Phase Accumulation Model Research Articles

Quantum well electronic states in spatially decoupled 2D Pb nanoislands on Nb-doped SrTiO3(0 0 1)

Two-dimensional (2D) Pb nanoisland has established an ideal platform for studying the quantum size effects on growth mechanism, electronic structures as well as high-temperature superconductivity. Here, we investigate the growth and quantum well electronic states of the 2D Pb nanoislands on Nb-doped SrTiO3(001) by scanning tunneling microscopy and spectroscopy. In contrast to Pb/Si(111), Pb/Cu(111) and Pb/Ag(111), there is no wetting layer of Pb formed on the Nb-doped SrTiO3(001) surface, resulting in isolated Pb nanoislands with an apparent height of 4 atomic layers as the building blocks for the island growth. According to the thickness-dependent quantum well states resolved in both occupied and unoccupied energy regions, the constant group velocity vg = 1.804 × 106 m/s and the Fermi wavevector kF = 1.575 Å−1, have been extracted from a linear fit of the Pb(111) band dispersion along the Γ-L direction. In addition, the energy-dependent scattering phase shift φ(E) obtained by means of the phase accumulation model shows a metallic-like scattering interface analogous to Pb/Ag(111). These spatially decoupled 2D Pb nanoislands thus realize an opportunity to explore the intrinsic quantum confinement phenomena in nanoscale superconductors on the doped titanium-oxide-type substrate.

Quantum-well states in fractured crystals of the heavy-fermion material CeCoIn5

Quantum well states appear in metallic thin films due to the confinement of the wave function by the film interfaces. Using angle-resolved photoemission spectroscopy, we unexpectedly observe quantum well states in fractured single crystals of CeCoIn$_5$. We confirm that confinement occurs by showing that these states' binding energies are photon-energy independent and are well described with a phase accumulation model, commonly applied to quantum well states in thin films. This indicates that atomically flat thin films can be formed by fracturing hard single crystals. For the two samples studied, our observations are explained by free-standing flakes with thicknesses of 206 and 101 \r{A}. We extend our analysis to extract bulk properties of CeCoIn$_5$. Specifically, we obtain the dispersion of a three-dimensional band near the zone center along in-plane and out-of-plane momenta. We establish part of its Fermi surface, which corresponds to a hole pocket centered at $\Gamma$. We also reveal a change of its dispersion with temperature, a signature that may be caused by the Kondo hybridization.

Electron quantum interference in epitaxial antiferromagnetic NiO thin films

The electron reflectivity from NiO thin films grown on Ag(001) has been systematically studied as a function of film thickness and electron energy. A strong electron quantum interference effect was observed from the NiO film, which is used to derive the unoccupied band dispersion above the Fermi surface along the Γ−X direction using the phase accumulation model. The experimental bands agree well with first-principles calculations. A weaker electron quantum interference effect was also observed from the CoO film.

Quantum-well states in thin Ag films grown on the Ga/Si(111)- 3×3 surface

Silver thin films have been created by room temperature deposition on a Ga/Si(111)-$\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ surface and their valence band structures and core levels have been measured by angle-resolved photoelectron spectroscopy (ARPES). Discrete quantum-well states (QWSs) quantized from the Ag $sp$ valence band are observed already at 3 monolayers (ML). The characteristics of the QWSs have been examined in the phase accumulation model for thicknesses between 3 and 12 ML. The phase shift and QWSs binding energies dependence with Ag film thicknesses have all been consistently derived. In-plane energy dispersion follows a parabolic curve, and the effective mass of the QWSs shows an increasing trend with binding energies as well as with reduced film thicknesses. Furthermore, the ARPES measurements reveal umklapp mediated QWSs around the $\overline{M}$ points of the Si(111) $1\ifmmode\times\else\texttimes\fi{}1$ surface Brillouin zone. The study confirms that the Ga/Si(111)-$\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ surface is a good substrate for growing uniform ultrathin Ag films in room temperature conditions.

Quantum size effect in an Fe quantum well detected by resonant tunneling carriers injected from a p-type Ge semiconductor electrode

We report the clear observation of the quantum size effect in an Fe quantum well (QW) detected by resonant tunneling carriers injected from a p-type Ge semiconductor electrode in fully epitaxial double-barrier magnetic tunnel junctions, which are composed of Co/Fe/MgO/Fe QW/MgO/Ge:B grown on a p+-Ge(001) substrate. A large tunnel magnetoresistance (TMR) ratio up to 137% (237%), which is comparable to that in Fe/MgO/Fe, is obtained at 297 K (3.5 K). The quantum oscillations are clearly observed in the dI/dV–V and d2I/dV2–V curves of our devices, and the resonance voltages are in good agreement with the resonant levels calculated by the phase accumulation model. Following these oscillations, the TMR is modulated by the quantum size effect. Our results are promising for realizing future quantum spintronics devices based on semiconductor/metal hybrid heterostructures with advanced functionalities.

