Plasmon features in electron energy loss spectra from carbon materials

Carbon 45 (2007) 1410–1418 www.elsevier.com/locate/carbon

Plasmon features in electron energy loss spectra from carbon materials L. Calliari a

a,*

, S. Fanchenko b, M. Filippi

a

ITC-irst (Centro per la Ricerca Scientifica e Tecnologica), 38050 Povo (Trento), Italy b RRC Kurchatov Institute, Moscow, Russia Received 23 October 2006; accepted 19 March 2007 Available online 28 March 2007

Abstract A phenomenological approach is proposed to derive plasmon energies for C materials as a function of mass density q and sp2 fraction f. It is shown that the energy of the graphite in-plane (p and p + r) and out-of-plane plasmons, as well as the energy of the diamond bulk (r) plasmon are correctly reproduced by the model. Crucial factors in this respect – and responsible for the deviation from a free electron picture – are the energy of interband transitions associated with plasmon excitation and the screening of p electrons due to r electron polarization. Plasmon energies, derived as a function of q and f, are discussed.  2007 Elsevier Ltd. All rights reserved.

1. Introduction Using a dielectric formulation, we previously proposed [1] a phenomenological approach to understand electron energy loss (EEL) spectra from carbon materials. We focused in particular on deriving the energy of plasmon excitations in the hypothesis of independent p and r electrons. As a result, only plasmon excitations associated with p ! p* and r ! r* interband transitions were obtained. However, among p bonded materials, the assumption is valid only for in-plane excitations in graphite and for tangential excitations in materials (e.g., fullerenes, nanotubes, onions, . . .) where curved graphitic sheets are involved [2–4]. Purely in-plane excitations take place when the material response only involves the component of the dielectric tensor normal to the c-axis, e?. We remember that, for a uniaxial anisotropic material like graphite, the dielectric function is a tensor, diagonal in a proper reference frame, with two independent components, e? and ek, perpendicular and parallel to the graphite c-axis, respectively. A description in terms of e? and ek locally holds also for

*

Corresponding author. Fax: +39 0461810851. E-mail address: [email protected] (L. Calliari).

0008-6223/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.03.034

curved graphitic sheets [2] and for amorphous carbon (a-C) materials, described as an ensemble of randomly oriented graphitic clusters [5]. Under the assumption of independent p and r electrons, we could understand the basic features of EEL spectra from C materials [1]. In particular, we showed how interband transitions and screening by valence electrons affect the energy of p and p + r plasmons in C materials. The relevant energies of graphite and diamond EEL spectra were fairly well reproduced and a relation linking plasmon energies to the material mass density and sp2 fraction for a generic C material was derived. However, a realistic description of C materials requires discarding the assumption of independent p and r electron systems, leading to additionally consider p ! r* and r ! p* interband transitions. Transitions like these, characterizing graphite out-of-plane excitations (described by ek), are expected not only for disordered C systems [6] but also for ordered materials whenever the acquisition geometry causes all the components of the dielectric tensor to be at play. This is, for example, the case of EEL spectra acquired in the reflection geometry, where, due to the role played by elastic scattering, the angle between momentum transfer in the inelastic scattering event and the material relevant axes is undefined [7].

L. Calliari et al. / Carbon 45 (2007) 1410–1418

The present paper considers reflection EELS from carbon materials. With respect to Ref. [1], a more detailed presentation of the results is provided and the approach is generalized to deal with both in-plane and out-of-plane excitations. 2. Experimental All spectra were acquired in a PHI 545 system operating at a base pressure of 2 · 1010 mbar. The instrument is equipped with a double-pass cylindrical mirror analyzer (CMA), a coaxial electron gun, a non-monochromatic Mg Ka (hm = 1253.6 eV) X-ray source and a He discharge lamp. EEL spectra at the C K edge were generated by a primary electron beam of 1 keV, while valence band (VB) photoemission spectra were excited by the HeI (hm = 21.2 eV) and HeII (hm = 40.8 eV) lines. Conversely, EEL spectra in the plasmon region were generated by primary electron beams of 2 keV and 250 eV, at a constant analyzer energy resolution of 0.6 eV. Highly Oriented Pyrolytic Graphite (HOPG) was cleaved ex situ before inserting into Ultra High Vacuum (UHV), where it was annealed at 550 C for 10 min. The polycrystalline diamond film did not undergo any surface cleaning procedure before acquisition of photoemission and C K edge EEL spectra. Beside C, the O signal was observed at its surface in a wide scan X-ray photoemission spectrum, its intensity being below 5% of the C intensity. On the other hand, the diamond surface was cleaned by heating at 900 C in UHV before acquiring EEL spectra in the plasmon region. This procedure leads to desorbing the H which terminates the surface of CVD polycrystalline diamond [8,9] and stabilizes sp3 bonding at the surface. Once H is desorbed, the diamond surface reconstructs by p bonding formation [9].

