Oscillator strengths, intracenter absorption and photoionization cross sections of optical transitions of shallow donors in silicon

Infrared photoionization and intracenter cross sections as well as oscillator strengths of intracenter transitions in silicon doped with single-electron, shallow donors are determined in research-grade crystals and compared with the corresponding values calculated by various theoretical models. The float-zone grown crystals were doped with substitutional group-V, interstitial group IA lithium and lithium-oxygen complex at concentrations of ${10}^{12}–{10}^{17}\mathrm{atoms}/\mathrm{c}{\mathrm{m}}^3$. The concentrations of electrically active impurity centers in the samples were determined from resistivity measurements. Experimentally integrated cross sections were obtained from low-temperature absorption spectra of impurities. For an isocoric substitutional donor, the oscillator strengths for intracenter transitions into the lowest odd-parity states were compared for two main crystal growth and doping methods: the float-zone and the Czochralski techniques. The applicability of the obtained calibration coefficients for various ranges of impurity concentrations is discussed. Recommendations are given for the optimal selection of optical transitions for quantifying the density of shallow donors in silicon along with experimental values for each shallow center. The oscillator strengths of transitions of shallow impurities were estimated for almost all observed donor transitions, including those into high excited, Rydberg-like atomic states, as well as for the intracenter transitions into several even-parity excited states.

Figure 1

The types of intracenter (between discrete states in the band gap) and photoionization (from the impurity ground state into the conduction band continuum, gray arrows up) optical transitions in the energy spectrum of shallow donors in silicon, addressed in this study, are schematically displayed for (a) hydrogen-like isocoric substitutional phosphorus; (b) interstitial lithium. Symmetry allowed $1s\to \mathrm{n}p$; $\mathrm{n}f$; $\mathrm{n}h$ (blue arrows up) and parity-forbidden $1s\to \mathrm{n}s$; $\mathrm{n}d$ transitions (purple arrows) were studied at low temperatures ($\sim 5$ K) while the $1s\to \mathrm{n}p$ transitions from the $1s$ excited, valley-orbit-split states (orange) were measured at elevated temperatures.

Figure 2

Examples of calibrated FZ-Si:P absorption spectra: (a) samples with low- to medium- and high phosphorus concentration at $\sim 4$.6 K. Note the concentration broadening of intracenter transitions; (b) absorption spectra of Si:P sample with $N_{\mathrm{P}}\approx 2.9\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$ at different temperatures. The absorption axis is cut at $\mathrm{OD}=$α ($v$) $d=1$.

Figure 3

(a) Integrated absorption vs donor concentration for the most intense intracenter transitions - from the impurity ground state $1s\left({\mathrm{A}}_1\right)$ into the lowest odd-parity excited states (phosphorus in silicon). The dashed lines mark the obtained values ${\sigma }_i^{-1}$ for donor transitions. Circles mark the values for FZ-Si:P, while stars are for CZ-Si:P. Open circles/stars correspond to ${\alpha }_i$ values obtained by the modeled line shapes. (b) Experimental ratio of oscillator strengths for pairs of the most intense transitions, on the scale of optical density for the $1s\left({\mathrm{A}}_1\right)\to 2p_0$ transition. Open circles/stars correspond to the ratios of the modeled line shapes. The dashed lines indicate the theoretical values for the selected ratios in Ref. [17]. (c) Calibration factors for phosphorus in FZ-Si:P for the most intense transitions taken for the integral absorption under the modeled transition lines. Crossed cycles mark $\mathrm{OD}=1$ for a 0.5-mm-sample thickness. The filled symbols in all plots are values obtained directly from the absorption spectra.

Figure 4

Absorption spectra of Si:P grown by FZ and CZ techniques, the phosphorus concentration in both is about $2\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$; moderate carbon ($2.2\times {10}^{16}\mathrm{c}{\mathrm{m}}^{-3}$) and oxygen ($6.9\times {10}^{17}\mathrm{c}{\mathrm{m}}^{-3}$) in CZ-Si:P vs lower (below the limits of detection) content of these impurities in FZ-Si:P, $N_{\mathrm{P}}\approx 2.0\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$. Integrated absorption (values in plot are in $\mathrm{c}{\mathrm{m}}^{-2}$) obtained for several energy transitions by direct integration over the line profile are given for spectrally resolved transitions. The impurity-induced broadening of intracenter transition in CZ-Si:P “compensates” for the loss in absorption (peak values) when compared with FZ-Si:P: this is valid only for the lowest-energy donor transitions.