Co and Phthalocyanine Overlayers on the Quantum-Well System Co(001)/Cu: Spin-Polarized Electron Reflection Experiments

The influence of a Co or phthalocyanine (Pc) molecular overlayer on the properties of quantum-well resonances (QWR) in Cu layers atop Co(001) is studied by means of spin-polarized electron reflection. For Co atoms and Pc molecules, an energy shift of the QWR-induced signal is observed with increasing coverage and is attributed to a variation of the electron reflection phase at the Cu/Co and Cu/Pc interface. For Co we find a linear energy shift in the Cu QWR energy position with increasing coverage down to the sub-monolayer regime. This shows that the phase accumulation model remains accurate within the sub-monolayer regime of a discontinuous interface. An opposite sign in the energy shift between Co and Pc overlayers could reflect an opposite impact on the Cu surface work function of overlayer adsorption.

Quantum confinement effect in armchair graphene nanoribbons: Effect of strain on band gap modulation studied using first-principles calculations

The quantum confinement effect may play an important role in the gap modulation of armchair graphene nanoribbons (AGNRs) under strain. Using the phase accumulation model, we have investigated the energy-dependent phase shift $\ensuremath{\phi}(\ensuremath{\varepsilon})$ at the $\ensuremath{\Gamma}$ point of AGNRs under various strains using first-principles calculation. The calculation results show that although the energy dispersion of the phase shift is modified by strain, the phase shift near the Fermi level is close to $0.75\ensuremath{\pi}$, indicating that strain has little effect near that energy level. We can approximate the energy-dependent phase shift by a constant, $\ensuremath{\phi}(\ensuremath{\varepsilon})=0.75\ensuremath{\pi}$, for AGNRs under various $x$ strains. Due to the structural similarity between AGNRs and zigzag carbon nanotubes (ZCNTs), the electronic properties of AGNRs should be similar to those of ZCNTs. The quantization condition of the wave vector of ZCNTs governed by the periodic boundary condition along the circumference direction is similar to that of AGNRs except that the phase shift is equal to zero, $\ensuremath{\phi}(\ensuremath{\varepsilon})=0$. Using the zone-folding (ZF) method, we can calculate the band gap of any strained AGNR (ZCNT) from the phase shift $\ensuremath{\phi}=0.75\ensuremath{\pi}$ $(\ensuremath{\phi}=0)$ and the electronic structure of the strained graphene. The AGNR shows a zigzag behavior of the dependence of the band gap on strain which is very similar to the ZCNT. The zigzag patterns are significantly shifted by different phase shifts. The peak value of the band gap and the period of the pattern decreases as the width of the ribbon increases. For a given AGNR, the peak value and the period of the pattern increase as the strain increases. A flattening of the peaks appears at the strain where the maximum band gap occurs due to large compressive strain. All these observations can be understood easily from our ZF calculations. The agreement between our model and real local-density approximation calculations indicates that our model can provide an efficient and accurate method to estimate the band gap of AGNRs and ZCNTs under strain, and therefore can provide a better understanding of the effect of quantum confinement on the electronic properties of AGNRs.

Observation of spin-dependent quantum well resonant tunneling in textured CoFeB layers

We report the observation of spin-dependent quantum well (QW) resonant tunneling in textured CoFeB free layers of single MgO magnetic tunnel junctions (MTJs). The inelastic electron tunneling spectroscopy spectra clearly show the presence of resonant oscillations in the parallel configuration, which are related with the appearance of majority-spin Δ1 QW states in the CoFeB free layer. To gain a quantitative understanding, we calculated QW state positions in the voltage-thickness plane using the so-called phase accumulation model (PAM) and compared the PAM solutions with the experimental resonant voltages observed for a set of MTJs with different CoFeB free layer thicknesses (tfl = 1.55, 1.65, 1.95, and 3.0 nm). An overall good agreement between experiment and theory was obtained. An enhancement of the tunnel magnetoresistance with bias is observed in a bias voltage region corresponding to the resonant oscillations.

Spin polarized low energy electron microscopy of quantum well resonances in Fe films on the Cu-covered W(110) surface

Spin polarized low energy electron microscopy has been used to investigate the quantum size effect (QSE) in electron reflectivity from Fe films grown on a pseudomorphic Cu layer on a W(110) surface. Intensity oscillations caused by the QSE as functions of Fe film thickness and incident electron energy identify quantum well resonance conditions in the film. Evaluation of these intensity oscillations using the phase accumulation model provides information on the unoccupied spin polarized band structure in the Fe film above the vacuum level. We also find evidence that the presence of the non-magnetic Cu layer shifts spin polarized quantum well resonances in the Fe layer uniformly downward in energy by 1.1eV compared to Fe/W(110) films without an interface Cu layer, suggesting that the Cu layer gives a small degree of control over the quantum well resonances.

