- Email: [email protected]
Electron-energy-loss spectroscopy of the C60 molecule
a+ __
25 November
-_ i_ B!!l
1996
PHYSICS
LETTERS
A
Physics Letters A 223 ( 1996) I 16- 122
ELSEiVIER
Electron-energy-loss
spectroscopy of the C60 molecule
D.A. Gorokhov, R.A. Suris*, V.V. Cheianov lofe
Physical
Technical
Institute.
Received
Politehnicheskaya
Street 26, 194021 Saint Perershurg.
15 August 1996: accepted for publication Communicated
15 September
Russiun Federation
1996
by V.M. Agranovich
Abstract In this work we present a theoretical study of EELS (electron-energy-loss spectroscopy) experiments on the CM, molecule. Our treatment of the problem is based on the simple two-fluid model originally proposed for the description of plasma
oscillations in graphite and fullerenes [J. Cazaux, Solid State Commun. 8 ( 1970) 545; Opt. Commun. 2 ( 1970) 173; G. Barton and C. Eberlein, J. Chem. Phys. 95 ( 1991) 15121. It is shown that in spite of the simplicity of the model the calculated intensities of the EELS peaks are in good agreement with experimental data which may indicate that the model can be used as a simple and effective tool for the investigation of the collective behaviour of electrons in fullerene systems. PACS: 31 .SO.+w; 61.46.S~: 78.90.+t Keywwds: Fullerenes; Electron-energy-loss
spectroscopy;
Plasma oscillations
1. Introduction Investigations of plasmons in small clusters in the nanometer size range have a long history. Originally plasma oscillations were studied in small (R = 2-10 nm) metal particles. It turned out that plasmons in metal clusters can be well described by linearized hydrodynamic equations [l-3]. The positions of plasmon peaks, their number and fast electron crosssections calculated within these equations are in good agreement with experimental data. New possibilities of the investigation of plasmons in small objects appeared after the discovery of fullerenes and the cheap methods of their synthesis. Plasmon oscillations were observed in fullerene molecules [ 461, carbon nanometer size tubes [ 71 and multishell fullerenes [ 81.
* Corresponding
author. E-muil: [email protected].
0375-9601/96/.$12.00 Copyright SO375-960 1(96)00707-4
PII
An important feature of fullerenes is the fact that these molecules are more or less of a spherical form and their electrons can be considered to be confined to the surface of the sphere [ 91. Hence, it seems to be a good starting point to consider the Che molecule as a rolled up graphite plane. This approximation is expected to work well for the description of collective excitations. The graphite plane consists of regular hexagons formed by carbon atoms. Each carbon atom is connected by 3 u-bonds with its three neighbours, meanwhile 7r-electrons form a common z--system consisting of orbitals perpendicular to the plane. In linear response theory r- and a-electron subsystems can be considered to be independent with respect to the electrical field parallel to the graphite plane because dipole transitions from rr- to c-orbitals are prohibited. This means that we can describe the plasmons in graphite whose wave vector is parallel to the plane as the plas-
0 1996 Published
by Elsevier Science B.V. All rights reserved.
D.A. Gorokhou et d/Physics
mans in a “two-liquid” electron system; it is proposed that both the liquids move in a common average potential. The graphite loss function - Im[ 1/EL(W)] with respect to the external electric field perpendicular to the c-axis shows two peaks at 7 and 28 eV (see Refs. [ 1O- 13 ] ) . They are called rr- and (T- (or “r + u”-) plasmons, Strictly speaking such a division is rather conventional. In fact, both 7~- and a-electrons make a nonzero contribution to (T- and r-plasmons respectively. In Refs. [ 14,151 Cazaux used the two-fluid model in order to explain the results of EELS studies on graphite. The same idea was used in Ref. [ 161 for the description of plasmons in the Ccc molecule: it was proposed that r-electrons be affected only by induced electric potential, and g-electrons be governed also by a restoring force. This force appears in the model via an effective description of the contribution of valence electrons to the polarizability [ 17-191. Using the two-fluid model Barton and Eberlein predicted the existence of plasma oscillations in the C60 molecule which can be classified by the spherical quantum numbers 1 and m and the index j (j = 1,2) which corresponds to rr- and c-plasmons [ 161. However, the plasmons with large 1 cannot exist because of their strong damping. EELS studies on Cbn (see Refs. [ 451) show a peak at 6.5-7 eV (also called “7r-plasmon”) and a rather broad “cr-plasmon” peak at 15-16 eV. The former peak is identified usually as the I = I r-plasmon, while the latter peak is considered to be a combination of the 1 = 1 a-plasmon and probably the plasmons with 1 > 1. However, the existence of multipole plasmons in C60 is still not clear. In Ref. [6] a small peak at 28 eV was observed and identified as the I = 1 cT-plasmon. The main goal of this work is to calculate the crosssection of fast electrons scattered by a C60 molecule and to show that the contribution of the plasmons with I= 2 to the cross-section is sufficiently large to be observed in EELS experiments. In the model we use it is possible to separate contributions of different plasmons, i.e. one can obtain relative cross-sections. Using this approach we cannot calculate plasmon damping. Widths of plasmon peaks will be taken from experimental data.