a

27 6.5

19.5

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12.5

b

33.8 23

12.8

10

20

30

40

50

ENERGY LOSS (eV) Fig. 1. EEL spectra of HOPG (panel a) and polycrystalline diamond (panel b). The energy of plasmon features and interband transitions is shown in the figure.

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3. EEL spectrum from graphite and diamond To illustrate the main features of the EEL spectra from C materials, we show, in Fig. 1, the spectra of two crystalline forms of C, graphite (panel a) and diamond (panel b). Both spectra are fitted in terms of Voigtian components and the zero loss peak is subtracted after fitting to a Gaussian function. Before considering the graphite spectrum, it is worth remembering that, since graphite is anisotropic, its EEL spectrum depends on the orientation of the momentum transfer ~ q with respect to the graphite c-axis and hence on the acquisition conditions. In particular, using the transmission geometry, spectra associated with different orientations of the momentum transfer can be acquired and, in principle, the two components of the graphite dielectric tensor, e? and ek, can be independently measured. In contrast, an average over all possible momentum transfers is obtained in the reflection geometry [7]. As a consequence, both components of the graphite dielectric tensor are involved [10–12], leading in-plane and out-of-plane excitations to coexist. Considering the spectrum of Fig. 1, panel a, we see the two main features characterizing the graphite EEL spectrum, namely the p (involving only p electrons) and p + r (involving all valence electrons) plasmon peaks at 6.5 eV and 27 eV, respectively. In addition, a small peak is observed at 12.5 eV. This peak is hardly observed on the graphite spectrum, so that no assignment for it can be found in the literature, to the best of our knowledge. However, its intensity and energy position lead to assign it to single particle interband transitions. More specifically, we ascribe it to r ! r* transitions between high density of p-states regions in the graphite valence and conduction bands. It corresponds to the maximum observed around 14 eV in graphite optical absorption spectra [13,14]. The energy difference between the two (EEL and optical absorption) spectra is possibly due to the non-zero momentum transfer in EELS. At higher energy loss, a shoulder is noticed at 19.5 eV, assigned to the graphite out-of-plane plasmon [15], while components at higher energy loss than the main peak are assigned [15] to r ! r* transitions involving deep lying s-states in the valence band. The diamond spectrum, on the other hand, exhibits a broad, two-component peak, due to r valence electrons and generally understood in terms of a bulk and a surface plasmon, respectively at 33.8 eV and 23 eV on panel b. The spectrum intensity should be zero from the zero loss peak up to the energy gap of diamond (5.45 eV), but it is not in the present case due to the p states existing at the reconstructed surface. At 12.8 eV, a component is resolved which we assign, as for graphite, to single particle r ! r* transitions between high density of states regions in the diamond valence and conduction bands. Diamond optical absorption spectra show indeed a sharp absorption peak, associated with a maximum in the Joint Density of States, at 12.5 eV [16].

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L. Calliari et al. / Carbon 45 (2007) 1410–1418

Other forms of carbon exhibit EEL spectra which can be discussed starting from the basic features observed on the graphite and diamond spectra. 4. Phenomenological approach 4.1. Plasmon energies 4.1.1. Independent p and r electron systems Let us briefly remember the approach, based on a dielectric formalism, proposed in Ref. [1]. Since EEL spectra   are proportional to the ‘‘energy loss function’’, Im 1 , the e material dielectric function, e, is the key ingredient for any discussion about such spectra. In the hypothesis of independent p and r electron systems, the dielectric function of a generic C material is written [1,13] as eðxÞ ¼ 1 þ vp ðxÞ þ vr ðxÞ

ð1Þ

where vp and vr are the electron susceptibility of p and r electrons, respectively. Electron susceptibilities are in turn written as Drude–Lorentz functions in the optical limit (i.e., for zero momentum transfer ~ qÞ: eðxÞ ¼ 1 þ

