The physical principles of terahertz silicon lasers based on intracenter transitions

Authors


Abstract

The first silicon laser was reported in the year 2000. It is based on impurity transitions of the hydrogen-like phosphorus donor in monocrystalline silicon. Several lasers based on other group-V donors in silicon have been demonstrated since then. These lasers operate at low lattice temperatures under optical pumping by a midinfrared laser and emit light at discrete wavelengths in the range from 50 to 230 µm (between 1.2 and 6.9 THz). Dipole-allowed optical transitions between particular excited states of group-V substitutional donors are utilized for donor-type terahertz (THz) silicon lasers. Population inversion is achieved due to specific electron–phonon interactions of the impurity atom. This results in long-living and short-living excited states of the donor centers. Another type of THz laser utilizes stimulated resonant Raman-type scattering of photons by a Raman-active intracenter electronic transition. By varying the pump-laser frequency, the frequency of the Raman intracenter silicon laser can be continuously changed between at least 4.5 and 6.4 THz. The gain of the donor and Raman-type THz silicon lasers is of the order of 0.5 to 10 cm−1, which is similar to the net gain realized in THz quantum cascade lasers and infrared Raman silicon lasers. In addition, fundamental aspects of the laser process provide new information about the peculiarities of electronic capture by shallow impurity centers in silicon, lifetimes of nonequilibrium carriers in excited impurity states, and electron–phonon interaction.

1 Introduction

Silicon is an elemental semiconductor that has been the object of intense fundamental research for decades and that dominates the semiconductor industry. Some of the most attractive features of silicon are its high thermal conductivity, the excellent dielectric properties of its oxide, the low reverse leakage current in silicon electronic devices. Silicon can be produced in large amounts with high quality and it can be doped with different species with high precision. Silicon-based microelectronic technology for integrated circuits is cheap and relies on the practically unlimited reserve of the material – silicon is the second most abundant element in the Earth's crust, making up 25.7% of its mass. Pure silicon can be doped with other elements to adjust its electrical conductivity by controlling the concentration and charge of the dopant. Such control is necessary for many electronic and photonic semiconductor devices. Besides that, it is worth noting that high-purity, isotopically enriched silicon has been used as material for the redefinition of the mass because of its high purity and its almost perfect lattice after growth 1.

While electronics is dominated by silicon, silicon-based photonics has not been developed up to the same stage. The main reason is the indirect bandgap in the momentum space between the conduction and valence bands that forbids direct optical recombination of a free electron between the bands of the semiconductor. Hence, nonradiative recombination dominates over the luminescence probability by a factor of ∼106 2.

Many different strategies have been developed to overcome the inefficient luminescence in silicon in order to enable a silicon laser (for a comprehensive review see Ref. 3). Some approaches rely on the enhancement of the quantum efficiency at bandgap transitions due to quantum confinement effects in low-dimensional silicon-based materials, such as nanometer-scaled silicon crystals dispersed in a dielectric matrix (see e.g. 4), light-emitting diodes containing SiO2 nanoclusters 5, or nanopatterned silicon-on-insulator emitters 6. Others consider band-structure engineering that allows overcoming the indirect bandgap in Si/SiO2 7 and Si/SiGe 8, 9 superlattices, where intersubband optical transitions are used 10. A silicon Raman laser emitting pulses of infrared radiation was reported for the first time in 2004 11 and soon after continuous-wave operation was reported 12, 13.

Another approach for the realization of a silicon laser is to use optical transitions between localized states of impurity centers (intracenter transitions) in silicon, as provided by native (isocoric and similar) dopants for silicon or rare-earth ions 14. While the latter falls into the infrared range (operating wavelength ∼1.5 µm), intracenter silicon lasers based on shallow impurity centers operate in the far-infrared, also called terahertz (THz), wavelength range and can potentially reach the midinfrared. Large optical cross-sections as well as long lifetimes of the laser levels are the specific advantages of the impurity-based silicon lasers.

The first silicon laser was reported in 2000 15. It operates on optical transitions between excited states of the isocoric phosphorus donor (emission wavelength: 55.2 µm) in pulsed mode under optical pumping by a midinfrared laser at low lattice temperatures. In 2002–2004 several donor silicon lasers utilizing intracenter optical transitions of all group-V donors in silicon were demonstrated 16–18. The first THz Raman-type intracenter silicon laser was reported in 2006 19. In the following years this type of Raman laser has been extended to silicon doped with other shallow donors 20–22. Although only pulsed operation has been achieved until now and relatively high optical pump powers are required for the laser operation, impurity-based silicon lasers demonstrate the possibility to obtain population inversion in silicon. This is an approach that might be developed for other semiconductors either as bulk or heterostructure materials. A variety of significant physical data concerning intracenter-related phenomena in silicon have been obtained from the analysis of the laser process. These are the basis for revisiting the cascade model of intracenter electronic relaxation, resulting in an improved understanding of the impurity–phonon resonant interactions as well as leading to a determination of the binding energies for different even-parity excited states of shallow donors in silicon 23, 24.

Here we review the major physical aspects and the performance of THz intracenter silicon lasers as well as report on new theoretical and experimental results. The review is organized as follows: We start with the theoretical introduction and theoretical predictions for the performance of THz silicon lasers. Then, the experimental results regarding silicon lasers based on optical transitions between excited shallow donor levels are described and compared with the theoretical predictions. This is followed by the demonstration of the Raman-type silicon laser based on light scattering by donor electronic resonances. Specific laser materials, including monoisotopic 28Si, multicrystalline and codoped silicon crystals, will be briefly mentioned. We discuss the features of nonradiative relaxation of bound electrons between low-energy electronic states. Finally, we discuss the directions for future research and the potential of intracenter silicon lasers.

2 Theory of intracenter silicon lasers

Solid-state intracenter lasers are well known. Transition-metal-doped and rare-earth-doped crystals are among the most effective bulk gain media, because of the extremely long lifetimes of particular excited impurity states. The first laser of this type was made from ruby, or chromium-doped sapphire (Cr3+:Al2O3). Widely used pulsed laser sources are titanium-doped sapphire (Ti3+:Al2O3) and neodymium-doped yttrium aluminum garnet (Nd3+:Y3Al5O12) infrared lasers. While in semiconductor lasers electronic injection schemes for achieving population inversion dominate, optical excitation of doped regions in a semiconductor has not been thoroughly considered. However, the frequency coverage of injection-populated bandgap semiconductor lasers is limited at the long-wavelength side by the condition of the thermal equilibrium EG = kBT, where EG is the semiconductor bandgap, kB is the Boltzmann constant and T is the temperature of the laser medium. Extending the wavelength range toward the mid- and far-infrared requires other laser principles. Intersubband p-Ge lasers (for a review see Refs. 25, 26), impurity-based silicon lasers and unipolar heterostructure-based quantum-cascade lasers 27 have extended the wavelength coverage to the far-infrared or THz region (40–300 µm or 1–7 THz frequency range). THz quantum-cascade lasers have made tremendous progress in filling the 1–5 THz range with continuous wave powerful coherent emission sources (for a review see Ref. 28). Silicon intracenter lasers offer emission lines in from 1 to 7 THz (Fig. 1) and remain a promising base material for semiconductor lasers in the entire THz range due to their low lattice absorption losses and large thermal conductivity.

Figure 1.

(online color at: www.pss-b.com) Frequency range of intracenter silicon lasers. Discrete lines are donor intracenter transitions, colored areas indicate Raman-type lasing. Different colors indicate different group-V donors.

The first proposal to use the optical transitions between the levels of an impurity atom for a laser has been made by Shastin in 1995 29. It was proposed to obtain population inversion between particular excited states, the relatively long-living 2p0 state and the lower 1s(E) state of the phosphorus donor embedded in a silicon crystal (Si:P), for instance, by photoionization of the donor atoms with radiation from a midinfrared CO2 laser 30. A similar four-level laser scheme was expected for silicon doped by bismuth, where the laser levels should be determined by resonant interaction with an optical phonon 31, 32. Spontaneous 32, 33 and stimulated emission from Si:P crystals were obtained in 1999 15. In 2000 Pokrovskiĭ et al. proposed using the forbidden optical transitions between 1s-type donor states with the same parity for achieving far-infrared laser action in silicon 34. This three-level laser scheme, however, has not been realized.

Although silicon has been thoroughly studied theoretically and experimentally since the 1950s, many important problems have not been completely addressed. One of the reasons is the complexity of the conduction (multivalley) and valence (degenerated and split-off) band structure that requires a rather complex analysis of impurity states and impurity–phonon interactions. For a long time simplified theoretical models, such as the cascade-dominated intracenter electron relaxation, or inaccurate relaxation rates and lifetimes derived from indirect experimental investigations have been accepted. Achieving laser action in silicon resulted in new insights about fundamental properties of n-type silicon and provided a significant feedback for the laser theory. New theoretical considerations have been triggered by the new and partly unexpected experimental data.

2.1 Shallow donors in silicon

Substitutional group-V atoms behave as hydrogen-like donors in silicon. The energy spectrum of localized impurity states originates from the six equivalent valleys of the conduction band localized on the Γ–X axes (〈100〉) in the Brillouin zone (for a review see Ref. 35). The conduction band minima are located at wave vector k0 = 0.855K〈100〉 36, where K〈001〉 = (2π/a0){1;0;0} is the reciprocal lattice vector from the center (Γ point) to the end (X point) of the first Brillouin zone along the 〈100〉 direction (Fig. 2), a0 = 5.431 × 10−10 m is the size of the unit cube in silicon. The surfaces of constant energy are ellipsoids of revolution with major axes along 〈100〉 (see Fig. 2).

Figure 2.

(online color at: www.pss-b.com) Schematic diagram of the first Brillouin zone of silicon. K010 is the reciprocal lattice vector between the high-symmetry points, the center (Γ) and the end (X) of the zone in 〈100〉 direction. The conduction band minima are represented by surfaces of constant energy, which are located at 0.855 × K〈100〉 from the Γ point. The dashed green arrows indicate two possible types (g- and f-) of intervalley scattering in silicon.

The hydrogen-like donors form discrete levels below the bottom of equivalent conduction band valleys. Each of the six valleys contributes to the formation of the eigen-wavefunction for the excited states. In the single-valley effective mass theory (EMT) the donor eigenstates are at least six-fold degenerate due to the six conduction band minima. The deviation from the spherically symmetric potential in the immediate vicinity of the donor site due to the chemical nature of the donor lifts the degeneracy of the ground 1s(A1 + E + T2) state and resolves it into 1s(A1), 1s(E), and 1s(T2) states. This decomposition is called valley–orbit or chemical splitting. Because of the valley–orbit splitting, all even-parity states, having a large amplitude at the donor site, deviate from the energy spectrum of a hydrogen-like atom, taking the conduction band bottom as zero energy, by the central cell correction value Δccc:

equation image((1))

where Ry = 13.6 eV is the Rydberg constant, ε the relative dielectric constant of silicon, mn and m0 the effective and free electron masses, respectively, and n is the main quantum number of the state 37. Due to a large ε ≈ 12 and a small effective mass mn/m0 ≈ 0.3, the binding energies of excited donor states En correspond to frequencies falling into the THz range.

The 1s(A1) state is the most downshifted. For the odd-parity p-like excited states the wavefunction yields a small probability near the donor site (Δccc ≪ n) and consequently the splittings are too small to be experimentally observable. Their spectra correspond well to the single valley EMT spectrum (Eq. (1)) with Δccc = 0 and prolate spheroidal electron mass 38. In general, the binding energies of particular states are taken from the comparison of the theoretically calculated spectra and low-temperature impurity-related absorption spectra by fitting the energy differences between observed and calculated donor transitions, originating from the ground state and terminating in higher p-type excited states. Usually, the measured energy of the 3p± state is assigned to be the same as the calculated value of 3.12 meV 38. With this assignment the values of all other states are defined. The energies of even-parity states, to which optical transitions are dipole-forbidden and whose energies are therefore not accessible by absorption spectroscopy, are usually derived by complementary methods such as Raman spectroscopy.

The ionization energies of the group-V donors are in the range from 42 to 71 meV 35. Since they are much smaller than the semiconductor bandgap, they are usually called shallow impurities. At the temperatures of interest (T < 20 K) shallow donor centers in silicon are filled by electrons and remain neutral atoms.

2.2 Intracenter relaxation of donor electrons in n-silicon

Being excited, the donor electrons occupy the higher, excited states of the donor or populate the conduction band. The first excitation process is called photoexcitation and the latter is called photoionization. After excitation, the electrons relax down toward the impurity ground state. The electronic capture by a donor center in silicon has been considered in a number of experimental and theoretical investigations, because of an anomalous fast relaxation rate, even at a low temperature of the lattice. Band-impurity luminescence (radiative relaxation) is a low-probability process. Inelastic nonradiative relaxation due to interaction with optical zone-centered (OZC) phonons suffers at low temperatures, because of the absence of a significant number of high-energy phonons in the silicon lattice. It has been observed experimentally only for p-type silicon where the OZC phonon–electron interaction is enhanced by the degenerated valence band 39. It was suggested 40 that fast intracenter relaxation in silicon is due to a cascade-type process accompanied by interaction with intravalley acoustic phonons mostly. In this concept free electrons are first captured into one of the dense highly excited states of the donor atom. Then they undergo a gradual relaxation through the lower excited impurity states with a characteristic energy step δE = (8|En|mns2)1/2 (1 − s(mn/(2En))1/2), where s is the velocity of sound, determined by energy and momentum conservation rules 40, with an emission of one acoustic phonon at each relaxation step. Transitions between neighboring excited states have, therefore, the largest probability (Fig. 3a). The rate of acoustic-phonon-assisted transitions decreases significantly for the lower excited states separated by energy gaps ΔEn greater than δE as (ΔEnE)−5 41. Particular excited states, such as the 2p0 state, are separated by large energy gaps from the ground state and appear to be relatively long living.

