To synthesize the sample, a 700 nm Cu film was deposited on the substrate via e-beam evaporation, with a 5 nm Ti layer in between the Cu and the substrate to increase adhesion. The Cu layer was oxidized in synthetic air for several hours to ensure its full oxidation41. Figure 1b,c show a microscope image of the finished sample as well as a scanning electron microscope (SEM) image, respectively.
Optical density measurement
As the sample is on a transparent substrate, Rydberg excitons can be examined through optical density (OD) measurements. The excitation light from a broadband white light source (Thorlabs SLS201L) was focused onto the sample with an objective lens with NA = 0.42 (Mitutoyo Plan Apo 20x). The transmitted light was collected with a similar objective and sent to a spectrometer (Princeton Instruments HRS-750, 1200 gratings/mm, 50 \(\upmu \hbox {m}\) slit width, resolution of 0.036 nm). The measurement was repeated at temperatures from 5–150 K with a precision of 0.25 K. Figure 2a shows the results from this measurement for the yellow exciton series. At low temperatures, resonances up to \(n=4\) were observed. As the temperature was increased, the exciton resonances were red-shifted and broadened until they could no longer be resolved. At temperatures higher than 150 K, no distinct exciton resonances could be observed. The energy shift can be attributed to changes in the bandgap energy, binding energy, and quantum defects of \(\hbox {Cu}_2\hbox {O}\) as a function of temperature. The change in the bandgap energy, \(E_g\), arises from the thermal expansion of the crystal lattice and phonon-electron interactions46. The binding energy, Ry, being proportional to the reduced mass of the exciton, is also temperature-dependent due to changes in the electronic bands’ curvatures with the temperature24. Finally, the quantum defect, which arises in part due to the non-parabolic bands, will likewise vary with temperature as the band structures are modified47. In the remainder of this section, we present the first analysis of temperature-dependent exciton behavior in synthetic \(\hbox {Cu}_2\hbox {O}\) and compare it to previous results on natural samples36.
The changes in the first two parameters can be summarized by Elliott’s model (2), which assumes that shifts are dominated by continuous absorption from \(\Gamma _3^-\) phonons46,
$$\begin{aligned} \begin{aligned} E_g(T) = E_{g0} + E_{gT} \Bigg [\coth {\Bigg (\frac{\hbar \omega _3}{2 k_B T}\Bigg )} – 1\Bigg ] ,\\ Ry(T) = Ry_{0} + Ry_{T} \Bigg [\coth {\Bigg (\frac{\hbar \omega _3}{2 k_B T}\Bigg )} – 1\Bigg ] , \end{aligned} \end{aligned}$$
(2)
with \(\ \hbar \omega _3 = 13.6\) meV.
\(E_{g0}\) and \(Ry_0\) are temperature-independent terms that represent the low-temperature limit of the bandgap and binding energies, respectively, and \(E_{gT}\) and \(Ry_T\) capture the temperature effects. To fit this model to our data, the Rydberg energies were extracted for each temperature using least squares fitting to an asymmetric Fano lineshape48
$$\begin{aligned} \alpha _n(E) = f_n \frac{\frac{\Gamma _n}{2} + 2 q_n (E – E_n)}{\big (\frac{\Gamma _n}{2}\big )^2 + (E – E_n)^2} , \end{aligned}$$
(3)
where \(E_n\) is the nth exciton energy, \(\Gamma _n\) is the corresponding linewidth, \(f_n\) is proportional to the oscillator strength, and \(q_n\) is an asymmetry factor modeling the interference between narrow optical transitions and the phonon continuum49.
As is visible in Fig. 2a, there is a background from continuum absorption which increases at higher energies. This background, known as the Urbach tail, signifies an increase in absorption near the bandgap energy caused by an increasing density of states in that energy range. This increase is described by an exponential function,
$$\begin{aligned} \alpha _{U}(E) = \alpha _0 \exp {\Bigg (\frac{E – E_g}{E_u}\Bigg )} , \end{aligned}$$
(4)
where \(E_g\) is the bandgap energy, \(\alpha _0\) is the magnitude of the continuum absorption, and \(E_u\) is the Urbach energy50,51.
