Atomic-Orbital Analysis of ZrB 2 Valence-Band by Two-Dimensional Photoelectron Spectroscopy

To investigate the electronic state of Zirconium diboride (ZrB2) (0001) clean surface, we measured twodimensional photoelectron intensity angular distribution (PIAD) patterns with a display-type spherical mirror analyzer (DIANA). As a result, we obtained two-dimensional band dispersion cross section patterns of ZrB2. In these patterns, we observed the transition-matrix-element effect of linearly-polarized light excitation. Comparing the experimental data to the calculation results, we concluded that the band at the K̄ point at a binding energy of 3.4 eV consists of B 2pz, Zr 4dyz, and Zr 4dzx orbitals. This suggests that this band is the bonding state between the B layer and the Zr layer. We confirmed that this hybrid orbital, where the d electrons of Zr are donated to the pz orbital of B, makes the Dirac-cone-like band of ZrB2. [DOI: 10.1380/ejssnt.2015.324]


I. INTRODUCTION
Zirconium diboride (ZrB 2 ) is an attractive material for wide applications.It has a high melting point, high electric conductivity, and high corrosion resistance.The ZrB 2 (0001) clean surface is terminated with a Zr layer in ZrB 2 [1].This surface is a promising substrate for GaN(0001) epitaxial thin film growth [2] and a substrate for sillicene [3].Although this material has been studied by angle-resolved ultraviolet photoelectron spectroscopy [2,4], the orbital characteristics of its energy band have not been studied.
Dirac-cone-like band structures at K point are commonly seen in graphene, and boraphene in MgB 2 (0001), ZrB 2 (0001) [5].MgB 2 is a novel superconductor with exceptionally high critical temperature (39 K) [6], which has the same crystal structure as ZrB 2 , but ZrB 2 shows no superconductivity at such a high temperature.Therefore, investigating the difference in the electronic structure between ZrB 2 and MgB 2 is very important to understand the properties of superconductors.The investigation of the orbital character of the electronic states is our original and it allows our deeper understanding of the electronic structure of ZrB 2 .This Dirac-cone-like band is the π-bond band in a layer of atoms arranged in a honeycomb structure.Because a B atom has only three valence electrons, a boraphene layer does not have π-bond electrons and this Dirac-cone-like band has been thought to be produced by charge transfer from metal atoms to the π-bond band of boraphene, which is the origin of the bonding between a metal layer and a boraphene layer.We can estimate the atomic orbitals constituting this π-bond band of ZrB 2 (0001) from the angular distribution of photoelectrons from this band.The purpose of this study is to investigate the atomic orbitals constituting the interlayer bonding.
We investigated the atomic orbitals constituting this Dirac-cone-like band structure in ZrB 2 (0001) at a binding energy about 3 eV by measuring two-dimensional photoelectron intensity angular distribution (PIAD) using horizontally linearly-polarized synchrotron radiation (SR).PIAD shows specific symmetry originating from the transition-matrix element which enables us to analyze atomic orbitals.

II. EXPERIMENTAL
The experiment was performed at the linearly polarized soft X-ray beamline BL-7 of SR center, Ritsumeikan University [7].The electric vector of the linearly polarized SR light was in the horizontal plane.The ZrB 2 (0001) sample was heated up to 1000 • C at first for degassing in the preparation chamber.After recovery of the vacuum, the sample was flash-annealed at 1500 • C for a few seconds several times.The surface quality was checked by low energy electron diffraction (LEED) and Auger electron spectroscopy.The two-dimensional photoelectron spectroscopy (2D-PES) measurements [8] were performed at room temperature under the ultrahigh vacuum of ∼1×10 −8 Pa by using a display-type spherical mirror analyzer (DIANA) [9] dow of 300 meV.The total energy resolution was about 400 meV and the angular resolution was about 1 • .The typical acquisition time for one PIAD measurement was 20 s.The Fermi level was determined by measuring the photoelectron spectra of gold.The bright areas correspond to the iso-energy cross sections of the three-dimensional band structures in these patterns, and their shapes change with the binding energy.The hexagon-like structure inside the BZ in Fig. 1 (a) spreads toward the outside of BZ with the increasing binding energy as shown by the yellow circles in Figs. 1  (b)-(f).This motion of the bright spots in the yellow circles corresponds to the thick red arrow in Fig. 1 (g), which shows the band dispersions calculated with firstprinciples calculation software, WIEN2k code [10].The dispersion of the red and green arrows in Fig. 1 (g) shows the Dirac-cone-like band.The apex of the Dirac-cone lies at K point at E B = 3.0 eV in the calculation (Fig. 1 (g)).