Figure 3.

(online color at: www.pss-b.com) Schematic energy diagram of antimony (Sb) donor states and intracenter relaxation in silicon. The parabolic continuum of states represents the bottom of the silicon conduction band. Horizontal lines mark the positions of donor states. The length of each line indicates approximately the relative localization of the state in momentum space. The donor states are labeled according to the nomenclature in Ref. 35. The colored curved arrows are acoustic-phonon-assisted nonradiative electron transitions. (a) Relaxation according to the cascade model of intracenter relaxation as proposed in Ref. 40. The upper excited states are step-by-step passed by the relaxing electron. The last steps across large energy gaps are accompanied by multiphonon or photon emission (bold arrow down). (b) Selective channels of intracenter relaxation as calculated in Ref. 60. The relaxation through the upper excited states is dominated by acoustic-phonon accompanied transitions between particular excited states. The large energy spacings between the lower states are passed due to interaction with intervalley acoustic phonons (straight arrows downwards).

In the cascade relaxation theory 40 the energy step δE is the only parameter characterizing the relaxation rate between a pair of interacting donor states. An additional differentiation comes from the selection rules for nonradiative transitions between donor levels. Transitions involving a triplet state 1s(T2) and doublet 1s(E) or singlet 1s(A1) states are forbidden, because of the symmetry of their wavefunctions 42, 43.

For nonradiative electron transitions with a large energy relaxation, interaction with the intervalley phonons (see Fig. 2) becomes essential. The large momentum of the phonons participating in intervalley scattering is compensated in the Umklapp process. For instance, for the transfer of an electron in a 〈100〉-type direction to the equivalent valley (g-scattering), the sum of the wave vector, qg, of the g-phonon and the change of the wave vector of the electron, Δk, is equal to a principal vector of the reciprocal lattice, K〈100〉. Momentum and energy conservation for this process can be written as

equation image((2))
equation image((3))

where ħωg is the energy of the intervalley g-phonon and ħ is the Planck constant divided by 2π. A similar equation can be written for f-phonons, which support the transfer between the equivalent valleys along the orthogonal axes in the Brillouin zone. The strongest scattering is provided by the intervalley phonon modes g-LO (fg-LO ≈ 15.3 THz), f-TO (ff-TO ≈ 14.3 THz) and f-LA (ff-LA ≈ 11.2 THz), which are allowed by group theory 44. These phonons exhibit a relatively low dispersion, so that one can neglect the dependence of their frequency on the wave vector 45. Thus, under the condition of Eq. (2), the electron scattering due to both acoustic and optical intervalley phonons is very similar to that due to intravalley optical phonons. The coincidence of the energy gap ΔE = En − En between particular excited donor states with an energy of an intervalley phonon according to Eq. (3) results in a resonant donor–phonon interaction with a relatively fast energy relaxation time. For instance, for the Bi donor in silicon, the 2p0 state is short living, a few ps, due to its resonant coupling to the ground donor state via interaction with the intervalley transverse optical (f-TO) phonon. This can be directly observed in the broadening of the 1s(A1) → 2p0 impurity absorption line 46.

2.3 Optical losses in the terahertz region

Moderately doped silicon offers low optical losses in the entire THz region. The losses can be intrinsic and induced by optical excitation of impurities inside the silicon crystal. Intrinsic ones occur mainly from optical absorption by lattice vibrations, impurity, and free carriers as well as due to the optical scattering on lattice vibrations and defects, or impurity atoms. Additional losses may appear due to optical excitation of negatively charged donor centers (D centers), which absorb at THz frequencies.

2.3.1 Lattice, impurity, and free-carrier absorption

The silicon crystals in our experiments are monocrystals doped to a level of up to 1016 cm−3. At low lattice temperatures (T < 10 K) and relatively low photoionization rates, the free electron concentration does not reach 1015 cm−3. Simulations show a concentration of ∼1013 cm−3 in Si:P at the lasing threshold 15. In this case the free-carrier absorption coefficient at frequencies above 5 THz, estimated by the classic Drude formalism, does not exceed 0.01 cm−1. Free-electron absorption becomes important only at donor concentrations above 1016 cm−3 and optical pump powers far above the observed laser thresholds 47.

The low lattice (phonon) absorption in silicon is caused by the covalent bond structure that prohibits even-parity OZC phonons being infrared active 48. In crystals with diamond structure the first-order terms of the phonon displacement vanish, because of the parity selection rule and only the second-order terms contribute to the infrared absorption. This gives rise to multiphonon (two or three phonon) absorption bands centered at sum 49 or difference 50 phonon frequencies, also in combination with optical and acoustic ones, both at the edge of the Brillouin zone. In the frequency range 2.4–12 THz (photon energy 10–50 meV) only the difference two-phonon processes contribute. This results in a low (∼0.1 cm−1 at 6 K) lattice absorption. At high frequencies (24–34 THz, 100–140 meV) the sum phonon processes dominate and result in absorption coefficients of ∼0.2–0.5 cm−1 at 6 K 50.

Impurity-related absorption includes those processes originating from intracenter (between localized donor states) and bound-to-continuum (between donor states and the conduction band) optical transitions. The intracenter absorption spectrum has a discrete line structure with absorption cross-sections as high as σD ∼ 10−14 cm2, while bound-to-continuum transitions are broad bands with peak cross-sectional values usually not exceeding σion ∼ 10−15 cm2 35. The absorption coefficients α = σ ΔN are determined by the difference in population, ΔN, of the initial and final states of the optical transition.

2.3.2 Terahertz light scattering in n-silicon

Inelastic light scattering by atomic vibrations or impurity atoms in doped silicon also causes THz optical losses. The probability of the first process is defined by changes of the electric susceptibility of the lattice caused by the atomic displacement. The latter can be associated with a set of phonons characterized by a wave vector q and a frequency ωq = 2πqs. The infrared spontaneous scattering efficiency in silicon at room temperature by zone-centered optical phonons (fOZC ≈ 15.53 THz) has been found to be only ∼10−7–10−6 cm−1 sr−1 in the infrared range 51. Since it depends on the frequency as ∼ω4, the scattering can be neglected in the THz range, especially for low temperatures, where the phonon-induced scattering suffers from the low occupancy of phonon modes given by Nq = [exp(ħωq/kBT) − 1]−1. The intensity of the Stokes scattering process, i.e., the scattering emission frequency ωS is less than the frequency ωi of the incident light (ωS = ωi − ωq), is proportional to Nq, while for the anti-Stokes process (ωAS = ωi + ωq) it is proportional to (Nq + 1). Therefore, the probability for anti-Stokes scattering at low lattice temperatures is even lower than that for the Stokes one. The efficiency of the Raman scattering in silicon can be significantly enhanced by electronic resonances 48, if the energy of a particular phonon is close to the energy of some electronic resonance, as for example noted in Eq. (3). However, the efficiency of the scattering on donor centers in the THz range does not exceed 10−9 cm−1 sr−1 52 and can be neglected at concentrations of scattering centers around the optimal doping for silicon lasers (∼1015–1016 cm−3).

2.3.3 Absorption related to D centers

Apart from the intrinsic THz absorption, additional absorption can be induced in optically excited n-silicon due to the formation of negatively charged donor centers (D centers), in which two electrons with opposite spin are bound to a positively charged ion. The binding energy, equation image, of such a center is small in comparison with the ionization energy of neutral donors, Ei (equation image ≈ 0.055Ei 53). Due to the specific structure of the eigenfunction, D centers have an extremely broad absorption band (equation image < ħω < 10 × equation image 53) with a maximum cross-section of σD ∼ 10−14 cm2 at 2 × equation image ≈ 4–7 meV for different group-V donor species. D centers have a relatively long lifetime of ∼10−8 s 53. Due to the long lifetime the population of the D centers can reach ∼1014 cm−3 at pump photon flux densities of ∼1023 cm−2 s−1 even though the capture rate for electrons on a donor center is quite low. Therefore, one can expect from D centers broadband (0.02–17 THz) absorption with an absorption coefficient equation image up to 1 cm−1. Using quantum cascade lasers in the range between 2.5 and 3.1 THz we have found that typical values of the CO2 pump laser induced losses are in the range from 0.1 cm−1 up to 3 cm−1 depending on the donor type, its concentration and the pump wavelength 54. These losses have been attributed to D center absorption. At the optimal doping concentration for intracenter silicon lasers the calculated loss due to D centers is between 0.02 and 0.03 cm−1 for Si:P lasers (operating frequency 5.4 THz) and for Si:Bi lasers (5.8 and 6.2 THz) 55.

2.4 Intracenter population inversion and gain

In intracenter silicon lasers, the population of the principal levels is mainly determined by phonon-assisted relaxation of donor electrons. At low lattice temperatures (T < 10 K), donor electrons occupy the ground state. Being excited by an external source, the nonequilibrium electrons are spread across excited donor states as well as in the conduction band. Optical excitation of donor electrons does not disturb localized states up to relatively large intensities and it is the most effective excitation of shallow impurities in silicon at low temperatures. One can distinguish two major types of optical excitation: intraband excitation (also called photoionization) and intracenter excitation (also called photoexcitation). Under intracenter excitation, the electron does not leave the donor center and does not contribute to the electric conductivity. This excitation requires a coincidence of the photon energy with a dipole-allowed optical transition originating from the even-parity ground state, 1s(A1). Photoionization of an impurity results in a nonequilibrium population of free electrons in the conduction band. The only requirement for this excitation is that the photon energy is larger than the donor binding energy Ei. We divide the observed laser mechanism into two classes: silicon lasers doped by donors with binding energies below the energies of the intervalley phonons exhibiting the strongest electron–phonon interaction (minimal energy of these phonons is ħωf-LA ≈ 46.3 meV) and silicon lasers with dopants having Ei > ħωf-LA. Lasers of the first class (Si:P, Si:Sb) have a long living donor state as the upper laser level while lasers of the second class (Si:As, Si:Bi) are dominated by a resonant interaction of certain states with intervalley phonons. In fact, this separation is quite formal and emphasizes the different efficiency of the laser schemes.

For the calculation of populations in donor states a rate equation model was developed 56. The relaxation rates for the lower excited states were estimated from the linewidths of intracenter transitions measured by absorption spectroscopy (see Section 3). The rates of a few highly excited states have been approximated by an extrapolation from the rates of lower states assuming a monotonous dependence of the relaxation time as a function of the binding energy 41.

2.4.1 Long-living upper laser level

Due to the large energy spacing between the 2p0 state and the lower-lying 1s split-off states the 2p0 excited state has a relatively long lifetime for all group-V donors in silicon except for bismuth. Calculations and experiments 33, 57–62 indicate that the lifetime of the 2p0 state, equation image, lies in the range from 10−10 to 10−7 s with the most accurate experimental value of 205 ps 61. Despite the large uncertainty for this time, even for the smallest value donor lasing seemed feasible because of the even shorter lifetimes for the donor states below the 2p0, in particular the 1s(E) and 1s(T2) states. Calculations in the framework of the isotropic multivalley model, which take into account inter- and intravalley acoustic-phonon-assisted relaxation, yield a lifetime equation image ∼ 10−11–10−10 s for the 1s(E) and 1s(T2) states of some group-V donors in silicon 63, 64.

Population inversion is most easily achievable in silicon crystals doped by phosphorus (Si:P, Ei = 45.53 meV) and antimony (Si:Sb, Ei = 42.74 meV), since their donor state populations are not directly affected by high-energy intervalley phonons and their 2p0 state exhibits the longest lifetime (∼200 ps) of all donor states 61. Optical pumping can be performed resonantly into the 2p0 state, or in any other impurity state at allowed optical transitions. Also, pumping into the conduction band state continuum is possible. In all resonant pump cases for Si:P and Si:Sb lasers, the 2p0 state serves as the upper laser level.

Resonant pumping into the upper laser state (Fig. 4a) is the most efficient excitation. It yields the highest quantum efficiency, lowest optical threshold and largest laser gain. This is due to the large photoexcitation cross section (∼7 × 10−15 cm2 35) as well as the fact that all states above the upper laser level are excluded from the laser process. The lower laser level, 1s(E), is significantly less populated. The lower laser level is determined by the selection rules caused by the specifics of the wavefunction of the donor states involved in the intracenter pump process (see Section 3).

Figure 4.

(online color at: www.pss-b.com) Relative populations of the donor states in Si:P as a function of their energy. The pump flux density is 1025 photons cm−2 s−1. The notations for the impurity states are the same as in Fig. 3. Straight arrows up indicate optical pumping and bold arrows down are for stimulated emission. (a) Intracenter resonant pumping (photoexcitation) into the 2p0 state: The laser transition is 2p0 → 1s(E). (b) Photoionization pumping (into the conduction band): The laser transition is 2p0 → 1s(T2). Note the larger relative population of the upper laser state, 2p0, in the case of resonant excitation.

