Theory of plasmon excitations in (SN)x

Solid State Communications,Vol. 23, pp. 243—246, 1977.

Pergamon Press.

Printed in Great Britain

THEORY OF PLASMON EXCITATIONS IN (SN)x*t J. Ruvalds Physics Department, University of Virginia, Charlottesville, VA 22901, U.S.A. F. Brosens and L.F. Lemmen4 Departement Natuurkunde, Umversiteit Antwerpen (U.I.A.), Universiteitsplein 1, B-2610 Wilrijk, Belgium and J.T. Devreese~ Leerstoel Toegepaste Wiskunde voor de Wetenschappen Universiteit Antwerpen (R.U.C.A.), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Received 26 March 1977 by A. Zawadowski) The anomalous anisotropy and dispersion of plasmons in polymeric sulfur nitride (SN)~are explainedwithin a model of an electron gas with an anisotropic effective mass ratio. Calculations in the random phase approximation yield very good agreement with the electron loss data. Exchange corrections for the model are shown to be small for systems of the (SN)~, type which exhibit reduced dimensionality. THE ELECTRONIC PROPERTIES of the polymer (SN),, have been the subject of intensive study in recent

response properties of an anisotropic electron gas. Introducing the effective mass m11 parallel and the

years. In particular, the discovery of superconductivity of (SN)~[1] stimulated theoretical interest since this polymer was considered to be a quasi one-dimensional conductor on the basisband of the highly anisotropic ductivity [2]. Several structure calculationscon[3—5] were performed using one-dimensional chain structures to represent this system. Electron loss measurements [6] revealed some

effective mass m1 perpendicular to the strand axis, the kinetic energy for a charge carrier with momentum p is given by: 2/2’n,~ (1) E(p) = (p~+ p~)/2m1+ P~ The momentum distribution function f(p) for the carriers m the ground state is equal to 2/(2~th)3when E&) is less than or equal to EF. For E(p) > E~the distribution function f(p) equals zero. The masses m 1, mj.(m~)”2 3ir2h~n. m~,the312 density n and=the Fermi energy are related by (2E~) The polarization of the cores can be taken into

anomalous features of the plasmon anisotropy and dispersion in (SN),, which be reconciled with the expected behavior for acannot one dimensional conductor. For example, the plasmon frequency should vanish at a direction of propagation perpendicular to the strand axis [7], whereas the data on (SN),, show a substantial value w~(0 = 90°)= 1.5 eV. This indicates a striking departure from the relation ~pI = cos 0 expected for a one dimensional electron gas [7J in the random phase approximation.

_.

account by considering that the electron is imbedded in a dielectric medium characterized by a gas dielectric constant e.. [8] . The Fourier transform of the effective Coulomb potential is then given by v(q) = 4ire2 /(e..q2). The dielectric function for the system is given by e(q, ~

—

e

—

Q 0(q, ~

In the present letter, we calculate the dielectric *

Supported in part by the National Science Foundation Grant no. DMR 72-02972-A02.

t Work partially performed in the framework of the project E.S.I.S. (Electronic Structure in Solids) of the University of Antwerpen and liege. ~ Aangesteld navorser of the N.F.W.O. §

Also at U.I.A.

(2)

1 + ~ — G(q, w)Qo(q, ci,)

where Q0(q, ci.,) is the anisotropic equivalent of the Lindhard polarizability and G(q, ~ is a measure for the deviation from the Hartree approximation (RPA) for e(q, a.,) [9J. The anisotropy in the masses enters in the Lindhard expression through the energy difference (p~q~ + p~q~)/m~ + p~q~/m11. A scale transformation reduces the expression for the Lindhard polarizability to its well known form, 112, provided q~(m~)~”2, a vector q~(mwith112 and components q~(m1) 11~

243

244

THEORY OF PLASMON EXCITATIONS IN (SN)~ ~

Vol. 23, No.4

0.3

-0.01

0.1

I.

10

100 R

Fig. 1. The exchange contribution to the long wave length plasmon dispersion of equation (8). The value of I(m1, m11, 0

=

0°)is shown for different mass ratios R

1.

-.~

=

mJm11 in units 1/rn11. 4.

.

-

.8

—

0:0

——

0:50

-

-

3.

-

-

I

.6

/

-

cos 0

-

/

/

/~,//~

~ 2.-

/

•2

0

~ ///,,,////

‘¼

-

L

1.

30°

‘

‘:~

60°

-

~‘..

0

q (A

(3a) =

(a2

—

b2) in

b—a

.

/

1)

.6

Fig. 3. Dispersion of plasmons for selected angles of propagation as a function of momentum q. The theoretical curves are calculated using the parameters obtained from analysis of the anisotropy and are denoted by a solid line for 0 = 0°and a dashed curve for 0 = 50°.The open circles are the experimental points of reference [6]. For the imaginary part of Qo(q, c~)one obtains Im Q

3s)1[F(sE,~,—h~2/2)—F(s~,c~+ h~2/2)]}

F(a b)

.

