Topmost Valence Band Research Articles

Evidence for absence of metallic surface states in BiO2-terminated BaBiO3 thin films

We investigated the atomic configuration and the electronic structure of a BaBiO3 (BBO) thin film and its (001) surface. It was theoretically predicted that two-dimensional electron gases would be formed when the film is BiO2-terminated. We deduced depth-profile information for a BBO thin film using angle-dependent X-ray photoemission spectroscopy. Analysis of the spectral weights of the Ba 3d and Bi 4f core levels confirmed that the BBO film should have a BiO2-terminated topmost layer. We used in-situ angle-resolved photoemission spectroscopy to experimentally determine the electronic structure of a BBO thin film and found no metallic surface state. We distinguished surface states from bulk states by evaporating potassium atoms in-situ on the surface. A surface state near the bottom of the topmost bulk valence band, which was predicted by the DFT calculations, was identified. However, other surface states well separated from the bulk states were not observed. Our results provided evidence that descriptions of the BBO electronic structure require more detailed and elaborate approaches.

Direct Measurement of Nanoscale Lateral Carrier Diffusion: Toward Scanning Diffusion Microscopy

A multimode scanning near-field optical microscopy technique that allows the mapping of surface morphology, photoluminescence (PL) spectra in illumination and illumination-collection modes, and PL dynamics, all in one scan, has been developed along with a method to use it for evaluation of carrier diffusion. The method allows measuring diffusion lengths as small as ∼100 nm and their anisotropy and spatial distribution, parameters remaining inaccessible to conventional far-field techniques. The procedure has been applied to study ambipolar carrier diffusion in a nonpolar m-plane InGaN/GaN quantum well. The diffusion was found to be highly anisotropic with diffusion coefficients along and perpendicular to the wurtzite c axis equal to 0.4 and 1.9 cm2/s, respectively. The large diffusion anisotropy confirms band structure calculations that suggest that the topmost valence band in an m-plane InGaN quantum well is highly anisotropic.

Exploring the electronic and optical properties of Cu2Sn1−xGexS3 and Cu2Sn1−xSixS3 (x = 0, 0.5, and 1)

To accelerate environmental friendly thin‐film photovoltaic technologies, earth‐abundant, non‐toxic, and low‐cost materials are demanded. We study the compounds of Cu2Sn1−xGexS3 and Cu2Sn1−xSixS3 (x = 0, 0.5, and 1) employing first‐principles method within the density functional theory. The compounds have comparable band dispersions. The band‐gap energies Eg can be tailored by cation alloying the Sn atoms with Ge or Si. The gap energies of Cu2Sn1−xGexS3 and Cu2Sn1−xSixS3, with x = 0, 0.5, and 1, vary almost linearly from 0.83 to 1.43 eV and 2.60 eV, respectively. However, the gap energy of Cu2SiS3 does not follow the linear relation for x > 0.8. The effective electron masses at the Γ‐point of the lowest conduction band are almost isotropic for all materials, which are between 0.15m0 and 0.25m0. On the other hand, the effective hole masses of the topmost valence band show very strong anisotropy for all compounds. In the (010) direction, the hole masses are estimated to be between 1.01m0 and 1.85m0, while between 0.11m0 and 0.41m0 in the (001) direction. Calculations reveal that all compounds have high absorption coefficients that are comparable with that of Cu2ZnSnS4. The absorptions in the energy region from Eg + 0.5 eV to Eg + 1.0 eV are even higher for Ge‐ and Si‐alloying of Cu2SnS3, compared with Cu2ZnSnS4. The high‐frequency dielectric constants of the compounds are between 6.8 and 8.9. Cu2Sn1−xGexS3 and Cu2Sn1−xSixS3 can be considered as potential candidates for absorber materials in thin‐film solar cells.

