Evidence for terahertz acoustic phonon parametric oscillator based on acousto-optic degenerate four-wave mixing in silicon doping superlattice

We report first evidence for a 1.0 terahertz (THz) self-starting mirrorless acoustic phonon parametric oscillator (MAPPO) produced from acousto-optic phase-conjugate degenerate four-wave (D4WM) mixing in a THz laser-pumped silicon doping superlattice (DSL). The DSL was grown by molecular beam epitaxy on a (100) boron-doped silicon substrate. A superconducting NbTiN subwavelength grating was used to couple the THz laser radiation into the DSL. Superconducting granular aluminum bolometric detection, coupled with Si:B piezophonon spectroscopy, revealed excitation of THz coherent compressional and shear waves, along the<111>direction only. The Bragg scattering condition for distributed feedback, and the energy conservation requirement for the D4WM process, were both verified. A THz MAPPO could provide a testbed for studies of non-classical acoustic phonon fields.

Phonons, the quanta associated with the vibrational waves of a crystalline lattice, are among the most important excitations in condensed matter. Major advances in our understanding of the physics of phonons during the past several decades have included first principles calculations of the phonon lifetimes in simple crystals, their interactions with other phonons, defects, boundaries and impurities, and in controlling thermal transport at the nanoscale 1 . A coherent source of acoustic phonons could find widespread applications in science and technology. The stimulated emission of transverse acoustic (TA) phonon avalanches at 50.4 gigahertz (GHz) has been observed in dilute centers in ruby 2 . Current phonon studies use visible-light pulse (VLP) transducer techniques 3 to produce and detect coherent near-THz acoustic phonons. In most VLP-based experiments, the generation of coherent phonons was limited to longitudinal acoustic (LA) phonons because of the lateral symmetry of the excited crystal, although TA phonons at 0. 4 THz have also been produced by piezoelectric coupling in GaN multiple quantum wells 4 . A (femto-or picosecond) VLP produces, by electronic excitation and relaxation, a stress as a source of near-THz coherent acoustic phonons. Stress joined with an LA or TA wave can be monitored by a pump-probe technique measuring changes in reflectivity. A VLP transducer can consist of a metal film, a surface region of a crystal, or a superlattice (SL). More recently, the sound amplification by stimulated emission of radiation (saser) in semiconductor superlattice structures has been described in connection with the VLP transducer technique. One type of saser uses an active medium, a population inversion in a voltage-biased SL containing free electrons 5,6 . Gain is due to phonon stimulated electronic transitions between next nearest energy levels of a Wannier-Stark ladder; the gain coefficient for LA waves at 650 GHz was 3 × 10 3 cm −1 . Another type 7 utilizes a miniband current in a SL (at room temperature) showing gain due to the acousto-electric effect; its gain coefficient for LA waves near 400 GHz was 8 × 10 3 cm −1 .
In this Rapid Communication, we report the first experimental evidence for a mirrorless acoustic phonon parametric oscillator (MAPPO) in a doping superlattice structure operating at one terahertz. It is based upon acousto-optic (AO) phase-conjugate degenerate four-wave mixing (4WM) 8 of counter-propagating fields. The MAPPO is optically-pumped by nano-second (ns) pulsed 1.04 THz plane-polarized laser radiation. Coherent light sources based upon 4WM in mirrorless laser oscillators, in both hot and cold atoms 9 , are known to have some unique properties such as cavity-free alignment and large tunability. We have designed the SL period to provide spatially-distributed feedback. Distributed feedback occurs when the Bragg scattering condition is satisfied for the signal and phase-conjugate acoustic waves propagating normal to a gain grating 10 associated with the 4WM process. We have not however, made any first-principles calculations, and more data is necessary to further validate, and more fully understand the phenomenon.
