Terahertz stimulated emission from silicon doped by hydrogenlike acceptors

Stimulated emission in the terahertz frequency range has been realized from boron acceptor centers in silicon. Population inversion is achieved at resonant optical excitation on the 1Γ+8 → 1Γ−7, 1Γ−6, 4Γ−8 intracenter transitions with a midinfrared free-electron laser. Lasing occurs on two intracenter transitions around 1.75 THz. The upper laser levels are the 1Γ−7, 1Γ−6, and 4Γ−8 states, and the lower laser level for both emission lines is the 2Γ+8 state. In contrast to n-type intracenter silicon lasers, boron-doped silicon lasers do not involve the excited states with the longest lifetimes. Instead, the absorption cross section for the pump radiation is the dominating factor. The four-level lasing scheme implies that the deepest even-parity boron state is the 2Γ+8 state and not the 1Γ+7 split-off ground state, as indicated by other experiments. This is confirmed by infrared absorption spectroscopy of Si:B.


I. INTRODUCTION
Transitions between energy levels of an impurity in a host crystal are the basis of many solid-state lasers [1]. Examples range from the ruby laser, the first-ever-made laser, to the very powerful Nd:YAG laser. The very long lifetimes of particular impurity levels, which can be as long as a few milliseconds, enable efficient laser schemes where population inversion is obtained between a long-living upper level and a lower level with a much shorter lifetime. The concept of using optical transitions between excited states of impurity atoms in semiconductors with indirect band gap, such as silicon, has been explored in recent years. These approaches are based on substitutional hydrogenlike impurity centers in silicon [2,3] and rare-earth atoms embedded in a silicon crystal [4,5]. Although the impurity states, which are involved in dipole-allowed optical transitions in silicon, have typical lifetimes on the picosecond to nanosecond scale, population inversion can be achieved due to large optical cross sections that enable efficient optical pumping. This compensates for the short lifetime of the upper laser level. Together with near-infrared Raman-type lasers [6,7] and hybrid III-V silicon evanescent lasers [8], these are the only lasers that are based on silicon.
Stimulated terahertz (THz) emission from dipoleallowed intracenter transitions as well as Raman lasing has been demonstrated for all group-V substitutional donors in silicon [9]. These lasers operate at low temperatures (<30 K) either with optical excitation directly into one of the excited states or by photoexcitation into the conduction band. All laser schemes in n-Si are based on the specific electron-phonon interaction, which leads to different relaxation rates for the excited impurity states. Evenparity states always relax much faster into the even-parity ground state than into odd-parity states [10]. Therefore, at nonequilibrium conditions, odd-parity states are more populated than the adjacent lower-lying even-parity states and laser emission occurs from dipole-allowed odd → even intracenter optical transitions. Unless particular donor states are in resonance with intervalley phonons [11], the upper level of the intracenter laser is always the odd-parity state with the longest lifetime, which is the state with the largest energy gap to its neighboring even-parity state. As a result, only four-level laser schemes can be realized in n-Si and the laser transition is between the lowest odd-parity and the lowest even-parity state [11]. Single-phonon electronic interaction with a zone-centered optical phonon (phonon energy hω LTO ∼ 64 meV) is weak in silicon and does not contribute to the nonradiative relaxation of bound and free electrons. This allows use of radiation sources with photon energies exceeding hω LTO , such as a CO 2 laser, to populate excited states of hydrogenlike impurities [2]. Resonant pumping in excited impurity states (photon energy less than hω LTO ) results in even more efficient lasing schemes [12]. While n-Si lasers have been realized with a number of dopants [9], a p-Si laser has not yet been demonstrated. Only spontaneous emission under CO 2 -laser pumping as well as electroluminescence has been observed from electrically excited Si:B [13].