Figure 5

Oscillator strengths vs photon energy hν for phosphorus intracenter transitions [from the ground state $1s\left({\mathrm{A}}_1\right)]$: (a) for the $\mathrm{n}{p}_0$ ($\mathrm{n}{f}_0$) series; (b) for the $\mathrm{n}{p}_{\pm }$ ($\mathrm{n}{f}_{\pm }$) series. Empiric asymptotics (shown as dashed lines, where $E_i=45$.59 meV is the phosphorus ionization energy) of these dependences enable selection of series of excited phosphorus states, which serve as terminating levels for intracenter transitions from the donor ground state. Stars mark experimental values for high purity silicon in Ref. [21]. Theoretical models are represented by Refs. [17] ([C]) and [16] ([B]).

Figure 6

Calibrated absorption spectra for moderately doped: (a) Si:Sb sample with $N_{\mathrm{Sb}}\approx 7.1\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$ and (b) Si:As sample with $N_{\mathrm{As}}\approx 3.6\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$ at different temperatures. The residual oscillatory background in the far-infrared part of the spectra is due to the nonvanishing interference of light in the samples.

Figure 7

Examples of calibrated Si:Bi absorption spectra in different spectral bands and temperatures: (a) heavily doped sample ($N_{\mathrm{Bi}}\approx 1.3\times {10}^{17}\mathrm{c}{\mathrm{m}}^{-3}$) at different temperatures. Thermally induced transitions from valley-orbit-split states into odd-parity states are detectable above 50 K (low-frequency part). Parity-forbidden $1s\to 1s$ transitions between valley-orbit-split states become detectable, together with several defect-induced phonon bands (high-frequency part). (b) Concentration broadening and related spectral shifts of intracenter transitions originating from the ground state. Parity-forbidden $1s\to ns, nd$ ($n>1$) transitions occur in the spectra of heavily doped samples. The absorption axis is cut at the $\mathrm{OD}=$α ($v$) $d=1$ for the sample ($N_{\mathrm{Bi}}\approx 7\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$).

Figure 8

(a) Integrated absorption vs concentration for the most intense bismuth transitions, FZ-Si:Bi. The dashed lines mark the obtained calibration coefficients. The circles mark the ${\alpha }_i$ values obtained directly from experimental spectra, while the open circles are from the modeled line shapes. (b) Oscillator strength ratios vs OD for $1s\left({\mathrm{A}}_1\right)\to 2p_0$ transition. The dashed lines are the theoretical ratios as from Ref. [17]. (c) Calibration factors obtained from ${\alpha }_i$ for the modeled line shapes. The crossed cycles mark the optical density equal to unit. For the case of the transition into $2p_0$ Bi state: ${\alpha }_i$ and $f_{2{\mathrm{p}}_0}$ are obtained by integration over the experimental line’ shape.

Figure 9

Dependences of the oscillator strengths of donor intracenter transitions [from the ground state $1s\left({\mathrm{A}}_1\right)]$ on the transition’ photon energy hν for the $\mathrm{n}{p}_{\pm }$ (left) and $\mathrm{n}{p}_0$ series. Open circles are experimental values for substitutional group-V donors and the Li-O complex, stars are for the interstitial Li donor. Solid circles are values calculated in Ref. [17].

Figure 10

(a) Absorption cross section spectra of group-V substitutional donors around their ionization energy, $E_i$. For Si:Bi, which transitions overlap with the two-phonon absorption background, also a differential spectrum Si:Bi* (see text for description) is given. (b) Determination of the photoionization cross sections from the absorption spectrum of Si:Li with accompanying Li and Li-O centers follows the modeling of additive photoionization spectra bands. Arrows show the binding energies of donors.