Critical role of parallel momentum in quantum well state couplings in multi-stacked nanofilms: an angle resolved photoemission study

We use angle resolved photoemission spectroscopy (ARPES) to investigate the coupling of electron quantum well states (QWS) in epitaxial thin Pb and Ag films. More specifically, we investigate the Ag/Si, Pb/Si, and Pb/Ag/Si systems. We found that the parallel momentum plays a very profound role determining how two adjacent quantum wells are coupled electronically across the interface. We revealed that in the Pb/Ag bimetallic system, there exist two distinctly different regimes in the energy versus momentum (E vs. k) space. In one regime the electronic states in Ag and Pb are strongly coupled resulting in a new set of quantum well states for the bi-metallic system. In the other regime the electronic states in individual metallic layers are retained in their respective regions, as if they are totally decoupled. This result is corroborated by calculations using density functional theory. We further unravel the underlying mechanism associated with the electron refraction and total internal reflection across the interface.


Introduction
Quantum well systems have played an important role in contemporary condensed matter physics and have been a key enabler for modern electronics. In a quantum well system, its electronic states can be designed through the control of the boundary conditions [1][2][3]. Coupling different quantum well systems together further add to the design complexity [4][5][6][7][8]. This approach has enabled researchers to create "designer electronic structures" providing a rich quantum material platform for realizations of novel physical phenomena and new technology [9][10][11][12][13][14][15].
In designing coupled quantum well systems, the most critical factor is to understand how the quantum well states (QWS) in different layers are coupled to each other and how this coupling leads to the resulting electronic structures of the composite as a whole. For example, some aspects of bi-quantum well systems (e.g. Pb/Ag and Ag/Si) have been studied previously. In the case of Ag/Si, several earlier studies reported observation of the coupling between the Schottky barrier (SB) confined Si hole-like QWS and the QWS of thin Ag films [7,[16][17][18][19][20]. In the case of Pb/Ag system, it was discovered that the electrons in the buried Ag layer can transmit coherently across the interface through the Pb overlayers and be detected as photoelectrons (even though the overlayer thickness is larger than the electron escape depth) [8]. Nevertheless, the role of parallel momentum in a coupling of these systems has rarely been discussed in previous ARPES (angleresolved photoemission) experiments, despite its importance in governing the electrical properties of materials.
Here we use angle-resolved photoemission spectroscopy (ARPES) to investigate the behavior of asymmetric double quantum well systems Pb/Ag. Several new aspects have been uncovered in the current study. More specifically, we unveil the critical role of the parallel momentum, ∥ , in determining the coupling of adjacent quantum well systems. Especially, we show that near the Brillouin zone (BZ) center the Ag QWS are well coupled to the Pb QWS, forming a composite Pb/Ag QW system. On the contrary, states in the BZ regions with large parallel momentum are completely decoupled from the Ag layer, retaining the original E vs k dispersion of the Pb quantum well system. Similarly, for Ag QWS with a large crystal momentum, they are retained in the original Ag layer and could not transmit through the Pb overlayer. We attribute these behaviors to the electron refraction across a potential step at the Pb-Ag interface. These findings are further corroborated using first-principles calculations within density functional theory (DFT).

Experimental details
ARPES measurement was performed with the sample held at liquid nitrogen temperature. Helium lamp light (21.2eV and 40.8eV) was used as a photon source and the spectra was acquired using a Scienta R3000 analyzer. The pressure was maintained under 2 × 10 "# torr during Helium lamp operation.
The samples were grown in MBE chamber under 2 × 10 "$% torr pressure by evaporating Ag, Pb source from K -cell and transferred to ARPES chamber via our Ultra High Vacuum(UHV) transfer vessel (based pressure < 1× 10 "# torr). All metal overlayers are grown on an atomically clean Si(111) 7x7 substrate (prepared by typical Si-flashing procedure). For epitaxial Ag/Si and Pb/Si growth, 2 step method is used. In the case of Ag/Si, clean Si(111) 7x7 substrate is cooled down to 100K and Ag source is evaporated on it with 0.3-0.5ML/min deposition rate. And next, it is annealed at room temperature for one hour. Lastly, it is cooled down to 100K for the measurement.
In the case of Pb/Si, a stripped incommensurate (SIC) wetting layer is first prepared on a clean Si(111) 7x7 substrate by depositing over 1.3ML Pb at room temperature and post-annealing it at 400-450℃ for 4 mins. And next, the sample is held at 100K and Pb is deposited on it with 0.3-0.5ML/min deposition rate. Lastly, the sample is annealed at 250K for 15min and then cooled down to 100K immediately to prevent the de-wetting process of Pb layers. In the case of Pb/Ag/Si, we used the ARPES measured Ag/Si sample made by the above procedure. The sample is held at liquid nitrogen temperature and Pb is deposited on the Ag/Si with 0.3-0.5ML/min deposition rate, the sample is post-annealed at 250K for 3 hours. Lastly, it is cooled down to 100K for measurements [13,15,[21][22][23].
The crystal orientations are determined using low energy electron diffraction. In all cases, the epitaxial metal films (Ag, Pb or Pb/Ag) are all rotationally aligned with the Si(111) substrate. The lattice constant ratio for Pb:Si (Ag:Si) is 10:11 (3:4). The ratio between Pb:Ag is 40:33, which is close to 6:5 (to within 1%).