Quantum well states in Ag thin films on MoS2(0001) surfaces

In-plane dispersions of quantum well (QW) states originating from the electron confinement of Ag sp electrons within the MoS2 band gap region are investigated by means of angle-resolved photoemission spectroscopy (ARPES). A number of QW resonances have been observed in the ARPES spectra in a binding energy range lying outside the MoS2 energy gap which is required for full confinement of the Ag sp electrons. In spite of having the expected free electron-like behavior, these QW states show a significant increase of in-plane effective mass with increasing binding energy due to the hybridization of Ag sp electrons with the MoS2 valence bands. The binding energy dependence of the bottom of the QW states (k‖ = 0) as a function of the Ag film thickness has been analyzed. The well-established phase accumulation model has been applied for calculating the phase shifts of electrons at the boundaries. Our observations show that the total phase shift behaves differently for energies above and below the MoS2 valence band maxima, due to the hybridizations being different in nature. The structure plot calculated considering the different quantum number dependent total phase shifts provides a good description of the experimental observations.

Determining the Thickness of Pb Film Similar to Bulk with Energy Dispersion Derived from Quantum Well States

It is known that the energy spacing between adjacent empty quantum well (QW) states in Pb islands on Cu(111) would reveal the shrinking characteristic originating from the effect of the image potential. Using the phase accumulation model, including a phase factor contributed from the image potential, the shrinking energy spacing can be quantitatively explained with the assumption of the parabolic energy versus wave vector (E–k) dispersion. However, an experimental dispersion acquired from analyzing the energies of the QW state reveals a linear E–k relationship corresponding to the Pb bulk band structure, implying the assumed parabolic dispersion is not appropriate. By combining the linear dispersion with the image potential effect in the calculation, it is found that the calculated values of energy spacing of island thickness below eight atomic layers are not in agreement with the experimental measurements. This implies that the electronic structure of Pb islands would be similar to that of the bulk when their thicknesses reach eight-atomic layers.

Modifying the Cu(111) Shockley surface state by Au alloying

The deposition of submonolayer amounts of Au onto Cu(111) results in a Au-Cu surface alloy with temperature- and thickness-dependent stoichiometry. Upon alloying, the characteristic Shockley state of Cu(111) is modified, shifting to 0.53 eV binding energy for a particular surface Au${}_{2}$Cu concentration, which is a very high binding energy for a noble-metal surface. Based on a phase accumulation model analysis, we discuss how this unusually large shift is likely reflecting an effective increase in the topmost layer thickness of the order of, but smaller than, the value expected from the moir\'e undulation.

Quantum-well states with image state character for Pb overlayers on Cu(111)

We study theoretically the quantum well states (QWSs) localized in Pb overlayers on Cu(111) surface. Particular emphasis is given to the states with energies close to the vacuum level. Inclusion of the long-range image potential tail into the model potential description of the system allows us to show the effect of hybridization between QWSs and image potential states (ISs). The particle-in-a-box energy sequence characteristic for QWSs evolves into the Rydberg series converging towards the vacuum level. The electron density of the corresponding states is partially moved from inside the metal overlayer into the vacuum. The decay rates due to the inelastic electron-electron scattering decrease with increasing energy, opposite to ``conventional'' QWSs and similar to the ISs. Many-body and wave packet propagation calculations of the inelastic decay rates are supplemented by simple analysis based on the phase accumulation model and wave-function penetration approximation. This allows an analytical description of the dependence of the QWS/ISs hybridization on different parameters and, in particular, on the overlayer thickness.

Interlayer exchange coupling in multilayered Co/Os

We have studied the interlayer exchange coupling (IEC) in Co/Osn/Co in trilayer geometry using the quantum well (QW) model, RKKY theory in the framework of ab-initio total energy calculation. Also we have used the phase accumulation model (PAM) to describe the thickness dependence of QW states energies. We predict the presence of a short period oscillation in Os with 3.5–4ML in addition to the long one of 7ML which has been measured experimentally. Our results of PAM relate the oscillation periods of 6.29ML and 3.69ML to Γ¯(k||=0) and M¯(k||≠0) direction, respectively. On the other hand, using our calculated Fermi surface including spin–orbit coupling of bulk Os we identified the two periods as the spanning vectors at Γ-centered nested electron surfaces e9, 6.86ML for long period, and inner hole ellipsoid h7, 3.80ML for short period. These periods are very comparable to those periods of 6.8ML and 3.7ML obtained by RKKY-fitting of the IEC. The principal behavior of the IEC found experimentally is reproduced by the theory. We have found in our calculations of IEC in the multilayer geometry Com/Osn a shift of the AF coupling peaks when we varied the thickness of the FM layers which might be interpreted as due to QW interactions.