117
Letters A 223 (1996) 116-122
2. Quantization of plasma oscillations in the two-fluid model
on a sphere
We shall consider electrons in the Cen molecule to be distributed over the surface of a uniformly charged sphere - the positive molecular core in the model. The Fourier transformation of the charge density &l(w) induced by the potential @ including both the external and induced potentials is given by the expression
(1) where n is the surface rr- and g-electron density, fa and w, are respectively the oscillator strength and electron energy levels. The index a enumerates the electron states. All denotes the angular part of the Laplace operator: in the model we use electrons which cannot move perpendicularly to the surface. As the r-electron subsystem of graphite is semimetal, we can use the following approximation for the n-electron contribution to the induced charge density, &l,(W)
(2)
= ___
We suggested w, in Eq. (2) to be equal to zero and used the sum rule c, far = l/4 for the oscillator strength of r-electrons in COO.N = 60 and R = 0.4 nm are the number of r-electrons and the effective radius of the sphere respectively. On the other hand, the g-electron subsystem of graphite is dielectric with the gap of order 12 eV (see Ref. [ 201). The oscillator strength of the m-electron subsystem has a maximum at o, N 16 eV. Hence we can use the following approximate expression for the cr-electron contribution, &l,(W)
3e2N = 4rmR2
1 co2 - co,=
QWJ).
Here we used the sum rule C, fan= 314 for the uelectron subsystem. w, = 16 eV is the characteristic transition frequency of a-eiectrons. ls2-electrons of carbon atoms are not taken into account in the model. The Q-field is connected with the total induced charge density Su = Sn,, + Sn, through the Gauss theorem,
Z/,-,,, - :I,;,_, =4da.
(4)
D.A. Gorokhov rr ol./Physic.t
118
Lerrers A 223 (19961 116-122
It is convenient to expand the density fields in Fourier series on the sphere,
The coefficients in Eq. ( 10) are chosen so as to keep it equal to the hydrodynamic Hamiltonian
an, = C %zYr,,(4,@ In1
H = ;rnq -
(T=
9
an, = C bJh,(~, in1
c QJi”l(4.~>. In1
8, (5)
Then, using the multipole
expansion
of the potential
0, for r < R,
we can rewrite (4) in the following 4rR &?I! = - ---Sim 2z+ 1
= -$$(airn
form, + br,,,),
(6)
which gives for (2) and (3) (see also Refs. [ 16,2 1] ) , &n, = -@hi, + W,), . b/, = -3f22(a~,, + b/m) - ~,%,,
J
k:dS + ;rnuo
+ +maow,* /rjdi+
-/ ;
+ imcro( 3f$* + w,,~) B* + 3ma&*AB,
(8)
JrnaoA
(9)
It can be easily verified by straightforward calculation that equations (7) and (8) are the equations of motion corresponding to the Hamiltonian
= Qlcosq$
- Q2sin&,
2n,* + WC2 2X& 012
The eigenfrequencies equation,
+ AI,-~BI,,),
icro-AI,,
brn, = igohB,,,,
R*
’
transformation, (14) (15)
(16)
01,j are given by the following
(10)
where the amplitudes Al,,, and Bt, are the (I,m)components of the displacement fields rt and r2 of the electron liquids which are connected with al,,, and bl, by the following equations, arm =
(13)
where the angle 41 can be found from the condition of the disappearance of the nondiagonal term in Hamiltonian ( 13) expressed in terms of the new coordinates Qt and QT. The substitution of (14) and (15) into Eq. ( 13) gives cot(2C#J1) = -
+ imao c ~Q*(AI,BI,-, Ill!
( 12)
~rncqpi* + $mao~* + ~mqfI~2A2
which can be done by the following
N go==.
$&dS.
Hamiltonian (10) is nondiagonal in the basis consisting of the functions &,( 4,O) because of the last term. On the other hand, this term does not mix amplitudes with different 1 and (ml because of the spherical symmetry of the model, hence the above Hamiltonian can be diagonalized by the diagonalization of each “block” with a number lm. We have shown that the problem can be reduced to the diagonalization of the Hamiltonian
v’%&B=Qtsin&+Qzcos+~. 1)
dS
(7)
where n2 = 47re*go Z(l+ 1 --3T-r’ mR
Jf;
(11)
One can clearly see that charge oscillations can be classified by the numbers 1 and m and the index j corresponding to .the lower and upper roots of Eq. ( 17). The fact that the eigenfrequencies do not depend on the number m is due to the spherical symmetry of the considered model. Using ( 11) we can find quantum mechanical operators for amplitudes Al, and Bl,. Substituting them
DA. Gorokhov
into the equation ations we obtain quantity,
et d/Physics
Letters A 223 (1996) 116-122
119
for classical charge density Auctuthe operator corresponding to this
J~~Jl(i+l)y,,,(9,8)
=(dJ,e) = -jg
lm
Sl(b)
(dz
x
,. (&?Ll +
(22)
s:_,,,, )
the angle 41 is chosen to be the minimal positive root of Eq. (16).
We shall consider only the case of fast scattering electrons, i.e. the only relevant processes are those in connection with the creation of one plasmon. Hence we can use the Born approximation. The quantitative criterium of the applicability of the Born approximation will be discussed further. In the initial state all the plasmon occupation numbers are equal to zero. Let us suppose that after scattering only the occupation number of the state Im, j is equal to one. Defining the initial and final momenta of the scattering electron k and k’ respectively we obtain the following expression for the modulus squared of the matrix element of fii”i,
3. Fast electron scattering
I(k;01iCiintlk’;Im,j)12
+g2(41)(& 7+$+_ pq
“‘.-
)
I, nr,2
>
(18)
’
1
where &,,,J is the annihilation
operator of the plasmon with numbers Im, j. and the functions gl (41) and g? (r#+) are given by the equations Sl(4i)
= hCOSq&
g2(41)
=COsh
+
sin#q,
(19)
V%in&,
In this section we will obtain the expression for the scattering amplitude of fast electrons due to charge density fluctuations in c60. In the secondary quantization representation the electron-plasmon interaction can be written as follows,
I?,“,=
J
(20)
@+(-e&)!kdr.
is the electron field operHere @ = xk e ik’r&/& ator, & is the potential corresponding to the electron surface density distribution S&( 4, #), i.e. 6 is the solution of the Poisson equation with the charge density S(r - R) 6&(4,6). Hence & and 65 are connected by the following equation, 6( r’ - R) i%((b’, 8’)
&,= J
Ir-r’l
dr,
.