III. RESULTS AND DISCUSSION
Note that the dispersion of the green arrow was not observed experimentally, which is discussed later.
We focus on the photoelectron intensity at K points in Fig. 1 (e) which are just below the apexes of the Diraccone-like band.We can see the transition-matrix-element effect in this 2D-PIAD pattern.That is, at K points in Fig. 1 (e), the intensity of photoelectron at K2 is stronger than that at K1.This difference is due to the transitionmatrix-element effect when the atomic orbitals constitut-ing each band are excited by linearly-polarized light.In Fig. 1 (e), the photoelectron intensity inside the second BZ is stronger than that of the first BZ due to the photoemission structure factor (PSF) [11,12].PSF is a phenomenon caused by the interference effect of photoelectrons emitted from each atom when a unit cell has two or more atoms.The intensity of photoelectron peak is different in each BZ.
In the atomic orbital analysis of ZrB 2 valence band, we focused on a PIAD pattern of Fig. 1 (e) at K point at E B = 3.4 eV.It is necessary to find the intensity anisotropy of PIAD at a specific binding energy for some particular energy band to identify the atomic orbitals constituting the band.The polar angle of the circumscribed circle of the first BZ is θ = 23 • in Fig. 2 and Fig. 3 (a).To increase the number of data points two times we measured the PIAD for the sample azimuthal angle not only ϕ = 0 • but also 90 • .These measurements correspond to the condition where the electric vector of linearly-polarized light parallel and perpendicular to [2-1-10] direction, respectively.From the 0 • pattern we can compare data at every 60 • , but when we include 90 • data we can compare data at every 30 • .Then we estimated the photoelectron intensity at K points.
Next, we made simulation patterns of photoelectron angular distribution from atomic orbitals (ADAO) [13].We calculated the angular dependence of photoemission intensity from tight-binding initial states within the dipole approximation.Using a Golden Rule formula [14], the intensity distribution I (θ, ϕ) is expressed as [11]: where D 1 (k // ) is the one-dimensional density of states in (k x − k y ) plane [13], and M is the transition matrix element of photoelectron.I (θ, ϕ) can be transformed into the formulae (2) [11].
where the PSF, F , is defined as follows.
G is the reciprocal lattice vector (G = 0 in the first BZ), q is a wave vector of the initial state, and τ i is the position vector of the i-th atom.a iν is a coefficient of the ν-th atomic orbital in the i-th atom.This PSF originates from the interference of the waves from different atoms in a unit cell [11,16].The PSF determins the appearance and disppearance of photoelectron intensity in the first and second Briliouin zones depending on the bonding characteristics [11,17].Although |A iν | 2 and PSF cannot be separated in the case of hybridization, we used separated |A iν | 2 to make a simple discussion because we have no information about the coefficients.In the present study this band is composed of not only one-kind of atomic orbital; however this idea of structure factor is useful to understand why the photoelectron intensity in Fig. 1 (e) is observed only outside of the first Briliouin zone.
|A iν | 2 is the ADAO from the ν-th atomic orbital of the i-th atom [11].ADAOs for each atomic orbital were calculated for each (θ, ϕ) as shown in Fig. 2. We considered the rotation of the atomic orbital except those being symmetric about the z-axis, because the wavefunction at K2 is considered to be rotated by 60 • with respect to he wavefunction at K1.We used the formula of rotation by α for the atomic orbital p y as p α y = −p x sin α + p y cos α.The simulation patterns are shown in Fig. 2, where the hexagons indicate the first BZ of ZrB 2 (0001).The circles show the polar angle of θ = 23 • .The intensities along this circle are plotted in Fig. 3 (c) with dotted lines.The experimental data do not agree well to these simulated ADAOs although some part agrees.
Then we examined which atomic orbitals constitute the band wavefunction at K points theoretically by using WIEN2k.The calculated band structure agrees well with that reported by I.R. Shein and A.L. Ivanovskii [5].We calculated the component ratio of atomic orbitals at K points at E B = 3.0 eV; B 2p z , Zr 4d yz , and Zr 4d zx are about 12%, 27%, and 9%, respectively.The remaining percentage results from the density outside the muffin tin sphere.We made a simulation pattern for this hybrid orbital with this ratio as shown in Fig. 3 (b).The intensities at K points in the hybrid orbital simulation pattern are similar to those of Fig. 3 (a) in the experiment.We plotted the intensity of this pattern along θ = 23 • in Fig. 3 (c) including simulations for other atomic orbitals and the experimental data.The circles, solid lines, and dotted lines in Fig. 3 (c) show the angular distribution of experimental data, the calculated intensity of the hybrid orbital and atomic orbitals, respectively.The experimental data are not symmetric with respect to 180 • because we used the data before two-fold symmetric operation to analyze the experimental data faithfully.We could not obtain data good enough to analyze at 270 • because of the high background due to the effect of reflected light.Hence we did not show the data at 270 • .Some of the ϕ dependence of ADAO overlap in Fig. 3 (c), and cannot be distinguished.Hence the experimental PIAD cannot be assigned to only one of them.Although there are possibilities of atomic orbitals other than p z , d yz , and d zx orbitals to make the hybrid band, we used the ratio of the atomic orbitals calculated with WIEN2k in the hybrid orbital at K point at E B = 3.0 eV.Experimental data are approximately reproduced by the calculation for the hyblid orbital as shown in Fig. 3 (c).We concluded experimentally and theoretically that this Dirac-cone-like band is the bonding state between a B layer and a Zr layer consisting of B 2p z , Zr 4d yz , and Zr 4d zx , all of which have a lobe perpendicular to the layer (z-direction).These atomic orbitals in the hybrid orbital, B 2p z , Zr 4d yz , and Zr 4d zx , for interlayer bonding will be exposed to vacuum when the surface is created between Zr and B layers.Hence the atomic orbital composition obtained in this study is a key knowledge to consider the initial stage of epitaxial growth on ZrB 2 (0001) surface.
The upper part of the Dirac-cone-like band (red and green bands) in Fig. 1(g) corresponds to the antibonding band of boraphene, and the lower part of the Dirac-conelike band corresponds to the bonding band of boraphene.The antibonding band was observed only in the first BZ as shown by yellow circles in Figs.1(a)-(c), which correspond to the upper part of the red arrow in Fig. 1(g).This phenomenon that the appearance in the first BZ and the disappearance in the second BZ is the effect of PSF.It means that the coefficients of the atomic orbitals, a iν in eq. ( 2) for one Zr and two B atoms in a unit cell have the same sign.The bonding band was observed only in the second BZ in Figs.1(e) and (f), which corresponds to the lower part of the red arrow in Fig. 1(g).This phenomenon that the appearance in the second BZ and the disppearance in the first BZ is also the effect of PSF.It means that the coefficients of the atomic orbitals for one Zr and two B atoms in a unit cell have different signs.This finding of the sign relations of the coefficients of the atomic orbitals originated from our original experiment using a display-type spherical mirror analyzer (DIANA) and linearly-polarized SR.