Elliott’s model was used to fit the center energy as a function of temperature, taking into account the 2p, 3p, and 4p peaks simultaneously, as shown in Fig. 2b. For simplicity, we ignored the quantum defects for all three resonances at all temperatures. This is just an approximation since the quantum defects vary with n and are expected to be temperature-dependent as well. Typical methods to extract the quantum defect as a function of n, such as those used in36, require the observation of high energy peaks whose quantum defects approach a constant value (see Sect. S3 of the supplementary materials for more details). Theoretical calculations show that this trend does not emerge until \(n \gtrapprox 10\)47. However, in36 the authors note that while assuming \(\delta _n(T) = 0\) is simplistic, it yields results that agree with both the literature and more detailed analyses which include the quantum defect corrections. From this fit, we have extracted the parameters shown in Table 1.
The values of \(E_{g0}\), \(E_{gT}\), and \(Ry_0\) are in agreement with the literature, but the extracted \(Ry_T\) is different33,36. It must be noted that ignoring \(\delta _n\) is valid for describing the temperature dependence of the bandgap energy but breaks down for the binding energy, so we attribute this particular discrepancy to the errors caused by this assumption. Overall, the accurate observations of \(E_{g0}\), \(E_{gT}\), and \(Ry_0\) confirm that the observed absorption lines indeed arise from Rydberg excitons in the synthetic \(\hbox {Cu}_2\hbox {O}\).
Photoluminescence measurement
Non-resonant photoluminescence (PL) measurements were performed with a 532 nm laser to excite electrons above the bandgap and create free electrons which form bound states as they relax to the lower exciton levels. The laser was focused down to a spot size of \(\hbox {FWHM}=3.1\) \(\upmu \hbox {m}\) using an objective lens (Mitutoyo Plan Apo 20x). The PL from the excitons was collected with the same objective and sent to the spectrometer after a long-pass filter (Semrock LP03-532RE-25) and a dichroic mirror (Thorlabs DMLP567) were used to block the reflected pump laser. Temperature-dependent data was taken at an incident laser power of 50 \(\upmu \hbox {W}\) at temperatures from 5 to 150 K. Power-dependent data was taken with the sample held at a constant temperature of 5 K at laser powers ranging from 50 \(\upmu \hbox {W}\) to 2 mW.
Figure 3a,b show the PL spectrum at the brightest point on the sample at a temperature of 20 K. Rydberg states up to 7p are clearly visible, but it is visually apparent if there are contributions from higher-lying states with weaker transition moments. Using Bayesian analysis, we have verified the presence of the \(n = 8\) state as well. Proof of this is shown in Fig. 3c, which compares the effective free energy, F, for a 5-peak, 6-peak, and 7-peak fit of the data (cf. “Methods” for a detailed explanation).
The higher number of observed Rydberg states in the PL measurement compared to the OD measurement can be attributed to the spatial non-uniformity of the sample and the broader spot size of the excitation. Another curious feature of the PL spectrum is the deviation of peak heights from the expected \(n^{-3}\) trend, which is made evident by comparison to the reference spectrum acquired from a bulk sample (cf. gray line in Fig. 3a). This deviation could arise from a variety of sources, such as the collisional population of Rydberg states during non-resonant PL excitation, which may not follow the standard scaling law, as well as scatterings from vacancies, the presence of the green 1S exciton, which can influence the matrix elements at lower n values, and the susceptibility of higher-n states to the effects of charge impurities and local electric fields52.