(21)
Substituting Eq. (2 1) and the explicit expressions for the electron field and the electron density fluctuation operators (see ( 18)) into Eq. (20), we obtain the expression for the electron-plasmon interaction
x gj2M)
= “~~~v~ L (a)
jam
y
Y,;,(i),
(23)
where q = k’ - k. As each eigenfrequency fil,j is (21f 1 )-fold degenerated, we shall calculate the total cross-section for the transferred energy fi0j.j. Summing up the contributions of plasmons with given 1 and j over m, we get
I(kOIAintlk’;ltn, j)12)
dgl j
A = J(c dii?kr x
‘%@k’
“t
+ fh
-
@k)
27rV k’= dk’V -zi2u c2Tj3
In the above equation Ok = fik*/2m, velocity of the electron. Using the equation
.
(24)
u is the initial
(25) (see Ref. [ 221)) we get after some algebra
DA. Gorokhou
et d/Physics
Letters
A 223 (1996)
116-122
Table I Frequencies and intensities of plasmon peaks calculated formulae
(26)
4. Results
and discussion
In experiments one can measure the number of electrons reaching a detector, i.e. one measures the function (26) integrated over a certain solid angle. Let this solid angle be restricted by two cones of angles 01 and 02 respectively. In this case we can write for the total cross-section C[,j(
ar.,(h
02) = -l(l+ fiQ,jE
1)(21-t-
X
lh$4r) (27)
Ik-“k;lR
where E = rnu2/2, 1k- k:l = Jk2 + k” - 2kk’cos O,, i=
1,2.
We shall consider the case of small transferred energy, i.e. @k - wk’ < Wk. In principle, several different situations are possible. If both the angles 81 and 82 are small, the integration limits in the integral in (27) are also small and we can expand the spherical Bessel function into a Taylor series and take only the first term of the expansion. As j,(x) g xl/(21 + I)!! if x --f 0, we get lk-k;lR crl.,,(4.&)
-
and (27)
for
I
using
keV incident electrons at 1.S” i
Type of
Plasmon
plasmon
frequency
/=I,
j=l
/=I,
j=7_
5.95 3731 __._
1=2. I=?,
j= I j=2
6.68
0.69
26.49
5.16
1=3. 1~3,
j= I j=2
7.03
0.09
30.12
1~4,
j=l
7.23
1~4,
j=2
33.35
1=S,
j=l
7.36
1~5,
j=2
36.29
lo
Cross section
(cV)
(nm’/sr)
3x 9.56
1.3.5 8.4 x IO-’ 0.22 5.6 x 10-j 0.024
01,02)
2n-e4N
9
( 17)
x2’-? s Ik-k;lR
dx,
(28)
and one can clearly see that the contribution of the plasmons with 1 = 1 is dominant. In the oppositesituation )k - k:J R >> 1, i = 1,2 we can use the asymptotic expression for the spherical Bessel function j,(x) Z sin (x - ~1/2)/ x i f x + 03. As the integration limits do not depend on 1 significantly, we can see that the values of the integral in (27) are of the same order for different 1. Hence the relative contributions of plasmons with different 1 depend only on the factor g~,j(~~)/fl~,j. In thecase [k - kj\R1, i= 1,2 it is
possible to estimate the integral in (27) by the substitution j,(x) N w//(21 + l)!!. On e can clearly see that only the contribution of the plasmons with small 1 is relevant because of the rapidly increasing multiplier ( (21+ 1) !!)’ in the denominator. Strictly speaking the above analysis is not applicable for the plasmons with 1 > 3. The CGOsymmetry group is icosahedral and it does not have irreducible representations with a dimension of more than five. Hence the plasmons with 1 > 3 in the spherical model split [ 23,241 and the classification of plasmons in the real C6a molecule is different. We will not discuss this difference further because in the experiments which we will analyse only the contribution of the plasmons I = 1,2 is relevant. On the other hand, the small size of C60 leads to the strong damping of plasmons. The estimation for the spherical mode1 in Ref. [ 16 J shows that the plasmons with 1 > 3 could hardly exist. We shall compare our results to the experiment from Ref. [S]. In this work the number of electrons with E = 1 keV scattering in the solid angle 0.5” < 0 < 2.5” (i.e. t9t = 0.5”, 82 = 2.5”) per unit time was measured. The average cross-section
m.,(h,&) */,jt@l,&)
=
27r(cos 6, - cos 02) ’
(29)
obtained using the spherical mode1 for the plasmons with 1 = 1.. ,5, is shown in Table I. We used the value w, = 16 eV for the characteristic g-electron transition frequencies and R = 0.4 nm for the radius of the sphere in the model. For the conditions of the analysed experiments Ik - kij R - 1, i = 1,2 and one
D.A. Gorokhov et al./ Physics Letters A 223 (1996) 116-122
0.0’.’