X

vj ¼ 1 þ x2p

j

X j

X2j

fj  x2 þ ixcj

ð2Þ

0:5

xp ¼ ðnNe2 =e0 mÞ is the plasma frequency for a free electron gas (n = number of valence electrons per atom; N = atomic density; nN = electron density); vj is the electron susceptibility associated with the jth oscillator; Xj, cj and fj are, respectively, the characteristic excitation energy (we take  h = 1), damping coefficient and oscillator strength for the jth oscillator. Assuming the outer shell electrons contribute to the material response with the same probability, fj is the fraction of outerP shell electrons associated with Xj. In this way, the sum rule j fj ¼ 1 is satisfied. It ensures the correct behaviour for the dielectric function at external frequencies much higher than any characteristic frequency of the material, i.e., when the electrons should be regarded as free: eð1; 0Þ ! 1  ðx2p =x2 Þ A discussion about damping, basically responsible for the shape of plasmon peaks, is beyond the scope of this work which is focused on the evaluation of the energy of plasmon peaks only. Since, however, high damping affects peak positions in addition to their shape [17], we assume that the damping constant is much less than the plasmon energy, so that it can be neglected. A second approximation comes from considering the optical limit of the dielectric function which amounts to neglect the momentum dependence of plasmon energies. Within the above approximations (independent p and r electrons, negligible damping, negligible energy vs. momentum dispersion), and denoting the characteristic excitation energy for p and r electrons with Xp and Xr, respectively,

the dielectric function component responsible for in-plane oscillations is written, for a generic C material, as " # x2p f 4f q þ ð3Þ e? ðxÞ ¼ 1 þ 4 X2p  x2 X2r  x2 where f is the sp2 fraction and q = q*/qg is the material relative density, q* and qg being the mass density of an unknown C material and of graphite, respectively. f and 4  f represent the fraction of valence electrons involved in p ! p* and r ! r* transitions, respectively. The energy of plasmon peaks, given by singularities in the energy loss function, is determined by solving the equation: e? ðxÞ ¼ 0

ð4Þ

For the dielectric function (3) and for f 5 0, i.e., for C materials exhibiting some degree of p bonding, this is a second order equation in x2, whose two solutions represent the energy of the p and p + r plasmons typically observed on EEL spectra from C materials: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2p;pþr ¼ B  B2  C ð5Þ where 1 B ¼  fq  x2p þ ðX2p þ X2r Þg 2 x2p C ¼ q  ½f X2r þ ð4  f ÞX2p  þ X2p  X2r 4 On the other hand, for f = 0, i.e., for purely r bonded materials, Eq. (3) becomes of first order in x2, with the solution x2r ¼ q  x2p þ X2r

ð6Þ

4.1.2. Mixed p and r electron systems Being derived in the assumption of independent p and r electrons, solutions (5) represent, among p bonded systems (f 5 0), the energy of plasmon peaks only for in-plane (graphite) or tangential (fullerenes, nanotubes) plasmon modes. As stated above however, understanding graphite reflection EEL spectra requires that out-of-plane plasmon excitations are taken into account in addition to in-plane excitations. A fortiori, this applies to spectra from a-C materials, irrespective of the acquisition geometry. According to Robertson’s cluster model [5] in fact, the sp2 bonded fraction of a-C is given by a collection of randomly oriented small graphitic clusters, so that in-plane and outof-plane collective excitations are generally induced by an incident electron beam. Accounting for the two types of excitations amounts to extend the above treatment from the isotropic to the anisotropic case. In this respect, let us consider the differential inelastic scattering cross section per unit solid angle and o2 r unit energy, oXoE , as given by the Bethe theory, in the first Born approximation, within a single-electron model, and

L. Calliari et al. / Carbon 45 (2007) 1410–1418

taking into account that screening in anisotropic media is described by the dielectric tensor eij [10,17,18]:

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o2 r 4 ¼ P oXoE a20 j i;j qi eij qj j2 *   + 2   X  X    expði~ q ~ rj i  dðEf  Ei  DEÞ ð7Þ  f     r f

ferential inelastic scattering cross section (12) has to be averaged over all angles:

2  or 1 1 1 ¼ 2 2 oXoE ave a0 pne q2 4p Z p  1  Im 2p sin # d# e¼ cos2 # þ e? sin2 # 0 ð13Þ

Here a0 = Bohr radius, rj = position vector of the electron participating in the transition. jii and jfi are the initial and final states, with energy Ei and Ef, respectively. In the dipole approximation, for a given angle h between the momentum transfer ~ q and the crystal c-axis, equation (7) is rewritten [10] as

It is worth noting that, for weak anisotropy (i.e., ee¼? ffi 1Þ, the integrand can be expanded into a Taylor series, giving, for the integration: !

