Absolute binding energies of core levels in solids from firstprinciples
The original has been published in Phys. Rev. Let. 118, 026401 (2017)
Taisuke Ozaki and ChiCheng Lee
Institute for Solid State Physics, The University of Tokyo, Kashiwa 2778581, Japan
January 10, 2017
Abstract
A general method is presented to calculate absolute binding energies of core levels in metals and insulators,
based on a penalty functional and an exact Coulomb cutoff method in a framework of the density functional theory.
The spurious interaction of core holes between supercells is avoided by the exact Coulomb cutoff method,
while the variational penalty functional enables us to treat multiple splittings due to chemical shift,
spinorbit coupling, and exchange interaction on equal footing, both of which are not accessible by previous methods.
It is demonstrated that the absolute binding energies of core levels for both metals and insulators
are calculated by the proposed method in a mean absolute (relative) error of 0.4 eV (0.16 %) for eight cases
compared to experimental values measured with Xray photoemission spectroscopy
within a generalized gradient approximation to the exchangecorrelation functional.
Since the pioneering works by Siegbahn et al. [1,2], the Xray photoelectron spectroscopy (XPS)
has become one of the most important and widely used techniques in studying chemical composition
and electronic states in the vicinity of surfaces of materials [3].
Modern advances combined with synchrotron radiation further extend its usefulness to enable
a wide variety of analyses such as core level vibrational fine structure [4],
magnetic circular dichroism (MCD) [5], spinresolved XPS [6],
and photoelectron holography [7].
The basic physics behind the still advancing XPS measurements is dated back to the first interpretation
for the photoelectric effect by Einstein [8].
An incident Xray photon excites a core electron in a bulk, and the excited electron with
a kinetic energy is emitted from the surface to vacuum. The binding energy of the core
level in the bulk can be obtained by measuring the kinetic energy [1,2].
Theoretically, the calculation of the binding energy involving evaluation of
the total energies for the initial and final states is still a challenging issue
especially for insulators, since after the emission of the photoelectron the system is
not periodic anymore and ionized due to the creation of the core hole.
The violation of the periodicity hampers use of conventional electronic structure methods
under a periodic boundary condition, and the Coulomb potential of the ionized bulk
cannot be treated under an assumption of the periodicity due to the Coulombic divergence.
One way to avoid the Coulombic divergence is to neutralize the final state with a core hole
by adding an excess electron into conduction bands [9,10,11,12,13]
or to approximate the bulk by a cluster model [14].
However, the charge compensation may not occur in insulators because of the short escape time
of photoelectron ( sec.) [15], while the treatment might be
justified for metals.
Even if we employ the charge compensation scheme, the screened core hole pseudopotential
which has been widely used in pseudopotential methods allows us to calculate only the chemical
shift of binding energies, but not the absolute values [9].
In spite of the long history of XPS and its importance in materials science,
a general method has not been developed so far to calculate the absolute binding energies
for both insulators and metals, including multiple splittings due to chemical shift, spinorbit coupling,
and exchange interaction, on equal footing [16].
In this Letter we propose a general method to calculate absolute binding energies of core levels
in metals and insulators, allowing treatment of all the issues mentioned above and the direct
comparison to experimental results,
in a single framework within the density functional theory (DFT) [19,20].
Let us start to define the absolute binding energy of core electrons in bulks measured
by a XPS experiment, based on the energy conservation. The energy of the initial state is given by
the sum of the total energy of the ground state of electrons and an energy of
a monochromatic Xray photon.
Figure 1:
Schematic energy diagram for a sample and a spectrometer in the XPS measurement.