A "n-i-p-i" DSL 11 is formed by a periodic n-and p -doping structure in an otherwise homogeneous semiconductor crystal with undoped (i-) regions in between. In its ground state the sample thus consists of a single crystal (in contrast to compositional SLs) with alternately positively and negatively charged sheets of ions. Given the backfolded acoustic phonons (see Fig. 1 of Ruden and Dohler 12 ) under the superlattice aspect are optical modes, doping SLs could be expected to be excitable by FIR electromagnetic fields. Our delta-doped doping SLs were grown by MBE on (100) float-zone 150 mm diameter silicon wafers and consisted of 30 sequential n-i-p-i' s with period d = 8.1 nm and dopant (B and Sb) concentrations: N 2D A = N 2D D = 3.3 × 10 13 cm −2 . The smallest subband energy level spacing in the V-shaped potential wells for both electrons and holes is of order 50 meV in our doping SL 13 , much larger than the 4.3 meV pump radiation. Intense THz fields on the order of MV/cm have been reported to induce carrier generation by impact ionizations associated with impurity states in undoped multiple-quantum wells 14 , however, in our experiments we use quite modest THz fields on the order 10 V/cm. Accordingly our AO 4WM process should lie in the reactive regime 15 where the FIR light frequency is below the onset of absorption by electronic transitions.
Acoustic phase conjugation has been observed in piezoelectric semiconductors at microwave frequencies 16 resulting from a nonlinear piezoelectric coupling between photons and phonons (or electric field E and elastic strain S ) of the form γE 2 S 2 , with γ the coupling constant. This four-wave coupling is of the same form as that found in optical 4WM giving rise to optical phase conjugation. We postulate that electrostriction coupling of this form is present in our experiments due to the direct action of the electric field acting upon the ion sheets of the DSL. Recently, electrostriction has also been reported to have produced an extraordinarily strong D4WM response yielding THz acoustic breathing modes in gold nanoparticles 17 . Figure 1 illustrates schematically our posited 1.04 THz MAPPA process (energy conservation and distributed feedback have been verified as described below). In AO D4WM one of the optical pump waves and the acoustic probe wave couple to produce an index grating 18 . This grating coherently scatters the counterpropagating optical pump wave to produce a phaseconjugate acoustic wave. Distributed feedback coupled with parametric gain, can then resonantly generate a standing-wave pattern along <111>. The index gaingrating planes are normal to the acoustic wavevector. In contrast to OPOs, for MAPPOs the difficulties of phase matching due to dispersion, are largely circumvented 15 . Figure 2 illustrates our experimental arrangement. A silicon crystal (size 7 mm × 5 mm × 0.60 mm) is immersed in vapor-pumped liquid helium (1.66 K) and carries on one surface a superconducting NbTiN gratingcoupler patterned on top of the doping SL, and on the opposite crystal surface a superconducting Al bolometer. The grating-coupler is illuminated by nanosecondpulsed polarized 1.04 THz laser radiation. We chose to use NbTiN for the grating-coupler (500 nm thick, 10 µm period, 6 µm line, 4 µm space) due to its high frequency gap 1.2 THz 19 and low absorptivity, to preclude the generation of broadband incoherent acoustic phonons. The subwavelength grating-coupler converts the incident FIR electromagnetic wave into counter-propagating evanescent surface waves within the DSL as shown in Fig.  1(a). Grating-coupler theory 20 shows that 85% of the incident FIR laser intensity is converted into the surface waves with 15% remaining in the propagating field. The evanescent field, with a 1.7 µm decay length, can be assumed to be uniform over the 0.24 µm thick DSL, and does not extend appreciably into the 0.60 mm thick Si:B substrate. The 100 nm thick, serpentine bilayer granular aluminum/palladium bolometer 21 with active area A = 0.53 mm 2 , was used for the detection of both pho- tons and phonons. The phase-conjugate acoustic wave would not be detected. The bolometer's responsivity R = 900 V/W, was obtained through a combined dc, and ns-pulsed, characterization study 22 . The bolometer was biased at a constant current of 40 µA, and the voltage signal was amplified 40× using 1-GHz bandwidth electronics, and recorded using a fast real-time digital sampling oscilloscope.
The Si substrate was boron-doped at a density of N B = 10 15 cm −3 in order to use Si:B piezo-phonon spectroscopy 23,24 . The resonant phonon scattering by the stress-split acceptor ground state can be used for phonon spectroscopy. We chose the boron doping density N B appropriate for our sample thickness, according to an scaling law found in the Appendix of Schwarte and Berberich 25 : N B ∼ N SB 4mm 0.6mm where N SB = 4.8 × 10 13 cm -3 and 4 mm, were their doping level and sample thickness, respectively. The scattering rate for TA phonons was shown 25 to be larger than for LA phonons, although the respective scattering rates approach each other near 5 meV. We used a uniaxial stress apparatus designed after Fröhlich and Nieswand 26 , and calibrated up to 3 kbar to apply the stress via a spring balance mounted above the cryostat.
