On the optical infrared antiresonance absorption in TCNQ solids

Chemical Physics 23 (1977) 237-242 Q North-Holland Publishing Company

ON THE OPTICAL INFRARED

ANTIRESONANCE

ABSORF’TION

IN TCNQ SOLIDS

K. KRAL Institute of Physics, Czechoslovakrim

Academy

of Science, Prague, C’SSR *

Received 4 January 1977 Revised manuscript received 15 Llarch 1977

Infrared antiresonance optical absorption of TCNQ salts is theoretically studied. Formulas relating the geometrical parameters of the antiresonar.ce absorption curve to the physical parameters of the crystal are derived and an earlier optical absorption experiment on TEA-TCNQ2 crystals is discussed.

1. Introduction Rice has recently published [I] a rather general formulation of infrared optical spectra of linear organic conductors such as TCNQ salts. He considered the unpaired electrons of the one-dimensional band of the TCNQ chains interacting with the intramolecular vibrations and showed that these interactions lead to the presence of a gap in the spectrum of the electronic excitations being due to the formation of the intramolecular-multiphoton-stabilized Frohlich CDW (charge density wave) state.of the system. Tn the same paper it has been shown that the interactions considered lead to strong infrared optical absorption polarized along the chain axis of the crystals, in the course of which the fully symmetric intramolecular vibrational modes, optically inactive in the case of isolated molecules, are excited. More recently Rice et al. [2] building on ref. [l] , have studied numerically the infrared activity of these materials by calculating the frequency dependent conductivity a(w) using the values of coupling constants of electrons with symmetric intramolecular vibrations, as calculated by T_.ipariet al. [3]. The fit of the energies of the resonant maxima of u(o) and of its overall shape, with a(w), measured formerly [43 in TEA-TCNQ2 crystals, is very good. It nicely confirms the explanation of the nature of the strong *

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infrared activity of TCNQ salts in the region of intramolecular vibrations [1,4-g] and also supports the explanation [2,10] of the nature of the broad infrared maximum present in the spectra of the conductive TCNQ salts in the range from about 2000 to 5000 cm-t. An important feature of ref. [2] is that the authors calculate the conductivity a(o) in the case when the region of energies of the electronic excitations over the gap of the Frohlich CDW conductor overlaps with the energy of the CZN vibrational group of the TCNQ molecule. Especially significant is the rather good agreement of the shape of u(w) in TEA-TCNQ, in the vicinity of the C=- frequency of about 2200 cm-t with the theoretical curve. This part of the experimental u(w), or of the imaginary part .ez(w) of the dielectric constant is not discussed in much detail in ref. [2]. According to the opinion of the present author this part of the spectrum is a remarkable manifestation of the cooperation of two rather different absorption channels, one being the light absorption due to the creation of the phase phonons and the other due to the excitations of the electrons over the PeierlsFriihlich gap resulting in the “antiresonance” shape of u(w), ~~(0) or of the extinction coefficient of the materiat. An effect of this kind was first studied by Fano [ 1 l] . It is the purpose of the present paper to study the antiresonance shape of the infrared absorption in TCNQ salts and to find some links between the geometrical characteristics of the antiresonance

238

K. Knil/Antiresonance

line on the one hand and the significant physical parameters of the TCNQ salts on the other. Co11and Beni [ 121 studied the influence of the polaronic motion on the conductivity of TCNQ salts. Their theory, however, applies only to the case of the half-filled band. Similarly, as Hinkelman and Reik 191, they did not consider the effect of the electronic motion, unbound to the polaronic hole, on the conductivity and the effect of the coexistence of the electronic free and polaronic motions either. This effect, as studied in ref. [2] and further discussed below, is rather important in TCNQ salts.

absoqhon

in TCNQ iolfds

free motion of an electron, being not captured within a potential well created by the vibrational polarization of a molecule, is described by the wavevector

where k is the wavenumber determined k=%m(dV)-l

n= 1, . .. . N,

,

and a is the magnitude of the lattice vector along the chain. The energy of state (2) is E(k) = 2A cos(ka) .

2. The effective crystal hamiltonian In contrast to the theory of Rice [I], here it is assumed that the TCNQ chain contains only one unpaired electron. It appears that the anomalously strong infrared absorption in the range of the intramolecular vibrational energies, its polarization, the broad absorption maximum at 2000 cm-l to 5000 cm-l [IO] and the antiresonance shape observed in TEA-TCNQ? can be explained in the most simple way without taking more than one unpaired electrons into account. In the hamiltonian

by the’formula

(3)

Among the states the physical nature of which is that the electron is captured by the polarization well created on a molecule, the ground state is as follows

IO=.

