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Core-level spectra from graphene

We calculate core-level spectra for pristine and doped free-standing graphene sheets. Instructions for how to perform the calculations are given in detail. Although pristine graphene is not metallic the core-level spectrum presents low-energy tailing which is characteristic of metallic systems. The peak shapes vary with doping level in a characteristic way. The spectra are compared to experiments and show good agreement. We compare to two different pristine samples and to one doped sample. The pristine samples are one with quasi-free-standing epitaxial graphene on SiC obtained by hydrogen intercalation and one with a suspended graphene sheet. The doped sample is a gold supported graphene sheet. The gold substrate acts as an acceptor so the graphene sheet gets p-doped.

The cb (conduction-band) electrons can have many different effects on the XPS peak from the core level. If the excitation could be considered adiabatic the screening of the core-hole by the cb electrons would lead to a shift of the peak to higher kinetic energy (lower binding energy of the core level). Now the excitation is not adiabatic, the excitation is swift. The excitation frequency is large compared to the frequency components taking part in the screening of the core-hole. This has the effect that the cb electrons are shaken up and are left in a non-equilibrium state. This is analogous to the following case: Assume that a cork is floating in a glass of water. If we remove the cork very slowly the water will be left very quiet and without any ripples. If we on the other hand remove it very briskly the water will be left in an upset state with many surface waves excited. In a metal the water waves correspond to plasmons, collective excitations of the cb electrons.
The shake-up effects will show up as plasmon replicas in the spectrum; the main peak will be at the adiabatic position; the first replica, a smaller copy of the main peak, will be at a shifted position to lower kinetic energy where the shift is equal to the plasma frequency; the second replica is even smaller and shifted with two plasma frequencies; this goes on until further peaks are absorbed by the background. The main peak corresponds to the system left in quasiequilibrium(fully equilibrium would demand that the core-hole were filled by one of the electrons), the second to a state where one plasmon is excited, the third to a state where two plasmons are excited and so on. Since all plasmons do not have identical energies, the plasmon dispersion curve shows dispersion, the replicas are not completely identical smaller versions of the main peak.
In a metal also single-particle excitations can take place. These electron-hole pair excitations form a contin-uum, starting from zero frequency and upwards. These excitations lead to a deformation, including a low-energy tail, of both the main peak and the plasmon replicas. Furthermore the finite life-time of the core-hole causes a lorentzian broadening of all peaks and experimental uncertainties give a gaussian broadening.
The low-energy tailing is a characteristic of a metallic system, i.e. a system where the chemical potential is inside an energy band and not in a band-gap, and can e.g. be used by experimentalists to find out where in a complex sample the core-hole is situated. Several parameters were introduced by Doniach andSunjić [6] to characterize the XPS line shapes and are still broadly used by experimentalists in the fitting procedure for XPS spectra.
In the present work we address pristine and doped graphene. In the pristine case the chemical potential is neither inside an energy band nor in a band-gap. The Fermi surface is just two points in the Brillouin zone. This makes this system special. As we will see there is still a low-energy tailing. In the doped case the chemical potential is inside an energy band and we would expect to find, and find, a tailing. However, the 2D (twodimensional) character of the system means that the collective excitations are 2D plasmons with a completely different dispersion than in the ordinary 3D metallic systems. The 2D plasmons give contributions that start already from zero frequency and upwards. This means that they contribute to the tail and no distinct plasmon replicas are distinguishable. Making any quantitative interpretation of graphene core-hole spectra using Doniach andSunjić fitting of the spectra is not feasible. It is more reasonable to calculate the spectra. One purpose of this letter is to provide the reader with the tools needed for such a calculation.
The XPS spectrum can be written as [8]  where and All variables have been scaled according to and are now dimensionless. The two last relations will be used below. The quantity d is the energy shift of the adiabatic peak. We have taken the finite life-time of the core-hole into account by introducing the first factor of the integrand of Eq.(1), where lw is the FWHM (Full Width at Half Maximum) of the Lorentz broadened peak.
We have here assumed that the core-hole potential can be represented by a pure Coulomb potential. The particular system enters the problem through Im ε(Q, W ) −1 . For pristine graphene the dielectric function is [9] where we have used the values [10] 8.73723 × 10 5 m/s for the fermi velocity v and 2.4 for κ, respectively. The background dielectric constant κ is the result of high frequency electronic excitations to higher lying empty energy bands and from lower lying filled energy bands. When the graphene sheet is doped the dielectric function becomes much more complicated. However it has been derived by several groups. [10][11][12] The dielectric function in a general point in the complex frequency plane, W , away from the real axis is [13] ε (Q, where D 0 = 4n/π 2 v 2 is the density of states at the Fermi level and n is the doping concentration. In both Eqs. (6) and (7) we let W be real valued but add a small imaginary part to get the retarded forms of the dielectric functions.
In Figs. 1 and 2 we show the contour plot and surface plot, respectively, of −Im ε(Q, W ) −1 for doped graphene.
The integrands in Eqs.
