagnitude[27],and,con- ~ωmin ≈2∆−~Γianter to 2∆. The cut-off energy ~ωmin is
sequently, the analyses based on the numbers (24) have associatedwith the zero conductivity interception of the
to be taken with reservation. For the parameters used lineobtainedbythelinearextrapolationoftheinterband
in Fig. 5, we obtain a factor of 25. This disagreement conductivity. The square-root singularity at ~ω = 2∆ is
is easily understood on noting that the semiconducting absent here, and, according to Ref. [18], the same shape
approach,Eq.(24),completelyneglectsthecoherenceef- of σxinxter(ω) is expected down to zero temperature.
fectsintheinverseeffective-masstensor. Inotherwords,
byreplacingtheverticesγLL(k)byunity,whenthenum-
xx
bersneff(C)andneff(C)reduceton andn indeed,this B. Overdamped regime
xx xx h e
approach does not take carefully into account the fact
thatthethermallyactivatedelectronsarerelatedonlyto A completely different situation appears in the over-
thestatesinthethermalwindowaroundtheFermilevel. damped regime where the damping energy ~Γinter be-
a
Noticeinthisrespectthesingularbehaviorofthevertices comes comparable to 2∆ and where the real part of the
γLL(k) at 2k = 0.5π/a, shown in the inset of Fig. 4, electron-holeself-energystartsplayinganimportantrole.
xx x
which dominates the behavior of neff for 2k 0.5π/a. Typical physical situation corresponding to this regime
xx F ≈
On the contrary, the common approaches based on the istheCDWinstabilityofthemagneticmetals,wherethe
surface-integralrepresentationofneff usuallygiveriseto widebandelectronsexhibittheCDWinstability,andare
αα
the correct results. usually scattered by the strongly correlated electrons in
8
0.25 (a)
0.2
0.2
0.1
T)0
C(0.15
Dσσ}/xxxx 0.1 20 m e4V0 00 0en.0e2rgy (0e.V04) 0.06 (b) (c)
{
Re 60
0.05 80
FIG.8: (a)Thecollectivecontributiontotheopticalconduc-
tivity in the common 1D models with the forward-scattering
0
0 0.02 0.04 0.06 0.08 0.1 short-range interactions. (b) The collective contribution in
energy (eV) the present model. (c) The Hopfield-like processes neglected
in the present approach.
FIG. 7: Main figure: The optical conductivity as a function
ofdampingenergy~Γinter atT =42K.Dotted,dashed,long-
a
dashedanddot-dashedlinescorrespondto~Γinter =20,40,60
a
and80meV,respectively. Theotherparametersarethesame V. COMPARISON WITH THE COMMON CDW
as in Fig. 6. The solid line in the total optical conductivity THEORIES
at T =67 K.Inset: Therelated interband contributions(the
parameters are thesame as in the main figure). In the low-energy examinations of the CDW systems
itiscommonto usedifferentstrictly 1Dmodels withthe
non-retarded short-range interactions in which not only
the forward-scattering but also the backward-scattering
one or more narrow bands. As the present discussion processes in the electron-hole self energy are neglected
of the general expression (16) neglects the real part of [15, 16]. These approaches are usually based on the
the self-energy, it turns out to be inappropriate for the bosonization procedure. For example, for the simplest
detailedquantitativeanalysisofthe overdampedregime, casewithonlytheforward-scatteringinteractiong taken
4
but it satisfactorily explains the interplay between two intoaccount,theopticalconductivityisgivenbytheRPA
mutuallycompetingenergyscales,~Γinter and2∆. Fig.7 diagrams in Fig. 8(a). The first diagram is nothing but
a
shows typical results. The position of the MIR peak is theidealintrabandconductivityofEq.(20). Theinterac-
at ~ω 4∆2+(~Γinter)2. In this case, ~ω is a tiong leadstotherenormalizationoftheelectrongroup
MIR ≈ a min 4
rathercomplpicatedfunctionofthemodelparameters,but velocity in Eqs. (20) and (21) in the way that
it is close to the value obtainedin the narrowband limit
of σxinxter(ω), where all interband excitations are assumed v0(k) u = m v0(k ),
toappearclosetothe thresholdenergyE (k,k) 2∆. x → ρ rm∗ x F
CC
≈
In this case, one obtains