(4)

In this state the electron is localized at the site i = q and no vibrational quantum is excited. Its energy is EC=-EB.

(5)

The states with the electron vlbrationally localized at site i and with the vibrational quantum of the branch (Yexcited at the jth molecule, are described by the wavevectors lU>~li,i,PL)=~~~‘i+~~lO);

i,j=

with u being an abbreviation

1, .__,N,

(6)

for the three indices i, j,

a. The energies of these states are simply E, = -EB + Ro,,,, _

in which the electron is allowed to interact with only intramolecular vibrations, but not with electronic excitations (see ref. ilO]), the first term describes the electronic charge transfer interactioninoving the electron along the molecular chain ofNmolecuIes, characterized by the parameter A _The second term is the hamiltonian of the s modes of the free intramolecular Fully symmetric vibrations. The last term of (1) is the electron-vibrational interaction operator. The energies of the eigenstates of(l) and the corresponding eigenfunctions approximately valid in the case when 2lA I = EB, EB E ZZ& (~&JZO~), which is probably the case of TCNQ salts, were discussed in ref. [IO]. Thus, according to ref. [IO], the

(7)

In eqs. (4) and (6) the particle operators c5 and 3: are related to the “bare” operators I$ and Q;& by means of the Lang-Firsov [ 131 transformation: ZZ?= exp(-is,)

CT exp(is,)

3: = exp(-is”)

Q&

, G-3)

exp(iSv) ,

where N

s

8) States (4) and (6) are analogical to the Friihhch CDW ground state and the states with the phase phonon

K. Knil/Antiresonance absorption in TCNQ solids

excited, respectively, in the theory of Rice [I]. The “free motion” state (2) is the analogy of the electronic excited states (without any vibrational quantum excited) above the Peierls-Frijhlich gap. Similarly, as in Rice’s theory, states (4) and (6) are multiphonon states. Confining ouselves to the space of states (2), (4) and (6) we take:

239

tors (8) the interaction operator IVof an unpaired electron in the ground state, with the incident fight is taken from ref. [IO] to be:

(14) P=

F

Ik>(kl+lO(GI+

c Iu)(uI=l. ”

(10) g = e(2rr9fiw

Then, using (10) we write: H=PHP=H

0 +H’iH”.

01)

Here we define

+

F

(12)

Ik)uclHlk)(kl,

and ff’=

$J c (IkwclHlu)(ul+ u

IuNJlHlk)(kl)

-

(13)

The operator Ho describes the motion of the electron-vibrational system in the unperturbed ground state and in the unperturbed states 1.Uand 1u). We identify (GIHIGI with EC given by the formula (5). Also, we take (ulHlu) and (klHlk) equal to E, and E(k), given by the formulas (7) and (3), respectively. The operator H’ contains the part of the interaction of the unperturbed states (2) and (6), which is the most important for the purpose of the present paper, namely the transformation of the states Ikl into Iv) and vice versa. In terms of Rice’s theory these trans-

Y-91’2

u(r) .

u is the electric field unit vector of the light beam with energy fiw per photon. e(r) is tE: dipole-moment operator matrix element of the eiectronic transition from one TCNQ molecule to the nearest neighbour

one, V is the volume of the system and ne is the number of photons on the oscillator with the wavenumber QThis interaction allows for transitions to both states Ik) and the vibrationally excited Iv>states [lo]. The optical transitions from IG> to the states Ik) and Iv) were considered in ref. [IO] with the interaction H’ completely neglected. This means that only the direct transitions from the ground state Ic) to these excited states were taken into account. When we take into account the effect of the operator H’ then we dis-

tinguish the following four lowest-order processes in the perturbation theory expansion with respect to H’: (a) direct optical transitions of an electron to the vibrtionally excited states I u), (b> transitions to the same states via the optical excitation of the virtual states IN, with the assistance of the interaction H’, (c) direct optical transitions

to the states I k>, (d) transitions to the states Ik) via the optical excita-

formations correspond to the decay of the Friihlich CDW state with one phase phonon excited, into the state with only one electron excited over the gap. The operator HN contains ali the other terms contained in Ph!P. They are assumed to cause only unimportant changes in the effective parameters of the hamiltonian Ho + H’ and are actually neglected here.

tion of the vibrationally excited localized states Iv), with the assistance of the interaction H’. Process (b) contributes obviously to the changes in the intensities of the relatively narrow absorption lines, of which the zeroth order (in H’) values of the absorption intensities are given by process (a). Although these intensity changes might be quite significant, we are not going to treat process (b) here. The attention

3. Optical absorption

will be concentrated on the continuous part of the absorption spectrum, represented here by both processes (c) and (d)Following velicf$ and Sak [ 141 we take the perturbed wavefunction of the electron with energy E(k)

In terms of the transformed

E and ii particle opera-

K.

