Quantum confined Rydberg excitons in reduced dimensions

In this paper we propose first steps towards calculating the energy shifts of confined Rydberg excitons in Cu 2 O quantum wells, wires, and dots. The macroscopic size of Rydberg excitons with high quantum numbers n implies that already μm sized lamellar, wire-like, or box-like structures lead to quantum size effects, which depend on the principal Rydberg quantum number n . Such structures can be fabricated using focused ion beam milling of cuprite crystals. Quantum confinement causes an energy shift of the confined object, which is interesting for quantum technology. We find in our calculations that the Rydberg excitons gain a potential energy in the μeV to meV range due to the quantum confinement. This effect is dependent on the Rydberg exciton size and, thus, the principal quantum number n . The calculated energy shifts in the μeV to meV energy range should be experimentally accessible and detectable.


Rydberg excitons
Rydberg atoms build an important platform for quantum applications because of their long lifetimes and large dipole moments, allowing for the formation of long-range interactions and rendering them highly sensitive for external fields on a quantum level [1][2][3][4][5][6][7]. They feature many interesting properties, such as the generation of nonclassical photonic states [8][9][10], or the dipole blockade [11,12], which can be used for optical switching and other quantum information processing applications [13][14][15][16][17][18]. However, their incorporation into the solid state remains difficult.
Rydberg excitons are the solid-state analog of Rydberg atoms. Due to the attractive Coulomb interaction between the conduction electron and the valence hole in a semiconductor, a quasiparticle called exciton can be formed as an excited electronic state of the crystal. This electron-hole pair is weakly bound and extends over many thousands of lattice unit cells, thus its interaction is screened by the static permittivity of the semiconductor (see figure 1(a)). Its energetic levels E n lie within the semiconductor band gap E , ( ) e is the background dielectric constant, and d P the quantum defect for P-excitons. This is in analogy to Rydberg atoms, where the outer electron orbits around the nucleus, following the hydrogen formula = -E n Ry , n 2 / with Ry being the Rydberg constant. Now, electron and hole orbit around each other and are termed Rydberg exciton, when being in quantum states with large principal quantum number  n 10. The total exciton energy, including its center-of-mass kinetic energy, is The center-of-mass kinetic energy of the exciton usually vanishes for direct band gap semiconductors with band gap energies in the visible due to momentum conservation. Therefore, only a series of sharp lines corresponds to an exciton with vanishing momentum [19]. The crystal ground state is a state with no excited electron-hole pairs; it is totally symmetric with no angular momentum and positive parity.
Excitons are decisive for semiconductor optical properties [36]. The electron-hole relative motion is on a large scale compared to interatomic distances. Therefore, in contrast to Rydberg atoms, which are described by a modified electron wave function only, the exciton wave function F exc is given by the electron wavefunction f e times the hole wavefunction f h times an envelope wave function f : The envelope wave function describes the electron-hole relative motion, or the angular momentum quantum state l, and can be expressed in spherical harmonics as The exciton wave function obeys the two-particle Schrödinger equation (also known as Wannier equation) with the reduced exciton mass m r [37].

Quantum confinement
Rydberg excitons in cuprous oxide quantum wells have not yet been studied, although the concept of the related problem is well known. The problem can be described via potential well calculations. These have already been extensively performed for electrons and holes in semiconductors [38].
Mesoscopic spatial confinements of the order of several μm, also known from quantum dots [39,40], are still large in comparison with the lattice constant of the material. The band structure of a semiconductor is therefore only weakly changed when being spatially confined compared to the bulk material [41]. This assumption allows investigating solely changes in the envelope part of the wave function caused by the confinement potential and is called envelope function approximation. However, in spatially confined structures, surface polarization effects may play a role due to the different dielectric constants of the material and its surrounding. This becomes in particular important when investigating Rydberg excitons, which are quasiparticles bound by the threedimensional Coulomb interaction. Quantum confinement effects arise as soon as the spatial confinement is comparable to the quantum object's Bohr radius. Rydberg excitons have Bohr radii up to μm size. This allows to confine them in so-called mesoscopic structures, the dimension of which is large compared to the lattice constant but comparable to the exciton Bohr radius (see figures 1(b), (c)).
We want to confine a whole Rydberg exciton into a quantum well, and expect large energy shifts of this giant quantum object. In excitons the Coulombic electron-hole attraction gives rise to bound states of the relative motion of the exciton. The excitonic bound levels in quantum wells are, in many respects, analogous to the Coulombic impurity bound states, meaning that the electron and hole relative motion is described by a Hamiltonian which is similar to that of an impurity [42]. We intuitively associate an extra kinetic energy with the localization of a particle in a finite region of space, i.e. the impurity binding energy increases when the quantum well thickness decreases.

