Surface plasmons in graphite intercalation compounds

Solid State Communications, Vol. 63, No. 3, pp. 241-244, 1987. Printed in Great Britain.

0038-1098/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.

SURFACE PLASMONS IN G R A P H I T E INTERCALATION COMPOUNDS P. Hawrylak Brown University, Providence, RI 02912, USA

(Received 14 November 1986 by J. Tauc) Collective charge density excitations in a semi-infinite Graphite Intercalation Compounds are studied. The general theory is formulated and applied to intraband surface plasmons in stage 1 compound and interband surface plasmons in stage 2 compound. It is found that the surface plasmons can exist for the wavevector q larger than a critical value q*, in analogy with semiconductor superlattices. Analytical expressions are derived for plasmon frequency and decay length. THE S E M I C O N D U C T O R SUPERLATTICE con- and In, k, i) are the tight binding wavefunctions on sists of quantum wells filled with free carriers. If the layer i. The layer functions (i, l(z) now play the role barriers are sufficiently wide, electronic motion along similar to the subband electronic wavefunctions in the superlattice direction is forbidden while the mo- quantum wells, but reflect the discrete nature of every tion in the perpendicular direction is allowed. The stage unit. Following [2] one can expand induced charge density fluctuations in different quantum wells charge and induced potential of the form V(r, z) = couple via Coulomb interaction so that density per- V(q, z)e~'r~e-i'°' (q is the in-plane wave-vector, co is the turbation can move along the superlattice axis even frequency) in terms of layer functions ~i, t(z) and derive though electrons do not tunnel between quantum the selfconsistent equation for the induced potential wells. The linear response of this system to a scalar matrix elements ~ ( l ) = Sdz~i.t(z)V(q, z)~j,t(z) in a perturbation has been studied recently, yielding bulk semi-infinite GIC (Einstein summation over repeated [1] and surface collective modes [2] and their coupling indices is assumed): to experimental probes [3]. One expects similar results ~ ( l ) = VqGo.,iT(l, 1')TZcj, ,rs(q, co)V~s(l'). (1) for Graphite Intercalation Compounds (GIC) which are another example of a superlattice [4]. The superlat- Here gv, cj,(q, co) is the frequency and momentum tice is formed by inserting intercalant layers between dependent layer polarizability tensor defined as every N carbon layers (N is a stage number). The hopping of electrons across intercalant layers is negli- nU,iT'(q, co) = A 2 .., ~ ~ (k + q, n, ileiq'lk, n,j) gible and the motion in the basal plane is governed by a tight binding wave functions in a given stage unit. j,)f(E~+q,.,) - f(Ek,.) (k, n, i'[e-~qr[k + q, n', Ek+q,., -- EL,. -- co' These wavefunctions are usually constructed out of ~b(rlr, z) i.e. P~ orbitals localized on carbon atom sites (2) [5]. The band structure can be modeled by a tight binding Hamiltonian (including Hartree contribuwhere A is the area of the sample, f is the Fermi tions) which depends on few phenomenological parfunction and E.. k are band energies (there are 2N ameters irrespective of the choice of the basis bands in every stage unit due to two nonequivalent wavefunction. Because the direction along the supercarbon atoms on every carbon layer). The Go, ~, (/, 1') lattice axis (z axis) and in the basal plane rll are so is the electron-electron interaction tensor, dissimilar it is convenient to work instead with wavefunctions which are separable in the z and rpl G,,.,.,.. (t,r) = f dz f coordinates i.e. ~b(rtl, z) = ~(z)~b(rll). This allows us to [~qlz-z'l _[_ cte-q(z+z')]~rr(2,)~j,r(Z,). (3) expand the wavefunction in terms of the carbon layer functions ~i. l(z). Vq = 2zce2/eq, e = (~ - e0)/(~ + eo), ~ is the background dielectric constant due to a bands and e0 is the II, n, k) = Z~=,~,,(z)ln, k, i), vacuum dielectric constant. The second term in equawhere ~g,~(z) is the i-th layer function in the/-th stage tion (3) corresponds to electrons interacting with unit, k is the inplane wavevector, n is the band index 241

242

image charges due to the discontinuity of the background dielectric constant across the GIC-vacuum interface. It is possible to simplify equation (1) by noting that V~j = Vji and G¢,.s(l, l') = Gji,~.~(l, l') = Gii' ~(l, f ) so that there are N ( N + 1)/2 independent potential matrix elements. It is best to focus on stage 2 i.e. N = 2. We have 3 independent matrix elements. One can assign new indices to each pair of indices; e.g. let us have (1, 1) = 1, (2, 2) = 2, (2, 1) = (1, 2) = 3. Using this notation we define a symmetrized polarizability tensor ~ti.j(i,j = 1, 2, 3) e.g. ~1, 3 = ~'CII.12 + 7/711.21, ~1.1 = ~11,11 and 7~3, 3 = 7['12,12 + ~21,12 "~ //~12,21" The similar procedure can be applied to an arbitrary stage. This allows us to write equation (1) in more compact form:

Vii(l) =

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GRAPHITE INTERCALATION COMPOUNDS

~, vqGi.j(l, l')?tj, ,(q, (o)V~(l'). l,, jr

(4)

Nontrivial solutions of equation (4) give collective modes of the semiinfinite GIC. Note that in calculating the polarizability tensor, equation (2), we have neglected local field effects. These effects become important when the induced potential varies appreciably over the in-plane distance comparable with the nearest neighbor carbon atom separation b. But since the repeat distance of the superlattice a is much larger than the nearest carbon-carbon distance b and coulomb matrix elements decay exponentially as exp ( - qa) we can neglect local field effects as long as qa is not too large compared to unity. Collective modes (plasmons) of the GIC can qualitatively be described in the same way as plasmons in semiconductor superlattices. The surface plasmons correspond to an induced potential decaying away from the surface at z = 0 with inverse decay length/~, while bulk plasmons can be characterized by Bloch wave-vector k~ and correspond to induced potential on subsequent stage units differing by a phase factor e zk~ (a is the repeat distance of the superlattice). Much the same way as other excitations in periodic systems (electrons, phonons, etc.) bulk plasmons form bands and plasmon frequency ~o depends on two wavevectors, the in-plane wave vector q and the wavevector k~ along the superlattice axis. One can also classify collective bulk and surface excitations into interband and intraband plasmons. The intraband plasmons correspond to the charge fluctuation along the basal plane while interband plasmons correspond to charge fluctuations between carbon layers in a given stage unit. The method of solving equation (4) has been given in detail in [2] for the case of semiconductor superlattice, so we omit it here for the sake of brevity. As an illustration I consider here two simple examples: bulk and surface intraband plasmons in stage 1 and interband bulk and surface plasmons in stage 2 compound.

The band structure of Blinowski [5] is used, and calculations are done in the long wavelength limit. For stage 1 the dispersion of the surface plasmon can be obtianed by solving equation (4) i.e.

l = Vq~(q, co)S(q, [3), with e l~a

=

( e -q'~ - - o~eqa)/(1

q-

~),

(5)

and

S(q, fl) =

sinh (qa) cosh (qa) - cosh (fla)"

The single layer polarizability ~(q, co) has both the intraband (valence --+ valence) contribution ~a and interband (valence ~ conduction) contribution ~i. One can write equation (5) so as to explicitly show both contributions:

~c"(q, ~ ) S ( q , t6) 1

=

vq

~e~(q,(o, fl)

'

(6)

Eeff = 1 - Vq~i(q, ~o)S(q, fi) is the interband frequency and wave-vector dependent screening constant which has to be evaluated at the surface plasmon energy and wave-vector. The solution to equation (6) is valid only when the imaginary parts of ~a and ~i are zero i.e. outside single particle intraband and interband excitation spectrum. The bulk plasmon frequency og(q, kz) can be obtained by the replacement fi --+ - i k z in equation (6) (0 < kz < ~/a). Since the plasmon frequency (o(q --+ 0, kz = 0) has to equal to the bulk 3 dimensional screened plasma frequency ~op as measured in experiment, the surface plasmon frequency can be approximately written as: here

2 qa S(q, fl) + 3/4~2q. ~o2(q) = ogp-~

(7)

where eq = (3/2)7obq and ~0 is the inplane resonance integral. The dispersion of bulk and surface plasmons as given by equation (7) is shown in Fig. l(a). The band of bulk plasmons is indicated by a shaded region. Note that co(q --+ 0, kz = 0) = % and a~(q ~ 0, kz = re~a) = 0. The surface plasmon has frequency above the bulk band and can exist for wave-vectors q larger than a critical wave-vector q* = (l/a) In Ie + e0/e - e0I. For a stage 1 graphite - SbC15 we have [6] a = 9.24A ~,alsoe = 2.4 ande0 = l, givingq* = 0.094 A 1. Note that the effect of interband scattering is to renormalize the surface plasmon frequency but not the decay length fl-1. This can be seen from the screened plasma frequency which contains the effective dielectric constant ee~ = 6.0 while the surface plasmon decay length is determined by the background dielectric constant E = 2.4.

G R A P H I T E I N T E R C A L A T I O N COMPOUNDS

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STAGE I INTRABANDPLASMONS

a)

L5 SURFACE P L A S M O ~

OJp I.O

0.5

~

-- BAND

0.5 0 . 6 6 7 ~

w[eV] ~

1.0

1.5

qa

ERBAND PLkAS:O:S

~

~

0.5

20 b)