The other “boundary” case is pumping far into the c.b. (Fig. 4b). This is the most inefficient excitation. The reasons are: lower photoionization cross sections at photon energies ≥2 × Ei, accumulation of nonequilibrium electrons in high excited states and in the conduction band, capture of electrons by neutral donors (forming D centers), and excitation of nonequilibrium phonon modes (that ends by their decay in low-energy thermal phonons and Joule heating). This results in an almost one order of magnitude smaller gain for silicon lasers under photoionization pumping in comparison with the resonantly pumped lasers. Initial calculations assuming a lifetime equation image = 3 × 10−7 s ≫ equation image and a doping concentration ND = 1015 cm−3 yielded a small signal gain α ∼ 0.3 cm−1 for Si:P at T < 10 K pumped by 10 W cm−2 intensity of a CO2 laser operating at 10.6 µm (117 meV) 32. However, the experimentally determined laser thresholds are several orders of magnitude larger, in agreement with the 200 ps lifetime of the 2p0 state in Si:P 61.

2.4.2 Resonant donor–phonon interaction

For the relatively deep group-V donor centers, arsenic (Si:As, Ei = 53.76 meV) and bismuth (Si:Bi, Ei = 70.98 meV), interaction of particular excited donor states with high-energy intervalley phonons is inherent. It couples the 2p0 and 2s states to the ground 1s(A1) state in Si:Bi via f-TO and g-LO phonons, respectively 46, and the 2s to the 1s(A1) state in Si:As via f-LA intervalley phonons 18. This coupling leads to relaxation rates of ∼1012 s−1, and, therefore, to short lifetimes for the excited states.

Fast depletion of the 2p0 and 2s levels in Si:Bi has been considered as a laser scheme with these states as lower laser levels, because they have very short lifetimes 32. This proposal is somewhat similar to those later employed in quantum cascade lasers, using the optical phonon for depopulation of the lower laser level 65. The laser emission was estimated to be in the frequency range 0.7–3 THz on transitions originating from the excited states close to the conduction band bottom and ending in the 2p0 and 2s states. The fast depletion of the lower laser state by intervalley phonons was expected to result in laser operation from Si:Bi at relatively high lattice temperature. Larger optical thresholds (above 100 W cm−2) and a gain of ∼2 cm−1 were also expected for Si:Bi crystals doped to 1016 cm−3 at T < 60 K.

2.5 Lasing without population inversion

Raman-type lasing from n-silicon has been experimentally observed prior to any theoretical model of this effect. Raman lasing is known in the visible and near infrared (pump wavelengths λPUMP ≤ 1.064 µm), but has not been observed in the far-infrared (20 µm ≤ λPUMP ≤ 40 µm) due to the strong decrease of the Raman scattering efficiency toward longer wavelengths that scales as equation image. This limitation can be overcome due to resonant effects and large pump powers making Raman lasing feasible at long wavelengths.

Raman-type scattering in n-silicon utilizing electronic resonances between impurity states appears predominantly via intracenter electron excitation. Jain et al. 66 have identified a Raman active donor 1s(A1) ↔ 1s(E) transition in the Stokes-shifted spectra under infrared laser excitation of silicon doped by phosphorus, antimony and arsenic. The following description of intracenter Raman silicon laser is based on experimental results and the formalism described in Ref. 48. Pumping of the donor electrons bound to the ground state (|GS〉 in Fig. 5) at low temperature brings an electron into the virtual state |v〉 with a complex energy Ev − ihΓS, here Ev is the energy of the pumped virtual state, h the Plank constant, and ΓS is the linewidth of the Stokes transition. This first process is caused by the intracenter absorption of the pump radiation ħωPUMP via bound electron–pump radiation interaction that is described by the Hamiltonian He-PUMP. The electron in the virtual state emits a photon ħωS (bound electron–Stokes radiation interaction He-S) and is transferred to the intermediate excited state |ES〉 with an energy EES (in our case this is the 1s(E) state). Further nonradiative relaxation of the electron, |ES〉 → |GS〉 occurs via the interaction of a bound electron with a phonon (or impurity–phonon), He-ph (Fig. 5), in which an electron is scattered in momentum space in a [100]-type direction to the equivalent valley (g-scattering). Taking into account monochromatic pump radiation, i.e., considering the excitation into the virtual state as a delta-function δ[ħωPUMP − ħωS − (EGS − EES)], the dominant contribution to the scattering probability P(ωS) 48 in the vicinity of the state |v〉 will be

equation image((4))

where the energy spacings in the denominators of Eq. (4) are defined as ΔEV = (EGS − Ev + ihΓS) and ΔEES = (EGS − EES). The scattering is significantly enhanced in the case of the “outgoing” electronic resonance ħωPUMP − ħωS = ΔEES. An additional enhancement follows from the “incoming” resonance ħωPUMP = EGS − Ev, i.e., when the virtual state coincides with one of the donor excited states. In the latter case Raman and intracenter lasing processes merge in pump and sometimes in emission frequency.

Figure 5.

(online color at: www.pss-b.com) Schematic energy (a) and Feynman (b) diagrams for the lasing mechanism without population inversion in Si:Sb. In this case pumping occurs into a virtual state, which may not coincide with any donor state. The experiments show that the virtual state must be located between the 2p0 and 3p0 states as indicated in (a). (b) The filled circles (vertices) indicate electron-radiation intracenter interaction followed by the emission of a Stokes photon. The open squared vertex is for electron–phonon intracenter interaction accompanied by emission of an intervalley phonon (g-TA in the case of the Si:Sb Stokes laser).

For both types of lasing, donor intracenter and Raman intracenter, the intervalley scattering of an electron that is accompanying its decay into the ground state is an Umklapp process (Eq. (2)). At the end of the process the participating electron is again in the impurity ground state.

The THz Stokes silicon laser differs strongly from the infrared Raman silicon laser since it employs the intracenter electronic transfer in n-type doped crystals, while Raman scattering of the photons in the infrared laser is manifested by the interaction with an electron–hole pair in the undoped silicon. The “zero”-momentum OZC phonon relaxes naturally in this Raman process; the momentum conservation in THz silicon lasers occurs inside the impurity atom, by nonradiative intervalley scattering of the bound electron accompanied by emission of an intervalley acoustic g-TA or g-LA phonons.

3 Lasers based on optical transitions between donor states

Silicon lasers based on optical transitions between donor states operate under optical pumping by midinfrared laser sources at lattice temperatures below 30 K. Intracenter pumping of the donors can be provided by an infrared (20–40 µm) free-electron laser (FEL) while photoionization of donors can be achieved by transverse excited atmospheric pressure CO2 lasers (9–11 µm). These silicon lasers emit discrete lines in the frequency range of 1–7 THz.

3.1 Experimental details

Silicon laser crystals doped by different species (phosphorus, antimony, arsenic, and bismuth) with a concentration of ND = (0.5–12) × 1015 cm−3 were grown by the float zone, the Czochralski and the pedestal techniques in [111] or [100] directions. The samples were shaped into rectangular parallelepipeds with typical dimensions of 7 × 7 × 5 mm3 with the 7 × 7 mm2 facet orthogonal to the growth axis. Similar but undoped (residual doping ∼1013 cm−3) silicon samples were used for reference measurements. Compensated Si:P crystals (compensation or ratio of acceptor (A) and donor (D) centers in the crystals NA/ND ∼ 0.01–0.35) were obtained by neutron irradiation of the float-zone-grown silicon crystals (see Ref. 67 for the procedure) that in turn were initially doped with boron to different NA concentration. These samples have a net donor concentration Nnet = ND − NA ∼ 3.0 × 1015 cm−3. The facets of all samples were optically polished parallel to each other with an accuracy of 1 arcmin in order to provide a resonator on total internal reflection modes.

For the experiments the crystals were usually cooled to ∼5–7 K in a liquid helium (lHe) transport vessel or in a flow cryostat. For higher laser operation temperatures a mechanical cryocooler was used as well. For photoionization pumping a TEA CO2 laser was used. It generates ∼1-µs long pulses with peak powers up to 1 MW (photon flux density ∼5 × 1025 cm−2 s−1). The FEL at the infrared user facility FELIX at the FOM Institute for Plasma Physics in Rijnhuizen, the Netherlands, which was used for intracenter silicon pumping, generates ∼6 µs long macro-pulses at a 5 Hz repetition rate in the range from 20 to 40 µm (21–43 meV) with a maximum power up to ∼5 × 1025 photons cm−2 s−1 at longer wavelengths. The FELIX macropulse consists of a train of micropulses with a duration between 3 and 10 ps separated by 1 ns. If not otherwise stated the repetition rate of micropulses was 1 GHz and that of the macropulses was 5 Hz throughout the experiments. The output emission from the silicon samples was collected orthogonally to the pump direction and detected by a lHe cooled Ge:Ga detector sensitive at wavelengths shorter than 140 µm (photon energy ≥ 9 meV). Combinations of a cold sapphire, room-temperature z-cut quartz or CaF2 infrared filters were used to protect the THz detector from the scattered pump light. For the spectral analysis a Fourier transform spectrometer (FTS) with a resolution of ∼0.1 cm−1 (0.012 meV) was used.

3.2 Identification of the laser transitions

Almost 20 discrete emission lines are available from silicon lasers doped by group-V shallow donor centers (Fig. 1). More than ten different variations of the laser scheme can be obtained by pumping at different frequencies and when external uniaxial stress is applied to the silicon crystal. In the following we review some common cases that are realized under intracenter and photoionization pumping and we discuss the experimental results with respect to theoretical models of the silicon laser schemes. All silicon lasers are based on four-level schemes on optical transitions within a donor center. There is no lasing found from continuum-to-bound transitions. The realized laser schemes have been identified from the emission spectra (e.g., Fig. 6). An analysis of the laser thresholds and its temporal dynamics completes the study.

Figure 6.

(online color at: www.pss-b.com) Typical emission spectra of silicon lasers under photoionization pumping with a CO2 laser operating at 10.59 µm (Si:P, Si:Sb, Si:As) and at 9.6 µm (Si:Bi). The pump intensity is ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1 (Si:P, Si:Sb, Si:As) and ∼80 kW cm−2 = 3.6 × 1024 photon cm−2 s−1 for Si:Bi. Laser action occurs on the 2p0 → 1s(T2) transitions in Si:P and Si:Sb and on the 2p± → 1s(E) and 1s(T2) transitions in Si:As pumped by any wavelength in the range 9.2–10.8 µm. The Si:Bi emission spectrum depends on the pump wavelength. The emission lines are from the 2p± → 1s(E), 1s(T2) transitions.

3.2.1 Photoionization pumping

Under photoionization pump conditions, the excited electrons relax first in the conduction band with emission of an intervalley g-LO optical phonon (fg-LO ≈ 15.3 THz) or a few high-energy acoustic intervalley phonons. Being captured by the donor center, a portion of the captured electrons relaxes to a relatively long-living excited donor state, where the electrons accumulate and give rise to population inversion (Figs. 3 and 7). As was found experimentally, the laser schemes depend strongly on the donor–phonon interaction with the intervalley phonons g-LO, g-TO, f-LA, and f-TO (phonon energy ħωip: 46.3–63.2 meV; fip ≈ 11.2–15.3 THz).

Si:P and Si:Sb lasers (Ei < ħωip), which are based on the long-living 2p0 state, exhibit emission from the 2p0 → 1s(T2) optical transition (Fig. 7). From theoretical analysis the upper laser level, 2p0, has been expected to be long living 15, 32, 60, 62. Its lifetime, has been experimentally determined from time-resolved analysis of optically induced transmission in Si:P (205 ± 18 ps) and Si:As (104 ± 11 ps) crystals 61. The lower laser level, 1s(T2), has been determined from the emission spectra. It is preferred over the close-by 1s(E) state because of a number of factors. The deciding one is the larger optical cross-section for transitions terminating in the 1s(T2) state, determined by the degeneracy ratio of the triplet 1s(T2) and doublet 1s(E) states equation image/gE = 3/2 68. The g-TA intervalley phonon couples the ground state 1s(A1) to one of the valley–orbit-split 1s states, 1s(E) for Si:Sb and 1s(T2) for Si:P, and provides a fast depletion of the lower laser level, transition rate >1011 s−1 69.

Figure 7.

(online color at: www.pss-b.com) Laser schemes realized in Si:P, Si:Sb, Si:As and Si:Bi when pumped with a CO2 laser into the conduction band. The straight blue arrows up indicate the optical pumping and the bold red arrows down are for stimulated emission. Resonant coupling with intervalley phonons is shown by green vertical arrows down. The dashed green arrows indicate relaxation with low probability. The laser transitions are 2p0 → 1s(T2) for Si:P and Si:Sb, 2p± → 1s(T2), 1s(E) for Si:As and Si:Bi. While multiphonon relaxation to the upper laser level is necessary in Si:P and Si:As lasers, resonant single g-LO phonon relaxation from the conduction band directly into excited states is possible in Si:Bi.

Si:As and Si:Bi lasers (Ei > ħωip) are characterized by short-living excited states (2s for both donor centers and, additionally, 2p0 for Si:Bi). They exhibit emission from the 2p± → 1s(T2) and 1s(E) optical transitions (Fig. 7). In the case of Si:Bi, emission is also observed from some transitions between higher excited states and the 2p0 state. However, the former mechanism dominates over laser transitions, which end in the short-living 2p0 state. The upper laser level, 2p±, is the last one above the 2s and 2p0 states, which both have fast relaxation rates because of the resonance with the g-TO, g-LO, and f-TO phonons. Due to this resonance all states below the 2p± state are weakly populated. As a result, lasing occurs on transitions originating from the 2p± state. There are no phonons providing a resonant depletion of the low laser level in Si:As and Si:Bi, whose ground states are more downshifted than those of Si:P and Si:Sb (Fig. 7). As a result, the Si:Bi and Si:As lasers have a relatively low efficiency. The weak competition between donor optical transitions in this case also facilitates the appearance of multiline emission spectra.