240

Gi~fAntiresonanceabsorption in TCNQsolids the operator H’, which-transforms the two kinds of excitations mutualIy one into another. FolIowing ref. [IO], it is finally found that

(3), in the form of an outgoing wave !k, +> of an electron scattered by the interaction potential H’, Ik,+)=Ik)+[~(~)+ie--Hg]-lN’lk,+~,

RlHln, e=O+.

I, cr) = N-“2

W) Xx$,

[2/I cos(ka) - Es + fiq,,J

t ewikan eeFf2 .

(21)

Approximating the full wavevector Ik, + ) of (15) by only the first two terms of the perturbation expansion, we find:

Substituting the matrix elements (19), (20) and (2X) into formuIa (18) for M, we find that

~k,+)=Ik)+[E(k)+ic-_~]-~H’Ik).

MZ~N-~/*

06)

e-F/* E(k)/-1

Using formulas (12) and (13), the wavevector (16) takes the following form: lk,

+) = lk) +

nFaIn,

[E(k) - EB + F2wa,lx~ e-F

E(k)+EB--tiwou-ie I, 03 [E(k) + ie + EB - fiw~l-’

II

x 01,I, alH’lk) .

07)

The interesting quantity appears to be the transition matrix eIement M from the ground state I G) into the fina state lk, +) under the action of the operator 10.

X-(klfflrt, 1, ar)
(18)

The matrix element CRlwlG) was already calculated in ref. [lo]. The result was (il-l,v(G) = 2gV-‘/*

e-F&os(ka)

,

(19)

with s

F= ,T;, (Y()&&

.

Similarly the absoIute values of the matrix elements (n, 1, alwIG) were calculated in ref. [lo] . We get them here in full detail,

ma eeF,

x, = yoJ!iwocr .

(20)

We have now to calculate the matrix element of

.

cm

Thus the optical absorption diverges at the band energies E,=-EB+tiwoa, for each fully symmetrical vibrational mode IYcoupled to the unpaired electrons. In the present approximation the energies of the resonance absorption coincide with the unperturbed energies of the free vibrations Trwoa. The matrix element (22) is defined in the range of energies E(k) CZ(-2/A I, 214). There is naturally no resonance absorption for any vibrational energy i%uQ, which does not fall into the latter energy interval. Thus, in this interval the vibrational “narrow” absorption lines should not be symmetrical, according to formula (22). The absorption intensities due to the transitions to the unperturbed states (2) and (6) do not simply add one to the other. Instead, the transition matrix elements of these two absorption channels cooperate according to formula (IS). On one side of the resonance maximum the absorption is reduced to zero by the cooperation of the two transition matrix elements proportionai to the two main terms contained in the braces in formula (22). Experimentally, the positions of these zeros should be marked by dips in the absorption curve. From (22) it is seen that the energies of the dips are given by the solutions E of the equation e (E-EB+fio&xae 1+X (E+EB--fiq,J a=1

=

I

2-F =O.

(23)

The effect of the resonance term in (22), given by the second term in the braces, is the most significant in the

K. Knil]Anriresonance abso@on

vicinity of the resonance frequencies Woa. Let us suppose that one of these frequencies, Wan, is s~ificiently far from the other resonance frequencies. Then, near the frequency wo6 the dip equation approximately reduces to

In the case when x2 evF 4 1, the expression for the position Ed, on the6energy s&e, of the dip associated to the vibrational mode 6 is particular1 simple, namely, up to the first order of xz e- Eyit holds, that

Ed=(-EB

+Bwo6)+2(EB

-hq-,,)x~

e -F .

(25)

The relative position of the dip and spike of the antiresonance absorption line determines the sign of the difference EB - h~o&. It is seen from (25) that when the dip is located on the high energy side of the resonance peak, then AE Z Ed - E, = 2(EB - fiwos)xs Z e -%o,

(26)

and thus EB > trwoB. When the conditions for the validity of the decomposition of the full formula (23) into the independent expressions (24) are fuhilled, then the number of resonance peaks (spikes) and the number of dips are equal, unless it incidentaly happens that Es - fiw~~ = 0 for a certain (Y. Another parameter of the antiresonance absorption curve is its “halfwidth” l?. It is defined here by the formula r=maxlf+

-Eri,

for

$= 1,2.

