Calculation of the graphene C 1 s core level binding energy
Toma Susi, Duncan J. Mowbray, Mathias P. Ljungberg, Paola Ayala
CCalculation of the graphene C 1 s core level binding energy Toma Susi, ∗ Duncan J. Mowbray,
2, 3
Mathias P. Ljungberg,
3, 4 and Paola Ayala University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre,Departamento de F´ısica de Materiales, Universidad del Pa´ıs Vasco UPV / EHU, E-20018 San Sebasti´an, Spain Donostia International Physics Center, Paseo Manuel de Lardizabal, 4. E-20018 Donostia-San Sebasti´an, Spain Deparment of Physics, Phillips-University Marburg, Renthof 5, 35032 Marburg, Germany (Dated: August 16, 2018)X-ray photoelectron spectroscopy (XPS) combined with first principles modeling is a powerful tool for deter-mining the chemical composition and electronic structure of novel materials. Of these, graphene is an especiallyimportant model system for understanding the properties of other carbon nanomaterials. Here, we calculate thecarbon 1 s core level binding energy of pristine graphene using two methods based on density functional theorytotal energy di ff erences: a calculation with an explicit core-hole ( ∆ KS), and a novel all-electron extension of thedelta self-consistent field ( ∆ SCF) method. We study systematically their convergence and computational work-load, and the dependence of the energies on the chosen exchange-correlation functional. The ∆ SCF method iscomputationally more expensive, but gives consistently higher C 1 s binding energies. Although there is a signif-icant functional dependence, the binding energy calculated using the PBE functional is found to be remarkablyclose to what has been measured for graphite. PACS numbers: 31.15.ag, 73.22.Pr, 79.60.-i, 81.05.ue
X-ray photoelectron spectroscopy (XPS) is a powerful toolfor studying the surface composition of materials. Morerecently, it has emerged as a particularly useful probe forlow-dimensional carbon-based nanomaterials such as carbonfibers , thin films , nanotubes , and graphene . Measuredbinding energies are often compared to molecular referencevalues to identify the corresponding atomic structures. How-ever, for novel nanomaterials, appropriate references are ofteneither not available, or it is unclear if they are directly appli-cable. Together with increases in computational power andmethod development, first principles modelling has gainedmore applicability for directly calculating the binding ener-gies — or at least the chemical shifts — of desired atomicconfigurations .The photoemission process can be conceptually dividedinto three basic steps. First, an X-ray photon is absorbed andtransfers its energy to a single core electron, creating a pho-toelectron. Then, this electron makes its way to the surfaceof the material. Finally, the electron escapes from the surfaceinto the vacuum. Experimentally, the need for knowing thework function of the material in the last step is bypassed byreferencing the binding energies to the Fermi level of the ma-terial, which is a well-defined procedure for systems withouta band gap.For calculating core level binding energies, two types ofmethodologies are typically applied: the so-called initial stateand final state methods . In the initial state methods, only theenergy level of the core electron before ionization is consid-ered, often by simply calculating its Kohn–Sham (KS) orbitaleigenvalue using density functional theory (DFT), referencedto the Fermi level. This is typically accomplished by explic-itly including the core level via an all-electron (ae) calculation.Initial state methods have the advantage that the KS eigenen-ergies may be calculated for all atoms of the system withina single calculation. The justification for this procedure isa linearization around the ground state of Janak’s theorem , which states that the orbital energy is the derivative of the to-tal energy with respect to the orbital occupation. However,the absolute values of carbon core levels are typically under-estimated by about 10% by DFT , partly because core-holerelaxation is disregarded within this approximation .In the final state methods, the core-hole is explicitly in-cluded in a second calculation, and the electronic structure re-laxed in its presence. The binding energy of the core electronis then computed from the total energy di ff erence between theexcited state with the core-hole ( E ex ) and the initial groundstate configuration ( E gs ). Since only total energy di ff erencesare used in the calculation, final state methods take advantageof DFT’s high level of accuracy with respect to total ener-gies, and avoid the well-known problems of describing energylevels using KS eigenvalues. However, a separate calculationmust be performed for each atom of interest. The Slater tran-sition state method should also be mentioned, where the exci-tation energy is calculated from the orbital energy