Another peculiarity of the laser scheme has been found in Si:Bi when pumped with a CO2 laser and when the pump photon energy differs from the bismuth binding energy by about the energy of the g-LO intervalley phonon (E1s(A1) + ħωg-LO ≈ 134 meV). Pumping of Si:Bi by low-energy, 10P-band photons from the CO2 laser (ħωPUMP ≈ 117 meV) is followed by multiphonon intraband relaxation and results in a relatively high population of the excited states above the 2p± state (Fig. 8a). This results in laser emission on transitions originating from the 4p0, 4p±, 5p0, and 6p± states and ending in the short-living 2s state 70. This was predicted in Ref. 32 and originally thought of as the most important laser scheme in Si:Bi. However, experimentally it was found that this low-frequency emission around 1.5 THz has a low efficiency and the highest laser threshold. Pumping by high-energy, 9R-band photons (ħωPUMP ≈ 130 meV) enables resonant relaxation of the free electron into the 2p± state that shortcuts the population of the bismuth states above the 2p± state (Fig. 8b). Pumping with this wavelength results in lasing on the 2p± → 1s(T2), 1s(E) transitions at frequencies between 5 and 7 THz.

Figure 8.

(online color at: www.pss-b.com) Energy diagrams (left parts of a, b) and relative populations of the states (right parts of a, b) for Si:Bi. The pump flux density is 1025 photons cm−2 s−1. The straight arrows up indicate optical pumping and the bold arrows down are for stimulated emission. The green vertical arrows down are for f-TO (2p0 → 1s(A1)) and g-LO (2s → 1s(A1)) intervalley-phonon-assisted relaxation. (a) Pumping with a CO2 laser (line 10P20, 117 meV, ∼400 kW cm−2 = 2 × 1025 photons cm−2 s−1). Many of the higher excited states are more populated than the 2p0 and 2s states. The laser transitions originate from a number of high excited states and terminate in the 2s state. Other laser transitions are 2p± → 1s(E), 1s(T2). (b) Pumping with a CO2 laser (line 9P22, 129 meV, ∼85 kW cm−2 = 3.8 × 1024 photons cm−2 s−1). Only the 2p± state has a significant population and the laser transitions are 2p± → 1s(E), 1s(T2).

Drastic changes of the laser schemes occur in Si:As and Si:Bi when uniaxial stress is applied to the crystal, because this changes the impurity–phonon resonant interaction 71, 72. At stress values below 1 GPa the excited donor states already undergo a splitting that results in changes of the interstate energy gaps relative to the ground state, ΔEES. In contrast, the phonon energies remain almost unchanged. This leads either to vanishing or to enhanced phonon–impurity resonances. In the case of Si:As and Si:Bi the ΔE2s and equation image resonances with the intervalley f-LA and TO phonon vanish. The corresponding reduction of the loss of excited electrons on phonon-mediated shortcuts to the ground state immediately leads to an increasing population of the 2p0 level (see Fig. 9). As a result, the original zero-stress laser scheme with the 2p± state as the upper laser level transforms into the standard donor laser scheme with the equation image state as the upper level 71. The original zero-stress transitions 2p± → 1s(E), 1s(T2) are changing when the stress reaches a value of approximately 25 MPa. Only a single laser transition (equation image → 1s(B2)) occurs. At higher stress (∼25–50 MPa) a couple of lines (equation image, equation image → 1s(B2)) appear until finally (stress >50 MPa) only a single line (equation image → 1s(B2)) remains. The laser mechanism becomes similar to that of the Si:P and Si:Sb lasers where it is based on the long-living 2p0 state. These changes indicate a reduction of the influence of the electronic f-scattering. With increasing stress the threshold of the Si:As laser slowly but continuously decreases, apparently due to a further reduction of the f-scattering. The laser efficiency is then dominated by the ΔE1s(B2) ↔ g-TA interaction (Fig. 9), which determines the rate at which the lower laser level is depopulated. In the vicinity of the ΔE1s(B2) = ħωg-TA resonance (stress ≈ 200 MPa) the Si:As laser output power reaches its maximum, indicating a large gain.

Figure 9.

(online color at: www.pss-b.com) Modification of the laser scheme in Si:As under photoionization pumping and uniaxial stress in the [100] direction. The stress splits the 6Δ equivalent conduction band valleys in lower 2Δ valleys (in stress direction) and upper 4Δ valleys. As a result the interstate energy gaps ΔEES decrease in case of the lower valleys. This leads to a vanishing ΔE2s ↔ ħωf-LA resonance, which significantly reduces the population of 2p0 state when no stress is applied. As a result the laser transitions switch from 2p± → 1s(E), 1s(T2) (unstressed laser) to equation image → 1s(B2) (stressed laser) when the stress becomes larger than 50 MPa. An additional enhancement of lasing occurs at stress values around 200 MPa when the depletion of the lower laser level becomes more effective due to an appearing resonance ΔE1s(B2) ↔ ħωg-TA.

3.2.2 Intracenter pumping

Intracenter pumping offers a number of various laser schemes depending on the pumped excited state and on the donor element. In general, stimulated emission appears (Fig. 10, upper graphs) when the pump photon energy is equal to the energy gap between the ground state and one of the excited donor states, i.e., ħωPUMP = ΔEES. Major features of the pump spectra for silicon lasers (we shall call the dependence of the silicon laser emission on the pump photon energy pump spectra) coincide with low-temperature impurity absorption spectra of the corresponding impurities (Fig. 10, bottom graphs) that contain dipole-allowed optical transitions originating from the impurity ground state. Deviations of the pump spectra from the energies of the impurity transitions in the absorption spectra such as for the 2p0 state occur, for instance, when an impurity transition is broadened by resonant interaction with a principal phonon of the silicon lattice. The peak of the absorption line in vicinity of the 2p0 bismuth state (Fig. 10b) is shifted by about 0.5 meV from the energy of the photon required for pumping the intracenter laser.

Figure 10.

(online color at: www.pss-b.com) Pump spectra for intracenter pumping (top) and transmission spectra (bottom) for Si:P (a) and for Si:Bi (b). The donor concentration of both samples is ∼3 × 1015 cm−3. The signal is integrated over the macropulse duration. The macropulse pump energy is frequency dependent with mean values 15 and 22 mJ for Si:P and Si:Bi, respectively. The black vertical lines mark the positions of the states as obtained from the impurity absorption spectra 35. The green lines in the bottom diagram of (b) mark the centers of the intervalley optical phonon energies. Note that the centers of the absorption lines coincide with the laser peak intensity in the pump spectra, except for the 1s(A1) → 2p0 absorption line in Si:Bi which is shifted by ∼0.5 meV from the peak in the pump spectrum. This is caused by line broadening due to the equation image ↔ ħωf-TO resonance. All measurements were done at T < 10 K.

Pumping in the 2p0 state always results in laser action from the 2p0 → 1s(E) optical transition for all donors (Fig. 11). This follows from the symmetry of the involved donor states that causes the specific selection rules for intracenter pumping and emission. In the group theory, the Td site symmetry 35 of the 2p0 state is (A1 + E + T2). From the transitions between the ground state and the singlet (A1), the doublet (E) and the triplet (T2) components of the 2p0 state, those involving the triplet state are forbidden, because of the symmetry of its wavefunction 42, 43. Only the (A1 + E) components of the 2p0 state wavefunction can be photoexcited from the ground state 1s(A1). Therefore, only the 2p0(A1 + E) → 1s(E) transitions appear in the silicon laser emission spectra 73. The resonant pumping in the upper laser state represents the case of a specific three-level laser scheme.

Figure 11.

(online color at: www.pss-b.com) Laser schemes realized in Si:P under intracenter pumping. The straight hollow arrows up indicate optical pumping and the bold arrows down are for stimulated emission. The resonant coupling of states with the intervalley g-TA phonon, equation image ↔ ħωg-TA, is shown with slim vertical arrows down. The laser transitions are 2p0 → 1s(E) when directly pumped into the upper laser level 2p0 and 2p0 → 1s(T2) when pumped above the 2p0 state 73.

When pumped above the 2p0 state, the relaxation from the pumped state into the upper 2p0 laser level is added. Because of the multichannel relaxation process (see Section 6.1) some relaxing electrons will not go through the 2p0 state. This leads to a partial loss of efficiency of the selective pumping of the upper laser level. The decay of photoexcited electrons from higher states with a contribution of intervalley phonons is not restricted by the above-mentioned selection rules and the components of the 2p0 state become equally populated. The 2p0 state serves as upper laser level for all pump photon energies ħωPUMP < ħωf-LA. For Si:P and Si:Sb, this condition is satisfied for practically all intracenter pump transitions. The lower laser level, 1s(E) or 1s(T2), is determined by a competition of different factors, such as the degeneracy of the states and the resonance with intervalley phonons. For Si:P, the ratio of the cross-sections of the 2p0 → 1s(T2) and 2p0 → 1s(E) optical transitions is the same as the ratio of the degeneracy factors of the final states, equation image/gE = 3/2. Furthermore, the 1s(T2) state is resonantly coupled to the phosphorus ground state, E1s(A1) − equation image ≈ ħωg-TA = 11.2 meV (Fig. 11). As a result, the 2p0 → 1s(T2) laser transition becomes preferable and dominates for all pump energies with ħωPUMP > equation image (Fig. 11). In the case of Si:Sb, the g-TA phonon couples the 1s(E) state to the antimony ground state, E1s(A1) − E1s(E) ≈ ħωg-TA. As a consequence, the 2p0 → 1s(E) laser transition dominates and remains in the emission spectra up to pumping in high excited states. The 2p0 → 1s(T2) transition is added to the 2p0 → 1s(E) line in the emission spectra when Si:Sb is pumped in the 3p± state and it is the only one when pumped in the conduction band.

More variations of the laser scheme are obtained in silicon crystals with deeper donors, where Ei > ħωf-LA. The strong electron–phonon interactions in Si:As and Si:Bi terminate particular relaxation channels. This results in additional excited states that can act as upper laser level. It also affects the laser gain and leads to a competition of laser transitions. For Si:As when pumped above the 2p0 arsenic state, five different laser schemes have been observed (Fig. 12). When pumped in the 2p± state, the weak relaxation rate to the 2p0 state does not support sufficient population of this state. The two-step relaxation via the 2s state, 2p± → 2s → 2p0, is not important because of the 2s → 1s(A1) f-LA phonon-assisted coupling, ΔE2s(E) ≈ ħωf-LA = 46.3 meV. As a result, the 2p± → 1s(E) transition dominates in the laser emission spectra. When pumping in the 3p0 state, the 2p0 state is efficiently populated by the strong direct 3p0 → 2p0 link. However, a small part of the photoexcited electrons escapes in the 2p± state, therefore leading to three lines in the emission spectra (Fig. 12). Pumping in higher excited states leads to different populations of the upper states and a variety of laser lines, until photoexcitation in the conduction band leads to relaxation channels that only provide population inversion for a single laser line, 2p± → 1s(T2). Increasing the photon energy adds the 2p± → 1s(E) line to the Si:As laser spectrum 18. The changing as well as the coexistence of laser lines with different upper laser levels indicate that several separate relaxation channels are present, and that the classical relaxation cascade theory 40 does not describe the involved processes completely. The size of the energy step appears not to be the dominating factor in the relaxation between two neighboring excited donor states. A recent theoretical model confirms this 60.

Figure 12.

(online color at: www.pss-b.com) Laser schemes realized in Si:As under intracenter pumping. The resonant coupling with the intervalley f-LA phonon is shown with slim vertical arrows down. The laser transitions are (from left to right): 2p0 → 1s(E) when pumped in the upper laser level 2p0, 2p± → 1s(E) when pumped in the 2p± state, 2p0 → 1s(E) and 2p± → 1s(T2), 1s(E) when pumped in the 3p0 state, 2p± → 1s(T2), 1s(E) when pumped in the 3d0 state; 2p± → 1s(T2), 1s(E), 3d0 → 1s(E) and 4p0 → 1s(E) when pumped in the 4p0 state. The changes of the laser scheme with changing pump transition indicate that specific relaxation paths exist that are not described by a step-like relaxation from one to the next lower state (see the discussion in Section 6.1).

In the case of Si:Bi, the number of the laser schemes is less than for Si:As, because its 2p0 state cannot be sufficiently populated: the resonant coupling with intervalley optical phonons is too strong (equation image ≈ ħωf-TO = 59.1 meV and ΔE2s(E) ≈ ħωg-LO = 62.0 meV, ħωg-LO = 63.2 meV) 46. For pump photon energies ħωg-LO < ħωPUMP < ΔE3p±, the emission spectra consist of a single line from the 2p± → 1s(E) transition. The 2p± → 1s(T2) transition appears in the emission spectra when Si:Bi is pumped in the 3p± state and higher, including pumping in the conduction band.