’
’
’
10
’
20
oj(eV)
’
o
30
’
’
40
Fi g. I. Inelastic cross-section with incident I keV electrons at 1.5’ & 1O: measured 1.51 and calculated using Eq. (27) (solid line).
can clearly see that the contribution of the plasmons with 1 > 2 decays rapidly. This is in good agreement with the qualitative consideration. The 1 = 1 T- and g-plasmons and the quadrupole cT-plasmon make the most significant contribution to the cross-section. Plasmons in C60 are characterized by strong damping due to the small size of the molecule: in small objects the energy of plasma oscillations per one electron fifi/N is strongly enhanced and leads to strong electron-electron scattering, while for bulk plasmons this energy is negligible. As energies of r- and (+plasmons do not depend on 1 significantly (see Table I), the parameter /%2/N is approximately the same for n-- and g-plasmons respectively and we can expect that the damping of plasmons does not depend on t strongly. In the above calculation of the cross-section we did not take into account plasmon damping. The characteristic inverse life time vj taken from the experiment is equal to 1.5 and 8 eV for rr- and g-plasmons respectively. The function u(w)
= c I..i
5L,(&>~2)
IT
1
v.i
(W -
Q,j>'
+
vj2'
(30) is plotted in Fig. I together with the experimental curve. One can see that our results are in good agreement with the experiment. As can be seen from Table 1 and Fig. 1 the first peak on the graph corresponds to the dipole n--
121
plasmon; the second peak corresponds to the dipole and quadrupole cT-plasmons. The contributions of the other plasmons are small. It has already been discussed that ratios U,,I/LT~,Z are proportional in our case to ratios (g2,/nr.l)/(g:2/~~,~). Note that although 01.1 < fl1.2. we have for the cross-sections (~1.1 > u/,2 and the ratio (T~.J/(T,,~ decreases if 1 increases. This is the consequence of the fact that gr,i i 0 if I -+ 00 : for large 1 we can write using Eq. (16) cot+l + -l/v’?, hence g/,1 ---f 0 (see ( 19)) and we obtain an interesting result: the ratio c~1,I l(~l.2 tends to zero if 1 -+ cc. Let us discuss the condition of the applicability of the Born approximation. The Born approximation is valid if lCrlu << Tuj (see Ref. [ 22]), where IUI and n are the characteristic value of the potential and its radius respectively, ~1 is the velocity of the scattering particle. In our case the characteristic value of charge density fluctuations Su is ( e/n2) dm, where 12 N 10 eV is the characteristic oscillation frequency. Hence IUI N (e’/a) VW and the condition of the applicability can be written as follows,
lZ2 s /-
<< nv.
(31)
Substituting ua = N/4rR’, n N R, R = 0.4 nm into (3 1) we see that the Born approximation is applicable if the velocity of the scattering electron satisfies the condition u >> 2 x 10’ cm/s. In the analysed experiments the electron velocity is 1.88 x lo9 cm/s. i.e. the Born approximation is applicable. Briefly summarizing we have shown that the simple two-fluid model used by the authors of Ref. [ 161 for the predictions of the collective spectrum of Cca turned out to be good for the quantitative calculations of the intensities of the plasmon peaks in EELS spectra. The advantage of the description based on this model is that it is physically and technically simple and the model itself does not include any fitting parameters (with the exception of the decay rates which however cannot be properly calculated in more sophisticated techniques [ 51). All this makes the model useful for the quantitative consideration of the collective behaviour of electrons in more complicated fullerene systems.
122
D.A. Gorokhm et d/Physics
Acknowledgement This work was brought to completion within the Russian research and development programme “Fullerenes and Atomic Clusters” project N 94014.
References 111 V.J. Crowell and R.H. Ritchie, Phys. Rev. 172 ( 1968) 436. 121 N. Barberan and J. Bausells, Phys. Rev. B 31 (1985) 6354. r31 M. Michalevich, Phys. Rev. B 45 (1992) 13664. [41 J.W. Keller and M.A. Coplan. Chem. Phys. L.&t. 193 ( 1992) 89. [51 N. Ju, A. Bulgac and J.W. Keller, Phys. Rev. B 48 ( 1993) 9071. 161 H. Tanaka, L. Boestem, K. Onda and 0. Ohashi, J. Phys. Sot. Jpn. 63 (1994) 485. 171 PM. Ajayan, S. Ijima and T. Ichihashi, Phys. Rev. B 47 (1993) 6859. I81 A.A. Lucas, L. Henrard and Ph. Lambin, Phys. Rev. B 49
Letters A 223 (1996) 116-122 (1994) 2888. [9] J. Cuo, D.E. Ellis and D.J. Lam. Chem. Phys. Lett. 184 (1991) 418. [ IO] K. Zeppenfeld, Z. Phys. 243 ( 1971) 229. [ II] R. Klucker, M. Skibowski and W. Steinmann, Phys. Stat. Sol. 65 (1974) 703. 121 H.H. Venghaus, Phys. Stat. Sol. B71 ( 1975) 609. 131 U. Biichner, Phys. Stat. Sol. 8 I (1977) 227. 141 J. Cazaux, Solid State Commun. 8 (1970) 545. 151 J. Cazaux, Opt. Commun. 2 (1970) 173. 161 G. Barton and C. Eberlein, J. Chem. Phys. 95 (1991) 1512. 171 U. Fano, Phys. Rev. 103 (1956) 1202. 181 U. Fano, Phys. Rev. 118 (1960) 451. 19) U. Fano, Rev. Mod. Phys. 64 (1992) 313. t 201 R.C. Tatar and S. Rabii, Phys. Rev. B 25 (1982) 4126. [21 ] V.V. Rotkin and R.A. Suris, Fizika Tverdogo Tela (Sov. Phys. Sol. State) 36 ( 1994) 1899. [ 22 1 L.D. Landau and E.M. Lifshitz, Quantum mechanics, Course in theoretical physics, Vol. 3 (Pergamon, London, 1958). 1231 N.V. Cohan. Proc. Cambridge Sot. S4 (1958) 28. 1241 A.G. McLellan, J. Chem. Phys. 34 (1961) 1350.