The energy structure of hydrogenlike acceptors in silicon has some peculiarities that are related to the strong spinorbit (SO) coupling of the valence subbands (labeled Γ þ 8 and Γ þ 7 [14,15], Fig. 1). The SO coupling breaks the parabolicity of the Γ þ 7 subband and the spectrum of acceptor states related to the Γ þ 7 valence band (VB) is modified. But it remains resonant to the continuum of the states in the Γ þ 8 VB. Theoretical calculations suggest a strong admixing of hydrogenlike wave functions from the lower 1Γ þ 8 VB to the resonant states bound to the bottom of the upper 1Γ þ 7 VB. This induces large shifts of the resonant state energies and it makes the entire impurity spectrum nonhydrogenlike [15,16]. For Si:B, for instance, the calculated shift is in the order of the spin-orbit energy of Δ SO ≈ 44 meV that would result in the 1Γ þ 7 state being localized in the silicon band gap with a binding energy of E Γ7þ ≈ −22 meV [15,16]. This is larger than the binding energy of any other excited boron state (Fig. 1).
However, a controversy exists about the ground state of boron in the Γ þ 7 subband, which, according to some experiments [17][18][19], is located in the energy gap of silicon at an energy of E S ≈ −23 meV, while other experiments [13,[20][21][22] do not show any evidence for this. Since optical transitions between the subband's ground states are forbidden in the dipole approximation, it is not possible to directly measure the 1Γ þ 7 state energy E 1Γ7þ by lowtemperature absorption spectroscopy. Wright and Mooradian [17] and Cherlow et al. [18] attributed a feature at E S ≈ 23 meV in the Raman spectrum of a Si:B sample (doping concentration N B ∼ 2 × 10 16 =cm 3 ) to the resonant 1Γ þ 7 → 1Γ þ 8 transition resulting in E 1Γ7þ ≈ 23 meV, which is in the band gap (lower dashed line in Fig. 1). Chandrasekhar et al. assigned the only spectral feature at 22.77 meV in the absorption spectrum of a heavily doped (N B ∼ 10 17 =cm 3 ) Si:B sample as a candidate for a transition ending in the localized 1Γ þ 7 state [19]. Thewalt, in contrast, did not observe any line with an energy around 23 meV in photoluminescence spectra obtained from floatzone-refined Si:B (N B ∼ 2.2 × 10 15 cm −3 ). He could not confirm the existence of the 1Γ þ 7 state in the band gap [20]. However, he observed a full set of transitions to other evenparity states. As an alternative explanation, Jones et al. attributed the 22.81 meV line in the absorption spectra of Si:B to a carbon-boron pair (B-X) center with a binding energy of −37.1 meV [21]. Also, electroluminescence spectra observed for boron acceptors [13] and phosphorus donors [22] indicate that the 1Γ þ 7 state is not in the band gap. Luminescence lines of Si:P end in the same split-off ground state, which is part of the four-level laser scheme in optically pumped Si:P [2]. In contrast, all electroluminescence lines observed from Si:B [13] end in the boron ground state 1Γ þ 8 . This indicates that in Si:B population inversion might be achieved in a three-level scheme, not requiring a 1Γ þ 7 state in the band gap. The controversy about the 1Γ þ 7 state has some implications for potential laser mechanisms in Si:B. Assuming that the 1Γ þ 7 state is localized in the silicon band gap, a fourlevel laser scheme with the 1Γ − 8 → 1Γ þ 7 laser transition can be expected in optically pumped Si:B. In this case, optical pumping is possible with photons of energy equal to or larger than E 1Γ8þ − E 1Γ8− ≈ 30.4 meV and the most effective resonant pumping should occur for the 1Γ þ 8 → 1Γ − 8 transition at 30.4 meV, which has a large oscillator strength (1.77 × 10 −2 [16]) and the longest lifetime of all excited states [23]. If the 1Γ þ 7 state is not in the band gap, this laser mechanism is not possible.  The narrow arrows up show dipole-allowed boron absorption transitions (ground state → Γ þ 8 subband: lines 1, 2, 3, 4, 5) (ground state → Γ þ 7 subband: lines 2 0 , 3 0 , 4 0 ), which are regularly observed in low-temperature absorption spectroscopy [14]. The 1Γ þ 7 ground state of the split-off state series (Γ þ 7 band) is expected to occur in between the two horizontal dashed lines: between resonant to the Γ þ 8 band location [14] and the theoretically predicted in the band gap [15]. The light blue dashed line is the position where the 1Γ þ 7 ground state should appear according to Raman spectroscopy [17,18]. (b) Si:B laser scheme. The bold straight arrows indicate the transitions for stimulated emission (red downward arrows) under optical pumping (bold blue upward arrow). Note that the laser transitions end in the lowest even-parity state. The curved green arrow indicates nonradiative relaxation.