3.3 Laser threshold

Stimulated emission appears in silicon crystals pumped by laser radiation with photon energies equal to donor resonances between ground and p-type excited states (ħωPUMP = ΔEp-ES) with intensities above 400 W cm−2 (∼1022 phonons cm−2 s−1) for intracenter pumping (Fig. 13) and for pump photon energies above the donor binging energy (ħωPUMP > Ei) above 40 kW cm−2 (∼1024 phonons cm−2 s−1) (Fig. 14). For both types of pumping the laser threshold appears to be pump-frequency dependent. This dependence has two reasons: the frequency-dependent lattice loss for pump photons as well as the different initial electronic populations formed by the pump radiation in silicon. At the frequencies of a CO2 laser the typical losses in silicon are ∼0.3 cm−1 for the 9 µm radiation band and ∼1.2 cm−1 for the 10 µm band 50. Although this difference seems to be small it results in a significantly more efficient pumping in the case of the Si:Sb laser at 9.6 µm 17. Pumping by a 9.6-µm line has also advantages with respect to the phonon-resonant electronic relaxation in Si:Bi 70 (see Section 3.2.1). In general, the intensity of the laser signal is a factor 103–104 larger than the spontaneous emission signal. The lowest laser thresholds are achieved in optimally doped and uniaxially stressed silicon crystals. This will be discussed in the following sections.

Figure 13.

(online color at: www.pss-b.com) Laser thresholds of Si:P (FZ grown, NP ∼ 3 × 1015 cm−3) for different pump photon energies under resonant pumping in excited states and in the continuum of the conduction band. The optical pumping originates from the ground state, 1s(A1) (see top of figure). Laser emission occurs on the 2p0 → 1s(T2) transition except for resonant pumping in the 2p0 state, when 2p0 → 1s(E) is the laser transition. The inset on the right shows schematically the pump scheme of the silicon sample. Note the increasing laser threshold with increasing pump photon energy.

Figure 14.

(online color at: www.pss-b.com) Comparison of laser thresholds for different donor species with about the same doping concentration (∼3 × 1015 cm−3) and the same pump photon energy (10.59 µm line of a CO2 laser, 100 kW cm−2 = 5 × 1024 photon cm−2 s−1). Pumping is into the conduction band. Laser action occurs on the 2p0 → 1s(T2) transition in Si:P and Si:Sb and on the 2p± → 1s(E), 1s(T2) transitions in Si:As and Si:Bi.

3.3.1 Donor photoionization and photoexcitation pumping

The lowest optical thresholds are achieved by intracenter pumping 74 of Si:P, Si:Sb and Si:As samples in the 2p0 state at optimal doping levels (Fig. 13). In general, the smaller the binding energy of the pumped excited state, the larger is the laser threshold. For pumping in the conduction band, the laser threshold is a factor of ∼100 larger than in the case of resonant pumping into the upper laser level (2p0 state). These results quantitatively correspond to the theoretical predictions: intracenter pumping has advantages caused by exclusion of nonradiative electron relaxation via above-lying states as well as larger pump cross sections. Above the threshold, laser emission tends to saturate at high pump power. This indicates the depletion of neutral donor centers by the pump radiation.

Silicon lasers, employing the scheme based on the long-living 2p0 state, i.e. Si:P and Si:Sb, have the lowest thresholds. Lasers based on resonant interaction with strong intervalley phonons as it is the case for Si:As and Si:Bi are less-efficient laser schemes under photoionization pumping (Fig. 14).

The predicted values for the laser thresholds 32 appeared to be about an order of magnitude smaller, than what is observed in the experiments. The main reasons for this discrepancy are underestimation of the rates for intervalley-phonon-assisted intracenter relaxation that results in a shorter lifetime for the upper laser level as well as an underestimated absorption by D centers at pump powers at about and above lasing threshold 55. Additionally, a reduction of the population of the upper laser state is caused by electrons stored in the higher excited states, which in turn is due to selective intracenter relaxation channels of donor electrons 23. In the case of the long-wavelength lasing additional losses caused by Drude absorption should be taken into account.

3.3.2 Compensation versus D centers

Compensation of the dominant dopant by acceptor centers has a positive effect on the laser threshold. It reduces one channel of losses that arises from the absorption of THz radiation by D centers. In a compensated n-silicon crystal the acceptor centers attract the electrons bound to the donor centers leading to a certain concentration of positively charged donor centers D+. These centers serve as effective traps for the photoionized electrons, since the capture of electrons by charged donors dominates over the capture by neutral donor centers. At these conditions, the formation of D centers is negligible. The laser photoexcitation threshold of Si:P is about a factor of two lower for crystals compensated up to ∼30% than for uncompensated ones 55.

3.3.3 Optimal doping

The optimal net doping concentration, i.e., the doping concentration for the lowest laser threshold of the silicon crystals, is about Nnet = (3–5) × 1015 cm−3 for lasers based on the long-living 2p0 state (Fig. 15). Below this concentration the gain decreases due to the decreasing number of donors, while above this impurity broadening of the energy levels starts to become important. This lowers the cross-section for particular intacenter transitions and results in lower gain and a higher laser threshold. It is worth noting that under external stress as well as in the case of resonant pumping in the 2p0 state, the doping range where laser action is achieved expands by about 10–15% in both directions toward lower and higher doping levels.

Figure 15.

(online color at: www.pss-b.com) Laser threshold as a function of net doping concentration. The symbols are experimentally obtained threshold values for pumping with a CO2 laser (Si:P and Si:As: pump wavelength 10.59 µm, intensity ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1; Si:Sb and Si:Bi: pump wavelength 9.6 µm, ∼80 kW cm−2 = 3.6 × 1024 photon cm−2 s−1). Si:Sb lasers operate across the widest doping range and have the lowest laser threshold. For all lasers the optimum doping is at about (3–5) × 1015 cm−3.

3.3.4 Influence of uniaxial stress

Uniaxial stress applied to the silicon crystal can significantly influence the intracenter laser threshold. When an external compressive force F is applied to a crystal, the energy of the conduction-band extrema shift by a value Δ〈100〉 = Ξu F/(c11 − c12) 75, where Ξu = 8.77 eV is the shear deformation potential constant of silicon 35 and c11 = 167.6 GPa and c12 = 65.0 GPa are the second-order elastic moduli at low lattice temperature. These shifts reduce scattering by f-intervalley phonons already at moderate stress values 76. The high-symmetry p- and s-type excited donor states split and shift together with the corresponding conduction band valleys. The low-symmetry singlet and doublet s-states exhibit a nonlinear shift in response to uniaxial stress (Fig. 16).

Figure 16.

(online color at: www.pss-b.com) Reduction of the laser threshold when uniaxial stress is applied to the Si:Sb laser crystal. The silicon sample has dimensions of 5 × 7 × 2 mm3 and the stress F is applied along the 〈110〉 crystal axis (to the 5 × 2 mm2 sample facet). Pumping is with a CO2 laser operating at 10.59 µm with an intensity of ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1.

Applying stress changes the donor–phonon resonances and changes the lifetimes of particular donor states. This leads for example to new electron–phonon resonances for a fast depletion of the lower laser level. A significant reduction, up to one order of magnitude, of the laser threshold has been observed for Si:P and Si:Sb lasers under uniaxial stress of ∼108 Pa (1 kbar) (Fig. 16) 69. This has been explained by a reduced relaxation rate of the upper laser level, 2p0, in both media, due to a vanishing 2p0 ↔ 1s(E) or 2p0 ↔ 1s(T2) state coupling via intervalley f-TA phonons (ff-TA ∼ 4.45 THz) and g-LA phonons (fg-LA ∼ 4.53 THz). The laser frequency is not changing under stress in Si:P and Si:Sb, since the upper and lower laser levels, 2p0(A1 + B2) and 1s(B2), shift similarly when stress is applied.

Even larger changes of the laser threshold occur for Si:As 71 and Si:Bi 72 lasers. This is because the resonant interaction of the 2p0 and 2s (Si:Bi) and the 2s (Si:As) states with the phonons disappears. When the stress is sufficient the laser mechanism for both materials becomes similar to the scheme with the long-living 2p0 donor state in Si:P and Si:Sb. Obviously this is accompanied by a step-like change of the laser frequency (Fig. 9, Section 3.2.1).

3.4 Laser temperature

Intracenter silicon lasers operate at temperatures that ensure that the thermal population of the split-off impurity ground state is small, i.e. kBT ≪ equation image. The deeper the donor ground state, the higher the laser operation temperature. Measurements made with the silicon laser in a temperature-variable mechanical cryocooler show that the optimal temperature of the crystal lattice is below 10 K (Fig. 17) but higher operation temperatures up to 30 K are possible with reduced output power. Slightly higher (T < 20 K) optimal operation temperatures can be obtained under resonant intracenter pumping of the silicon crystals. When immersed in lHe, the duty cycle of the pulsed silicon laser was determined by the repetition rate of the pump laser (≤10−5). When the silicon crystal was mounted on the cold finger of the cryocooler, which has a limited heat-extraction capacity, the maximal achieved duty cycle was 2 × 10−5 for pumping with a CO2 laser. Lasing from Si:P silicon crystals with natural isotopic composition ceases at a repetition rate of the CO2 laser higher than 10 Hz, while it remains constant up to the highest repetition rate of the CO2 laser (20 Hz) for lasers made from isotopically enriched 28Si:P crystals, apparently due to higher thermal conductivity of monoisotopic silicon crystals 77. The roll-off of the laser emission with increasing temperature occurs due to an increase of the lattice temperature caused by a slow extraction from the silicon of low-energy thermal phonons, which in turn is the result of the decay of nonequilibrium high-energy phonons.

Figure 17.

(online color at: www.pss-b.com) Laser emission as a function of the heat sink temperature. The laser crystals are mounted to the cold finger of a closed-cycle cryocooler. They are optically excited by radiation from a CO2 laser operating on the 9P20 line (130 meV) in the case of Si:Bi (peak pulse intensity ∼80 kW cm−2 = 3.6 × 1024 photon cm−2 s−1) and on the 10P20 line (117 meV) for all other lasers (intensity ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1).

3.5 Laser dynamics and gain

THz silicon lasers exhibit a specific emission pulse structure. Intracenter donor-related spontaneous emission always coincides with the optical pump pulse: It starts simultaneously with the pump pulse and ceases at the end of the pumping (Figs. 18a and b). The intensity of the spontaneous emission is ∼103–104 times smaller than the stimulated emission just above pump threshold. The Si:P and Si:Sb laser pulses start also simultaneously with the pump pulses and reach the peak value at the maximum of the pump intensity. Their emission ceases shortly after the 100-ns peak of the CO2 laser pulse has passed (Fig. 18c). There is no laser emission at the 900-ns pump pulse tail, even if its intensity exceeds the pump threshold. The laser emission of Si:Bi and Si:As lasers always has a delay of up to 200 ns after the pump pulses (Fig. 18d). For Si:Bi, this delay is pump-frequency dependent. The laser emission pulses are as long as the pump pulses, even if the intensity in the pump pulse is below (pump pulse tail) the laser threshold. A similar temporal behavior has been found for Si:P and Si:Bi lasers under intracenter pumping (Fig. 19). In this case, the 6-µs macropulse from the FEL serves as a kind of “continuous” pumping 78.

Figure 18.

(online color at: www.pss-b.com) Typical pump (blue upward curves) and silicon emission pulses (for Si:P and Si:Bi lasers, magenta downwards curves). In a,b spontaneous emission signals are shown and in c, d laser emission pulses are shown. Pumping is done with a CO2 laser operating on the 10P20 line (117 meV, peak pulse intensity ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1). The CO2 laser pulses are recorded with a photon drag detector. The signals are normalized for clarity. The silicon laser pulses are recorded with a Ge:Ga photoconductive detector matched to a 50-Ω load (response time ∼200 ns). The spontaneous emission signals are recorded with a pump intensity just below the laser threshold. For clarity these signals are multiplied by 103.

Figure 19.

(online color at: www.pss-b.com) Typical dynamics of Si:Bi lasing under (a) photoionization (pump photon energy 71 meV) and (b) resonant intracenter excitation (64.8 meV, excitation in the 2p± state) of Si:Bi. The dB values are the attenuation from the maximum pump pulse energy. The maximal pump macropulse energy available from the FEL is (a) 20 mJ and (b) 30 mJ. The repetition rate of the FEL micropulses is 1 GHz and 5 Hz for the macropulses.

The duration of the laser pulses is not yet completely understood. It is evident that the formation of high populations in the high-energy phonon modes, assisted by the nonradiative relaxation of the donor electrons during each laser cycle, terminates the laser action in the lasers based on the long-living 2p0 state. On the contrary, this supports the laser process in the media with deeper donors. We assume that in Si:P and Si:Sb lasers these phonons lead to heating effects that reduce the laser gain within the first ∼100 ns, a time scale that is of the order of the characteristic time of the decay of high-energy phonons into low-energy thermal phonons in the silicon lattice, i.e., the characteristic “heating” time scale of silicon lasers. Since this heat cannot be extracted from the crystal during the next few hundred ns, the silicon laser pulse is shortened to the ∼100 ns value. For Si:Bi and Si:As lasers, whose operation is less affected by the crystal temperature, the laser action continues after the CO2 laser pump pulse peak as long as the pump intensity exceeds the laser threshold.

Assuming the low-gain linear mode of the development of the laser emission, the evaluation of the Si:Bi pulse delay yields a small signal gain between about 0.005 and 0.020 cm−1. The lowest gain value and largest Si:Bi laser pulse delay correspond to the low-efficiency pumping by the 10P20 line (the population inversion is as in Fig. 8a). The shortest delay (∼50 ns) and the highest gain are obtained for pumping with the 9R20 line (similar to that in Fig. 8b). These results indicate that the low gain is the reason for the relatively slow dynamics of the silicon lasers.