Computational details
Density functional theory (DFT) calculations were performed with the Vienna ab-initio Simulation Package (VASP) [24,25] using the plane wave basis and an energy cutoff of 500 eV. The projector augmented wave (PAW) [26] method and the exchange-correlation functional of the Perdew-Burke-Ernzerhof (PBE) form [27] in the generalized gradient approximation (GGA) are employed.
The supercell lattice constant of the Pb/Ag bimetallic structure was chosen to be an average supercell lattice constant based upon 6× 6 Pb(111) and 5×5 Ag(111) (i.e., &'()*+),, = (5 × -.($$$) + 6 × 12($$$) )/2), so that the strain levels are -0.3% and 0.3% for Pb and Ag, respectively. The self-consistent cycles were carried out until the energy difference between two cycles was less than 10 "3 eV. The supercell electronic structure was unfolded in reference to the Pb Brillouin zone using the BandUP code [28][29][30]. The fermi surface and E-k dispersion mapping images of Ag 9ML, Pb 10ML/Ag 9ML and Pb 10ML are represented in Fig. 1 taken from photoemission measurements. We first discuss results for individual Ag or Pb layers on Si(111). In Ag/Si, the QWS, well-resolved in spectra are obtained. The parabolic shape bands in E-k dispersion indicate quasiparticles in Ag films behave like free electrons. In addition, one also observes fringes due to the coupling of Ag QWS to the Si downward hole QWS confined by the Schottky barrier at the Ag/Si interface. These features are well-understood and have been reported several times previously [7,[16][17][18][19][20].