25 November
-_ i_ B!!l
1996
PHYSICS
LETTERS
A
Physics Letters A 223 ( 1996) I 16- 122
ELSEiVIER
Electron-energy-loss
spectroscopy of the C60 molecule
D.A. Gorokhov, R.A. Suris*, V.V. Cheianov lofe
Physical
Technical
Institute.
Received
Politehnicheskaya
Street 26, 194021 Saint Perershurg.
15 August 1996: accepted for publication Communicated
15 September
Russiun Federation
1996
by V.M. Agranovich
Abstract In this work we present a theoretical study of EELS (electron-energy-loss spectroscopy) experiments on the CM, molecule. Our treatment of the problem is based on the simple two-fluid model originally proposed for the description of plasma
oscillations in graphite and fullerenes [J. Cazaux, Solid State Commun. 8 ( 1970) 545; Opt. Commun. 2 ( 1970) 173; G. Barton and C. Eberlein, J. Chem. Phys. 95 ( 1991) 15121. It is shown that in spite of the simplicity of the model the calculated intensities of the EELS peaks are in good agreement with experimental data which may indicate that the model can be used as a simple and effective tool for the investigation of the collective behaviour of electrons in fullerene systems. PACS: 31 .SO.+w; 61.46.S~: 78.90.+t Keywwds: Fullerenes; Electron-energy-loss
spectroscopy;
Plasma oscillations
1. Introduction Investigations of plasmons in small clusters in the nanometer size range have a long history. Originally plasma oscillations were studied in small (R = 2-10 nm) metal particles. It turned out that plasmons in metal clusters can be well described by linearized hydrodynamic equations [l-3]. The positions of plasmon peaks, their number and fast electron crosssections calculated within these equations are in good agreement with experimental data. New possibilities of the investigation of plasmons in small objects appeared after the discovery of fullerenes and the cheap methods of their synthesis. Plasmon oscillations were observed in fullerene molecules [ 461, carbon nanometer size tubes [ 71 and multishell fullerenes [ 81.
* Corresponding
author. E-muil: [email protected].
0375-9601/96/.$12.00 Copyright SO375-960 1(96)00707-4
PII
An important feature of fullerenes is the fact that these molecules are more or less of a spherical form and their electrons can be considered to be confined to the surface of the sphere [ 91. Hence, it seems to be a good starting point to consider the Che molecule as a rolled up graphite plane. This approximation is expected to work well for the description of collective excitations. The graphite plane consists of regular hexagons formed by carbon atoms. Each carbon atom is connected by 3 u-bonds with its three neighbours, meanwhile 7r-electrons form a common z--system consisting of orbitals perpendicular to the plane. In linear response theory r- and a-electron subsystems can be considered to be independent with respect to the electrical field parallel to the graphite plane because dipole transitions from rr- to c-orbitals are prohibited. This means that we can describe the plasmons in graphite whose wave vector is parallel to the plane as the plas-
0 1996 Published
by Elsevier Science B.V. All rights reserved.
D.A. Gorokhou et d/Physics
mans in a “two-liquid” electron system; it is proposed that both the liquids move in a common average potential. The graphite loss function - Im[ 1/EL(W)] with respect to the external electric field perpendicular to the c-axis shows two peaks at 7 and 28 eV (see Refs. [ 1O- 13 ] ) . They are called rr- and (T- (or “r + u”-) plasmons, Strictly speaking such a division is rather conventional. In fact, both 7~- and a-electrons make a nonzero contribution to (T- and r-plasmons respectively. In Refs. [ 14,151 Cazaux used the two-fluid model in order to explain the results of EELS studies on graphite. The same idea was used in Ref. [ 161 for the description of plasmons in the Ccc molecule: it was proposed that r-electrons be affected only by induced electric potential, and g-electrons be governed also by a restoring force. This force appears in the model via an effective description of the contribution of valence electrons to the polarizability [ 17-191. Using the two-fluid model Barton and Eberlein predicted the existence of plasma oscillations in the C60 molecule which can be classified by the spherical quantum numbers 1 and m and the index j (j = 1,2) which corresponds to rr- and c-plasmons [ 161. However, the plasmons with large 1 cannot exist because of their strong damping. EELS studies on Cbn (see Refs. [ 451) show a peak at 6.5-7 eV (also called “7r-plasmon”) and a rather broad “cr-plasmon” peak at 15-16 eV. The former peak is identified usually as the I = I r-plasmon, while the latter peak is considered to be a combination of the 1 = 1 a-plasmon and probably the plasmons with 1 > 1. However, the existence of multipole plasmons in C60 is still not clear. In Ref. [6] a small peak at 28 eV was observed and identified as the I = 1 cT-plasmon. The main goal of this work is to calculate the crosssection of fast electrons scattered by a C60 molecule and to show that the contribution of the plasmons with I= 2 to the cross-section is sufficiently large to be observed in EELS experiments. In the model we use it is possible to separate contributions of different plasmons, i.e. one can obtain relative cross-sections. Using this approach we cannot calculate plasmon damping. Widths of plasmon peaks will be taken from experimental data.