(2), (3), and (4) all have an extra factor 1/W for small W which means that small energy transfers are important in the shake up structure of the spectra. In Fig. 3 we give the corresponding surface plot for pristine graphene. Here the excitations are of electron-hole pair type. We have taken the lorentzian broadening into account but how should we include the gaussian instrumental broadening? Here we use a trick. If all shake up processes were involving only one discrete frequency, ω 0 , the a-, b-, and D-functions would become where η is a strength constant and the spectrum would consist of a series of delta functions, The amplitudes of the delta functions form a poisson distribution. When the strength parameter η is large this turns into a gaussian distribution. It is close to a gaussian already at η = 5. The gaussian FWHM is given by gw = 2 ω 0 √ η2 ln 2. The trick is now to choose a small enough value for ω 0 so that η becomes large enough and add the resulting a-, b-, and D-functions to the original functions in Eq. (1). Thus we get the final spectrum with both lorentzian and gaussian broadened structures from In Fig. 4 we compare our results to an experimental XPS spectrum [14] represented by red circles. The experiment was performed on a single quasi-free-standing epitaxial graphene layer on SiC obtained by hydrogen intercalation [15]. This leads to a virtually undoped graphene  sheet. The photon energy used was 750 eV. The leftmost peak is from graphene C 1s and the rightmost from a Si core-level in the substrate. The dashed curve is the result one would get if there were no interaction between the electrons and the core-hole. Another way to express this is to say that the electron wave functions are frozen. The energies are given relative the position of this peak. The peak is symmetric and has the full broadening but no structure from shake up effects. The thick solid curve is the adiabatic peak. This is what the spectrum would look like if the excitation were adiabatic, i.e., the excitation were so slow that the electrons were left in a quasiequilibrium state; the core-hole potential were fully screened. This peak is just the dashed peak shifted by the energy d. The thin solid curve is the full result from Eq. 10. When we used the dielectric function from pristine graphene the curve emerged a little above the experimental tail. When we added a very small doping level, 10 11 cm −2 , the agreement with experiment was perfect. The reason a small amount of doping pushes down the tail a little is that there is some extra contributions at the top of the peak and the normalization then leads to a reduction away from the centrum of the peak.
In Figs. 5 and 6 we compare our results for pristine and doped free-standing graphene to the experimental results in Ref. [16]. The experiments were performed using a photon energy of 480 eV on a suspended single graphene sheet and on a gold supported single graphene sheet. In The experimental (µ-XPS) C 1s corelevel spectrum of a suspended graphene sheet obtained in [16] using the photon energy 480 eV, red circles. The solid curve is the theoretical result for an undoped free-standing graphene sheet. A linear background was subtracted from the experimental data. We used lorentzian and gaussian broadening of .4 eV and .666 eV, respectively the first sample the graphene is more or less undoped; in the second gold provides p-doping. We subtracted a linear background from both experimental sets of data and used the same broadening parameters for both spectra. The main peak in the doped sample is broader than in the undoped but this extra broadening comes from different shake-up effects in the two samples. The theoretical spectrum was obtained assuming a doping density of 3.2 × 10 13 cm −2 . This number 3.2 × 10 13 should not be taken too seriously. We varied the density in equidistant steps on a logarithmic scale and tried 10 13 cm −2 , 10 13.5 cm −2 and 10 14 cm −2 . The one in the middle gave the best fit with experiments.
Finally, in Fig. 7 we show the theoretical results for different doping levels. In order to more clearly see the effect doping has on the peak shape we have here removed all different shifts, d, for the curves. The solid black curve is the non-interacting curve or the adiabatic curve (they are identical now when the shift has been removed). The red dashed curve is the pristine result. The blue shortdashed curve is for the doping density 10 13 cm −2 , and the green dotted curve from a sheet with doping concentration 10 14 cm −2 . Important doping effects show up first at doping concentrations exceeding 10 12 cm −2 . The reason is that some electron-hole-pair excitation-channels are blocked with doping due to the Pauli exclusion principle; this is compensated by new electron-hole-pair excitationchannels and the new plasmon-excitation channel.
To summarize, we have calculated core-level spectra for pristine and doped free-standing graphene. Although, pristine graphene is not a metal its core-level spectrum shows a peak tailing, characteristic of metallic systems. . (Color online) The shape of the C 1s core-level spectra of free-standing graphene. To get a better view of the peak shapes we have removed the over all shift, d, from all spectra. The symmetric peak is the result from the fully adiabatic approximation and the result when there is no interaction between the core hole and the electron-hole system. The red dashed curve is the result from the undoped graphene sheet, the blue short-dashed from a sheet with doping concentration 10 13 cm −2 , and the green dotted curve from a sheet with doping concentration 10 14 cm −2 , respectively. The curve for doping concentration 10 12 cm −2 has been omitted since it is very close to the curve for pristine graphene.
The tailing increases with doping for doping concentrations exceeding 10 12 cm −2 . The peak shape changes with further increase of the doping concentration. This opens up for a complementary way to estimate the degree of doping of a sample.