Direct measurements of optically induced THz gain in silicon lasers doped by phosphorus and bismuth 79 have been carried out by a pump and probe technique similar to that used for measurements of the THz gain in intersubband p-germanium lasers 80. The pump radiation was split into two parts and guided to two different samples. One of these serves as a laser and the second one as an amplifier. To avoid the generation of laser emission, the amplifier sample had a specific shape with unpolished facets. The light generated by the laser is transmitted through the amplifier and results in a signal I on the Ge:Ga detector (inset in Fig. 20). This signal depends on the pump power P delivered to the amplifier as I(P) = I(0) exp(g(P)L), where g is the gain and L is the sample length.

Figure 20.

(online color at: www.pss-b.com) Typical dependence of the small signal gain on the pump intensity for Si:P lasers with CO2 laser excitation (10.6 µm). The points show the experimental values and the curve is a simulation result 79. The scheme of the measurement is shown in the inset.

The CO2 pump laser has a pulse duration of about 70 ns and pulse repetition rate of a few Hz. The transmitted signal is integrated by the Ge:Ga detector. This excitation can be considered as stationary on the time scale of the relaxation mechanisms and lifetime of the pump level (equation image ∼ 200 ps 61). These assumptions allow the direct derivation of the gain g(P) from the exponential factor, which describes the relative change of the transmitted light through the amplifier as a function of the pump power. In the case of an FEL as pump source that emits much shorter pulses (FEL micropulse duration, 6–10 ps < equation image) with a repetition rate f = 1 GHz that exceeds the speed of the Ge:Ga detector (∼1 MHz), averaging of the signal I(P) over many pump pulses as well as short-living amplification have to be considered. This results in the following expression for the “pulsed” gain: g(P) = ln{(I(P)/I(0) − 1)/(fequation image) + 1}/L 79.

As derived from the experimental data, the maximal gain for intracenter excitation with an FEL is about 5.3 cm−1 for the 2p0 → 1s(E) transition and ∼6 cm−1 for the 2p0 → 1s(T2) transition for the Si:P laser. For the Si:Bi laser and the 2p± → 1s(E) transition it is about 10 cm−1. Photoionization pumping with a CO2 laser has a significantly smaller gain, ∼0.5 cm−1 for 2p0 → 1s(T2) transition of Si:P. The gain is below the detection limit of this method for Si:Bi. It is worth noting that the obtained experimental values for the THz gain agree well with theoretical calculations based on balance equations 79.

3.6 Frequency tunability with an external magnetic field and uniaxial stress

To tune the frequency of an intracenter silicon laser, external fields that provide the shifts of the donor levels, can be applied. This can be an external uniaxial stress, electric or magnetic fields. The magnetic field appears to be the most practical way for frequency tuning. It provides reliable and reproducible frequency variation. By applying a magnetic field to the THz intracenter silicon laser the degeneracy of the donor states is lifted 81, which results in splitting and shifting of the states (Zeeman effect). At low magnetic fields (typically below 2–3 T) shift and splitting of a particular donor state grow linearly with the magnetic field. In this case the splitting, ΔE±, of np± states into np+ and np (n = 1, 2, 3, …) components is given by 82 ΔE± = ħωT cosθB, where ωT = eB/mT is the transverse cyclotron frequency, e the electron charge, B the magnetic induction, mT is the transverse effective mass of the electron, and θB is the angle between the magnetic field and the major axis of the conduction band ellipsoid. The splitting of the silicon laser transition frequency is expected to be ν±/B ≈ ±2.5 cosθB cm−1 T−1. The ns and np0 states are not affected by the linear Zeeman effect because of their symmetry.

In the case of a bismuth-doped silicon laser operating at 6 THz, it is demonstrated that this effect can be used to tune the emission frequency with a rate 40–60 GHz T−1 83 (Fig. 21), which is of the order of the expected linear Zeeman effect for the 2p± donor state, which is the upper laser level in the Si:Bi laser. The shifts at large magnetic fields cause quenching of stimulated emission, because the laser levels approach particular electron–phonon resonances. The frequency change depends on the relative orientation of the magnetic field and the crystallographic axis of the laser. Si:P and Si:Sb lasers do not exhibit changes of their operation frequency, because of the absence of the linear Zeeman effect for the states involved in the laser emission.

Figure 21.

(online color at: www.pss-b.com) Zeeman effect of the Si:Bi laser when pumped with a CO2 laser operating on the 9P20 line (9.55 µm). The magnetic field is applied along the [11–2] crystal axis. The laser transitions are 2p± → 1s(E) (on the left) and 2p± → 1s(T2) (on the right). The inset shows the scheme of the experiment.

Uniaxial stress applied to a silicon crystal lifts equivalent conduction band valleys relatively to the “center of mass”, as described in Section 3.2.4, forming upper (u) and lower (l) conduction band valleys. While most of the excited states of donors follow the bottom of the corresponding conduction band valley linearly, the energies of optical intracenter transitions, relevant to intracenter lasing, remain unchanged. Therefore, the emission frequency of silicon lasers does not change with stress 69 unless the entire lasing mechanism recovers 71. However, the spin–orbit split 1s state, such as in Si:Bi, exhibit a nonlinear variation of the state energy with respect to uniaxial stress. This feature can be used for changing the laser frequency, if this state acts as a laser level. Shifts up to 4 cm−1 (120 GHz) have been observed in Si:Bi lasers operating on equation image → 1s(T2:equation image) and equation image → 1s(T2:equation image) transitions by varying the stress along the [100] direction from 0 to 30 MPa 72.

4 Lasers based on resonant Raman-type scattering by donor-bound electrons

Donor electrons in silicon can not only absorb but also scatter the infrared pump radiation. Similar to conventional Raman scattering by the silicon lattice, macroscopic strains, associated with phonons, assist the intracenter scattering process. The relatively low scattering probability (see Section 2.5) can be enhanced significantly, if the phonon energy coincides or is very close to an electronic resonance such as described by Eq. (4). This phonon-assisted intracenter relaxation yields a new phonon, which contributes again to the scattering process. At sufficiently large pump intensities the number of phonons, assisting the intracenter donor relaxation, can increase to a level where stimulated Raman-type scattering occurs. In this section we describe the realization of a Raman-type THz silicon laser based on all-intracenter resonant stimulated scattering.

4.1 Pump spectra

At moderate pump power, which holds in general for photon flux densities in the FEL micropulse peak not larger than 1025 photon cm−2 s−1, the pump spectrum of THz silicon lasers is similar to its low-temperature absorption spectrum with discrete lines at energies corresponding to intracenter pumping (Figs. 10 and 22) and a continuous spectrum for photoionization ħωPUMP > Ei. At larger pump powers, the entire pump spectrum becomes quasicontinuous for ħωPUMP > equation image. Some of the intense peaks correspond to impurity transitions ending in the odd-parity states (s → p electron-dipole-allowed), and some others to those ending in the even-parity donor states (s → s electron-quadrupole-allowed).

Figure 22.

(online color at: www.pss-b.com) Pump spectra of a typical Si:P intracenter Raman laser (doping concentration 2.8 × 1015 cm−3). The attenuation from the maximum pump power is shown in dB on the right. The spectral resolution is ∼0.2 meV. For clarity, the baseline is shifted for each spectrum. All curves (except that for 0 dB attenuation, where the signal related to pumping into 2p0 state is saturated) are normalized with respect to the peak intensity at 34.11 meV that corresponds to pumping into the 2p0 state. The lines on the top indicate the positions of the excited donor states relative to the ground state.

These changes of the pump spectrum indicate the occurrence of Raman lasing for pump photons with energies in the range equation image < ħωPUMP < equation image. The closer the photon energy is to the incoming electronic resonance, the stronger is the Raman lasing. At photon energies beyond equation image the scattering loses its resonant character and Raman lasing ceases. Pumping higher than into the 3p0 state corresponds to a specific energy range where the density of donor states is large enough to support the population of a particular donor state(s) and to suppress the Raman emission in favor of the dominant donor lasing.

The Raman emission does not depend on the crystal orientation relative to the polarization of the pump radiation. For Si:Sb Raman lasers a narrower range of doping concentration was found as compared with Si:Sb lasers under intracenter resonant pumping 52. Also, no emission was observed when a pure, undoped silicon crystal was optically pumped. This underlines the importance of the electronic resonance.

4.2 Identification of the Raman resonance

The spectra measured for pump photon energies equation image < ħωPUMP < ΔE1s(A1) exhibit, except in the case of Si:Bi 22, two types of emission lines with different features. They belong to donor intracenter (D) or Raman Stokes (S) emission. Pumping into the 2p0 state results in spectrally almost merged D + S emission for all donors in silicon. The donor emission does not depend on the pump laser frequency while Stokes emission changes its frequency ωS linearly with a change of ωPUMP (Fig. 23):

equation image((5))

The intracenter 1s(E) → 1s(A1) Raman-active transition has been found to be responsible for stimulated Stokes emission in silicon doped by group-V donor centers. Thus, Stokes emission in intracenter Raman lasers has the following photon energies and wavelength ranges (in squared brackets) 52:

equation image

The gaps in the spectral coverage of the Raman lasers are either caused by insufficient pump powers at particular wavelengths (Si:As: strong water absorption of pump radiation 21) or by low Raman efficiency when pumping far from resonance (Si:P: in excited states 20). The Stokes linewidth ΓS is significantly broader (full width at half-maximum FWHM up to 0.76 meV for Si:Sb and up to 0.40 meV for Si:P) than that of the donor line (FWHM = 0.12–0.20 meV).

Figure 23.

(online color at: www.pss-b.com) Typical emission spectra of the Si:Bi laser (doping density 3 × 1015 cm−3) for different pump wavelengths (λFEL). The signal is taken by integration over the entire laser pulse.

It is worth noting that electronic scattering between 1s(E) and 1s(A1) donor states in Si:Sb is assisted by the interaction with the g-TA intervalley phonon. The magnetophonon resonance technique yields ħωg-TA values of 11.3 meV (for magnetic field direction 〈111〉), 12 meV (〈110〉) and 12.2 meV (〈100〉) for pure silicon at a lattice temperature of 65 K 36. Therefore, the Stokes emission in Si:Sb is always assisted by the emission of an intervalley g-TA phonon, such that the condition for the outcoming resonance for Raman scattering, ħωPUMP − ħωS ≅ ΔE1s ≅ ħωg-TA, is satisfied. The incoming Raman resonance appears when the pump photon energy coincides with the impurity transition originating from the ground state ħωPUMP ≅ ΔE1s(A1)→. This enhances the Raman emission in vicinity of resonant pumping into one of the excited states: 2p0, 2s, 2p±, 3p±, 3p0. When both conditions are satisfied, the lowest threshold for the Raman lasing is obtained. In the case of simultaneous donor and Stokes lasing (e.g., pumping into the 2p0 state) the emission spectra are hardly spectrally resolved without time gating 19, 20, 85 (see Section 4.5).

4.3 Optical thresholds

Raman laser action for the case of pumping out of resonance with an impurity state exhibits the highest pump thresholds (Fig. 24), but the emission does not saturate with increasing pump power as observed when pumping into the odd-parity excited donor states (compare with Fig. 13). This type of Raman lasing has a lower efficiency and requires peak pump intensities of ∼5–50 kW cm−2 compared to ∼0.1–10 kW cm−2 for Raman lasing, when pumping into the odd-parity donor states. When pumped into an even-parity donor state (dipole-forbidden optical transition), a characteristic step appears in some cases (Fig. 24). The reason for this feature is mixing of two contributions to the lasing mechanism: Raman-type lasing and lasing based on optical transitions between donor states. The first one has a larger laser threshold than the second one, therefore leading to the appearance of a step in the pump curve.

Figure 24.

(online color at: www.pss-b.com) Raman laser threshold for Si:Sb (doping density 4 × 1015 cm−3). Optical pumping is done with an FEL (photon energy shown in the graph).

4.4 Temperature dependences of Raman lasing

Raman lasers operate only at lattice temperatures below 10 K. Thermal depopulation of the ground state is apparently the reason that laser action ceases at increased temperatures.

4.5 Lasing dynamics and gain

The Raman emission pulse is delayed by about 0.5–1.5 µs with respect to the intracenter donor laser emission. The exact value depends on the donor species and the pumping frequency. The temporal separation of donor and Raman lasing is possible by temporally gated measurements of pump and emission laser spectra (Fig. 25). The emission signal from the front part of the laser signal originates solely from donor lasing, while Raman emission appears somewhat later and prevails until the end of the laser pulse. At resonant pumping into one of the donor states, such as into the 3p0 state, the donor emission pulse has the same duration as the pump pulse. But it is shortened in the case of simultaneous generation of donor and Raman lasing, i.e., when pumping into the 2p0 or 2p± states 19. This indicates a competition of these mechanisms, resulting in the suppression of donor lasing on the 2p0 → 1s(E) and 2p± → 1s(T2) transitions with respect to the Stokes emission.

Figure 25.

(online color at: www.pss-b.com) Temporal dynamics of the laser emission and of the pump spectra for a Si:Bi laser (doping density 3 × 1015 cm−3): (a) Si:Bi laser emission pulses when pumped at 14.4 THz (59.55 meV), for different pump powers (given in dB attenuation from the maximum pulse energy of 20 mJ); (b) pump spectra of the Si:Bi laser taken for different time gates as shown in (a); (c) enlarged part of the pump spectrum showing the coincidence of absorption in the transmission Si:Bi spectrum to a dip in the pump spectrum of the pulse end as well as to a peak of the front pulse. The lasing in the front of the laser pulse reproduces approximately the absorption spectrum and is mostly due to intracenter donor emission, while the delayed emission has contributions from both donor and Raman lasing.