while the electron dispersions in ∂f(k)/∂ε(k), together
with the related density of states, remain unaffected by
Re σinter(ω) ~ω~Γianter g4. The boson field ψ(q,ω) is found to satisfy the equa-
{ xx } ∝ (~ω 2∆)2+(~Γinter)2 tion of motion (19), while the optical conductivity is
×(~ω−+2∆8)~2ω+∆(~aΓinter)2. (26) [f9o,un16d].to be given by Eqs. (20), (21), with vx0(k) → uρ
a The present Q1D model differs from these strictly
1D models in three important points. First, the
As a result, in the overdamped regime ~ω increases
min electron-electron interactions in question are the re-
with increasing ~Γinter, for example, from the value
a tarded, phonon-mediated interactions [5, 6, 7], second,
~ω ∆ at ~Γinter = 5∆ to ~ω 2∆ at ~Γinter =
10∆mi.n ≈Knowing tahe experimentalmvinalu≈es of ~ω a and the collective infrared-active mode is the phase phonon
MIR mode, which participates in the total conductivity spec-
~ωmin,wecanagainestimate∆and~Γianter andcompare tral weight with usually less than 0.5 % (e.g., m∗ is of
∆ to the activation energy of transport coefficients.
the order of 400m in K MoO [28]), and, finally, the
0.3 3
As mentioned in the introduction, a clear evidence of external electric fields are assumed to be well below the
theoverdampedregimeisrecentlyfoundinthevanadium critical field required to give the non-linear conductivity
chaincompoundBaVS [3]. BaVS isthemultibandsys- [12, 13].
3 3
tem with the (commensurate) CDW instability in which In this section, we want to determine the structure of
one weakly correlated wide band and two strongly cor- the photon-phase phonon coupling, and to compare the
related narrow bands coexist at the Fermi level. Thanks obtainedresults withboththe textbook resultsobtained
to the extensive experimental activity in this system in within the same Q1D model [5, 6, 7, 8] and the results
recentyears[1,2,3,4],BaVS seemstobeagoodcandi- ofstrictly 1Dmodels [9, 16]. We will derivehere, for the
3
datefordetailedinvestigationoftheoverdampedregime. first time, a clear selection rule for the infrared-active
9
(a) (b) (Fig. 9(a)) is given by the ground-state average of the
coupling Hamiltonian
Hext = E ( q) δPL′L(k,q)c† c
1 − x − xλ L′kσ Lkσ
FIG. 9: (a) The direct photon-phonon coupling and (b) the Xλq LXL′kσ
iCnDdiWrecsty,setleemctsr.on-mediated photon-phonon coupling in pure δPxLλ′L(k,q) = √s2λNe gλL′L(k,k)u˜xλq (30)
(s = 1,s =1andtheindicesinthevertexgL′L(k,k)
A − ϕ λ
(and Raman-active) phonon modes. On the qualitative are A = ϕ, ϕ = A). Here u˜xλq is the amplitude of the
level, the arguments are the following. For the quarter- electronoscillationsandδPL′L(k,q)istherelateddipole
xλ
filled case shown in Fig. 1, the electric fields couple to vertex. Fortheincommensuratecase,thedirectcoupling
thedipolemomentofthephasephononmode(thedipole is given in a similar way in terms of the current vertex
moment is carried by the bound (condensed) electrons).
For ∆ comparable to the band width, this mode is the δJL′L(k,q) = sλe gL′L(k,k) ∂ u˜ . (31)
ordinary infrared-active phonon mode, characterized by xλ √2N λ ∂t xλq
the effective ioncharge q =q /√2, the reducedion mass
c
M and the concentration of ions 1/V . On the other Since the phase phonon field can be written as uxϕq =
0
hand, for ∆ small in comparison with the band width, 2i uA ϕq [5], we obtain again that the current vertex
h i
the electrons in question are only weakly bound and the is proportional to the time derivative ϕ˙q, and the re-
magnitude of the dipole is now proportional to e˜, which lated equation of motion is analogous to Eq. (19), with
e, m, neff,0 and ω (q) replaced by e˜, M, 1/V and ω .
is q multiplied by the ratio between the amplitudes of xx 0 0 ϕq
the electronandionoscillations. Inthe limit ∆ 0,the These parameters enter into the optical conductivity ex-
→ pressionthroughtwoadjustableparameters(m∗ andω
dipole moment vanishes. ϕq
To discuss this question quantitatively, we neglect in Eq. (6), for example).