6ire2n(q2e..hEp)’

x (1 +(2h where

/

/CONTINUUM ../

-/

.2

-~

=

//

__________________

Fig. 2. Anisotropy of the plasmon spectrum in the long wave length limit (q 0). Data for (SN)~are given by points which show the deviation from the cos 0 dependence expected for a one dimensional electron gas and denoted by the dashed line: the angle of propagation 0 is measured relative to the polymer-chain axis. Results of the present theory are indicated by the solid curve using a mass ratioR = mum11 = 1.9. 2 are introduced. The real part theQo(q, quantity (2EF)” of ~ iss =then given by Re Qo(q, ~)

/~/

(3b)

0(q, ~)

=

6ir2e2n(q2eJi2Eps~3)1

(4)

[(~2 s2 (c., + II~2/2)2 )~(~ + (~2s2 (~ n~ /2)2 )8(~ x

—

—

—

—

~ + !L~2/21) —

~ /2)].

Vol. 23, No.4

THEORY OF PLASMON EXCITATIONS IN (SN)~

The function 8(x) is equal to one for x positive and equal to zero for x negative. The G(q, w) factor for the anisotropic electron gas isgivenby [9] G(q, w) = v(q) 2ire2h2 e’ Q~2(q, w)A(q, ) (5) where A(q, w) is a sixtuple integral over products of displaced Fermi functions. The scale transformation used for the Lindhard expression does not reduce G(q, w) to isotropic form. In the smallq limit it turns out that G(q, w) can be written as follows: 4 7T2 h2n2 ~I(m 2w4e2 9e 1, 7~ij1,8) (6) G(q, w) — 2E~Qo(q,w)q —

—

where 8 is the angle between the q-vector and the strand axis. We have depicted the numerical value of I(m~,m 11, 8 = 0) for different values of the mass ratios R = mJm11 in Fig. 1. The maximal exchange correction to the plasmon dispersion along the z-axis occurs for an R value slightly less than one. In the case of extreme anisotropy the numerical value of I approaches zero. In order to estimate the importance of the exchange correction for the long wave length plasmon dispersion, it is sufficient to know the parameters e.., mj, mu and ~. For the carrier density of (SN)~one takes one ii electron per molecule; this yields a density ~ = 3 ~e.10~ [6, 10]. The electron dielectric constant is taken to be 6.5 from optical data. [8]. The measured value of the plasmon frequency w~ 1(0= 0°)= 2.48 eV gives a mass m11 = 1.03 times the free electron mass. This value of m11 is consistent with optical and reflectivity data provid. ing the core polarization is properly taken into account. Consider the long wave length expansion of the dielectric function: 2B(8) 6EB(8) 2 4irne e(q, ‘~‘ = ~ w2 + 5w2 q [1 _L(8)]} (7) ~

I

where B(0) = ~2/q2 = sin2 O/m~+ cos2 8/rn 11 andm the 2h2I(m~, exchange contribution is L(0) = l5nrre 11, 0)1 (l6E~.e,.).From this expression the following plasmon spectrum is obtained:

=

3E~e..q ~ k/, ~ + 4irne2 / 4irne2 B(8)

245

~,2

2

[1

—

L(8)]).

(8) The best fit of the long wave length data [6] to w,~= [4irne2B(0)e.:’] 1/2 is obtained for a mass ratio R = mJm 11 = 1.9, and is shown in Fig. 2. The remarkable agreement between the data and the theoretical curve for R = 1.9 casts doubt on the validity of the quasi one dimensional interpretations of (SN),,. Turning to the exchange contribution, we note that there are no latingL(O),adjustable it is found to be oforder for (SN),, remaining parameters in the0.01 model. Calcu-and therefore negligible in the low q regime. The plasmon dispersion is calculated using the full RPA dielectric function, but neglecting the exchange .

contribution G(q, w). it is gratifying to obtain good agreement with the electron loss data using the same value of n~ R, e.. and m11 as shown in Fig. 3 for the angles of propagation 0 = 0°and for 0 = 50°.The apparent decrease in the plasmon energy at large momentum (q ~ 0.4 A-’ ) may be attributed to the nearly degenerate continuum resonance peak structure in the energy loss function Im e’ (q, w’). Nevertheless, the persistence of a plasmon.like structure in the challenge, electron loss data till 1 poses a theoretical especially q = 0.55 A- behaviour has been reported in other since a similar metals. —

Although our model is qualitatively consistent with the reported three dimensional band structures [11, 121 the variations in the published structures near the Fermi energy. are substantial. A detailed investigation of the band structure in this region is clearly warranted and may be relevant to future studies of the superconducting properties of(SN)~.

Acknowledgement it is a pleasure for one of the authors (J.R.) to acknowledge the warm hospitality of the Universiteit Antwerpen during the summer of 1976. —

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THEORY OF PLASMON EXCITATIONS IN (SN)~

Vol. 23, No.4

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