Here, we report on the realization of population inversion and lasing in p-Si, namely silicon doped with boron acceptors. Lasing is achieved at a lattice temperature of approximately 5 K by resonant infrared optical pumping into excited boron states. The large oscillator strengths of the closely spaced boron absorption lines enable gain at transitions between particular excited states. The laser scheme is completely different from those known for n-type THz intracenter silicon lasers [9] and it does not involve emission lines observed in electrically excited Si:B [13]. This indicates that the 1Γ þ 7 state is not in the band gap.

II. EXPERIMENTAL PROCEDURE
Several Si:B crystals were grown by float zone as well as Czochralski techniques in h100i or h111i directions. The doping concentration is in the range of N B ∼ ð1-5Þ× 10 15 cm −3 . This concentration was chosen because it covers the optimal doping range of n-Si lasers [9]. The crystals are characterized by measuring their absorption spectra with a Fourier transform infrared (FTIR) spectrometer.
The Si samples for obtaining stimulated emission have dimensions of approximately 7 × 7 × 5 mm 3 . The surfaces are optically polished to form a high-Q resonator operating on total internal reflection modes. The samples are mounted in a cryogenic dipstick that is immersed in a liquid helium transport vessel. The beam from the pump laser is directed onto the (100) or (111) sample facets, while the emission from the sample is registered in the orthogonal direction by a Ge:Ga detector inside the dipstick as well as by unstressed and stressed Ge:Ga detectors that are sensitive from 40 to 120 μm and 120 to 210 μm, respectively. These detectors are mounted in a separate cryostat outside of the dipstick. The emission is analyzed with a FTIR spectrometer equipped with the Ge:Ga detectors. Combinations of infrared filters at liquid helium and at room temperature prevent pump radiation from reaching one of the detectors.
Photoionization pumping into the valence band is done with a transversely excited atmospheric CO 2 laser that provides 1 MW peak power in a 1 μs pulse at a 5 Hz repetition rate. The laser is tunable in the range 114-135 meV (9.2-10.9 μm). Intracenter pumping with photon energies in the range from 30 to 50 meV is done with the FEL FELIX (formerly at the FOM Institute for Plasma Physics, Rijnhuizen, The Netherlands). Its pump pulses consist of 6-μs-long macropulses at a repetition rate of 5 Hz and a peak power of up to 10 MW. Each macropulse in turn consists of ∼10-ps-long micropulses with a spectral width of Δλ=λ ≈ 0.6% (Δλ is the FWHM) that are separated by 1 ns. As has been shown for n-Si lasers, pumping with the frequency tunable FEL can be significantly more effective than photoionization pumping by a CO 2 laser since it allows resonant pumping of excited states. This reduces the number of levels involved in the process of generating population inversion and consequently reduces recombination losses, for example, by nonradiative processes. In addition, the optical absorption on 1Γ þ 8 → oddparity impurity transitions is more than an order of magnitude larger than that for photoionization, which enables more efficient pumping.
The relaxation time of the 1Γ − 7 and 1Γ − 6 states, which are, as we will see later, the most important states, are measured by a pump-probe technique. The Si:B sample for these measurements is cut from the same material as one of the Si:B laser samples. It has a boron concentration N B ∼ 2 × 10 15 cm −3 and a mean thickness of 400 μm. The Si:B surfaces, which face the pump and probe radiation, are polished and wedged in order to reduce interference effects in the sample. The experiments were carried out at the FEL of the Helmholtz Zentrum Dresden-Rossendorf. This FEL provides quasicontinuous radiation at a repetition rate of 13 MHz and a pulse duration of approximately 10 ps. The experimental setup is described in detail elsewhere [24]. The high signal-to-noise ratio achievable with this setup allow us to significantly reduce the pump energy (down to 70 pJ) and accurately detect pump-induced transmission changes at the level of 0.3%. Precise overlap of the FEL linewidth with the boron transitions 1Γ þ insures optimal pumping of the probed states.