117
Letters A 223 (1996) 116-122
2. Quantization of plasma oscillations in the two-fluid model
on a sphere
We shall consider electrons in the Cen molecule to be distributed over the surface of a uniformly charged sphere - the positive molecular core in the model. The Fourier transformation of the charge density &l(w) induced by the potential @ including both the external and induced potentials is given by the expression
(1) where n is the surface rr- and g-electron density, fa and w, are respectively the oscillator strength and electron energy levels. The index a enumerates the electron states. All denotes the angular part of the Laplace operator: in the model we use electrons which cannot move perpendicularly to the surface. As the r-electron subsystem of graphite is semimetal, we can use the following approximation for the n-electron contribution to the induced charge density, &l,(W)
(2)
= ___
We suggested w, in Eq. (2) to be equal to zero and used the sum rule c, far = l/4 for the oscillator strength of r-electrons in COO.N = 60 and R = 0.4 nm are the number of r-electrons and the effective radius of the sphere respectively. On the other hand, the g-electron subsystem of graphite is dielectric with the gap of order 12 eV (see Ref. [ 201). The oscillator strength of the m-electron subsystem has a maximum at o, N 16 eV. Hence we can use the following approximate expression for the cr-electron contribution, &l,(W)
3e2N = 4rmR2
1 co2 - co,=
QWJ).
Here we used the sum rule C, fan= 314 for the uelectron subsystem. w, = 16 eV is the characteristic transition frequency of a-eiectrons. ls2-electrons of carbon atoms are not taken into account in the model. The Q-field is connected with the total induced charge density Su = Sn,, + Sn, through the Gauss theorem,
Z/,-,,, - :I,;,_, =4da.
(4)
D.A. Gorokhov rr ol./Physic.t
118
Lerrers A 223 (19961 116-122
It is convenient to expand the density fields in Fourier series on the sphere,
The coefficients in Eq. ( 10) are chosen so as to keep it equal to the hydrodynamic Hamiltonian
an, = C %zYr,,(4,@ In1
H = ;rnq -
(T=
9
an, = C bJh,(~, in1
c QJi”l(4.~>. In1
8, (5)
Then, using the multipole
expansion
of the potential
0, for r < R,
we can rewrite (4) in the following 4rR &?I! = - ---Sim 2z+ 1
= -$$(airn
form, + br,,,),
(6)
which gives for (2) and (3) (see also Refs. [ 16,2 1] ) , &n, = -@hi, + W,), . b/, = -3f22(a~,, + b/m) - ~,%,,
J
k:dS + ;rnuo
+ +maow,* /rjdi+
-/ ;
+ imcro( 3f$* + w,,~) B* + 3ma&*AB,
(8)
JrnaoA
(9)
It can be easily verified by straightforward calculation that equations (7) and (8) are the equations of motion corresponding to the Hamiltonian
= Qlcosq$
- Q2sin&,
2n,* + WC2 2X& 012
The eigenfrequencies equation,
+ AI,-~BI,,),
icro-AI,,
brn, = igohB,,,,
R*
’
transformation, (14) (15)
(16)
01,j are given by the following
(10)
where the amplitudes Al,,, and Bt, are the (I,m)components of the displacement fields rt and r2 of the electron liquids which are connected with al,,, and bl, by the following equations, arm =
(13)
where the angle 41 can be found from the condition of the disappearance of the nondiagonal term in Hamiltonian ( 13) expressed in terms of the new coordinates Qt and QT. The substitution of (14) and (15) into Eq. ( 13) gives cot(2C#J1) = -
+ imao c ~Q*(AI,BI,-, Ill!
( 12)
~rncqpi* + $mao~* + ~mqfI~2A2
which can be done by the following
N go==.
$&dS.
Hamiltonian (10) is nondiagonal in the basis consisting of the functions &,( 4,O) because of the last term. On the other hand, this term does not mix amplitudes with different 1 and (ml because of the spherical symmetry of the model, hence the above Hamiltonian can be diagonalized by the diagonalization of each “block” with a number lm. We have shown that the problem can be reduced to the diagonalization of the Hamiltonian
v’%&B=Qtsin&+Qzcos+~. 1)
dS
(7)
where n2 = 47re*go Z(l+ 1 --3T-r’ mR
Jf;
(11)
One can clearly see that charge oscillations can be classified by the numbers 1 and m and the index j corresponding to .the lower and upper roots of Eq. ( 17). The fact that the eigenfrequencies do not depend on the number m is due to the spherical symmetry of the considered model. Using ( 11) we can find quantum mechanical operators for amplitudes Al, and Bl,. Substituting them
DA. Gorokhov
into the equation ations we obtain quantity,
et d/Physics
Letters A 223 (1996) 116-122
119
for classical charge density Auctuthe operator corresponding to this
J~~Jl(i+l)y,,,(9,8)
=(dJ,e) = -jg
lm
Sl(b)
(dz
x
,. (&?Ll +
(22)
s:_,,,, )
the angle 41 is chosen to be the minimal positive root of Eq. (16).
We shall consider only the case of fast scattering electrons, i.e. the only relevant processes are those in connection with the creation of one plasmon. Hence we can use the Born approximation. The quantitative criterium of the applicability of the Born approximation will be discussed further. In the initial state all the plasmon occupation numbers are equal to zero. Let us suppose that after scattering only the occupation number of the state Im, j is equal to one. Defining the initial and final momenta of the scattering electron k and k’ respectively we obtain the following expression for the modulus squared of the matrix element of fii”i,
3. Fast electron scattering
I(k;01iCiintlk’;Im,j)12
+g2(41)(& 7+$+_ pq
“‘.-
)
I, nr,2
>
(18)
’
1
where &,,,J is the annihilation
operator of the plasmon with numbers Im, j. and the functions gl (41) and g? (r#+) are given by the equations Sl(4i)
= hCOSq&
g2(41)
=COsh
+
sin#q,
(19)
V%in&,
In this section we will obtain the expression for the scattering amplitude of fast electrons due to charge density fluctuations in c60. In the secondary quantization representation the electron-plasmon interaction can be written as follows,
I?,“,=
J
(20)
@+(-e&)!kdr.
is the electron field operHere @ = xk e ik’r&/& ator, & is the potential corresponding to the electron surface density distribution S&( 4, #), i.e. 6 is the solution of the Poisson equation with the charge density S(r - R) 6&(4,6). Hence & and 65 are connected by the following equation, 6( r’ - R) i%((b’, 8’)
&,= J
Ir-r’l
dr,
.