di ff erencesin a state halfway between the initial and final states, that is,with a non-physical half-core-hole. However, this method isin general not as accurate as the final state methods, and itshares their complication of requiring an explicit core-hole.Modeling the final state with a core-hole is significantlymore challenging than a ground state calculation. If the core-hole is introduced via a projector augmented-wave (PAW)dataset (i.e., a PAW setup) or within an atomic pseudopoten-tial, the atom becomes charged in the final state. A periodic“bare core-hole” calculation would require a huge supercell toproperly include this charge distribution . Further, for low-dimensional materials, the long-ranged Coulomb interactionintroduces an additional slow convergence of the total energywith the amount of vacuum . These issues may be partly ad-dressed by explicitly including an extra electron charge withinthe conduction band of the material, resulting in a so-called“screened core-hole”. However, using a PAW dataset or apseudopotential does not allow the other core electron(s) to a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec relax, which may limit the accuracy of the absolute bindingenergies . Although a rigid shift can be applied to align thecalculated values with experiment, this assumes that the ef-fect of core-hole relaxation is of identical magnitude for everyatom of interest — which can be a priori uncertain for atomsof di ff erent elements. Thus, accurate absolute values from aphysically motivated calculation are of great practical interest.As the prototypical low-dimensional carbon nanomate-rial, graphene is useful for understanding the structureand often also the properties of other interesting materials,such as carbon nanotubes. Significant e ff orts have been di-rected towards modifying its properties, such as opening aband gap or tuning the carrier concentration, by chemicalfunctionalization or by heteroatom doping . For suchstudies, a chemically sensitive quantitative probe like XPS isa vital tool for discerning the amount and bonding of dopantatoms or functional groups.Here, we calculate the C 1 s core level binding energy ofpristine graphene using two methods based on DFT: a deltaKohn–Sham ( ∆ KS) calculation using a PAW-dataset includ-ing an explicit core-hole, and a novel application of the deltaself-consistent field ( ∆ SCF) method including the core levelswithin an all-electron calculation (see Ref. for a note on thenomenclature). We study the convergence and computationalworkload of both methods, the functional dependence of theenergies, and show how the magnetic moment a ff ects the ∆ KSresults.Our DFT calculations were performed with the grid-basedprojector augmented-wave simulation package gpaw . Ex-change and correlation were estimated by the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation , andthe LDA , PW91 , revPBE and RPBE functionals testedin selected cases. We applied periodic boundary conditionsin orthorhombic unit cells of 2 to 11 elementary lattice units,yielding supercells with 8 to 242 carbon atoms. Monkhorst-Pack × ×
1, 5 × × × × k -point meshes were applieddepending on the cell size (yielding 3, 5 and 8 k -points in theirreducible part of the Brillouin zone). The relaxed graphenelattice parameter was a = s was h ≈ .
19 Å (spacingsdown to 0.10 Å were tested).In the ∆ KS total energy di ff erences method , the corelevel binding energy is the total energy di ff erence between afirst core ionized state and the ground state in a spin-polarizedcalculation. To make the unit cell charge-neutral, a compen-sating electron charge is introduced into the conduction band.This is a good approximation for metals (including graphene)where core-hole screening is e ffi cient. We additionally inves-tigated the e ff ect of di ff erent magnetic moments of the finalstate. For a singlet, we initialized the magnetic moment ofthe core-hole atom to 1.0 Bohr magnetons (counting valenceelectrons, with the core-hole in spin up) and fixed the totalmagnetic moment, and also ran fixed calculations with -1.0Bohr magnetons (triplet). Otherwise the magnetic momentwas allowed to relax freely.We then turned to the delta self-consistent field ( ∆ SCF)method implemented in gpaw . As a modification to includecore levels in the calculation, we used so-called “pseudoatom” ∆SCF∆KS E vac E F e − s screening e − valencecore valence(no core) E B = E e x − E g s FIG. 1. A schematic illustration of the core level binding energy ( E B )of graphene, which we calculate as the di ff erence between the excitedstate ( E ex ) and ground state ( E gs ) total energies. In both the ∆ KS andthe ∆ SCF excited states, one electron ( e − ) is removed from the 1 s core state to vacuum ( E vac ), and a compensating electron charge isintroduced at the Fermi level ( E F ). However, in ∆ KS, the core isdescribed by a PAW dataset