The symmetry-resolved electronic structure of 2H-WSe2(0 0 0 1)

The orbital symmetry of the band structure of 2H-WSe2(0 0 0 1) has been investigated by means of angle-resolved photoelectron spectroscopy (ARPES) and density functional theory (DFT). The WSe2(0 0 0 1) experimental band structure is found, by ARPES, to be significantly different for states of even and odd reflection parities along both the – and – lines, in good agreement with results obtained from DFT. The light polarization dependence of the photoemission intensities from the top of the valence band for bulk WSe2(0 0 0 1) is explained by the dominance of W 5 states around the -point and W 5dxy states around the -point, thus dominated, respectively, by states of even and odd symmetry, with respect to the – line. The splitting of the topmost valence band at , due to spin–orbit coupling, is measured to be 0.49 ± 0.01 eV, in agreement with the 0.48 eV value from DFT, and prior measurements for the bulk single crystal WSe2(0 0 0 1), albeit slightly smaller than the 0.513 ± 0.01 eV observed for monolayer WSe2.

Structural, electronic, optical and elastic properties of the complex K2PtCl6-structure hydrides ARuH6 (A = Mg, Ca, Sr and Ba): first-principles study

We report a systematic study of the structural, electronic, optical and elastic properties of the ternary ruthenium-based hydrides A2RuH6 (A = Mg, Ca, Sr and Ba) within two complementary first-principles approaches. We describe the properties of the A2RuH6 systems looking for trends on different properties as a function of the A sublattice. Our results are in agreement with experimental ones when the latter are available. In particular, our theoretical lattice parameters obtained using the GGA-PBEsol to include the exchange-correlation functional are in good agreement with experiment. Analysis of the calculated electronic band structure diagrams suggests that these hydrides are wide nearly direct band semiconductors, with a very slight deviation from the ideal direct-band gap behaviour and they are expected to have a poor hole-type electrical conductivity. The TB-mBJ potential has been used to correct the deficiency of the standard GGA for predicting the optoelectronic properties. The calculated TB-mBJ fundamental band gaps are about 3.53, 3.11, 2.99 and 2.68 eV for Mg2RuH6, Ca2RuH6, Sr2RuH6 and Ba2RuH6, respectively. Calculated density of states spectra demonstrates that the topmost valence bands consist of d orbitals of the Ru atoms, classifying these materials as d-type hydrides. Analysis of charge density maps tells that these systems can be classified as mixed ionic-covalent bonding materials. Optical spectra in a wide energy range from 0 to 30 eV have been provided and the origin of the observed peaks and structures has been assigned. Optical spectra in the visible range of solar spectrum suggest these hydrides for use as antireflection coatings. The single-crystal and polycrystalline elastic moduli and their related properties have been numerically estimated and analysed for the first time.

Anomalous threshold reduction from uniaxial strain for a low-threshold Ge laser

We theoretically investigate the effect of <100> uniaxial strain on a Ge-on-Si laser. We predict a dramatic ~200x threshold reduction upon applying sufficient uniaxial tensile strain to Ge. This anomalous reduction is explained by how the topmost valence bands split and become anisotropic with uniaxial tensile strain. Approximately 3.2% uniaxial strain is required to achieve this anomalous threshold reduction for 1×1019cm−3 n-type doping, and a complex interaction between strain and n-type doping is observed. Achieving this critical uniaxial strain level for the anomalous threshold reduction is dramatically more relevant to practical devices than realizing a direct band gap.

Angle-resolved photoemission studies of the valence bands of ZrSxSe2−x

The electronic structure of the ternary layered transition metal dichalcogenide compounds of ZrSxSe2−x, where 0≤x≤2, has been studied by means of high resolution angle-resolved photoemission spectroscopy (ARPES) used in conjunction with synchrotron radiation facilities. The crystals were grown by the chemical vapor transport technique using iodine as a transport agent. They are found to be degenerate extrinsic n-type semiconductors with an indirect bandgap character. The experimental valence band structure of the complete series of ZrSxSe2−x is reported along the major symmetry azimuthal directions in the Brillouin zone parallel to the layers. The results show that the binding energies of the topmost valence band shift almost linearly with the composition parameter x. Further, an emission from the conduction band minimum observed just below the Fermi edge enabled us to estimate the energy gap values. The electronic structure deduced from the photoemission measurements are discussed and compared to band structure calculations.