Figure 4a shows the results from the phonon replica region of the spectrum, where the quadrupole-allowed and \(\Gamma _3^-\) phonon-assisted relaxations of the yellow 1s-orthoexciton state can be observed as Fano and Boltzmann-tailed peaks, respectively. At higher temperatures, one can see an anti-stokes phonon-assisted transition appearing at higher energies. As indicated by the black dotted line, these peaks red shift with temperature similar to the trend obtained from the OD measurement. The information from this spectral range is also useful for gauging the prevalence of metallic impurities in the sample. Excitons bound to these impurities fluoresce at energies between 1.99 and 2.01 eV, with intensities on the same order of magnitude as the \(\Gamma _3^-\) phonon-assisted transition53. As can be seen in Fig. 4a, such features are absent from the spectrum, even at low temperatures, demonstrating that the synthetic film is metallic impurity-free.
For the temperature and power-dependent analyses, the peaks were fit with Fano lineshapes using a least squares algorithm. Figure 4b shows the results from the temperature-dependent study. As in the OD measurements, the energy of the yellow exciton states observed in PL redshift as temperature increases. A similar trend can be observed in the power-dependent data shown in Fig. 4c. The change in the Rydberg energies can be attributed to two possible effects: temperature change due to laser absorption and exciton–exciton interactions caused by the high exciton density created at high laser powers. As already discussed in “Optical density measurement”, the former effect manifests as a redshift, while the latter is expected to manifest as a blueshift, caused by repulsive van der Waals interactions11.
To differentiate between these two effects, we performed an interpolation, using the center energies from the power-dependent measurement to extract an effective temperature as a function of the laser power. Separate interpolations were performed for the 2p, 3p, 4p, and 5p peaks, with the most contributions even at elevated temperatures. As can be seen in Fig. 4d, all four interpolations follow the same trend. In the presence of notable exciton–exciton interactions, however, one would anticipate that the interpolations from higher states deviate from the lower ones. This is because excitons in higher energy states exhibit stronger interactions, attributed to the overlap of their extended wavefunctions. Combined with the linear trend, this indicates that the energy shifts due to temperature change dominated over those from the exciton–exciton interaction, due to the strong above-the-bandgap absorption.
In Fig. 5a,b, we show the energy and linewidth of Rydberg excitons, respectively, comparing the synthetic and natural \(\hbox {Cu}{}_2\hbox {O}\) samples. These properties are plotted as a function of n for data obtained from PL and OD measurements of the synthetic sample at the lowest temperature of 5 K and the lowest laser power of 50 \(\upmu \hbox {W}\), as well as for data from OD measurements of a natural bulk sample (see Sect. S3 of the supplementary materials).
The black dashed line in Fig. 5a shows the ideal \(n^{-2}\) trend of the Hydrogenic Rydberg series. As can be seen, the Rydberg energies of the synthetic film and the natural bulk sample follow this trend, indicating that confinement effects are negligible for these states. (cf. Sect. S1 of the supplementary material).
As for the linewidths depicted in Fig. 5b, for ideal Rydberg states a \(n^{-3}\) scaling is expected, but as can be observed, here they reach a plateau which agrees with previous results33. The n-dependent behavior can be modeled as
$$\begin{aligned} \Gamma (n) = \alpha \frac{n^2 – 1}{n^5} + \beta , \end{aligned}$$
(5)
where \(\alpha\) is a proportionality constant and \(\beta\) represents a minimum value which is the linewidths for very large n54. This model was used to fit the data from each series shown in Fig. 5b. The high-n asymptotes (\(\beta\)) for each series are reported in Table 2. A similar linewidth plateau has been observed in ultra-cold Rydberg atoms interacting with a dense background gas55, a phenomenon attributed to the frequent scattering of high-lying Rydberg electrons off of the ground-state atoms56. By analogy with the atomic case, we hypothesize a similar phenomenon could affect Rydberg excitons in \(\hbox {Cu}{}_2\hbox {O}\), resulting from collisions with a background electron-hole plasma, phonons, or a high-density of 1s ground-state excitons57,58, which will be reported elsewhere.