On the other hand, that of the final state is contributed by the total
energy
of the excited state of electrons with a core hole, and the kinetic energy
of photoelectron placed at the vacuum level as shown in Fig. 1.
Therefore, the energy conservation in the XPS measurement is expressed by



(1) 
Noting that the chemical potential of sample is aligned with that of the spectrometer
by Ohmic contact, and that the vacuum level of the spectrometer is given by
using the work function of the spectrometer
,
Eq. (1) reads to



(2) 
The left hand side of Eq. (2) is the binding energy
measured
by the XPS experiment [21].
The right hand side of Eq. (2) provides a useful expression to calculate
the absolute binding energy
for bulks regardless of the band gap of materials.
Instead of using the experimental chemical potential , it is possible to rewrite
the total energies with the intrinsic chemical potential , where intrinsic means
a state which is free from the control
of chemical potential, by noting that what the shift of chemical potential
does is
only the constant shift of potential . Then, we rewrite them as
and
using the intrinsic total energies
and
by assuming the common chemical potential for both the initial and final states
due to a very large .
Inserting these equations into Eq. (2) yields



(3) 
Equation (3) is an important consequence, since only quantities which can be calculated from firstprinciples
are involved. Thereby, we use Eq. (3) to calculate
the absolute binding energy
.
Note that the chemical potential is calculated by assuming the Fermi distribution at a finite
electronic temperature.
It should be emphasized that Eq. (3) is valid even for semiconductors and insulators.
In an arbitrary gapped system, the common chemical potential is pinned at either the top of valence band or
the bottom of conduction band, or located in between them.
For all the possible cases, the exactly same discussion above is valid.
Thus, we conclude that Eq. (3) is a general formula to calculate
the absolute binding energy
of solids.
Especially for metals, Eq. (3) can be further reorganized by noting a rigorous relation derived with the Janak theorem [23]:
,
where is an occupation number of an oneparticle eigenstate on the Fermi surface, defined with the area of
the Fermi surface and an infinitesimal area , and the surface integral is performed over the Fermi surface.
By inserting the above equation into Eq. (3), we obtain the following formula:



(4) 
which allows us to employ the total energy of the neutralized final state
instead of that of the ionized state. For metals, Eqs. (3) and (4) should result in an equivalent
binding energy in principle, however, the convergence is different from each other as a function of
the system size as shown later on.
Since
and in Eqs. (3) can be calculated by a conventional approach
with the periodic boundary condition, we now turn to discuss a method of calculating
in Eq. (3)
based on the total energy calculation including many body effects.
Core electrons for which a core hole is created are explicitly included in the calculations
to treat multiple splittings due to chemical shift, spinorbit coupling, and exchange
interaction between core and spinpolarized valence electrons, and to take account
of many body screening effects. The creation of the core hole can be realized by expressing
the total energy of the final state by the sum of a conventional total energy
within DFT and a penalty functional as



(5) 
with the definition of :



(6) 
where
is the integration over the first Brillouin zone whose volume is ,
the Fermi function,
and
the KohnSham wave function of twocomponent spinor.
In Eq. (6) the projector is defined with
an angular eigenfunction of the Dirac equation under a spherical potential
and a radial eigenfunction obtained by an atomic DFT calculation
for the Dirac equation as



(7) 
with for
and
and for
and
where is a spherical harmonic function, and and spin basis functions.
The variational treatment of Eq. (5) with respect to leads to the following KohnSham equation:



(10) 
where is the kinetic operator, and the conventional KS effective potential
originated from .
If a large number, 100 Ryd. was used in this study, is assigned for in Eq. (7),
the targeted core state specified by the quantum numbers and is penalized
through the projector in Eq. (10),
and becomes unoccupied, resulting in the creation of a core hole for the targeted state.
Since the creation of the core hole is selfconsistently performed, the screening effects by both
core and valence electrons, spinorbit coupling, and exchange interaction are
naturally taken into account in a single framework.
It is also straightforward to reduce the projector to the nonrelativistic treatment.
After the creation of the core hole, the final state has one less electron, leading to charging of the system.
In the periodic boundary condition, a charged system cannot be treated in general because of the Coulombic divergence.
The neutralization of the final state may occur in a metal, and theoretically such a neutralization can be
justified as shown by Eq. (4). However, it is unlikely that such a charge compensation takes place in an insulator
during the escape time of photoelectron ( sec.) [15].
To overcome the difficulty, we propose a general method of treating the charged state
based on an exact Coulomb cutoff method [24].
It is considered that the created core hole is isolated in the sample, resulting in violation of
the periodicity of the system.
The isolation of the core hole can be treated by dividing the charge density
for the final state into a periodic part
and a nonperiodic part
which, when integrated over the unit cell,
is exactly 1, where
is the charge density for the initial state without the core hole.
Then, as shown in Fig. 2(a) the Hartree potential
in the final state is given by