Isovalent Ca and Ba substitutions in thermoelectric layer-structured oxyselenide Sr2CoO2Cu2Se2

Multilayered compounds typically present exotic functionalities, and some of them have been suggested as potential materials for thermoelectric conversion owing to their unique capability to decouple electronic and heat transport. Here we report new [CoO2] and [Cu2Se2] layered A2CoO2Cu2Se2 compounds in which Sr at the intervening alkaline-earth A site is partially replaced with Ca or Ba. The parent Sr2CoO2Cu2Se2 phase is a direct gap p-type semiconductor, and density functional theory (DFT) calculations indicate its topmost valence band consists of Cu 3d–Se 4p states. Upon the isovalent cation substitution the lattice modification in the ab plane is constrained by the stiff [CoO2] layer such that the lattice shrinkage/expansion mainly happens along the c axis. Substitution of Sr with the heavier and larger Ba significantly enhances the thermopower but more hole states would be required to optimize the thermoelectric performance. Thermal stability is related to the inter-oxide-selenide-layer interaction, and our thermogravimetric measurement data reveal that the A2CoO2Cu2Se2 materials could operate in the intermediate temperature region.

InGaN/AlGaN stress compensated superlattices coherently grown on semipolar () GaN substrates

The crystal growth and optical properties of semipolar () InGaN/AlGaN stress‐compensated superlattices (SCSLs) are investigated. X‐ray diffraction and transmission electron microscopy reveal that appropriately designed 125‐period In0.2Ga0.8N (1 nm)/Al0.2Ga0.8N (3 nm) SCSLs grow coherently on () GaN substrates. The refractive index of a fabricated SCSL is lower than that of GaN in the green spectral range; indicating that SCSLs have potential as the cladding layers in green laser diodes (LDs). Photoluminescence appears in the near ultraviolet (UV) range without significant in‐plane optical anisotropy because the two topmost valence bands are close together in coherent semipolar InGaN with an experimentally determined In composition of 28%. This nearly isotropic nature allows photo‐pumped lasing in a cavity with () cleaved mirrors; suggesting potential applications of SCSLs in UV LD active layers.Schematic of InGaN/AlGaN SCSLs on GaN substrates. Balanced tensile and compressive stresses in AlGaN and InGaN, respectively, realize coherent growth of thick SCSLs.

Electronic structures of [001]- and [111]-oriented InSb and GaSb free-standing nanowires

We report on a theoretical study of the electronic structures of InSb and GaSb nanowires oriented along the [001] and [111] crystallographic directions. The nanowires are described by atomistic, tight-binding models, including spin-orbit interaction. The band structures and the wave functions of the nanowires are calculated by means of a Lanczos iteration algorithm. For the [001]-oriented InSb and GaSb nanowires, the systems with both square and rectangular cross sections are considered. Here, it is found that all the energy bands are doubly degenerate. Although the lowest conduction bands in these nanowires show good parabolic dispersions, the top valence bands show rich and complex structures. In particular, the topmost valence bands of the nanowires with a square cross section show a double maximum structure. In the nanowires with a rectangular cross section, this double maximum structure is suppressed, and the top valence bands gradually develop into parabolic bands as the aspect ratio of the cross section is increased. For the [111]-oriented InSb and GaSb nanowires, the systems with hexagonal cross sections are considered. It is found that all the bands at the Γ-point are again doubly degenerate. However, some of them will split into non-degenerate bands when the wave vector moves away from the Γ-point. Although the lowest conduction bands again show good parabolic dispersions, the topmost valence bands do not show the double maximum structure. Instead, they show a single maximum structure with its maximum at a wave vector slightly away from the Γ-point. The wave functions of the band states near the band gaps of the [001]- and [111]-oriented InSb and GaSb nanowires are also calculated and are presented in terms of probability distributions in the cross sections. It is found that although the probability distributions of the band states in the [001]-oriented nanowires with a rectangular cross section could be qualitatively described by one-band effective mass theory, the probability distributions of the band states in the [001]-oriented nanowires with a square cross section and the [111]-oriented nanowires with a hexagonal cross section show characteristic patterns with symmetries closely related to the irreducible representations of the relevant double point groups and, in general, go beyond the prediction of a simple one-band effective mass theory. We also investigate the effects of quantum confinement on the band structures of the [001]- and [111]-oriented InSb and GaSb nanowires and present an empirical formula for the description of quantization energies of the band edge states in the nanowires, which could be used to estimate the enhancement of the band gaps of the nanowires as a result of quantum confinement. The size dependencies of the electron and hole effective masses in these nanowires are also investigated and discussed.