Quantum confined Rydberg excitons
In mesoscopic cuprite slabs surrounded by air or vacuum, the potential barrier accounts for 2.98 eV, which is given by subtracting the Rydberg binding energy (2.17 eV) from the work function energy of electrons in cuprite to air (5.15 eV). This finite potential well energy can, however, be treated as an infinite potential barrier due to the fact that the linear dimension of the confinement exceeds the lattice constant of the semiconductor [41].
In the following, we will focus on the weak confinement regime, where the confinement acts only on the center-ofmass motion of the exciton, i.e. the envelope of the Bloch functions, and does not interfere with the relative motion of the electron-hole pair. Here, the Rydberg exciton binding energies are larger than the confinement effects. In contrast, in the strong confinement limit, the picture of an exciton would be destroyed, as one would then treat electron and hole separately with their individual motions being quantized. In this case, the confinement energy dominates over the Rydberg exciton binding energy.

Methods
In order to calculate the energy shifts a Rydberg exciton experiences when being confined in a quantum well, we perform potential well calculations for the center-of-mass coordinate of a Rydberg exciton in a cuprous oxide quantum well (see figure 1(d)). The quantum well consists of a cuprite slab that is extended many micrometers in x and y direction but confined to a few hundreds of nanometers only in z direction. It is surrounded by air or vacuum. Usually, this quantum mechanical problem is not separable due to the confined geometry. In particular, the Coulomb interaction, causing the electron and hole relative motion, is always of three-dimensional nature and depends on the relative electron-hole distance. However, for large quantum wells (weak confinement regime) the confinement or perturbation acts only on the center-of-mass coordinate and is assumed to not disturb the relative motion. Then a separation into relative and center-of-mass coordinates is possible as an approximation.

Electron states for infinite potential barriers
When a quantum object, such as an electron, is spatially confined in one dimension, it gains potential energy V z , ( ) which can be expressed in the Schrödinger equation as: The electron wave function can be separated into ) ( ) which allows for solving the Schrödinger equation via the separation ansatz: which is comparable to equation (2) for the exciton, while equation (9) determines the particle's quantized energy eigenvalues due to the quantum confinement with j being the quantum state index in the quantum well and L z the well width. The quantized bound state energies increase with decreasing quantum well width and are proportional to the quantum state index j . 2

Weakly confined Rydberg excitons in cuprite quantum wells
Following the calculation scheme from the previous section, we calculate the Rydberg exciton energies in a cuprite quantum well with quasi-infinite potential well barrier The Hamiltonian describing this problem is given by This is similar to equation (1) but with an additional quantized energy term p D =  j m L 2 conf r z 2 2 2 2 ( ) due to quantum confinement in one dimension, with j being the quantum state index and L z being the quantum well width.   For both, the lowest ( = j 1) and third excited ( = j 3) quantum state index, and for all different well widths L z shown here, the energy shift D conf increases with increasing principal quantum number n. This increase in energy is absolutely larger and steeper the smaller the well widths. For the lowest quantum state index = j 1, D conf accounts for up to 14 μeV, while for the third excited quantum state index  = j 3 D conf reaches values of up to 140 μeV, which is one order of magnitude larger.