The delay is caused by the comparatively low gain of the Stokes-type process. Donor lasing dominates due to its larger optical cross section (∼(1–7) × 10−15 cm2 35) that results in a gain of ∼1–2 cm−1 already at moderate pump intensities P (see Section 3.5). A Raman Stokes “pulsed” gain per unit pump intensity γS, IS(P) ∼ exp(γSPL), where IS is the intensity of the scattered light, has been estimated by considering the radiation decay time in the cavity (∼15 ns), which is relatively long compared to the time between two micropulses (1 ns), and cavity losses of ∼0.01 cm−1. This yields a value of γS ∼ 2–5 cm MW−1. Therefore, the “pulsed” gain for Raman lasing is comparable with the gain for donor lasing, ∼(5–10) cm−1 79, at pump intensities above ∼1 MW cm−2. The intracenter Raman gain is significantly larger compared to the continuous-wave Raman gain obtained for room-temperature Stokes infrared silicon lasers 12.

5 Lasers based on specific materials

Natural silicon doped by group-V donor centers is a well-investigated semiconductor material, and the technology to produce this material is well established. In particular, sophisticated purification and doping procedures have been developed, and it is nowadays possible to design and manufacture silicon samples with high precision in terms of the specifications of the material properties. In this section, we consider briefly the main results of the influence of the lattice structure of silicon on the operation of these lasers as well as some features of codoped laser materials.

5.1 Monoisotopic 28Si:P laser and multicrystalline Si:P laser

The principles of population inversion in n-doped silicon rely on the specifics of the phononic spectra of the host lattice. In this context, changes of the phonon properties as a consequence of the modification of the silicon lattice structure can play a significant role in the performance of intracenter silicon lasers. These changes might be rather subtle such as lattice imperfections induced by silicon isotopes, for example natural silicon in comparison with monoisotopic silicon, or quite dramatic such as strains induced by adding germanium atoms to the silicon lattice or disorder in multicrystalline silicon.

The isotopically enriched 28Si is from the material that was produced in the frame of the international Avogadro project for the redefinition of the mass 1. The original material is dislocation-free float-zone grown 28Si (>99.994% of 28Si) with low concentrations of carbon (<4 × 1014 cm−3), oxygen (<2 × 1014 cm−3), and electrically active centers (<1 × 1013 cm−3). Multicrystalline silicon (mc-Si) ingots have been Czochralski-grown from polycrystalline seeds 85. The mc-Si ingot has monocrystalline grain dimensions in the range of 50–500 µm. The ingots were grown in the 〈100〉 direction. Growth, doping of the single crystals and preparation of the laser samples were provided by the Leibniz Institute of Crystal Growth in Berlin 86. For a direct comparison monocrystalline monosiotopic 28Si and natural Si (92.23% 28Si, 4.67% 29Si, 3.1% 30Si) samples have been prepared. The samples have the same phosphorus doping concentration (3 × 1015 cm−3) and the same dimensions (parallelepipeds of 7 × 7 × 5 mm3). In order to compare their performance with that of mc-Si crystals, Czochralski-grown monocrystalline silicon (c-Si) samples have been used. These samples had the same shape as the above-mentioned ones but had slightly different phosphor concentrations: ND ≈ 1.6 × 1015 cm−3 (mc-Si) and NP ≈ 2.3 × 1015 cm−3 (c-Si).

The impurity absorption spectra of all samples show a very similar structure of dipole-allowed optical transitions (1s → np and 1s → conduction band). The Raman spectra reveal a blueshifted frequency of the OZC phonon in the 28Si:P crystal in comparison to that in natural Si:P; while the OZC phonon frequency remains either unchanged or it is slightly redshifted in comparison to that of natural monocrystalline Si:P 85. We interpret the redshifted Stokes emission in mc-Si:P to be caused by boundary scattering on crystallite disoriented interfaces.

Intracenter lasing occurs in all types of silicon crystals. The emission frequency corresponds to the 2p0 → 1s(E) phosphorus transition under resonant excitation into the 2p0 state and to the 2p0 → 1s(T2) transition under resonant excitation into higher excited states as well as into the conduction band continuum. This indicates that the four-level lasing mechanism with the upper 2p0 level is not affected by lattice perturbations 86, 88. There is no significant difference in the lasing thresholds under photoionization pumping of 28Si:P, c-Si:P, and mc-Si:P. Significant changes, however, occur for Raman-type lasing, where the 28Si:P crystal exhibits a rather broad operation range (lasing occurs when pumping with any energy from the 2p0 to 3p± state). Natural c-Si:P shows Raman lasing only in the vicinity of resonances with 2p0 and 2p± states, while mc-Si:P does not exhibit Raman lasing at the available photon flux densities 87. When the lasers are operated in a mechanical cryocooler stimulated emission in mc-Si:P under CO2 laser pumping ceases at ∼10 K (maximal repetition rate 4 Hz), in natural c-Si:P at ∼14 K (12 Hz) and in 28Si:P at ∼16 K (>20 Hz).

For comparison, SiGe:P alloy crystals do not show stimulated emission under photoionization 88, apparently because of the smaller peak cross section for the impurity transitions caused by inhomogeneous line broadening due to induced lattice strain in the alloy 89.

Thus, weak lattice perturbations do not change dramatically the performance characteristics of intracenter donor silicon lasers. The low-temperature Raman scattering spectra in the range of interest from 5 to 6 THz, do not show a significant modification neither with respect to frequency nor with respect to line intensities. The fact that Raman lasing does not occur in the mc-Si:P crystals indicates a high lasing threshold, that is possibly due to a slight increase of interface scattering of phonons. A major change of the crystal lattice, such as germanium atoms on a silicon lattice site, produce a dramatic broadening of impurity optical transitions and, as a result, prohibit intracenter stimulated emission.

5.2 Codoped Si:P:Sb laser

Since monodoped silicon crystals are lasing on a single or a couple of strong impurity transitions, it is obvious to extend the operation frequency by using several dopants while keeping the doping concentration in the optimal range of n-Si lasers. There is no fundamental technological limit for this extension and codoping with two or three elements is relatively straightforward to achieve. The most difficult group-V donor for doping of silicon is bismuth, because of its low solid solubility and low vapor pressure. The elements that are most easily doped are phosphorus and antimony, because of their advantageous diffusion coefficient and solubility.

In order to obtain a codoped crystal, an initial phosphorus-doped silicon crystal with a donor concentration of 1.8 × 1015 cm−3 was grown by the float-zone technique using gas-phase doping. In a second step this crystal was used for pedestal growth in the [100] direction with simultaneous doping by antimony from the melt. A slight antimony gradient along the growth axis allowed fabrication of samples with different ratios of NSb/NP in the range of 0.8–1.2. The crystals used in these experiments had a total donor concentration of NP + NSb ∼ (2.5–4) × 1015 cm−3, which is close to the optimum for THz silicon lasers 84. Standard sample preparation and the experimental setup at the FEL IR-User Facility of the FOM Institute for Plasma Physics were used. Stimulated emission has been observed for all investigated Si:P:Sb samples under resonant photoexcitation pumping with the typical pump thresholds above ∼1023 photons cm−2 s−1. Samples with larger donor concentration also operate under photoionization pumping for pump photon fluxes above ∼8 × 1024 photons cm−2 s−1. These thresholds are larger than those obtained for single-donor doped silicon lasers, most likely because of the about two times reduced donor concentration in comparison with the optimal doping level. Time-resolved spectra allowed the separation of intracenter donor and Raman lasing. By varying the pump wavelength different emission frequencies have been obtained from the Si:P:Sb lasers (Fig. 26). Emission on the 2p0 → 1s(T2) transition of Sb has been observed for resonant pumping into the 3p0 state and the higher states of Sb including photoionization. Emission on the 2p0 → 1s(T2) of P has been observed for pumping into the 3p± state and into the conduction band. It should be noted that when the pump photon energy exceeds the ionization energy of the Sb donor, both emission lines occur in higher-doped samples 84. In the case of excitation corresponding to pump energies below the 3p0 state of Sb donors, Raman lasing occurs along with intracenter lasing on the 2p0 → 1s(E) transition of both donors. Raman lasing from P donors occurs only when pumped into the 2p0 and 2p± states or in close proximity to it. The observed Stokes shifts, 12.1 meV (Si:Sb) and 13.0 meV (Si:P) are in good agreement with those of single-donor silicon Raman lasers 52. Significantly weaker stimulated emission from codoped Si:Sb:P crystals has been observed when the sample was excited with a CO2 laser (into the conduction band).

Figure 26.

(online color at: www.pss-b.com) Typical emission spectra of a codoped Si:P:Sb laser for different pump photon energies. The total donor concentration is NSb + NP ∼ 4 × 1015 cm−3, NSb/NP ∼ 1.2. The labels on the right with an arrow in front correspond to the state that is pumped. The emission lines from the Sb donors are brown and the emission lines from P donors are green. For both donors intracenter emission occurs on the 2p0 → 1s(E), 1s(T2) transitions. There is a notable shift of some spectral lines from their positions in single-doped crystals (Si:P, Si.Sb, dashed vertical lines).

Spectra have also been obtained from Si:P:Sb samples when applying uniaxial stress in the [100] direction (Fig. 27). The most intense line in the spectra is the equation image → 1s(B2) antimony intracenter line. It dominates at lower stress values. Lasing from the 2pmath image → 1s(B1) phosphorus transition peaks at around 40 MPa and the lower-valley contribution from the equation image → 1s(B2) phosphorus transition remains the only emission line at stress values above 120 MPa. We attribute the observed dependencies on the stress to a reduced phonon-assisted f-scattering, that supports the population of the 2p0 state, together with the resonant character of depletion of the lower laser state which is due to its interaction with a g-TA phonon, similar to that observed for the singly doped silicon crystals under stress (Section 3). The change of the laser transition of phosphorus states allocated under the upper and lower shifted conduction band valleys indicate stress-related variations of the relative populations of the conduction band valleys. The lasing threshold decreases significantly, more than a factor of 20, with stress up to 20 MPa and slightly increases above this value.

Figure 27.

(online color at: www.pss-b.com) Emission spectra at different stress (left) and dependence of the observed laser lines on uniaxial stress from the Si:P:Sb sample under photoionization pumping (10.59 µm, intensity ∼400 kW cm−2 = 2 × 1025 photon cm−2 s−1). The total donor concentration is NSb + NP ∼ 4 × 1015 cm−3, NSb/NP ∼ 1.2. The labels on the right show the donor transition.

Thus, codoping of silicon crystals does not enhance the efficiency of intracenter lasing, but allows multifrequency stimulated emission, controllable by the pump frequency or the applied external stress. The relative line intensities in codoped crystals follow the relative concentration of the dopant.

6 Other results from the research on intracenter silicon lasers

Research on the physics of the intracenter silicon laser initiated various theoretical and experimental investigations concerning the capture and intracenter relaxation of nonequilibrium electrons into impurity centers in semiconductors, the lifetimes of localized impurity states, nonlinear spectroscopy of even-parity excited states of an impurity atom and others. We will briefly review some results in the following section.

6.1 Noncascade intracenter relaxation in silicon

Experiments with silicon lasers under resonant intracenter pumping (see Section 3.2) have shown that there are particular intracenter relaxation routes that dominate the capture process. The anisotropy of the wavefunctions of donor states results in an additional correction of the intravalley and intervalley scattering of donor electrons. The selection of particular relaxation routes, as reported in Ref. 23, is strongly dependent on the wavefunctions and chemical splittings of the even-parity states (Fig. 28b). For highly excited donor states with small energy spacing ΔEn < δE, the electronic relaxation is dominated by acoustic phonons. The largest relaxation rates have been found for the electron interactions with longitudinal acoustic (LA) phonons 60. Intervalley phonons assist electron transitions between large-spaced donor states.

Figure 28.

(online color at: www.pss-b.com) Schematic representation of electronic relaxation channels in Si:As. Bold curves indicate the observed dominant relaxation channels that bypass a few adjacent donor states. The width of the individual arrows indicates approximately the transition probability. The diagonal straight arrows down show the size of the possible relaxation step as estimated for the cascade capture model.

The wavefunction of donor states, involved in the phonon-assisted relaxation process, is essential for the capture process. For example, the nonradiative acoustic-phonon-assisted transition that spans over a few adjacent states, such as 3p0 → 2p0, ΔE = equation image − equation image = 6.0 meV, has been estimated to be a factor 30 larger than that of the 3p0 → 2p± transition, ΔE = equation image − E2p± = 0.9 meV. Although it does not increase significantly the particular relaxation rate coefficients, relaxation channels that bypass the long-living states, such as 2p0, result in a significant reduction of the population of the latter state. The actual state population becomes strongly dependent on the initial state from where the relaxation starts, i.e., finally on the excitation mechanism.