Large trigonal-field effect on spin-orbit coupled states in a pyrochlore iridate

The half-filled topmost valence band of ${\mathrm{Ir}}^{4+}$ in several iridates such as ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4},\phantom{\rule{0.16em}{0ex}}{\mathrm{IrO}}_{2}$, and ${\mathrm{CaIrO}}_{3}$ has been proposed to originate mainly from the spin-orbit coupled ${J}_{\mathrm{eff}}=1/2$ states. In pyrochlore iridates ${R}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$ ($R$: rare earth), some exotic electronic states are theoretically proposed by assuming ${J}_{\mathrm{eff}}=1/2$ states. However, the octahedral coordination around Ir is trigonally distorted, which may affect the energy level scheme of Ir $5d$ states. Here, we report spectra of resonant elastic and inelastic x-ray scattering in ${\mathrm{Eu}}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$ at the Ir $L$ edges. A large suppression of the magnetic scattering signal at the Ir ${L}_{\mathrm{II}}$ edge supports the ${J}_{\mathrm{eff}}=1/2$ picture rather than the $S=1/2$ one. The inelastic scattering spectrum indicates that the magnitude of the trigonal field on the ${\mathrm{Ir}}^{4+}$ states is evaluated to be comparable to the spin-orbit interaction. The energy diagram of the $5d$ state is proposed based on the simple cluster model.

First principles study of defect formation in thermoelectric zinc antimonide, β-Zn4Sb3

Understanding the formation of various point defects in the promising thermoelectric material, β-Zn4Sb3, is crucial for theoretical determination of the origins of its p-type behavior and considerations of potential n-type dopability. While n-type conductivity has been fleetingly observed in Te:ZnSb, there have been no reports, to the best of our knowledge, of stable n-type behavior in β-Zn4Sb3. To understand the origin of this difficulty, we investigated the formation of intrinsic point defects in β-Zn4Sb3 density functional theory calculations. We found that a negatively charged zinc vacancy is the dominant defect in β-Zn4Sb3, as it is also in ZnSb. This explains the unintentional p-type behavior of the material and makes n-doping very difficult since the formation of the defect becomes more favorable at higher Fermi levels, near the conduction band minimum (CBM). We also calculated the formation energy of the cation dopants: Li, Na, B, Al, Ga, In, Tl; of these, only Li and Na are thermodynamically favorable compared to the acceptor Zn vacancy over a range of Fermi levels along the band gap. Further analysis of the band structure shows that Li:Zn4Sb3 has a partially occupied topmost valence band, making this defect an acceptor so that Li:Zn4Sb3 is indeed a p-type thermoelectric material. The introduction of Li, however, creates a more orderly and symmetric configuration, which stabilizes the host structure. Furthermore, Li reduces the concentration of holes and increases the Seebeck coefficient; hence, Li:Zn4Sb3 is more stable and better performing as a thermoelectric material than undoped β-Zn4Sb3.

Connection of a topological surface state with the bulk continuum in Sb(2)Te(3)(0001).

The surface state of a Z(2) topological insulator connects with the conduction and valence band continua of the bulk, thereby bridging the band gap of the volume. We investigate this connection of the surface and bulk electronic structure for Sb(2)Te(3)(0001) by photoemission experiments and calculations. Upon crossing the topmost valence band the topological surface state (TSS) maintains a coherent spectral signature, a two-dimensional character, and a linear dispersion relation. Surface-bulk coupling manifests itself in the spectra through (i) a characteristic kink in the TSS dispersion as it crosses the topmost valence band and (ii) the appearance of hybridization gaps between the TSS and bulk-derived surface resonance states at higher binding energies. The findings provide a natural explanation for the unexpectedly weak surface-bulk mixing indicated by recent transport experiments on Sb(2)Te(3).