2  or 1 1 1 ¼ Im 1 ð14Þ oXoE ave a20 pne2 q2 e þ 23 e? 3 ¼

j

  o2 r 4 2 2 2 2 ¼   2 q¼ j M ¼ j þ q? j M ? j oXoE a2 e¼ q2 þ e? q2  ? 0 ¼

ð8Þ

where q2¼ ¼ q2 cos2 # and q2? ¼ q2 sin2 # are the squares of the momentum transfer components parallel and perpendicular to the crystal c-axis, ek and e? are the dielectric tensor components parallel and perpendicular to the crystal c-axis, and the functions jM=j2 and jM?j2 are determined via dipole matrix elements in the following way: XX jM ¼ j2 ¼ jhf jrj¼ jiij2 dðEf  Ei  DEÞ ð9aÞ f

2

jM ? j ¼

rj

XX f

jhf jrj? jiij2 dðEf  Ei  DEÞ

ð9bÞ

rj

For purely parallel and purely perpendicular orientation of the momentum transfer with respect to the crystal c-axis, the inelastic scattering cross section becomes, respectively:  o2 r  4 2 2 ¼  ð10aÞ  q jM ¼ j  2 oXoE ¼ a e¼ q2 2 ¼ 0 ¼  o2 r  4 ¼ q2 jM ? j2 ð10bÞ  2 oXoE ? a0 je? q2? j2 ? By comparing these two relations with their analogues derived within a dielectric approach, one obtains  1 1 jM ¼ j2 ¼ 2 2 Im ð11aÞ 2 4p ne e¼ je¼ j  2 1 1 jM ? j ¼ 2 2 Im ð11bÞ 2 4p e? ne je? j so that, for a given angle h, the differential inelastic scattering cross section is  o2 r 1 1 ¼ 2 2 2 Im ð12Þ oXoE p a0 ne q2 e¼ cos2 # þ q2 e? sin2 # in agreement with previous reports [2,14]. To account for graphite reflection EEL spectra or, in any case, for EEL spectra from a whole ensemble of randomly oriented graphitic clusters in a-C materials, the dif-

and leading to propose [2] an ‘‘effective dielectric function’’: 1 2 eeff ¼ e¼ þ e? 3 3

ð15Þ

One should note however that use of this effective dielectric function is really confined to the nearly isotropic case and it cannot be extended to the general anisotropic case. For the general case, integration of (13) gives

2  or oXoE ave 8 qffiffiffiffiffiffiffiffiffiffiffiffi9 > = < 1 þ 1  ee¼? > 4e0 1 1 1 q ffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 Im  ln > > 2e a0 pne2 q2 : 1  e¼ 1  1  e¼ ; ?

 4e0 1 1 ¼ 2 Im eeff a0 pne2 q2

e?

e?

ð16Þ

which can be used to define an ‘‘effective isotropic dielectric function’’ eeff. As mentioned in Section 4.1.1, plasmon energies corre  spond to singularities in the energy loss function Im e1 . eff Apart from the isotropic case e? = ek, singularities occur in (16) for e? = 0 and for ek = 0, where the relation reduces, respectively, to   1 p 1 Im ¼ Im pffiffiffiffiffiffiffiffiffi ð17Þ eeff 2 e? e¼ and 

1 1 1 4e? Im ¼ Im  ln eeff 2 e? e¼

ð18Þ

From (17) and (18) we see that, for uniaxial anisotropic materials, plasmon peaks occur at energies where they are foreseen either for purely in-plane or for purely out-of-plane excitations. Singularities given by (17) and (18) – inverse square root in one case, logarithm in the other – are however weaker than first order pole singularities encountered for purely in-plane or purely out-of-plane excitations. Determining plasmon energies for the anisotropic case thus amounts to solve the two equations: e? = 0 and

L. Calliari et al. / Carbon 45 (2007) 1410–1418

ek = 0. The first one has already been solved (Section 4.1.1), so we are left with equation ek = 0. We first write ek in terms of Drude–Lorentz functions within the same approximations used in Section 4.1.1, namely neglecting damping and the momentum dependence of the dielectric function: " # x2p f f q e¼ ðxÞ ¼ 1 þ þ ð19Þ 4 X2pr  x2 X2rp  x2 Xpr and Xrp are the characteristic excitation energies for p ! r* and r ! p* transitions, respectively. The fraction of electrons taking part in the transitions is calculated by noting that the p band is always involved (either in the initial or in the final state) in this case, so that the fraction of participating electrons is always dictated by the p band occupancy, namely f/4 of the total valence electrons. In contrast, the fraction of electrons taking part into p ! p* and r ! r* transitions is related to the fraction of p and r electrons, respectively. Assuming, as a first approximation, X2pr  X2rp  X2p$r , we have x2p 2f q 2 ¼0 e¼ ðxÞ ¼ 1 þ 4 Xp$r  x2