III. RESULTS AND DISCUSSION
In Fig. 2, a typical absorption spectrum of a Si:B sample is shown at a temperature of 5 K. A number of absorption lines are clearly visible and these can all be assigned to transitions in Si:B, namely, the 1Γ þ 8 → odd-parity impurity transitions (lines 1; 2; 3; 4; 4a; 4b; 5; 6; …; 10; nomenclature according to Ref. [14]). The closely spaced 4, 4a, and 4b lines are not resolved, because these lines are significantly concentration broadened in the Si:B sample. By tuning the frequency of the FEL, a similar absorption spectrum of the Si:B sample is obtained (Fig. 2), although with less spectral resolution because of the linewidth of the FEL. The slope in the spectrum is due to absorption by water vapor in the atmosphere and power variations of the FEL. Some of the absorption lines (marked with an asterisk) are caused by atmospheric water. Emission from the Si:B sample is observed only when pumping on the 4, 4a, 4b lines (Fig. 2), which correspond to the 1Γ þ 8 → 1Γ − 7 , 1Γ − 6 , 4Γ − 8 intracenter transitions (calculated oscillator strengths are 2.60 × 10 −2 , 3.76 × 10 −2 , 2.30 × 10 −3 , correspondingly [16]). This is different from n-Si lasers where pumping from the ground state into several lowest odd-parity impurity transitions leads to laser emission [9]. The FWHM of the Si:B emission signal is approximately the same as that of the absorption line measured with the FEL, which indicates that each of the three pump lines leads to laser emission.
The Si:B laser emission spectrum (Fig. 3) consists of two lines at 1.740 and 1.748 THz (172.3 and 171.5 μm), which are barely resolved due to the limited spectral resolution of the FTIR spectrometer (∼5 GHz). The analysis of pump and emission spectra and comparison with the known Si:B states allows identification of the involved boron states. The upper laser levels are the 1Γ − 7 , 1Γ − 6 states, which are not resolved in the laser spectrum, and the 4Γ − 8 state. The lower laser level for all emission lines is the 2Γ þ 8 state [ Fig. 1]. The binding energy of the 2Γ þ 8 state as derived from the laser spectrum is −13.44 meV, which agrees very well with the experimental (−13.44 meV [25]) and calculated (−13.41 meV [25], −13.34 meV [15]) values. The laser scheme is an indirect indication that the Γ þ 7 state is not the state with the largest binding energy, because in that case one would expect laser emission on the 1Γ − 8 → 1Γ þ 7 transition. In particular, absorption of pump radiation on the 1Γ þ . 2) and the relaxation time of the 1Γ − 8 state is longer than that of the 1Γ − 7 , 1Γ − 6 states [23]. The coexistence of two emission lines and the large linewidth are indicators of low competition between the lasing transitions. This is caused by the low gain and small population of the upper laser level.