(21)
Substituting Eq. (2 1) and the explicit expressions for the electron field and the electron density fluctuation operators (see ( 18)) into Eq. (20), we obtain the expression for the electron-plasmon interaction
x gj2M)
= “~~~v~ L (a)
jam
y
Y,;,(i),
(23)
where q = k’ - k. As each eigenfrequency fil,j is (21f 1 )-fold degenerated, we shall calculate the total cross-section for the transferred energy fi0j.j. Summing up the contributions of plasmons with given 1 and j over m, we get
I(kOIAintlk’;ltn, j)12)
dgl j
A = J(c dii?kr x
‘%@k’
“t
+ fh
-
@k)
27rV k’= dk’V -zi2u c2Tj3
In the above equation Ok = fik*/2m, velocity of the electron. Using the equation
.
(24)
u is the initial
(25) (see Ref. [ 221)) we get after some algebra
DA. Gorokhou
et d/Physics
Letters
A 223 (1996)
116-122
Table I Frequencies and intensities of plasmon peaks calculated formulae
(26)
4. Results
and discussion
In experiments one can measure the number of electrons reaching a detector, i.e. one measures the function (26) integrated over a certain solid angle. Let this solid angle be restricted by two cones of angles 01 and 02 respectively. In this case we can write for the total cross-section C[,j(
ar.,(h
02) = -l(l+ fiQ,jE
1)(21-t-
X
lh$4r) (27)
Ik-“k;lR
where E = rnu2/2, 1k- k:l = Jk2 + k” - 2kk’cos O,, i=
1,2.
We shall consider the case of small transferred energy, i.e. @k - wk’ < Wk. In principle, several different situations are possible. If both the angles 81 and 82 are small, the integration limits in the integral in (27) are also small and we can expand the spherical Bessel function into a Taylor series and take only the first term of the expansion. As j,(x) g xl/(21 + I)!! if x --f 0, we get lk-k;lR crl.,,(4.&)
-
and (27)
for
I
using
keV incident electrons at 1.S” i
Type of
Plasmon
plasmon
frequency
/=I,
j=l
/=I,
j=7_
5.95 3731 __._
1=2. I=?,
j= I j=2
6.68
0.69
26.49
5.16
1=3. 1~3,
j= I j=2
7.03
0.09
30.12
1~4,
j=l
7.23
1~4,
j=2
33.35
1=S,
j=l
7.36
1~5,
j=2
36.29
lo
Cross section
(cV)
(nm’/sr)
3x 9.56
1.3.5 8.4 x IO-’ 0.22 5.6 x 10-j 0.024
01,02)
2n-e4N
9
( 17)
x2’-? s Ik-k;lR
dx,
(28)
and one can clearly see that the contribution of the plasmons with 1 = 1 is dominant. In the oppositesituation )k - k:J R >> 1, i = 1,2 we can use the asymptotic expression for the spherical Bessel function j,(x) Z sin (x - ~1/2)/ x i f x + 03. As the integration limits do not depend on 1 significantly, we can see that the values of the integral in (27) are of the same order for different 1. Hence the relative contributions of plasmons with different 1 depend only on the factor g~,j(~~)/fl~,j. In thecase [k - kj\R1, i= 1,2 it is
possible to estimate the integral in (27) by the substitution j,(x) N w//(21 + l)!!. On e can clearly see that only the contribution of the plasmons with small 1 is relevant because of the rapidly increasing multiplier ( (21+ 1) !!)’ in the denominator. Strictly speaking the above analysis is not applicable for the plasmons with 1 > 3. The CGOsymmetry group is icosahedral and it does not have irreducible representations with a dimension of more than five. Hence the plasmons with 1 > 3 in the spherical model split [ 23,241 and the classification of plasmons in the real C6a molecule is different. We will not discuss this difference further because in the experiments which we will analyse only the contribution of the plasmons I = 1,2 is relevant. On the other hand, the small size of C60 leads to the strong damping of plasmons. The estimation for the spherical mode1 in Ref. [ 16 J shows that the plasmons with 1 > 3 could hardly exist. We shall compare our results to the experiment from Ref. [S]. In this work the number of electrons with E = 1 keV scattering in the solid angle 0.5” < 0 < 2.5” (i.e. t9t = 0.5”, 82 = 2.5”) per unit time was measured. The average cross-section
m.,(h,&) */,jt@l,&)
=
27r(cos 6, - cos 02) ’
(29)
obtained using the spherical mode1 for the plasmons with 1 = 1.. ,5, is shown in Table I. We used the value w, = 16 eV for the characteristic g-electron transition frequencies and R = 0.4 nm for the radius of the sphere in the model. For the conditions of the analysed experiments Ik - kij R - 1, i = 1,2 and one
D.A. Gorokhov et al./ Physics Letters A 223 (1996) 116-122
0.0’.’