ð20Þ

fq 2 x þ X2p$r 2 p

a

σ*

σ

0.8 0. 0.6

π

π*

0.4 0.22 0.0 1.0

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0.8 0.6 0.4 0.2 0.0 -15

-10

-5

0

5

10

15

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25

ENERGY (eV)

with the solution x2p$r ¼

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INTENSITY (arb. units)

1414

ð21Þ

It gives the energy of out-of-planeq plasmons. It is displaced ffiffiffiffi

above the free electron energy, xp f2q, by a term due to the characteristic energy of mixed p ! r* and r ! p* interband transitions. It is worth noting that no p plasmon is obtained in this case, but only a mixed p M r plasmon, involving half of all the valence electrons. 4.2. Characteristic excitation energies Solutions (5), (6) and (21) depend on the mass density, the sp2 fraction and the characteristic excitation energies of interband transitions associated with plasmon peaks in EEL spectra. These energies are usually taken as the Penn gap, namely the average bonding–antibonding splitting of electron states. As such, they can be measured by probing the occupied and empty states of the material in terms of suitable techniques, as shown in Fig. 2 for HOPG (panel a) and diamond (panel b). The occupied states are probed by UV (hm = 40.8 eV) excited VB photoemission, while the empty, conduction band, states are probed by EELS at the C K edge. A UV (hm = 21.2 eV) excited secondary electron spectrum is added to panel a, because it provides a reliable measurement for the position of the graphite most intense peak in the r* band [19]. (A 0.3 eV shift of the EEL spectrum was required to align it to the secondary electron spectrum). From panel a, the peak-to-peak p–p* and r–r* separation turns out to be 3.8 eV and 15 eV, respectively.

Fig. 2. Occupied and empty density of states for HOPG and polycrystalline diamond (—), as probed, respectively, by VB (hm = 40.8 eV) photoemission and EELS at the C K edge. (a) On the HOPG panel, the HeI (hm = 21.2 eV) excited VB photoemission spectrum and low energy secondary electron peak are also shown (  ). (b) On the diamond panel, states between 0 and 5 eV are brought about by electron irradiation.

Graphite optical absorption spectra [13,14] give 4 eV and 14 eV for p–p* and r–r* transitions, respectively. Always from panel a, the peak-to-peak r–p* and p–r* separation turns out to be 8 eV and 10 eV, respectively. For comparison, graphite optical absorption spectra exhibit a sharp peak around 11 eV when ek is probed [14]. Panel b shows that a r–r* separation around 15 eV can be assigned to diamond too. This suggests that Xr does not depend on the local hybridization of C atoms, which only defines whether p electrons are present or not to fill in the gap between the r ones. If we further assume that the characteristic energies do not depend on the medium/ long range order of the material, plasmon energies for C materials would only depend on mass density and sp2 fraction. To justify this hypothesis, in particular for Xp, we should note that we are dealing here with the average bonding–antibonding separation, rather than with the minimum gap, known to depend on the configuration of the sp2 sites [5]. Assuming constant (with respect to local hybridization and material structure) excitation energies, we previously [1] assigned the graphite peak-to-peak separations in Fig. 1 to Xp and Xr. These led, however, to slightly higher

L. Calliari et al. / Carbon 45 (2007) 1410–1418

than measured plasmon energies for graphite. For this reason, we take here a different approach. As mentioned above, features at 12.5 eV and 12.8 eV on the graphite and, respectively, the diamond EEL spectra are assigned to r ! r* transitions. This shows that, on the one hand, maxima in the r band Joint Density of States occur indeed at nearly the same energy for graphite and diamond and, on the other hand, that the r–r* peak-topeak separation of 15 eV could overestimate the average gap. We thus assume Xr = 12.5 eV which would correspond to measuring the band-to-band separation at 80% of the maximum intensity in Fig. 2. In an analogous way, we choose Xp = 3.4 eV (instead of the peak-to-peak separation, 3.8 eV), to better reproduce the p plasmon energy. For transitions involving bands of different symmetry, where we have made the approximation: X2pr  X2rp  X2p$r we take XpMr = 9 eV, i.e., the average between the two peak-to-peak separations measured in Fig. 2. 4.3. Approximate solutions for in-plane plasmons Relations (5), giving the energy of in-plane plasmons, do not readily provide the crucial factors affecting plasmon energies. For this reason, we considered [1] approximate solutions in addition to the exact ones. For the p + r plasmon, using the inequality Xp < Xr < x, we obtained x2pþr ¼ q  x2p þ