Electronic and optical properties of the LiCdX (X = N, P, As and Sb) filled-tetrahedral compounds with the Tran–Blaha modified Becke–Johnson density functional

The structural, electronic and optical properties of the LiCdN, LiCdP, LiCdAs and LiCdSb filled-tetrahedral compounds have been explored from first-principles. The calculated structural parameters are consistent with the available experimental results. Since DFT with the common LDA and GGA underestimates the band gap, we use a new developed functional able to accurately describe the electronic structure of semiconductors, namely the Tran–Blaha-modified Becke–Johnson potential. The four investigated compounds demonstrate semiconducting behavior with direct band gap ranging from about 0.32 to 1.65eV. The charge-carrier effective masses are evaluated at the topmost valence band and at the bottommost conduction band. The evolution of the value and nature of the energy band gap under pressure effect is also investigated. The frequency-dependent complex dielectric function and some macroscopic optical constants are estimated. The microscopic origins of the structures in the optical spectra are determined in terms of the calculated energy band structures.

Transition from parabolic to ring-shaped valence band maximum in few-layer GaS, GaSe, and InSe

By performing first-principles electronic structure calculations in frames of density functional theory we study the dependence of the valence band shape on the thickness of few-layer III-VI crystals (GaS, GaSe, and InSe). We estimate the critical thickness of transition from the bulklike parabolic to the ring-shaped valence band. Direct supercell calculations show that the ring-shaped extremum of the valence band appears in $\ensuremath{\beta}$-GaS and $\ensuremath{\beta}$-GaSe at a thickness below 6 tetralayers $(\ensuremath{\sim}4.6nm)$ and 8 tetralayers $(\ensuremath{\sim}6.4nm)$, respectively. Zone-folding calculations estimate the $\ensuremath{\beta}$-InSe critical thickness to be equal to 28 tetralayers $(\ensuremath{\sim}24.0nm)$. The origin of the ring-shaped valence band maximum can be understood in terms of $\text{k}\ifmmode\cdot\else\textperiodcentered\fi{}\text{p}$ theory, which provides a link between the curvature of the energy bands and the distance between them. We explain the dependence of the band shape on the thickness, as well as the transition between two types of extremes, by the $k$-dependent orbital composition of the topmost valence band. We show that in the vicinity of critical thickness the effective mass of holes in III-VI compounds depends strongly on the number of tetralayers.

GeTe Liner Stressor Featuring Phase-Change- Induced Volume Contraction for Strain Engineering of Sub-50-nm p-Channel FinFETs: Simulation and Electrical Characterization

We report the experimental demonstration of strained p-channel FinFETs featuring a GeTe liner stressor that exhibits very large volume contraction (~10%) during phase change. Conformally grown GeTe liner was formed on FinFETs with sub-50-nm gate length L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> . When the GeTe liner changes phase from amorphous (α-GeTe) to polycrystalline (c-GeTe) state, it contracts and imparts very high compressive channel stress. A finite element method simulation was performed to study the channel stress in FinFETs, followed by a k · p calculation of Si valence band structure with the simulated strain tensors. The effective mass of the topmost valence band is reduced and the band dispersion between heavy-hole and light-hole sub-bands at Γ point increases with the effect of the strain induced by the GeTe liner. Significant drive current I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Dsat</sub> enhancement of 96% was observed for FinFETs with 50-nm c-GeTe liner stressor over the control devices. I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Dsat</sub> enhancement increases as L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> reduces, showing good scalability of the GeTe liner stressor for possible application in future technology nodes.

First principles study of electronic structure, structural and optical properties of Mg3Si2O5(OH)4

We present first principles calculations of structural properties, electronic structure, and optical properties of hexagonal Mg3Si2O5(OH)4 by using the Vienna ab initio simulation package (VASP) within the generalized-gradient approximation (GGA). Using experimentally established lattice parameters as input, full optimization of the geometry of the lattice has been performed. A difference of 3.6% between calculated and experimentally measured volume of the unit cell was found in the present calculations. The total density of states was calculated using the optimized lattice parameters. From that Mg3Si2O5(OH)4 was found to be a wide-band gap material with the calculated band gap exceeding 4eV. The orbital and site projected density of states show the topmost valence band contains sharp peaks belonging to O atoms, which reveals the ionic nature of the chemical bonds in Lizardite. The lowest conduction band has a strong contribution from the Si 4s hybridized with O 2s electrons. In addition, reflectivity, absorption coefficient, electron–electron loss function, the extinction coefficient, and the refractive index are also studied. We show that the optical properties of Lizardite are isotropic and do not depend on direction. The absorption and reflectivity values are almost zero in the photon energy range 0–4eV, whereas the Refractive index is ~1.58. So, if combined with Si, Mg3Si2H4O9 will possess good antireflection properties while the present of hydrogen will be important to passivate dangling bonds which are at the surface and other bulk defects in Si.

