Midinfrared properties of cuprous oxide: High-order lattice vibrations and intraexcitonic transitions of the $1s$ paraexciton

This contribution addresses both the general mid-infrared (MIR) properties of ${\mathrm{Cu}}_2\mathrm{O}$ and induced absorption due to intraexcitonic $1s\to 2p$ transitions under simultaneous cw laser excitation. After an overview of the current state of knowledge and remaining open questions concerning excitons in ${\mathrm{Cu}}_2\mathrm{O}$ we will first discuss several absorption bands in the spectral range between approximately 100 and $150\mathrm{meV}$ in terms of multiphonon and biphonon features. The second, more extensive part will concern a pump-probe experiment in the limit of low densities of $1s$ excitons where we investigated intraexcitonic transitions appearing in the MIR spectral range as well. We find a signal which can be clearly assigned to a $1s\mathit{para}\to 2p\mathit{para}$ transition and present various data depending on excitation conditions and sample temperature. Applying the results of detailed theoretical considerations we can estimate the density and lifetime of the $1s$ para-exciton and at least speculate about its dispersion.

Figure 1

Temperature dependent MIR optical density $\alpha d$ of a $210\mu \mathrm{m}$ thick (111) sample. With regard to the pump-probe experiment a subset of the nominal $1s$ to $np$ transition energies for $T=10\mathrm{K}$ is indicated as well, whereas transitions to ∞ describe the ionization of the corresponding $1s$ exciton.

Figure 2

(a) Absorption band 1 for various temperatures, measured on a $210\mu \mathrm{m}$ thick (111) sample. (b) The corresponding area $\int \alpha d\bullet dE$ as a function of temperature where the reflection offset as well as the influence of the sidebands 1b and 1c were removed. The solid line shows a fit according to Eq. (3) resulting in $E_i\approx 19.2\mathrm{meV}$.

Figure 3

Upper left and upper right corner: sidebands 1b, 1c, and 2b for various temperatures where the contribution of the main bands 1 and 2 was removed. Lower left and lower right corner: temperature dependent position of the absorption maximum of the different sidebands.

Figure 4

Term diagram of the yellow series $1s$ and $2p$ states including the different irreducible representations with the corresponding degeneracies given in parentheses. States the energy of which is known from experiment are denoted with (∗). Dipole allowed transitions are indicated by arrows, whereas the energies on the right-hand side specify the nominal transition energies at $T=10\mathrm{K}$ and neglected $2p$ fine structure.

Figure 5

Ratio of transmission spectra with $\left({\mathrm{Tr}}^{*}\right)$ and without (Tr) cw laser excitation [$E_{\gamma }\approx 2.06\mathrm{eV}$, $650\mu \mathrm{m}$ (111) sample] for two temperatures. Note that the absorbed excitation density $I_{\mathrm{abs}}$ is considerably higher for $T=75\mathrm{K}$. The detection gap arises from MIR absorption band 2 (see Sec. 3D).

Figure 6

(a) Temperature dependent density of $1s$ para-excitons, extracted from the area of induced absorption via Eq. (12) ($650\mu \mathrm{m}$ sample). In order to create $1s$ ortho-excitons with constant excess energy, the laser was redshifted parallel to the $1s$ ortho-exciton with increasing temperature from $2.06\mathrm{eV}$ at $10\mathrm{K}\text{to}2.047\mathrm{eV}$ at $100\mathrm{K}$. The indicated densities are normalized to an absorbed photon flux density of $\approx 1.8\bullet {10}^{19}∕\left({\mathrm{cm}}^2\mathrm{s}\right)$ corresponding to $\approx 6.0\mathrm{W}∕{\mathrm{cm}}^2$ for $E_{\gamma }=2.06\mathrm{eV}$. (b) External photoluminescence quantum efficiency of the $1s$ ortho-exciton (data from Ref. 8). Since phonon assisted emission clearly dominates the PL spectrum, the quantum efficiency directly reflects the temperature dependence of the $1s$ ortho population.

Figure 7

Dispersions of generalized $1s$ and $np$ states including the notations which are used to deduce the theoretical line shape of induced $1s\to np$ absorption. Since the calculation will be done on the energy scale $E$ of the infrared photons which have their $k$-dependent energetical zero point in the $1s$ dispersion, the “old” energy scale is denoted by $E^{\prime }\left(k\right)$.

Figure 8

Theoretical line shape of induced absorption for a $1s$ $\mathit{ortho}\to 2p$ ortho transition. The dashed-dotted line corresponds to Eq. (21), while the solid line shows the numerical convolution of Eq. (21) with a Lorentzian (dotted line) using the temperature dependent width of the $2p$ ortho ${\Gamma }_4^{-}$ state.

Figure 9

Normalized induced absorption due to a $1s$ $\mathit{para}\to 2p$ para transition for excitation in the phonon assisted absorption continuum where the absorbed excitation density was $\approx 6\mathrm{W}∕{\mathrm{cm}}^2$. The indicated fit (solid line) was accomplished by using the homogeneous linewidth of the $2p$ ortho ${\Gamma }_4^{-}$ state and fixing the exciton temperature to the lattice temperature. The effective mass $m_{1s,\mathit{para}}$ as well as the transition energy $\Delta E_0$ (indicated by arrows) where used as fit parameters.

Figure 10

(a) Observed halfwidth of the $1s$ $\mathit{para}\to 2p$ para transition (data points) in comparison to the expected halfwidth (solid lines) for different effective masses of the $1s$ para-exciton. The dashed lines are calculated for neglected homogeneous broadening of the $2p$ para state. (b) Observed $1s$ $\mathit{para}\to 2p$ para transition energy as a function of temperature compared to the nominal $1s$ $\mathit{ortho}\to 2p$ ortho transition energy extracted from linear absorption and photoluminescence.

Figure 11

Density of $1s$ para-excitons as a function of absorbed excitation density for excitation in the phonon assisted absorption continuum at $E_{\gamma }\approx 2.06\mathrm{eV}$ [$650\mu \mathrm{m}$ (111) sample].

Figure 12

Pump-probe results for different excitation energies in the phonon assisted absorption continuum [$650\mu \mathrm{m}$ (111) sample, $T=30\mathrm{K}$]. (a) Absorbed laser photon flux density in the area of the phonon assisted absorption edge. (b) Corresponding integrated area of induced $1s$ $\mathit{para}\to 2p$ para optical density $\Delta \alpha d$.

Figure 13

$1s$ $\mathit{para}\to 2p$ para transition for different excitation conditions [$700\mu \mathrm{m}$ (100) sample]. Resonant excitation corresponds to an excitation energy of approximately $2.032\mathrm{eV}$; in this case the given excitation density is only an upper boundary.