f X2p þ ð4  f ÞX2r 4

ð22Þ

which shows that the p + r plasmon energy is increased pffiffiffi above the free electron energy, xp q, by a term containing the characteristic excitation energies of p and r electrons, their weight being modulated by the sp2 fraction f. For the p plasmon, we considered the expression for the graphite r electron polarizability: 3 2   x 4 p  vr xp-graphite ¼  X2r  x2p-graphite

ð23Þ

and, using x < Xr, we obtained x2p ¼

fq 2 xp 4 Þ 1 þ ð4f q  vr ðxp-graphite Þ 3

þ X2p

ð24Þ

The relation shows that the p plasmon qffiffiffiffi energy is increased above the free electron value, xp f4q, by the contribution of the p electron characteristic excitation energy, Xp. In addition however, it shows that xp is affected by the r electron polarizability, vr(x). Since the latter is positive at the frequencies of interest here, it is responsible for the wellknown marked decrease in the graphite p plasmon energy x below the free electron value 2p ¼ 12:5 eV [13,20]. The background polarization provided by r electrons is not constant, but depends on f. It increases as f decreases, this

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reflecting the increasing number of r electrons available for screening as the number of p electrons decreases. For graphite, Eq. (24) reduces to x2p

¼

x2p 4

1 þ vr ðxp Þ

þ X2p

ð25Þ

giving vr ðxp Þ ¼ 4:1 which is used to calculate xp from (24). 5. Results and discussion Though the mass density and sp2 fraction of a-C materials are interconnected [21], due to the growth conditions, it is useful to regard them as independent in the analysis of the plasmon energy formulas derived in the preceding section. The behaviour of plasmon energies is given in Figs. 3–5 as a function of q and f. Fig. 3 shows the p + r plasmon energy. Panel a provides the q dependence of xp+r at constant sp2 fraction (f = 1) while panel b provides the f dependence of xp+r at constant mass density (q = 1). On both panels, we plot the exact (full squares) and approximate (open circles) solution, and we see that the latter closely reproduces the former. Moreover, we see that xp+r strongly depends on the material density, while it is only slightly affected by the sp2 fraction (less than 1 eV over the entire range). For graphite (q = 1, f = 1), we have – see panel a – xp+r = 27.4 eV, to be compared with xp+r = 27 eV in Fig. 1, while for diamond (q = 1.55, f = 0) we derive – from Eq. (6) – xr = 33.5 eV, to be compared with xr = 33.8 eV in Fig. 1. Such an agreement between measured and calculated plasmon energies supports the assumption of a r–r* separation independent from the detailed structure of carbon systems. On panel a, the p + r plasmon energy for a free electron gas (open triangles) is added for comparison. The free electron energy lies, over all the mass density range, below the energies determined here. This corresponds to the wellknown fact that the free electron approximation is unable to account for the measured plasmon energies of graphite and diamond. Such a failure clearly comes from neglecting the contribution of the p–p* and r–r* splitting to the plasmon energy. It might be worth noting that, within the free electron approximation, plasmon energies depend on the electron density only and not on structural features of the material, such as, for example, the sp2 fraction. As a consequence, a comparison analogous to that shown on panel a, is not possible for panel b. In Fig. 4, exact (full squares) and approximate (open circles) solutions, are plotted for the p plasmon energy xp. As above, panel a provides the q dependence at constant sp2 fraction (f = 1), while panel b provides the f dependence at constant mass density (q = 1), the latter being meaningful

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L. Calliari et al. / Carbon 45 (2007) 1410–1418

Eπ+σ (eV)

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Fig. 3. p + r In-plane plasmon energy as a function of density with f = 1 (panel a) and sp2 fraction with q = 1 (panel b). Exact solutions are given by full squares (j), approximate solutions by open circles (s). The free electron energy (n) is added to panel a.