(11) 
where
is the periodic Hartree potential calculated using the periodic part
via a conventional method using a fast Fourier transform (FFT) for the Poisson equation.
On the other hand, the nonperiodic Hartree potential
is calculated
using
and an exact Coulomb cutoff method by



(12) 
where
is the discrete Fourier transform of
, and
is
given by
, which is the Fourier transform of a cutoff Coulomb potential
with the cutoff radius of [24].
Figure 2:
(a) Treatment of the Hartree potential in a system with a core hole under the periodic boundary condition.
(b) Configuration to calculate
the nonperiodic part of the Hartree potential
by the exact Coulomb cutoff method for .

If
is localized within a sphere of a radius as shown in Fig. 2(b),
the extent of the Coulomb interaction is at most in the sphere, which leads to .
In addition, a condition should be satisfied to avoid the spurious interaction
between the core holes. In practice, we set
, and investigate the convergence of
the binding energy as a function of . With the treatment the core hole is electrostatically
isolated from the other periodic images of the core hole even under the periodic boundary condition.
The proposed method has been implemented in a DFT software package OpenMX [25], which is based on
normconserving relativistic pseudopotentials [26,27] and pseudoatomic basis functions [28].
The pseudopotentials were generated including the state for a carbon, nitrogen, and oxygen atom, and
up to the states for a silicon atom [26,27], respectively, by solving the spherical Dirac equation.
In the pseudopotential generation, the occupation of the state for which a core hole is created in the final state calculations
is reduced by 0.5, and the remaining occupation is added to that of the valence states.
The pseudopotentials were commonly used in the calculations for the initial and final states.
The treatment for the occupation in generating the pseudopotentials might be a nonbiased way in applying
pseudopotentials for the initial and final states on equal footing. The KohnSham wave functions are expressed by
a twocomponent spinor, and the spatial part is expanded by a linear combination of pseudoatomic orbitals (PAOs).
As basis functions of PAOs variationally optimized double valence plus single polarization orbitals (DVSP)
and triple valence plus double polarization orbitals (TVDP) were used for bulks and gaseous molecules,
respectively, after careful benchmark calculations for the convergence [28,41].
The spinorbit coupling was taken into account selfconsistently through the jdependent relativistic pseudopotentials [26,27].
Neither a frozen core approximation nor any restriction for spin polarization were introduced for the core state
for which a core hole is created in the final state calculations.
A generalized gradient approximation (GGA) [29] to the exchangecorrelation functional was used.
It was assumed that electrons populate the KohnSham eigenstates according to the Fermi distribution at a finite temperature,
resulting the selfconsistent determination of the chemical potential depending on the band gap and shape of density
of states. For all the calculations in the study we used an electronic temperature of 300 K.
In all the cases we used the chemical potential calculated for the initial state without a core hole for Eq. (3).
The real space grid techniques were used with the energy cutoff of 350 Ry in numerical integrations and the solution of Poisson
equation using FFT [42].
Figures 3(a) and (b) show relative binding energies of core levels in gapped systems and metals
including a semimetal (graphene), respectively, as a function of intercore hole distance,
where all the molecular and crystal structures used in the study were taken from experimental ones.
For the gapped systems the convergent results are obtained at the intercore hole distance of
15, 20, and 27 Å for cubic boron nitride (diamond), bulk NH, and silicon, respectively.
Figure 3:
Calculated binding energies, relative to the most converged value, of (a) gapped systems
and (b) a semimetal (graphene) and metals as a function of intercore hole distance.
The reference binding energies in (a) and (b) were calculated by Eqs. (3) and (4), respectively,
for the largest unit cell for each system.