Strongly confined Rydberg excitons in cuprite quantum wells
Subject of this paper is the weak quantum confinement of Rydberg excitons in cuprite quantum wells. The relevant structures are of several hundreds of nanometers to a few micrometers in size, which can be procuded straight-forwardly using focused ion beams. Furthermore, we want to investigate a single, giant quantum object-the Rydberg exciton. In the strong confinement regime, the confinement energy would exceed the Coulomb energy and thus the binding energy of the exciton. In this case, electron and hole are confined separately in their respective confinement potentials. We believe that the following considerations regarding the strong confinement regime might be of interest for the reader.
3.3.1. Exciton binding energy in the strict 2D limit. The 3D Rydberg P-exciton energy is given by with E : g band gap energy, Ry : * Rydberg constant for excitons in cuprous oxide, n: principal quantum number, and d = 0.23: P quantum defect for P-excitons [43]. In strictly two dimensions, the Rydberg exciton binding energy E b becomes modified into 44]. This implies that the lowest 2D exciton energy ( = n 1) has a magnitude four times larger than the 3D exciton ground state, when neglecting the quantum defect: Thus, the exciton ground state is farther away from the bandgap in 2D and the 2D Bohr is half as big as the 3D value. The excitonic resonances as well as the exciton binding energies are stronger in 2D. Interestingly, the absorption strength decreases in this situation more rapidly with n. The transition from 3D to 2D would cause exciton energy shifts of even a few meV: The exciton binding energy in three dimensions decreases rapidly with n. So does the binding energy in two dimensions as well, however, at some larger values. The difference of both, defined as the energy shift D , 2D thus, follows the same trend. Note that a larger Rydberg binding energy implies a smaller Rydberg exciton energy as the binding energy ( meṼ ) is subtracted from the band gap energy (~eV) in order to yield the exciton energy = -E E E .
n g b Therefore, the effect the 2D confinement will have on the total exciton energy will be a red-shift toward lower energies.

3.3.2.
Influence of the permittivity of the surrounding material outside the quantum well. In the case of a very narrow quantum well, the permittivity inside and outside the well is different. The Coulomb interaction between electron and hole in an exciton is of three-dimensional character and, thus, not squeezed inside the well, but occurs primarily outside the well with less effective screening [45].

Discussion
The quantum confinement inhibits free motion and, thus, influences the kinetic energy of the quantum object. Only discrete values are allowed, leading to a series of quantized states [46]. Confined to a quantum well, Rydberg excitons gain energy, so they experience an energy blue-shift. Within the weak confinement regime, this energy shift accounts for a few and a few tens of μeV for the lowest and third excited quantum state index, respectively. The energy shifts are controllable via the three parameters principal quantum number n, quantum state index in the quantum well j, and quantum well width L z over a wide range. Such controllable energy shift could be used for realizing quantum technologies using Rydberg excitons in cuprous oxide [2]. In order to enlarge the range over which the energy can be shifted, one could go to higher confinements, meaning a more tight confinement along one dimension (intermediate or strong confinement regime in 2D).
The conditions for the strong confinement regime become, as shortly discussed in section 3.3, more complicated. Strictly speaking, for a very strong exciton confinement, electron and hole become quantized separately, so the quantization energy dominates over the Coulomb interaction energy and we cannot speak of an exciton any more. In order to get a feeling for how the energy shifts would develop when going towards the intermediate regime ( » L r 2 z ), we applied our weak confinement model to excitons in this intermediate confinement regime. The resulting energy blue-shifts D conf are depicted in figure 4 for the lowest quantum state index = j 1, for different well widths L , z and in dependence on the principal quantum number n.
The transition from the weak to the intermediate regime is smooth. As expected, the energy blue-shifts become significantly larger the narrower the quantum wells, and account for up to several tens of meV. In the strong confinement regime, the electron and hole confinement energies would exceed the exciton binding energy = Ry 96meV. *

Outlook
Higher quantum confinements can also be reached by confining a quantum object in two or three dimensions. This can be realized by confining Rydberg excitons into cuprite quantum wires or quantum dots (see figure 1(b)). In such structures the energy shift caused by the confinement geometry reads: The blue-shift could, thus, be enhanced by a factor of 2 and 3 compared to the quantum well structures. It remains, however, unknown how the Rydberg binding energy would change in such structures.
Recent advances in focused ion beam milling using Au + ions (Raith IonLine) make it possible to fabricate tailored quantum wells with widths in the hundreds of nm range, as shown in figure 5. Therefore one should in principle be able to weakly confine experimentally Rydberg excitons into quantum wells. The difficulty remains in that the well interface needs to be very flat, to not disturb the crystal lattice, its symmetry, and thus the exciton formation. The energy shifts could be measured in absorption or by pump-probe spectroscopy. This gives for the first time the opportunity to probe such large quantum objects inside the confined geometry of quantum wells.