including an explicit core hole, whilein ∆ SCF, all electrons are included in the valence and the core holedescribed by subtracting the density of a spin-up carbon 1 s orbital. all-electron datasets. In this recently implemented feature, thecore states are included in the valence, enabling an explicit aecalculation within the PAW scheme (note that this is di ff er-ent from the relaxed core method of Marsman and Kresse ).In a ∆ SCF calculation, the density of a specified orbital ϕ a ( r )(in this case a spin-up carbon 1 s orbital) is subtracted from thetotal density in each step of the self-consistency cycle. As inthe ∆ KS method, the missing core charge is compensated byan extra electron in the conduction band. Figure 1 illustratesthe methods schematically. Finally, we tested the influence ofusing ae datasets on other atoms in the system in both the ∆ KSand the ∆ SCF calculations.Turning now to our results, we first studied the influenceof the compensating charge in the ∆ KS method by calculatingthe C 1 s energy of a charged 9 × z -axis in our geometry). Wefound convergence to be very slow, not reaching a constantvalue even for a separation of 50 Å. Furthermore, the calcula-tions trended towards a significantly too high binding energy(288.15 eV). However, when the system was made charge-neutral, only 8 Å of vacuum was enough to converge the C1 s energies. (For the charge-neutral unit cell, non-periodicboundary conditions in the z -direction yielded no di ff erenceto a periodic calculation.)Concluding thus that the extra charge is needed, we con-sidered the convergence of the energies as a function of thesupercell size and the number of k -points in the calculation.For even the smallest 2 × k -point mesh of 7 × × ∆ k < − ) was enough to converge both theground and excited state energies to within 1 meV per atom.However, although the absolute changes in energy were notlarge, convergence of the excited state energy was found to berather slow as a function of system size. This is likely due tothe long-range Coulomb interaction between periodic images a e - D S C F f c + a e - D S C Fa e + f c - D K S f c - D K S
U n i t c e l l s i z e ( n · n ) C 1 s binding energy (eV) FIG. 2. (color online.) The graphene C 1 s binding energy as a func-tion of supercell size calculated with the ∆ SCF and ∆ KS methodsusing frozen core (fc) and / or all-electron (ae) PAW datasets as de-scribed in the text. A su ffi cient number of k -points were employedthroughout. Decaying exponential fits yield asymptotic limits (dot-ted horizontal lines) representing extrapolations for fully convergedvalues (Table I). of the core-hole, which destabilizes the final state and arti-ficially increases the excited state energies. Overall, for thelargest unit cells (9 × k -point mesh of3 × × ffi cient for full convergence.In Fig. 2, we have plotted the k -point converged bindingenergies for each unit cell size from the ∆ KS and the ∆ SCFcalculations. For each method, we have fitted the data with de-caying exponentials, whose y -o ff sets give estimates for fullyconverged C 1 s energies. We see that for the largest com-putationally tractable 11 ×
11 unit cell containing 242 carbonatoms, the ∆ KS values are converged to within 50 meV. Formore standard sizes like the 6 × + fc- ∆ KS), the converged value was raised by only30 meV compared to the all-fc calculation. Conversely, whenwe performed ∆ SCF calculations with an ae dataset just onthe target atom and normal fc datasets on other atoms (fc + ae- ∆ SCF), we see that the ae- ∆ SCF values are systematicallyonly 30 meV higher in energy. Thus the relaxation of coreelectrons on neighboring atoms does not appear to be signifi-cant. Furthermore, fixing the magnetic moment to the singletvalue in the ∆ KS calculation was found to raise the convergedvalue by about 0.1 eV, with the triplet being about 30 meVlower than this.Experimentally, the reference value of the C 1 s bindingenergy of graphite is 284.42 eV . For graphene, valuesfound in the literature range from 283.97 eV for grapheneon Pt(111) , 284.15 on Ir(111) , 284.2 eV on Au-intercalated Ni(111) , 284.47 eV for suspended few-layergraphene , 284.6 eV on hydrogen-intercalated SiC , 284.7eV on Ni(111) , to 284.8 eV on SiC . While it is thus clearthat charge transfer from and screening by the substrate a ff ectthe measurements significantly, the exact value for freestand- TABLE I. Converged graphene C 1 s binding energies calculated withthe methods described in the text using the PBE functional. Thelast two columns give the CPU time scaling α kN β prefactors andexponents. C 1 s ScalingMethod (eV) 10 − α β fc- ∆ KS 283.58 0.06 2.86ae + fc- ∆ KS 283.61 36 2.14fc + ae- ∆ SCF 284.29 14 drr2.31ae- ∆ SCF 284.33 7.2 2.59 ing single-layer graphene has not been fully established.Taking the graphite value as the experimental referenceagainst which to evaluate the data in Fig. 2, we