6.2 Determination of binding energies of the lower even excited states of donors in silicon

The EMT approximation is the main approach for calculating the binding energies of the ground state and the excited states of shallow hydrogen-like donors in silicon (see for example Ref. 38). The EMT describes very well the binding energies En for p-type states as well as for very shallow (En < 1 meV) donor states. The energies for the ground state as well as a few lower excited s-type states that are split by the valley–orbit interaction into singlets ns(A11), doublets ns(E:Γ3), and triplets ns(T25), n = 1,2,3,… have been calculated by taking into account the intervalley kinetic energies and mass anisotropy (see for example Ref. 90 and references therein). Since at low crystal temperature impurity-related electrons are bound to the ground state of the impurity center, impurity absorption spectroscopy yields the binding energies of the states into which optical transitions from the ground state are dipole allowed 35. Commonly, the assignment of the binding energy to a particular state is based on the assumption that the measured value of the 3p± state is the same as the one calculated by the EMT (E3p± = 3.12 meV). Major difficulties occur with the binding energies of even-parity excited states. Transitions from these states to the 1s(A1) ground state are dipole-forbidden. In general, the determination of the energies of these states requires either indirect spectroscopic methods or specific crystal conditions. Impurity absorption with the crystal at somewhat elevated temperature has been used in order to determine the energies of the lowest excited states, 1s(E) and 1s(T2), because their thermal population enables absorption spectroscopy 91, 92. The selection rules can be lifted in relatively heavily doped crystals due to perturbing effects by other impurity centers and therefore the 1s → 2s transition can be observed in absorption 93 as well as by Raman spectroscopy 66. However, it is important to note that both approaches result not only in a broadening of the investigated line, but also in frequency shifts, which might be significant 24. Another approach to derive the binding energy of s-states is based on two-electron transition spectra 94, where an exciton, which is bound to the ground state of the impurity, radiatively decays leaving the donor in the excited state. All these techniques have provided the binding energies for several s-type donor states in silicon with values that are in good agreement with the calculated ones 90.

The analysis of the pump and emission spectra of silicon lasers under intracenter resonant pumping allows determination of the binding energies of the lower excited states that are involved in the laser process for group-V donors in silicon. This is another method and it is particularly noteworthy that in this case disturbing effects that affect the binding energies such as heating are avoided. The determination is based on the analysis of the laser transitions in silicon, such as np → 1s(E), 1s(T2). Since the upper laser levels, the np states, are very accurately known from absorption spectroscopy, knowledge of the laser frequencies yields immediate information about the energies of the lower laser levels, 1s(E) and 1s(T2).

In general, the binding energies determined from the laser emission agree well with the values obtained with other spectroscopic techniques (see Table 1). An exception is Si:Bi which is the material with the deepest group-V donor. The advantage of the laser data is that they were determined under equilibrium conditions for a crystal lattice temperature below 10 K. These data provide independent evidence for the binding energies of even-parity excited states in silicon doped with group-V donors.

Table 1. Binding energy of some excited states for hydrogen-like donor centers in silicon as derived from an analysis of pump and emission spectra of silicon intracenter lasers under resonant photoexcitation.
donor1s(T2)1s(E)
n-Si lasingcal.spectroscopyn-Si lasingcal.spectroscopy
  • a

    Doublet 1s(T28);

  • b

    singlet 1s(T27).

    References: n-Si lasing: as deduced from pump and emission spectra of silicon intracenter lasers 24; spectroscopy: impurity absorption spectroscopy at elevated crystal temperatures 92; and for highly doped Si:BiHD at T ∼ 10 K 93; cal.: as calculated by EMT with an empiric model Hamiltonian 90; for Si:Bi: from the standard EMT 38.

P33.91 ± 0.02 (4 K)34.233.89 ± 0.01 (45 K)32.61 ± 0.02 (4 K)32.732.56 ± 0.01 (45 K)
Sb32.83 ± 0.02a (4 K)32.932.83 ± 0.01a (30 K)30.50 ± 0.02 (4 K)30.530.53 ± 0.01 (30 K)
 33.12 ± 0.01b (30 K)
As32.73 ± 0.02 (4 K)32.732.68 ± 0.01 (60 K)31.34 ± 0.02 (4 K)31.331.25 ± 0.01 (60 K)
Bi31.90 ± 0.02a (4 K)31.331.89 ± 0.01 (80 K)30.17 ± 0.02 (4 K)31.330.47 ± 0.01 (80 K)
32.63 ± 0.01b (4 K)32.89 ± 0.01HD (10 K)

6.3 Determination of resonant electron–phonon interactions

Several resonances in the electron–phonon coupling have been found by analysis of the linewidth of impurity transitions obtained by absorption spectroscopy. The anomalous broadening of the 1s(A1) → 2p0 line in Si:Bi was explained by a the strong coupling of the involved states to the f-TO phonon, ΔE2p0 ↔ ħωf-TO 46. A similar suggestion based on the results from absorption spectroscopy has been made for the broadened line #2 in Si:Ga 95. This reveals a strong coupling of the pair of gallium states, equation image and equation image, to an OZC phonon.

The analysis of lasing schemes in Si:As under photoionization 18 and resonant intracenter pumping 20 indicates the occurrence of another strong link between group theory allowed intervalley acoustic f-LA phonon to couple the ground and 2s states in this medium. This resonant interaction leads to the unexpected fast electronic capture in Si:As and the weakest lasing of all group-V donor lasers under photoionization pumping. This resonance can be lifted by applying external uniaxial stress to the silicon crystal 71. Another somewhat weaker interaction regarding the lower excited states 1s(E) and 1s(T2) has been found in Si:Sb and Si:P crystals. In these materials the intervalley acoustic g-TA phonon couples the 1s(E) and 1s(T2) states with the 1s(A1) ground state. This dominates the relaxation mechanism and finally makes Si:Sb the most efficient material for Raman lasing 19.

7 Future research

Group-V donors in silicon enable intracenter laser action. The laser lines fall into the 1–7 THz range. Most of these lines are above 5 THz, a frequency region that is rarely covered by other coherent radiation sources. However, for applications lower optical pump thresholds or electrical pumping is desired. Lowering of the pump threshold can be obtained by a combination of the appropriate doping and compensation with constant stress applied to the silicon crystal. From the gain measurement, one can see that the major factor that limits the efficiency of silicon intracenter lasers is the low pump efficiency. Absolute gain values of a few cm−1 for unstressed Si lasers are comparable to the net gain obtained for THz quantum cascade lasers 27. Therefore, the next step is the improvement of the pump efficiency. One way would be the replacement of the TEA CO2 laser (the optical cross section of the 1s(A1) → conduction band pump transition Si:P is only ∼4 × 10−16 cm2) with a more appropriate laser source, for example a midinfrared quantum cascade laser.

THz electroluminescence originating from particular excited donor and acceptor intracenter transitions has been reported in Refs. 96–98. However, the large ionization energies of impurities in silicon lead to a relatively small cross section for impact ionization and thus prevent the formation of significant population inversion for shallow donors and acceptors. In the case of optically pumped silicon lasers an electric field, which was applied to the laser during optical excitation, terminated the laser emission at field strengths above the impurity breakdown 99. This indicates that impact ionization negatively affects the population inversion. However, electrical pumping can be an option for a Si-based heterostructure laser 100 using intrasubband or bound-to-continuum transitions.

Another research direction is the extension of the laser concept to deeper donors, such as Mg (Ei = 108 meV). Beside an improved pump efficiency the operation temperature can be increased. The stronger resonant impurity–phonon interaction, which is inherent to deeper donor centers in silicon, can be utilized in order to improve the efficiency of the Raman intracenter scattering.

There is no straightforward transfer of the donor-related laser principles to acceptor-doped silicon. Although there are no intervalley-phonon-assisted relaxation resonances, only three-level or sophisticated four-level schemes are possible by group-III shallow acceptor centers.

One of the most intriguing research directions is the controlled manipulation of the phonon spectra. As the stress experiments indicate this will change dramatically the operational properties of n-Si lasers. In order to modify the phonon spectrum one may look into silicon-based semiconductors where the phonon spectra can be manipulated or even to some extent designed. SiGe alloys, e.g., exhibit lifetimes of the 2p0 state of about 100 ps 89. The phononic spectra of these alloys can be significantly changed by changing the relative germanium and silicon content. This might be used to some extent to optimize impurity phonon resonances. Another exciting option is the combination of phononic crystals and silicon lasers.

Besides bulk silicon other doped semiconductors may also be used for the development of intracenter and Raman-type lasers. Photoluminescence has been observed in a few materials, which have in common the existence of relatively long-living impurity states. Examples are ZnSe:Li, InSb:Cu, Ge:Te 70, 101.

The technical implementation of n-Si lasers has neither been investigated thoroughly nor optimized. The conventional resonators based on total internal reflection modes provide stimulated emission inside a high-Q cavity. But they do not provide an optimal outcoupling of the laser emission. Homogeneous pumping of larger silicon crystals in order to increase the volume of the active medium and the total gain in n-Si lasers could be investigated. Miniaturization of n-Si lasers down to submillimeter dimensions, along with increasing pump power densities may lead to unexpected results, because nonequilibrium phononic hot spots may not have enough space to exist.

8 Conclusions

A variety of THz semiconductor lasers based on intracenter transitions of shallow donors in silicon has been developed during the last decade. The lasers operate in a pulsed mode at low lattice temperatures. Their peak output power is approximately a few mW. The gain coefficients obtained for the donor and Raman-type THz silicon lasers are of the order of the net gain realized in THz quantum cascade lasers and infrared Raman silicon lasers. Despite practical limitations caused by the requirements of cryogenic cooling and relatively large optical pump powers, they provide new laser lines for applications in a frequency range, which is poorly covered by any other type of laser.

The research on the intracenter silicon lasers has triggered a series of theoretical and experimental investigations beyond pure laser physics. Examples are the capture processes of nonequilibrium electrons by impurity centers in semiconductors, the investigation of lifetimes of donor electrons in excited localized state, the spectroscopy of the even-parity excited states of an impurity atom, as well as Raman-type scattering by electronic donor resonances assisted by large-momentum intervalley acoustic phonons.

In summary, THz silicon lasers remain a fascinating object of research with impact on the physics of semiconductors as well on laser technology.

Acknowledgements

Over the years we have collaborated with many colleagues on the topic of THz silicon lasers. We gratefully acknowledge significant contributions to the experiments from M. H. Rümmeli, M. Greiner-Bär, R. Eichholz, and U. Böttger (DLR, Berlin, Germany), S. A. Lynch (University of Cardiff, UK), K. Litvinenko (University of Surrey, Guilford, UK), and M. F. Kimmitt (University of Essex, Colchester, UK) as well as contributions to the laser theory from E. E. Orlova, E. V. Demidov, and V. V. Tsyplenkov and contributions to the experiments on stressed silicon from K. A. Kovalesky and absorption spectroscopy from B. A. Andreev (all from the Institute for Physics of Microstructures of the Russian Academy of Sciences, N. Novgorod, Russia). We are grateful to H. Riemann and N. V. Abrosimov from the Leibniz Institute of Crystal Growth, Berlin, and to H.-J. Pohl (VITCON Project-consult GmbH, Jena) for providing silicon samples. We are pleased to acknowledge the assistance in the experiments at the FEL at FOM Institute for Plasma Physics in Rijnhuizen from J. N. Hovenier and T. O. Klaassen (Delft University of Technology, The Netherlands), D. A. Carder and P. J. Phillips (Herriot-Watt University Edinburgh, UK), and B. Redlich and A. F. G. van der Meer (IR User Facility at FELIX, Rijnhuizen, The Netherlands). Financial support for this research was provided from Deutsche Forschungsgemeinschaft and the Russian Foundation for Basic Research as well as Humboldt Foundation and the Investitionsbank Berlin.

Biographical Information

Sergey G. Pavlov received his Ph.D. degree (1995) and the doctor of science degree (2010) in physics and mathematics from the Russian Academy of Science. In 2000 he joined the Institute of Planetary Research of Deutsches Zentrum für Luft- und Raumfahrt (German Aerospace Center) in Berlin, Germany. His research interests include the infrared, Raman scattering and atomic emission spectroscopies of semiconductors and geologic materials, as well as the investigation of terahertz semiconductor lasers.

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Biographical Information

Roman Kh. Zhukavin received the M.S. degree in physics in 1995 from the Nizhny Novgorod State University, Nizhny Novgorod, Russia, and the Ph.D. degree in physics and mathematics in 2005 from the Institute for Physics of Microstructures of the Russian Academy of Sciences (RAS). He holds the position of a Senior Researcher at the Institute for Physics of Microstructures, RAS. He was a Fellow of the German Academic Exchange Service (DAAD, 2002).

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Biographical Information

Valery N. Shastin received the radio-physics diploma, the PhD and Doctor of Science degrees from Gorky State University in 1971, 1984 and 1995. Employment: 1971–1993 IAP (RAS), then Head of the Laboratory in IPM (RAS). His interests include hot-electron phenomena, nonlinear optics, and THz lasing. 1979–1995: investigation of the hot-hole THz lasing in germanium. Currently: THz lasing of shallow impurity centers in silicon. In 1987 he received the Soviet Union State Prize for Science and Engineering after pioneering work on terahertz p-Ge lasing.

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Biographical Information

Heinz-Wilhelm Hübers received a Dr. degree in physics (1994) and a habilitation degree (2009). Since 1994 he is with Deutsches Zentrum für Luft- und Raumfahrt (DLR) in Berlin, Germany, where he became head of department (2001). Since 2009 he is professor of experimental physics at the Technische Universität Berlin and head of department at DLR. His research interests include terahertz physics and spectroscopy. He has received the Innovation Award for Synchrotron Radiation and the Lilienthal Award.

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