Results and discussion
For Pb/Si, the spectra show rather flat QWS at 4~ -0.5 eV near the zone center with an effective mass of significantly larger than the result using DFT calculation. As previously reported, the strong mass enhancement can be attributed to two different factors; the coupling to the VBM of the underlying Si and electron localization induced by lateral atomic spacing of Pb layers [31,32].
Other QWS near the zone center can be better observed using He-II excitation as shown in Fig. S1 accompanied by the energy distribution curve (EDC) at the G point. The coupling of this QWS (-0.5 eV) with the SB confined hole QWS of Si can also be observed as fringes in Fig. 1g.
In addition to QWS near the zone center, we observe several well-separated bands with negative linear dispersion in the approximated region with k = 0.5 -0.9 Å "$ . These states correspond to the folded-back QWS (see also DFT calculation shown in Fig. 3a). The DFT calculation shows many closely spaced folded back quantum well states and many of them are nearly degenerate.
Experimentally, we can observe up to 6 well separated bands. with parabolic shape bands near the Γ point (Fig. 3a).
The most interesting features are in studies of the Pb/Ag layers. Note that the underlying Ag layer is precisely the same Ag layer used in measurements Fig. 1 (b, e) There are three key observations: (I) Observation of "Ag-like" QWS despite that the Ag layer is buried underneath the Pb layer which is thicker than the escape depth (first reported by Brinkley et al.) [8]; although within the same energy window, the number of such QWS are roughly doubled; (II) the total absence of "Aglike" QWS beyond a certain value of ∥ (~ 0.6Å "$ ); (III) observation of the folded back QWS that are the same as Pb/Si with the same Pb thickness.
The observation of "Ag-like" QWS near the zone center was first interpreted as coherent coupling of the QWS in the Ag underlayer to the Pb overlayer, thus enabling them to be observed despite being buried underneath in a depth more than the photoelectron escape depth (Fig. 1 (c, f)). We agree with the notion that the electronic states in Ag are coherently coupled to the Pb layers.
Nevertheless, we interpret that such a coupling creates a new set of QWS for the Pb-Ag bimetallic layer as a whole and such QWS extend throughout the whole bi-metallic system. Thus, they can be detected by ARPES because the wavefunction is extended to the surface (Fig. 4a). In addition, the fact that the number of QWS within the same energy window is roughly doubled is also consistent with the interpretation that these are new QWS which are now confined within the whole bi-metallic thickness.
On the other hand, observations (II) and (III) indicate that in a certain region of − space, the electronic states in the Ag and Pb layers stay within their respective regions without coupling. For example, for Ag QWS with large ∥ (in region with ∥ > 0.6Å "$ ) which can be clearly observed in the Ag/Si, are no longer detectable in the Pb/Ag system. Moreover, the folded back QWS in the Pb/Si system are now completely replicated in the Pb/Ag system. We have further carried out DFT calculations to investigate this contrasting behavior. Namely in some regions of − space, the electronic states in Pb and Ag layers are strongly coupled, forming a new set of QWS for the bi-metallic layer (observation (I)); whereas in other region of − space, the electronic states in Pb and Ag layers are totally decoupled(observation II and III). Shown in Fig. 3(a ,b c) are DFT band structure calculations for 10 ML Pb, 9 ML Ag, and the 10/9 combination of the bimetallic Pb/Ag layers. To obtain electronic structures of the bimetallic layer (Fig. 3c), the lattice constant ratio between Pb and Ag is approximated by 6/5 which is within 1% of the actual ratio. The calculated electronic structures are then projected on to the Pb BZ for comparison. Fig. 3d shows the integrated electron density over atomic planes layer-by-layer for selected states marked on the band structures in Fig. 3c. State 1-3 are the empty states and can be projected only onto the Pb layer because they fall into the region of the L-gap in the Ag band structure. State 4 is the Ag surface state which exists there because the calculation is based on a free standing bi-metallic system with vacuum on the Ag side. Corresponding to our observation (I) are states 5-8 whose wavefunctions extend throughout the Ag and Pb layers. This calculation confirms our interpretation that for small ∥ a new set of QWS are formed, which are confined within the Pb/Ag bi-metallic system as a whole. This interesting behavior can be understood by considering a simple model describing how electrons traverse across a potential step (Fig. 4a). For that, we use 5.5eV and 9.5eV as Ag and Pb fermi energy which are derived from free electron approximation (i.e. 6 = Though the fermi energy derived from DFT calculation and free electron approximation has small difference, it doesn't change the result. We can construct a model with a potential step % of about 4 eV (with region 1 for Pb and region 2 for Ag). For simplicity, we consider a free electron traversing across this potential step (different effective mass calculation also shows similar behavior). The kinetic energy in region 1 is $ = When the electron is moving normal to the interface (i.e. ∥ = 0) and with enough energy to overcome the barrier, the transmission probability across the interface is simply = Note that this transmission probability is rather large in most of the energy range. For example, at   These distinctly two different regions in k-space that influence the coupling between the QWS in Pb and Ag layers can also be viewed clearly in the constant energy surface (CES) mapping in kspace. Shown in Fig. 1 (b, c, d) are 2D k-space CES mapped near the Fermi energy for 9ML-Ag, 10ML-Pb/9ML-Ag, and 10ML-Pb on Si(111). For Ag, the CES near the Fermi energy appear as concentric rings extending to a maximum radius of kmax ~ 1.25 Å "$ . For Pb, near the Fermi energy the dominating CES are due to the folded back bands, which now appear as concentric hexagons.
In the case of Pb/Ag/Si, in addition to these hexagons, two center rings are observed resulting from the composite quantum well state due to the coupling of Ag and Pb electronic states. Outside the hexagon rings, however, no trace of Ag-like states can be found, indicating that these states stay in the buried Ag layer deeper than the photoelectron escape depth and cannot be detected.
In summary, we have used ARPES to investigate quantum well state couplings in Pb/Ag systems.
For the Pb/Ag bimetallic system, we discovered that there exist two distinct regimes in the . Similarly, the folded back QWS in the Pb/Si system are now completely replicated in the Pb/Ag system, indicating that these states are confined only in the Pb layer. When viewed in the extended zone, these folded back states correspond to states with very large ∥ , and experience total internal reflection. Our studies illustrate the important role played by the parallel momentum in determining the coupling of the electronic states in multi-layer quantum confined systems.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. Figure S1. ARPES measurements on Ag(9ML), Pb(10ML)/Ag(9ML), Pb(10ML) using 40.8eV photon energy (He II). He II data shows better sensitivity on certain bands than He I 21.2eV photon energy.