As a result, the direct photon-phonon coupling con-
again the Umklapp processes in the electron dispersions
and focus the attention on the α = 0 two-band model stantamountstothereplacementofc†L′kσcLkσinEq.(30)
of Eq. (4). In this case, all relevant vertex functions are by hc†L′kσcLkσi = δL,CδL′,CfC(k). This result shows
simple functions of the auxiliary phase ϕ(k) defined by that the coupling function of the (Raman-active) mode
u , gCC(k,k), vanishes, while the coupling function of
Aq ϕ
2∆
tanϕ(k)= . (27) the phase phonon mode uϕq is finite and, as mentioned
εcc(k) above, is proportional to the magnitude of the charge-
density wave q , which is given by
c
Inthelongwavelengthlimit,theelectron-phononvertices
and the charge vertices of Eqs. (A5) and (A8) become 1
q = e sinϕ(k)f (k). (32)
c C
− N
gCC(k ,k) = gCC(k ,k) Xkσ
A + − A +
= sinϕ(k+)+ϕ(k) sinϕ(k), NoticethatthefactthatthephononsuλqinEq.(30)cou-
− 2 ≈− ple to external electromagnetic fields through the func-
gCC(k ,k) eiφ(k)cosϕ(k), tiongλCC(k,k),ratherthangλCC(k,k),isaconsequenceof
A + ≈ the fact that the variations V (r ) in Eq. (2) are propor-
gCC(k ,k) = gCC(k ,k) λ i
ϕ + ϕ + tional to the derivative of uλi with respect to ri, rather
ϕ(k) ϕ(k ) than to the displacement vector u .
+ λi
= isin − (q ),
2 ≈O α In the Hopfield-like processes the photon momentum
gCC(k ,k) ieiφ(k), (28) is shared between an electron-hole pair and a infrared-
ϕ + ≈ activephononinthewaydescribedbythedipoledensity
qCC(k+,k) qCC(k+,k) 1, operator
≈ ≈
~ q JCC(k)
qCC(k+,k) ≈ eECαCα(k+,k). (29) LLX′q′kσδPxLϕ′L(k+q′−q,k,q′)c†L′k+q′−qσcLkσ
AccordingtoFig.9,thephoton-phononcouplingconsists
inEq.(30), with gL′L(k,k) inthe dipole vertexreplaced
of two contributions, the direct coupling and the indi- A
rect, electron-mediated coupling. In the commensurate by gL′L(k+q′ q,k). The sum overstates in the phase
case in which the Hopfield-like processes [29] of Fig.8(c) phonAon branch−(q′) is restricted to the region where the
areneglected,thelineardirectcouplingbetweenexternal the vertex gL′L(k+q′ q,k) of Eq. (A5) is non-zero.
A −
electromagneticfields (E ( q) isthe electric component The indirect, electron-mediated photon-phonon cou-
x
−
of the field polarized along the highly conducting direc- pling shown in Fig. 9(b), on the other hand, is given
tion) and two q 0 phonon modes under consideration bythe interbandcorrelationfunctionχinter(q,ω) defined
≈ 1,λ
10
in Appendix, or by Eq. (4.3) in Ref. [7]. According to APPENDIX A: VERTEX FUNCTIONS AND
Eqs. (28) and (29), the product of vertices qCC(k,k ) CORRELATION FUNCTIONS IN THE
+
and gCC(k ,k) in this correlation function is propor- TETRAMERIZED Q1D ELECTRONIC SYSTEM
λ +
tional to exp φ(k) JCC(k), i.e., it is proportional to
x
the bare elect{r−on gro}up velocity v0(k). v0(k) is an odd In order to fix the notation, here we rewrite the text-
x x
book expressions [5, 6, 7, 8] for the exactly solvable
function of k and, consequently, the coupling function
x
χinter(q,ω) vanishes for both phonon branches. The tetramerized CDW case in which the electron-phonon
1,λ
coupling is independent of the electron wave vectors
background of this result is the fact that the electron-
(Eq. (2) in the main text).