(27)

E,,, are the two energies, at which the absorption coefficient of the crystal reaches one half of the magnitude of the absorption to the unperturbed states of the band of the free motion (3). Without going into details we present the approximate formula for i’ valid when 1 %3x2 evF and the theoretical absorption curve to the unpe&rbed states of the band (3) is sufficiently smooth and nearly constant in the area of the antiresonance. Namely, 2 -F I’= 31(EB - fiwo6)lx6 e . (28)

in

TCNQ solids

241

4. Discussion and comparison with experiment In terms of this work the shape of the imaginary part of the dielectric constant Ed [4] obtained from reflectivity measurements on TEA-TCNQ, cr@ds, or the shape of the conductivity u(w), are interpreted as the antiresonance absorption of light in this crystal. The dip in the curve of E? lies on she high energy side of the resonance spike, which is found at about 2100 cm:1_We conclude, that the deptbof the polaronic hole EB is larger than the energy of the free CZN symmetric vibration, therefore’Eg> 0.28 eV Lipari et al. [3] calculated the contribution of the symmetric intramolecular vibrationai modes to EB, using quantum-mechanical methods. They obtained only 0.15 eV for this part of EB_ The conciusion is made that a!so other polarization mechanisms contribute significantly to the effective magnitude of EgAmong them there should be the lattice vibrational modes and also the electronic polarization of molecules. Within the present model E, can be estimated in the following way. In TEA-TCNQZ, where apparent the low energy edge of the band (3) is nearly coincid width the cyano-group frequency WeN, it holds th: EB = tlW&N + 21AI -

(2

Taking IA I Q 0.1 eV [ 151 one obtains from (29) th Es = 0.48 eV. Using the value of 0.48 eV for EB and taking for XGNesF the value calculated by Lipari.et al. [3] equat to 0.04 e-1.4 , the difference of the energy of dip and the resonance spike associated with the cy: frequency comes out equal to 0.004 eV. This value not in contrast to the experimental shape ofez(w’ in which the spike-dip distance AI? is rougitLy eqc to 0.01 eV. The shape of Q(W) near the cyano-fre quency also qualitatively agrees with the present ’ theoretical result

r=lsaE. Summing up, it should be emphasized that thf observed effect of the antiresonance absorption, gether with the quite good agreement of the expl mental shape of the absorption curve with theor supports the picture of the energy spectrum of 1 salts as containing the coexisting hands of polan

motion of the unpaired electrons together with the phase-phonon bands and the band of the free electronic motion_ It appears also that although the interaction of the electron with the fully symmetric intramolecular vibrations localizes the electron significantly, the interaction of the electron with other crystalline vibrational and electronic degrees of freedom are equally important.

(31 N.O. Lip&

141 [5] 161 [7] [S] [9]

Acknowlddgement The author would Iike to thank Dr. KS. Rice fur sending a preprint of his work prior to pubfication.

References 12) M.J. Rice,

Phys. Rev. Letters 37 (1976) 36. [Z?] M.J. Rice, L. Pietronero and P. Briiesch, preprint.

[ 101 111) [ 121 I131 (141 (151

C.B. Duke, R. Bozio. A. Gitlanda, C. PeciIe and A. Padva, C&em. Phys. Letters 44 (1976) 237. A. &au, P. Briiesch, LP. Farges, IV. ?liinz snd D. Kuse, Phys. Stat. SoL62b (1974) 615. K. Kril, Czech, J. Phys. B26 (I976) 226. K. I(&, Czech. J. Phys. B26 (1976) 660. G.R. Anderson and J.P. Devli, 1. Phys. Chem. 79 (197.5) 1100. M.D. Rarplnnov, T.P. Panova and Y.C. Borodko, Phys, Stat. Sol. 1324(1967) K67. H. Hinkebnan and H-G. Reik, Solid. State Commun. 16 (1975) 567. K. Kr;il, Czech. J. Phys. B, to be published. U. Fano, Phys. Rev. 124 (1961) 1866. CF. COB and G. Beni, Solid State Commun. 1.5 (1974) 997. LG. Lang and 1-A. Firsov, Zh. Exper’m. i Teor. Fiz. 43 (1962) 1843. B. treIicIrf and J. Sak, Phys. Stat. SOL 16 (1966) 147. A.J. Beriinsky. J.F. Carulan and L. We&r, Solid State Commun. IS (1974) 795.