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1. 0

0.0

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Fig. 4. p In-plane plasmon energy as a function of density with f = 1 (panel a) and sp2 fraction with q = 1 (panel b). Exact solutions are given by full squares (j), approximate solutions by open circles (s). The free electron energy (n) is added to panel a.

only for f 5 0. Again we see that the approximate solution closely reproduces the exact one. For graphite, we obtain xp = 6.5 eV, in agreement with Fig. 1. At variance however with respect to the p + r plasmon, xp strongly depends on the sp2 fraction, while it is only weakly affected by the mass density (0.5 eV for 0.5 < q < 1), unless the relative density drops to very low, unphysical values. It had already been observed [5,13] that this plasmon energy, being affected by r electron screening, does not provide a reliable measurement of the mass density of C materials. To this observa-

tion, we add that xp is rather insensitive to the mass density, this representing an additional reason why it should be disregarded as a mass density indicator for C materials. The free electron curve (open triangles), exhibited on panel a, is always higher than xp obtained in this work, at least in the mass density range of interest (q > 0.5). Neglecting – as the free electron approximation does – the role of Xp, would decrease xp (as it happens for xp+r). This has however only a minor effect in the present case, while a major effect comes from neglecting the screening due to

L. Calliari et al. / Carbon 45 (2007) 1410–1418 22 20 18 16

Eπ<->σ (eV)

14 12 10 8 6 4 2 0 0.0

0.2

0.4

ρ

0.6

0.8

1.0

Fig. 5. p + r Out-of-plane plasmon energy as a function of density with f = 1.

the r electron polarizability, a simplification which greatly increases the free electron p plasmon energy above the measured value. Fig. 5 presents the q dependence for the out-of-plane p M r plasmon energy. In this case, we only have the exact solution, whose functional dependence is symmetrical with respect to q and f, so that only one curve is plotted. We can observe that, while xp is mainly affected by f and xp+r is mainly affected by q, xpMr is affected by the two variables to the same extent. For graphite, we have xpMr = 19.8 eV, consistent with the shoulder one can observe on the low energy side of the p + r plasmon peak (see Fig. 1). It could be tempting to alternatively assign this shoulder to a surface plasmon. If this were the case, its intensity should increase by enhancing the spectrum surface sensitivity, as it occurs, for example, when the energy of the primary electron beam decreases. Fig. 6 compares two spectra, acquired at 2 keV and 250 eV, after subtracting a

INTENSITY (arb. units)

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linear background and normalizing the p + r plasmon peaks to a common height. Electron inelastic mean free ˚ and 7 A ˚, paths associated with the two energies are 32 A respectively [22]. Though the peak-shape changes at high energy, no gain in intensity is observed on the low energy side of the peak on going from the bulk sensitive (2 keV) to the surface sensitive (250 eV) spectrum. This observation prevents us from assigning intensity in this region to surface related features and, on the contrary, supports assigning it to out-of-plane plasma oscillations. It is worth remembering that this component to the EEL spectrum of p bonded C materials can only be expected when the momentum transfer ~ q has a non-vanishing component along the c-axis, as it occurs for a-C, for materials based on curved graphitic sheets, for polycrystalline graphite or, for HOPG, when the acquisition geometry leads to excite both in-plane and out-of-plane oscillations. Since however for polycrystalline materials plasmon damping increases as the average size of the crystals decreases, plasmon peaks substantially broaden on going, for example, from graphite to a-C, where the graphite-like clusters can be very small. Resolving several contributions to the p + r plasmon peak becomes therefore problematic for such cases. 6. Conclusions Within a phenomenological approach, assuming negligible damping and dispersion, we have derived the energy of plasmon features for carbon systems as a function of the material mass density and sp2 fraction. The proposed solutions satisfy the basic requirement of reproducing the energy of plasmon peaks for systems, graphite and diamond, whose mass density and sp2 fraction are known. We have shown that crucial factors to reproduce measured plasmon energies are the energy of interband transitions, Xp, Xr and XpMr, and the screening of p electrons due to the r electron polarizability. To make the approach applicable to p bonded systems in general, we have introduced an ‘‘effective isotropic dielectric function’’ and we have shown that the plasmon modes are given in this case by the modes associated with in-plane and out-of-plane collective oscillations in graphite. Though plasmon broadening, because of cluster size effects, is likely to prevent from observing all plasmon modes for a-C materials, we have shown that, on the graphite spectrum, out-of-plane plasmons are indeed observed. Future steps of this work will deal, on the one hand, with removing the approximations still involved, and, on the other hand, with applying the model to unknown C materials. Acknowledgements

ENERGY LOSS (eV) Fig. 6. EEL spectra of HOPG excited by 2 keV (—) and 250 eV (  ) electrons. After subtracting a linear background, the spectra are normalized to a common height of the p + r plasmon peak.