Electronic localization of quantum-well states in Ag/Au(111) metallic heterostructures

We report on a detailed analysis of the evolution and spatial localization of quantum-well states (QWSs) in Ag layers on a Au(111) substrate by means of high-resolution photoelectron spectroscopy combined with model calculations based on a simple particle-in-a-box picture, the phase accumulation model, and density functional theory-based slab-layer calculations. Due to the finite electron escape depth we could link the photoemission intensity of the QWS to the simulated charge-density distribution and therewith confirm the calculated localization of these states. The first QWS starts to be localized within the Ag film at layer thicknesses $>$7 ML.

Determination of spin-polarized quantum well states and spin-split energy dispersions of Co ultrathin films grown on Mo(110)

Co thin films were epitaxially grown on Mo(110) and investigated by spin-polarized low-energy electron microscopy. We find that the spin asymmetry of the electron reflectivity from the Co film alternates its sign as a function of both the electron energy and the Co film thickness as a result of spin-polarized quantum well states in the Co film. By measuring spin-dependent quantum well states, we are able to resolve the spin-split energy dispersions of the Co film precisely. The determined spin-resolved energy bands are further tested by fitting the quantum well states using the phase accumulation model, and the result shows an excellent agreement between the fitting and the experimental data.

Adatom-induced lateral inhomogeneity of quantum well states in metal multilayers

The influence of Co adatoms on the quantum well states (QWSs) existing in Cu/Co(100) multilayers is investigated by means of ab initio calculations. The typical oscillations of the density of states at the Fermi level as a function of the number of Cu layers are found to be strongly perturbed by the presence of adatoms on the surface. In a lateral direction, the QWSs exhibit atomic-scale variations, which depend on the number of Cu layers. These results suggest that the phase accumulation model, which is often used for analyzing QWS, is not sufficient to interpret electronic features near adatoms and call for experimental real-space investigations of QWS.

Phase Contribution of Image Potential on Empty Quantum Well States in Pb Islands on the Cu(111) Surface

We use scanning tunneling spectroscopy to explore the quantum well states in the Pb islands grown on a Cu(111) surface. Our observation demonstrates that the empty quantum well states, whose energy levels lie beyond 1.2 eV above the Fermi level, are significantly affected by the image potential. As the quantum number increases, the energy separation between adjacent states is shrinking rather than widening, contrary to the prediction for a square potential well. By simply introducing a phase factor to reckon the effect of the image potential, the shrinking behavior of the energy separation can be reasonably explained with the phase accumulation model. The model also reveals that there exists a quantum regime above the Pb surface in which the image potential is vanished. Moreover, the quasi-image-potential state in the tunneling gap is quenched because of the existence of the quantum well states.

Growth and electronic properties of ultra-thin Ag films on Ni(1 1 1)

We studied the growth mode and electronic properties of ultra-thin silver films deposited on Ni(111) surface by means of scanning tunnelling microscopy (STM) and angle resolved photoemission spectroscopy (ARPES). The formation of the 4d-quantum well states (QWS) was analysed within the phase accumulation model (PAM). The electronic structure of the 1ML film is consistent with the silver layer which very weakly interacts with the supporting surface. The line-shape analysis of Ag-4dxz,yz QWS spectrum support the notion of strong localization of these states within the silver layer. The asymmetry of the photoemission peaks implies that the decay of the photo-hole appears to be influenced by the dynamics of the electrons in the supporting surface.

D-band quantum well states in Ag(111) monolayer films; substrate-induced shifts

We report a study of 4d electronic states in monolayer silver films grown on Pd(111),Ni(111), Mo(110) and Cu(100) surfaces studied by means of high-resolution angle-resolvedphotoemission spectroscopy (ARPES). The Ag-4d states, when measured in the surfaceBrillouin zone centre (SBZ), show substrate-dependent shifts. Density functional theory(DFT) calculations for a free-standing silver monolayer provide evidence that theobserved shifts are not induced by lateral expansion, compression or distortion of thesilver unit cell. Using the phase accumulation model we show that 4d-derivedelectronic states in silver monolayers can be described in terms of quantum well statesand that the matching of the electron wavefunctions at the interface with thesubstrate is one of the important mechanisms that generates the Ag-4d energy shifts.The dispersion of the states around the SBZ centre is measured and discussed.