~

1.0

1.5

qa

2.0

Fig. l(a). The dispersion of bulk and surface intraband plasmons in stage 1 compound. The shaded area denotes bulk plasmon frequencies co(q, kz), the solid line denotes a surface plasmon which intersects the bulk band at a critical wave-vector q*a. Here cop is the 3 dimensional plasma frequency and a is the repeat distance. Fig. l(b). The dispersion of bulk and surface intraband plasmons in stage 2 acceptor compound graphite SbC15 [6]. The shaded area denotes bulk plasmons and the solid line denotes the surface interband plasmon. Note the shift of the interband plasmon frequency with respect to the single particle excitation energy ~h. In stage 1 compound charge density fluctuations take place along carbon layers. In stage 2 compound change density can fluctuate both along carbon planes and between two carbon layers in the direction perpendicular to carbon layers. The first type of fluctuation creates electric field parallel to the carbon layers and involves predominantly electron intraband transition while the latter one results in electric field perpendicular to carbon layers and involves interband electronic transitions. Hence the coupling between intraband and interband plasmon modes is very weak. In fact an anticrossing behavior of intraband and interband plasmons has been observed experimentally in 2D-systems [7] and in superlattices [8]. An explicit, detailed calculation of plasmons in semiconductor superlattices [9] verified that reasonable results are reached when interband and intraband plasmons are treated independently. This allows us to derive simple analytical formulas which should be a good starting

243

point for interpretation of experimental results and comparison with full solution of equation (4). Let us consider the polarizability tensor for stage 2 acceptor compound. Let Yl be the valence band splitting energy and let us take the Fermi energy E~ > Yl i.e. both valence bands are partially depopulated. For the frequency co ,-, Yl and qa ,~ 1 the largest contribution to layer polarizability tensor ~,j comes from valence --* valence bands transition. In fact, in this limit intraband contribution goes exactly to zero while the valence to conduction band contribution merely renormalizes the background dielectric constant which is included in Vq but not in ~. A good choice for the renormalized dielectric constant would be the value derived from the screened plasma frequency for stage 2. Hence we only retain the ~1, l( 0, O)) and ~2, 2(0, co) polarizability elements in equation (4) which are calculated using wavefunctions and energies of [5]. Moreover, in calculating coulomb matrix elements Gi, j(l, l ' ) we approximate the actual charge density ~ t(z) by a delta function localized on a layer i in the l stage unit. Following [2], we obtain analytic expression for the surface plasmon dispersion: 0ico)2 = 7~ + 2 v q B F ( q , fl) F(q, #)

=

sinh (qa) - sinh (qa - qc) - cosh (fla) sinh (qc) cosh (qa) - cosh (fla) e~,, =

(e q" + o~eq") - (e q(c-a) + o~eq(a-c)) (1 -- e qc) + 0~(1 - e -qc)

B = ~

471Ep- 7~lnk2E:--~

"

Here c is the carbon layer separation. The bulk plasmons are obtained by the substitution B ~ - i k z in equation (8). The dispersion of bulk and surface interband plasmons is shown in Fig. l(b) for parameters [6] corresponding to stage 2 graphite -SbC15. (E F = 0.79eV, a = 12.42•, 71 = 0.375, ee~ = 5.9, cop = 1.07eV). Note that the bulk plasmon frequency is shifted from the single particle excitation energy 71 = 0.375 eV. This shift, known as the depolarization shift in semiconductor superlattices, depends on the kz wave-vector and the Fermi energy. Also note the typical softening of plasmon frequency with increasing in-plane momentum q. The surface plasmon frequency lies above the bulk plasmon band. Surface plasmon intersects the bulk band at a finite critical wave-vector q*. It involves charge density fluctuations between carbon layers, which are out of phase on subsequent stage units and decay away from the surface.

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GRAPHITE INTERCALATION COMPOUNDS

At this level of approximation intraband plasmons would be given by equation (7) with a corresponding screened plasmon frequency cop = 1.07eV. Intraband and interband bulk plasmons can couple when their wave-vector (q, kz) and frequency co are identical. For cases of interest (COp >> ~1) plasmons with kz = 0 never couple, and equation (8) is exact. For finite kz there is an anticrossing of bulk intraband and interband plasmons within interband plasmon band. For kz = 7r/a we estimate the crossing to take place at qa ~ 1.2. Hence the surface interband mode is well defined as the critical wave-vector q*a ~ 1.3. In other cases a weak damping of a surface mode by interband plasmons is expected. The damping of plasmons due to single particle excitations has not been considered here. In general surface plasmons in superlattices are long lived excitations because both surface plasmons and electron-hole pairs have momentum parallel to the layers so there is no Landau damping. Damping due to valence ~ conduction band transitions is possible. This problem will be investigated in the future. In summary we have given a systematic approach to collective modes in Graphite Intercalation Compounds. The dispersion relation for bulk and surface modes are analogous to plasmons in semiconductor superlattices. They can be studied by Raman and Electron Energy Loss spectroscopy but not the Optical Reflectance at normal incidence, as they can exist only for finite wave-vectors. The basic assumption in this work pertains to the abrupt and good quality vacuum-GIC interface. We

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hope that experiments will verify this assumption. In this case theory developed here can be extended and compared with experiment.

Acknowledgements--I thank Professor J.J. Quinn for helpful discussion and critical reading of the manuscript and acknowledge the financial support of the U.S. Army Research Office, Durham, N.C. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9.

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