Electronic and optical properties of the SiB2O4 (B=Mg, Zn, and Cd) spinel oxides: An ab initio study with the Tran–Blaha-modified Becke–Johnson density functional

We report ab initio density functional theory calculations of the structural, electronic and optical properties of the spinel oxides SiMg2O4, SiZng2O4, and SiCd2O4 using the full-potential linearized augmented plane-wave method. The structural parameters calculated using both the local density and generalized gradient approximations to the exchange-correlation potential are consistent with the literature data. To calculate the electronic properties, the exchange-correlation potential is treated with various functionals, and we find that the newly developed Tran–Blaha-modified Becke–Johnson functional significantly improves the band gap. We predict a direct band gap in all of the considered SiB2O4 compounds, and the band gaps continuously decrease as the atomic size of the B element increases. The decrease in the fundamental direct band gap (Γ–Γ) from SiMg2O4 to SiZn2O4 to SiCd2O4 can be attributed to p–d mixing in the upper valence bands of SiZn2O4 and SiCd2O4. The lowest conduction band is well dispersive, similar to that found for transparent conducting oxides such as ZnO. This band is mainly defined by the s and p electrons of the Si and B (B=Mg, Zn, Cd) atoms. The topmost valence band is considerably less dispersive and is defined by O-2p and B–d electrons. The charge-carrier effective masses are evaluated at the topmost valence band and at the bottommost conduction band that were calculated. The frequency-dependent complex dielectric function, absorption coefficient, refractive index, extinction coefficient, reflectivity and electron energy loss function were estimated. We find that the value of the zero-frequency limit of the dielectric function ε(0) increases as the band gap decreases. The origins of the peaks and structures in the optical spectra are determined in terms of the calculated energy band structures.

Effect of co-substitution of nitrogen and fluorine in BaTiO3 on ferroelectricity and other properties

BaTiO3, with both nitrogen and fluorine substituted for oxygen, has been prepared and the properties of the doped material with the composition BaTiO2.8N0.1F0.1 have been studied. The color of the sample changes to light green on doping, accompanying a broad visible absorption band with its edge at 2.5 eV. The first-order ferroelectric transition changes to a broad transition on doping along with a decrease in the dielectric constant. We have examined the local structure, electronic structure and polar lattice dynamical properties of N and F co-substituted BaTiO3 using first-principles density functional theory-based calculations and derived the mechanism of the diffuse ferroelectric transition observed here with a broad peak in the temperature-dependent dielectric response. The calculated Born charges clearly reveal a strong disparity in the interaction of N, O and F with Ti: N being the most ferro-active, with a highly anomalous charge, and F being the least active, with an almost nominal charge. This originates from an electronic structure in which the top-most valence band is constituted of the 2p states of N, while the electronic states of F lie deep in the energy band, resulting in a local structure with short covalent Ti–N and long ionic Ti–F bonds. Disorder in these hetero ferro-active anions leads to a relaxor-like diffuse phase transition and a reduction in polarization. Calculations confirm the reduction of 0.65 eV in the bandgap of BaTiO3, which changes from indirect to direct type.

Energy offset between valence band anti-crossing and optical polarization switching in semipolar InGaN quantum wells

Light emitted from a semipolar InGaN quantum well parallel to the surface normal is partially polarized, in contrast to the unpolarized emission of c-plane quantum wells. Anti-crossing of the two topmost valence bands causes polarization switching between ordinary and extraordinary polarizations for certain semipolar quantum well orientations with increasing indium content. Two properties of the two measured spectra, their polarization resolved but spectrally integrated intensity and their measured energy splitting, are associated with anti-crossing. Here, we show that only the observable energy splitting coincides with band anti-crossing, while the switching point of the polarization can deviate from the anti-crossing point.