’
’
’
10
’
20
oj(eV)
’
o
30
’
’
40
Fi g. I. Inelastic cross-section with incident I keV electrons at 1.5’ & 1O: measured 1.51 and calculated using Eq. (27) (solid line).
can clearly see that the contribution of the plasmons with 1 > 2 decays rapidly. This is in good agreement with the qualitative consideration. The 1 = 1 T- and g-plasmons and the quadrupole cT-plasmon make the most significant contribution to the cross-section. Plasmons in C60 are characterized by strong damping due to the small size of the molecule: in small objects the energy of plasma oscillations per one electron fifi/N is strongly enhanced and leads to strong electron-electron scattering, while for bulk plasmons this energy is negligible. As energies of r- and (+plasmons do not depend on 1 significantly (see Table I), the parameter /%2/N is approximately the same for n-- and g-plasmons respectively and we can expect that the damping of plasmons does not depend on t strongly. In the above calculation of the cross-section we did not take into account plasmon damping. The characteristic inverse life time vj taken from the experiment is equal to 1.5 and 8 eV for rr- and g-plasmons respectively. The function u(w)
= c I..i
5L,(&>~2)
IT
1
v.i
(W -
Q,j>'
+
vj2'
(30) is plotted in Fig. I together with the experimental curve. One can see that our results are in good agreement with the experiment. As can be seen from Table 1 and Fig. 1 the first peak on the graph corresponds to the dipole n--
121
plasmon; the second peak corresponds to the dipole and quadrupole cT-plasmons. The contributions of the other plasmons are small. It has already been discussed that ratios U,,I/LT~,Z are proportional in our case to ratios (g2,/nr.l)/(g:2/~~,~). Note that although 01.1 < fl1.2. we have for the cross-sections (~1.1 > u/,2 and the ratio (T~.J/(T,,~ decreases if 1 increases. This is the consequence of the fact that gr,i i 0 if I -+ 00 : for large 1 we can write using Eq. (16) cot+l + -l/v’?, hence g/,1 ---f 0 (see ( 19)) and we obtain an interesting result: the ratio c~1,I l(~l.2 tends to zero if 1 -+ cc. Let us discuss the condition of the applicability of the Born approximation. The Born approximation is valid if lCrlu << Tuj (see Ref. [ 22]), where IUI and n are the characteristic value of the potential and its radius respectively, ~1 is the velocity of the scattering particle. In our case the characteristic value of charge density fluctuations Su is ( e/n2) dm, where 12 N 10 eV is the characteristic oscillation frequency. Hence IUI N (e’/a) VW and the condition of the applicability can be written as follows,
lZ2 s /-
<< nv.
(31)
Substituting ua = N/4rR’, n N R, R = 0.4 nm into (3 1) we see that the Born approximation is applicable if the velocity of the scattering electron satisfies the condition u >> 2 x 10’ cm/s. In the analysed experiments the electron velocity is 1.88 x lo9 cm/s. i.e. the Born approximation is applicable. Briefly summarizing we have shown that the simple two-fluid model used by the authors of Ref. [ 161 for the predictions of the collective spectrum of Cca turned out to be good for the quantitative calculations of the intensities of the plasmon peaks in EELS spectra. The advantage of the description based on this model is that it is physically and technically simple and the model itself does not include any fitting parameters (with the exception of the decay rates which however cannot be properly calculated in more sophisticated techniques [ 51). All this makes the model useful for the quantitative consideration of the collective behaviour of electrons in more complicated fullerene systems.
122
D.A. Gorokhm et d/Physics
Acknowledgement This work was brought to completion within the Russian research and development programme “Fullerenes and Atomic Clusters” project N 94014.
References 111 V.J. Crowell and R.H. Ritchie, Phys. Rev. 172 ( 1968) 436. 121 N. Barberan and J. Bausells, Phys. Rev. B 31 (1985) 6354. r31 M. Michalevich, Phys. Rev. B 45 (1992) 13664. [41 J.W. Keller and M.A. Coplan. Chem. Phys. L.&t. 193 ( 1992) 89. [51 N. Ju, A. Bulgac and J.W. Keller, Phys. Rev. B 48 ( 1993) 9071. 161 H. Tanaka, L. Boestem, K. Onda and 0. Ohashi, J. Phys. Sot. Jpn. 63 (1994) 485. 171 PM. Ajayan, S. Ijima and T. Ichihashi, Phys. Rev. B 47 (1993) 6859. I81 A.A. Lucas, L. Henrard and Ph. Lambin, Phys. Rev. B 49
Letters A 223 (1996) 116-122 (1994) 2888. [9] J. Cuo, D.E. Ellis and D.J. Lam. Chem. Phys. Lett. 184 (1991) 418. [ IO] K. Zeppenfeld, Z. Phys. 243 ( 1971) 229. [ II] R. Klucker, M. Skibowski and W. Steinmann, Phys. Stat. Sol. 65 (1974) 703. 121 H.H. Venghaus, Phys. Stat. Sol. B71 ( 1975) 609. 131 U. Biichner, Phys. Stat. Sol. 8 I (1977) 227. 141 J. Cazaux, Solid State Commun. 8 (1970) 545. 151 J. Cazaux, Opt. Commun. 2 (1970) 173. 161 G. Barton and C. Eberlein, J. Chem. Phys. 95 (1991) 1512. 171 U. Fano, Phys. Rev. 103 (1956) 1202. 181 U. Fano, Phys. Rev. 118 (1960) 451. 19) U. Fano, Rev. Mod. Phys. 64 (1992) 313. t 201 R.C. Tatar and S. Rabii, Phys. Rev. B 25 (1982) 4126. [21 ] V.V. Rotkin and R.A. Suris, Fizika Tverdogo Tela (Sov. Phys. Sol. State) 36 ( 1994) 1899. [ 22 1 L.D. Landau and E.M. Lifshitz, Quantum mechanics, Course in theoretical physics, Vol. 3 (Pergamon, London, 1958). 1231 N.V. Cohan. Proc. Cambridge Sot. S4 (1958) 28. 1241 A.G. McLellan, J. Chem. Phys. 34 (1961) 1350.