The work was funded by Fondo Unico per la Ricerca (Provincia Autonoma di Trento) in the frame of ‘‘Microcombi’’ Project. A. Tucciarone and G. Verona-Rinati

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(Universita` di Roma ‘‘Tor Vergata’’) are kindly acknowledged for supplying us with the diamond sample. References [1] Calliari L, Fanchenko S, Filippi M. On a dielectric approach to understand electron energy loss spectra from carbon materials. Diamond Relat Mater 2007;16:1316–20. [2] Sto¨ckli T, Bonard JM, Chaˆtelain A, Wang ZL, Stadelmann P. Plasmon excitations in graphitic carbon spheres. Phys Rev B 1998;57:15599–612. [3] Stephan O, Taverna D, Kociak M, Suenaga K, Henrard L, Colliex C. Dielectric response of isolated carbon nanotubes investigated by spatially resolved electron energy-loss spectroscopy: from multiwalled to single-walled nanotubes. Phys Rev B 2002;66:155422(6). [4] Delfs H, Henninger R, Ding AM. Electron energy loss spectroscopy of free fullerenes – a dielectric model of the collective excitations. Synth Metals 1996;77:169–72. [5] Robertson J. Diamond-like amorphous carbon. Mater Sci Eng R 2002;37:129–281. [6] Theye ML, Paret V. Spatial organization of the sp2-hybridized carbon atoms and electronic density of states of hydrogenated amorphous carbon films. Carbon 2002;40:1153–66. [7] Werner WSM. Differential surface and volume excitation probability of medium-energy electrons in solids. Phys Rev B 2006;74:075421(14). [8] Filippi M, Calliari L, Pucella G, Verona-Rinati G. Temperature evolution of the surface region of CVD diamond: an electron spectroscopy study. Surf Sci 2004;573(2):225–36. [9] Pate BB. The diamond surface: atomic and electronic structure. Surf Sci 1986;165:83–142. [10] Ohno Y. Kramers–Kronig analysis of reflection electron-energy-loss spectra measured with a cylindrical mirror analyzer. Phys Rev B 1989;62:8209–19.

[11] Caputi LS, Chiarello G, Santaniello A, Colavita E, Papagno L. Secondary-electron emission and electron-energy-loss results on graphite single crystals. Phys Rev B 1986;34:6080–4. [12] Dapor M, Calliari L, Filippi M. Computational and experimental study of p and p + r plasmon loss spectra for low energy (<1000 eV) electrons impinging on Highly Oriented Pyrolytic Graphite (HOPG). Nucl Instr Meth B 2007;255:276–80. [13] Taft EA, Philipp HR. Optical properties of graphite. Phys Rev 1965;138(1A):A197–202. [14] Daniels J, Festenberg Cv, Raether H, Zeppenfeld H. Springer Tracts Mod Phys 1970;54:77–135. [15] Leiro JA, Heinonen MH, Laiho T, Batirev IG. Core-level XPS spectra of fullerene, highly oriented pyrolytic graphite, and glassy carbon. J Electron Spectrosc Relat Phenom 2003;128:205–13. [16] Philipp HR, Taft EA. Optical properties of diamond in the vacuum ultraviolet. Phys Rev 1962;127:159–61. [17] Egerton RF. Electron Energy Loss Spectroscopy in the Electron Microscope. New York: Plenum Press; 1996. pp. 131–299. [18] Nozie`res P, Pines D. Electron interaction in solids. Characteristic energy loss spectrum. Phys Rev 1959;113:1254–67. [19] Oelhafen P, Freeouf JL. Accurate spectrometer calibration in electron spectroscopy. J Vac Sci Technol A 1983;1:96–7. [20] Marinopoulos AG, Reining L, Rubio A, Olevano V. Ab initio study of the optical absorption and wave-vector-dependent dielectric response of graphite. Phys Rev B 2004;69:245419(12). [21] Ferrari AC, Libassi A, Tanner BK, Stolojan V, Yuan J, Brown LM, et al. Density, sp3 fraction, and cross-sectional structure of amorphous carbon films determined by X-ray reflectivity and electron energy-loss spectroscopy. Phys Rev B 2000;62:11089–103. [22] Tanuma S, Powell CJ, Penn DR. Calculations of electron inelastic mean free paths. Surf Interface Anal 2004;36:1–14.