IV. CONCLUSION
We have successfully conducted atomic-orbital analysis by 2D-PES.We have measured 2D-PIADs from ZrB 2 (0001) valence band by using horizontally linearlypolarized SR and a display-type spherical mirror analyzer (DIANA).Comparing the experimental data to the calculation results, we concluded that the bonding band at the K points at E B = 3.4 eV in the experiment, which is the Dirac-cone-like band consists of B 2p z , Zr 4d yz , and Zr 4d zx orbitals.Hence we confirmed experimentally and theoretically that this Dirac-cone-like band is the bonding state between a B layer and a Zr layer consisting of B 2p z , Zr 4d yz , and Zr 4d zx , all of which have a lobe perpendicular to the layer (z-direction).Moreover the relations of the coefficients of atomic orbitals constituting this band have been revealed.

Figures 1 (
Figures 1 (a)-(f) show the PIAD patterns of ZrB 2 (0001) at binding energies E B from 1.8 to 3.8 eV by 0.4 eV step excited by horizontally linearly-polarized SR.The original patterns were averaged by horizontal and vertical mirror symmetry operation considering the symmetry of the crystal and the excitation light.The hexagons in Figs.1-3 indicate the first Brillouin zone (BZ) of ZrB 2 (0001).These patterns indicate the cross sections of valence band dispersion excited by horizontally linearly-polarized light.The bright areas correspond to the iso-energy cross sections of the three-dimensional band structures in these patterns, and their shapes change with the binding energy.The hexagon-like structure inside the BZ in Fig.1(a) spreads toward the outside of BZ with the increasing binding energy as shown by the yellow circles in Figs.1 (b)-(f).This motion of the bright spots in the yellow circles corresponds to the thick red arrow in Fig.1 (g), which shows the band dispersions calculated with firstprinciples calculation software, WIEN2k code[10].The dispersion of the red and green arrows in Fig.1 (g)shows the Dirac-cone-like band.The apex of the Dirac-cone lies at K point at E B = 3.0 eV in the calculation (Fig.1 (g)).Note that the dispersion of the green arrow was not observed experimentally, which is discussed later.We focus on the photoelectron intensity at K points in Fig.1(e) which are just below the apexes of the Diraccone-like band.We can see the transition-matrix-element effect in this 2D-PIAD pattern.That is, at K points in Fig.1(e), the intensity of photoelectron at K2 is stronger than that at K1.This difference is due to the transitionmatrix-element effect when the atomic orbitals constitut-