This implies that the difference charge
induced by the creation of the core hole
is localized within a sphere with a radius of , e.g., 7 Å for silicon.
In fact, the localization of
in silicon can be confirmed
by the distribution in real space and the radial distribution of a spherically averaged
as shown in Figs. 4(a) and (b). The deficiency of electron around 0.3 Å corresponding to
the core hole in the states is compensated by increase of electron density around 1 Å,
which is the screening on the same silicon atom for the core hole.
As a result of the short range screening, the nonperiodic Hartree potential
deviates largely from as shown in Fig. 4(c).
In Fig. 3(a) it is also shown that the binding energy of the bulk NH calculated
with Eq. (4) converges at 1.2 eV above, implying that Eq. (4) cannot be applied to gapped systems.
On the other hand, for the metallic cases we see that Eq. (4) provides
a much faster convergence than Eq. (3), and both Eqs. (3) and (4) seem to give
a practically equivalent binding energy, while the results calculated with Eq. (3) for
TiN and TiC do not reach to the sufficient convergence due to computational limitation [34].
Therefore, Eq. (4) is considered to be the choice for
the practical calculation of a metallic system because of the faster convergence.
By compiling the size of the unit cell achieving the convergence into the number of atoms
in the unit cell, the use of a supercell including 500 and 64 atoms
for gapped and metallic systems in threedimensions might be a practical guideline for achieving
a sufficient convergence by using Eqs. (3) and (4), respectively.
The calculated values of binding energies are well compared with the experimental
absolute values as shown in Table I for both the gapped and metallic systems,
and the mean absolute (relative) error is found to be 0.4 eV (0.16 %) for the eight cases.
We see that the splitting due to spinorbit coupling in the silicon states is well reproduced.
In addition, binding energies of a core level for gaseous molecules are shown in the supplemental material,
where the mean absolute (relative) error is found to be 0.5 eV (0.22 %) for the 23 cases.
Figure 4:
(a) Difference charge density
in silicon, induced by the creation of a core hole in the states,
where the unit cell contains 1000 atoms, and the intercore hole distance is 27.15 Å.
(b) Radial distribution of
, where
is a spherically averaged .
(c) Radial distribution of
being
a spherically averaged
.

In summary, we proposed a general method to calculate absolute binding energies of core levels
in metals and insulators in a framework of DFT. The method is based on a penalty functional
and an exact Coulomb cutoff method. The former allows us to calculate multiple splittings
due to chemical shift, spinorbit coupling, and exchange interaction,
while the latter enables us to treat a charged system with a core hole under
the periodic boundary condition.
It was also shown that especially for metals Eq. (4) involving the neutralized
final state is equivalent to Eq. (3) involving the ionized final state, and
that Eq. (4) is computationally more efficient than Eq. (3).
The remarkable agreement with the absolute binding energies measured in XPS
demonstrates the validity of the proposed method for a variety of materials.
For a better description of a case where an exchange interaction plays a dominant role
in the splitting, a good approximation to the exchangecorrelation functional should be
adopted and our method provides a natural way to examine the resulting total energies,
while a possible error by the pseudopotentials and the dependency of chemical potential
on surface structures should also be addressed in future work.
Considering the importance of the XPS measurement in materials researches,
the proposed method is anticipated to play an indispensable role
in quantitatively analyzing absolute binding energies of core levels in solids.
We would like to thank Y. YamadaTakamura and J. Yoshinobu for helpful discussions
on the XPS measurements.
This work was supported by Priority Issue (Creation of new functional
devices and highperformance materials to support nextgeneration
industries) to be tackled by using Post 'K' Computer, MEXT, Japan.

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