The lasing threshold is above ∼10 24 photons cm −2 s −1 and depends on the pump photon energy (Fig. 4). Pumping directly at the center wavelength of the boron absorption line (4 and 4a: pump photon energy 39.6 meV) has a lower threshold than pumping in the wings of the absorption line (4b: 40.0 meV) (Fig. 4), because the absorption cross section for the pump radiation is smaller in the latter case. The relaxation time of the upper laser states is derived from pump-probe experiments by fitting the decay of the photoinduced transmission with an exponential function (Fig. 5). The finite duration of the pump and probe pulses is taken into account by convoluting the exponential decay with a Gaussian pulse of ∼10 ps FWHM. The linewidth of the FEL is too large (Δλ=λ ≈ 0.6% ≈ 2 μm) compared to the separation of these states (∼0.2 μm) to allow measuring of the relaxation time of the 1Γ − 7 , 1Γ − 6 , and 4Γ − 8 states individually. The relaxation time is t ¼ 53 AE 2 ps. It is independent of the pump energy across the whole range  from 70 to 850 pJ (Fig. 6), because at these low pump energies only relaxation from the excited into the ground state is important. Recombination processes become significant at pump energies above 2 nJ [24]. The relaxation time is shorter than the lifetimes of the odd-parity excited states in Si:B with larger binding energies [23] and it is significantly shorter than those of n-Si lasers, where the upper laser level has a typical lifetime between 100 and 250 ps [26,27]. The short lifetime of the upper laser level is also the reason for the rather high laser threshold. Only samples with a net boron concentration in the range N B ∼ ð1-3Þ × 10 15 cm −3 exhibit laser emission while samples with N B ∼ ð4-5Þ × 10 15 cm −3 do not show emission. This is significantly lower than the maximum doping of n-Si lasers, which operate up to a concentration of ∼10 16 cm −3 [9]. Because the spacing of the energy levels around the states involved in Si:B lasing is smaller than in n-Si, concentration broadening leads to overlap of the upper laser level with other states at lower concentration. This results in a shorter lifetime of the upper laser level and in less gain. The low gain is one of the reasons for the absence of stimulated emission when the samples are pumped with a CO 2 laser and a photon flux density up to ∼2 × 10 25 cm −2 s −1 on the sample. The other reason is the lower pump efficiency of nonresonant pumping, because there are several relaxation paths for the excited carriers similar to those in n-Si [28]. Some of these do not involve the upper laser level, especially those utilizing even-parity excited states. Also, no Raman-type stimulated emission is observed from any of investigated Si:B samples, even for maximum pump power from the FEL. This is different from n-Si lasers, where Raman lasing has been observed from all donors [9]. Assuming that the scattering strength of the resonant electronic Raman process in Si:B is close to that of Si:P [17], one can expect stimulated Raman emission from the investigated Si:B samples with a Stokes shift of E S ≈ 23 meV, provided that the 1Γ þ 7 ground state has a binding energy of approximately 22 meV. The fact that Raman lasing is not observed is another indication that the 1Γ þ 7 state is very likely not located in the band gap of silicon. This is in contradiction to the Raman spectroscopy results discussed above [17,18]. Possibly, the observed spectral features in these spectra are a result of a resonance between the pump laser photon energy (λ ¼ 1064 nm, E 1064 nm ≈ 1.165 eV) and the sum of the indirect band gap of silicon (E g ≈ 1.12 eV) and the spin-orbit energy E g þ Δ SO ≈ 1.164 eV. As has been shown by Wright and Mooradian, the Raman spectra of Si:P and Si:B have the same lines at 37.9 and 64.8 meV [17]. While the line at 64.8 meV is related to the zone-centered optical phonon hω LTO , the line at 37.9 meV might be related to the indirect band gap E g ¼ E 1064 nm − 37.9 meV. Then the lowerenergy line in the Si:B spectrum at ∼23.4 meV evidently belongs to an electronic state that is not in the band gap E g , but is resonant to the 1Γ þ 8 continuum: E B ¼ E 1064 nm − 23.4 meV ≈ 1.142 eV > E g . An accurate interpretation of the spectra obtained by resonant Raman scattering requires experimental verification that would include both Stokes and anti-Stokes bands as well as a nonresonant excitation laser.
In order to investigate the issue of the 1Γ þ 7 state, absorption spectra are measured with a FTIR spectrometer at temperatures up to 180 K. Provided that the 1Γ þ 7 state is in the band gap at E 1Γ7þ ≈ −23 meV, it should be significantly populated at temperatures between 50 and 100 K. This is similar to the absorption spectra of n-Si:As, whose 1sðEÞ and 1sðT 2 Þ states are split off from the ground state by ∼ 21 meV and where transitions from 1sðEÞ, 1sðT 2 Þ states into the higher-lying odd-parity states are observed at temperatures between 30 and 160 K with a maximum at around 75 K. In Si:B, one would expect a