can see thatthe PBE ∆ KS underestimates the binding energy by about 0.8eV, as we observed before . However, when using the ∆ SCFmethod, the relaxation of the other core electron of the targetatom is included in the description, unlike with the frozen-core (fc) PAW datasets. With the fully ae ∆ SCF method, weget a converged C 1 s energy of 284.33 eV, constituting onlya 0.03% di ff erence to the experimentally reported graphitebinding energy. (Although currently only possible in the ∆ KSmethod, fixing the spin state of the extra charge to a singletwould likely have a similar magnitude e ff ect also for the ∆ SCFvalue, raising the C 1 s energy by a further 0.1 eV.)However, the near-perfect agreement with the graphitemeasurement that results should be considered fortuitoussince the choice of the exchange-correlation functional wasfound to a ff ect the energies by several tenths of an eV. Tosee this, we selected the fc + ae- ∆ SCF and fc- ∆ KS methods,and looked at the C 1 s values calculated for the 9 × . Thus, whilethe functional dependence can be used as an estimate for theuncertainty in our calculated values, the functional that repro-duces the experimental value best may be considered the mostuseful for core level calculations using this methodology. Weshould also note that the total energy (including atomic ref-erence energies) of the fc + ae- ∆ SCF ground state was consis-tently about 0.25 eV lower and the excited state about 0.3 eVhigher than the corresponding fc- ∆ KS ones. Although calcu-lations with a finer grid lowered both ground and excited stateenergies, this did not a ff ect the total energy di ff erences appre-ciably.We further reconstructed the all-electron densities for eachcalculation, and computed di ff erences between the ∆ KS (Fig-ure 3 a-c) and ∆ SCF excited states and ground states (Fig-ure 3 g-i), and between the two excited states (Figure 3 d-e).The isosurfaces displaying the di ff erences between the excitedstate and ground state charge densities in each method lookvery similar, confirming that the forced occupation of the coreorbital in the ∆ SCF method reproduces the general features ofthe better tested frozen core-hole dataset. Only by looking at
TABLE II. The functional dependence of our calculated C 1 s ener-gies with the fc- ∆ KS and fc + ae- ∆ SCF methods in the 9 × ∆ KS fc + ae- ∆ SCFXC C 1 s (eV) C 1 s (eV) Di ff erence (eV)LDA 280.90 281.32 0.42PBE 283.77 284.33 0.58PW91 284.02 284.69 0.71revPBE 284.15 284.84 0.69RPBE 284.30 284.99 0.69 a hgfedb c i FIG. 3. (color online.) All-electron charge density di ff erence isosur-faces calculated in the 6 × ∆ KS excitedand ground state (side view in b), (g-i) ∆ SCF excited and groundstate (side view in g), and (d-f) ∆ KS excited state and ∆ SCF excitedstate (side view in f). Positive values are denoted in red and negativein blue (isovalues ± ± ± ± / Å (e,f)). the di ff erence between the two excited state densities plottedat low isovalues (Figure 3 d-f), subtle di ff erences between thetwo methods can be seen near the core-hole atom.Finally, we considered the computational e ff ort required tocomplete each calculation (total running time multiplied bythe number of cores). The computational time scales theoret-ically with the number of atoms N in the supercell and with the number of k -points in the irreducible part of the Brillouinzone. We can thus model the CPU time data as α kN β and usethe scaling prefactors α and exponents β as given in Table Ito compare the di ff erent methods. As an example of actualtimes, for an 8 × ∆ SCF, fc + ae- ∆ SCF, ae + fc- ∆ KS, and fc- ∆ KS methodstook 20.7, 10.0, 11.4 and 0.76 CPU-hours to complete, re-spectively. Thus, we can see that the fc- ∆ KS calculations aremuch faster than the other methods.To conclude, our results indicate that prohibitively largeunit cells are required to completely converge the C 1 s corelevel binding energy of graphene using DFT calculations withperiodic boundary conditions. However, for larger systemsizes, convergence within 50 meV is reached and the under-estimation is systematic. Thus, when choosing a size forthe computational unit cell, one can balance considerationsof computational e ffi ciency (when a large number of systemsor target atoms need to be simulated) with for example therequirement of having a realistic concentration of defects ordopants. However, although computationally cheap, the ∆ KScalculations underestimate the experimentally expected valueby about 0.8 eV. By performing physically motivated ∆ SCFcalculations using all-electron datasets, systematically higherbinding energies were obtained, although the exact value wasfound to be sensitive to the chosen exchange-correlation func-tional. Nonetheless, the PBE functional gives a C 1 s bindingenergy that is remarkably close to the experimental value. ACKNOWLEDGMENTS
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