phonon coupling g is independent of the electron wave
vectors. A more complicated form of the infrared selec- The total Hamiltonian reads as H = H0 +H1′, with
tion rules and the photon-phase phonon coupling func- the bare Hamiltonian and the electron-phonon coupling
tionscanbeexpectedinthemodelsinwhichtheelectron- Hamiltonian given, respectively, by
phonon coupling is related to the variations of the bond
1
energies[6](namely,g dependsnowontheelectronwave H = ε(k)c† c + p† p
vectors,introducinganewchannelintheelectron-photon 0 kσ kσ 2M (cid:20) q+lQ q+lQ
Xkσ lX=±1qX≈0
coupling).
Finally,itisimportanttonoticethatthecommontext- + Mωq0+lQ 2u†q+lQuq+lQ(cid:21),
book analyses [5, 7, 8] explain the photon-phase phonon (cid:0) (cid:1)
geiφ
coupling in the way (see Eqs. (4.3) and (4.7) in Ref. [7]) H′ = u +iu c† c +H.c. .
which is formally equivalent to the replacement of the 1 (cid:20)√N Aq ϕq k+qσ k+Qσ (cid:21)
q≈X0kσ (cid:0) (cid:1)
bare electron group velocity in χinter(q,iν ) by v0(k ).
1,λ n x F (A1)
(That is, both the single-particle interband conductivity
andthephoton-phasephononcouplingaregiveninterms
Here ε(k) = 2t cosk a is the bare electron dis-
of the dimensionless function f(ω) defined in Ref. [5].) − α α α α
persion. pq+lQPis the phonon field conjugate to uq+lQ
Such correlation function is finite indeed and is of the and ω0 is the bare phonon frequency.
same order as the expression (32). However, when an- q+lQ
The result of the phase transformation of H, which
alyzing complicated CDW systems, this approximation
leads to the replacement u δ u +u , is the
has to be taken with reservation. Aq → q,0h Ai Aq
effective tetramerized single-electron Hamiltonian
H0e = H0ll′(k)c†lkσcl′kσ, (A2)
lXl′kσ
e
VI. CONCLUSION
with the index l = 1,0,1,2,and with running over
− k
the first Brillouin zone of the tetrameriPzed lattice. The
In this article, we have derived the infrared selection matrix elements in the Hamiltonian (A2) are Hll(k) =
0
rulesforthesingle-particleandcollectivecontributionsto ε (k)=ε(k+lQ),representingthedispersionsoffourar-
l
theopticalconductivityinthetetramerizedCDWsystem tificially tetramerizedbands,andHll+1(k)=∆exp iφ ,
with the electron-phonon coupling related to the varia- with ∆ = g u /√N being the inte0rband hybridiza{tio}n
A
tions of electron site energies. We have shown that the h i
term. The diagonalization of (A2) leads to
relationsbetweentheeffectivenumbersofchargecarriers
andtheactivationenergyoftransportcoefficients,orthe He = E (k)c† c . (A3)
MIR optical threshold energy, observed in the ordinary 0 L Lkσ Lkσ
LXkσ
Peierls systems are naturally explained in terms of the e
CDW coherence effects. The same coherence effects are
The E (k) are four Bloch energies of the present prob-
L
expected to be important in the correlated multiband
lem (Eqs. (4) in the main text), and they, together with
systems with the CDW instability, as well. The quan-
the total energy of the electronic system, depend on the
titative analysis of the latter systems is left for future phase φ in Hll+1(k).
0
investigations.
Theobservation(seethediscussionofFig.1inSec.2.1)
that, for u =0, the lattice vibrations associated with
A
h i6
u andu areRamanandinfraredactive,respectively,
Aq ϕq
means that there will be the direct photon-phonon cou-
pling similar to the photon-phonon coupling in ordinary
Acknowledgement
semiconductors. In principle, one can also expect the
electron-mediated photon-phonon coupling, and, as dis-
This researchwassupportedby the CroatianMinistry cussedindetailinSec.5,inthecommonmicroscopicap-
of Science and Technology under Project 119-1191458- proaches, exactly this coupling is regarded as the dom-
0512. inant coupling [5, 6, 7, 8]. It is given in terms of the