We used a previously described 27 custom cavitydumped, optically-pumped, FIR molecular gas laser to pump the DSL. The FIR laser resonator, filled with methyl fluoride at a pressure of 6 Torr and pumped by the 9R20 line of a multi-mode TEA CO 2 laser, yielded smooth cavity-dumped pulse envelopes [∆t 5 ns (FWHM)] (see Fig.2), as measured by a 1-GHz bandwidth pyroelectric detector, containing the linearlypolarized 1.04 THz radiation. Peak powers of order 10 kW were obtained at 10 Hz pulse repetition rates. The measured wavelength, λ = 288.2 ± 0.05 µm, obtained with a scanning metal-mesh (1000 lpi) Fabry-Perot inteferometer 28 , agreed with the published wavelength 29 for this FIR line. The energy contained in a FIR pulse fluctuated by≈ 50% from pulse to pulse, with frequent larger excursions. A waveguide 30 , used for FIR laser beam transport to the entrance window of the liquid helium cryostat, was interrupted to insert a pair of crossed wire-grid polarizers to reduce the pump power to ∼ 10 mW. Figure 3 shows a stacked sequence of digitized bolometer traces (signal versus time) in response to single laser pulses of progressively increasing power. Due to the nonlinearity of the bolometer response (described below) however, the listed powers are suspect. The first pulse observed (time 0 ns) in all traces is the bolometer response to the arrival of undiffracted grating-coupled FIR laser radiation. As the laser radiation pump power reaches a threshold power of 6.4 mW, ballistic acoustic phonons appear at a delay time of 110 ± 2 ns. As the incident FIR power increases further, the acoustic signal at the 110 ns delay grows progressively stronger, and at a 8.8 mW pump power, a second phonon signal appears at 210 ± 2 ns. The distinct onsets of the phonon signals with pump power may be misleading however due to the nonlinearity of the bolometer response. For determination of group velocities of waves consistent with observed signals, we use the equations of motion (Christoffel's equations 31 ) for sound waves in silicon (mass density ρ = 2.332 g/cm 3 , elastic constants c 11 = 167.8 GPa, c 12 = 65.2 GPa, c 44 = 80.01 GPa) 32 . Among the surveyed modes for <100>, <110>, <111>, <012>, and <122> directions, we find  (Fig. 3). The slight discrepancy with the TA transit time may be due to a change in the elastic constants due to the Si:B substrate doping. Identical arrival times however, would be expected for phonons produced along any of the four equivalent <111> forward directions. The Bragg scattering condition for distributed feedback [see Fig. 1(c)] is given by: with θ the angle between the phonon wavevector − → Q and the DSL reciprocal lattice vector − → G = 2π d [001] , and with N integer. For the observed <111> LA and TA modes at 1.04 THz (frequency verified below), with associated wavelengths Λ LA = 9.35 nm and Λ T A = 4.91 nm, respectively, we indeed find near integer values N for both LA and TA modes, i.e, 1.00 and 1.90, respectively. Thus, the experimental results are in excellent agreement with the distributed feedback requirement. A higher threshold observed for the TA mode could be expected since the Bragg scattering condition is not quite as well satisfied in this case, i.e., 1.90 instead of 2.00. Interestingly, a [001] LA mode at 1.04 THz would also satisfy the Bragg condition with N = 1.00 but this mode is not observed.
As shown in Fig. 3, as the incident FIR estimated power increases, the registered bolometer signals broaden in time and the amplitudes saturate. Thus, the bolometer unfortunately was operated in a nonlinear regime in these experiments. To continue with an analysis, we will assume that the integrated area of a given pulse is proportional to the absorbed energy. We accordingly assume that the absorbed signal power is proportional to its integrated pulse area, relative to that of the integrated weakest laser pulse response (lowest trace Fig. 3). Estimated absorbed powers were then obtained by multiplying the ratio of areas by the peak voltage of the weak laser pulse, and dividing by the bolometer responsivity R and the post-detection electronics amplification. The estimated incident pump powers are obtained (numbers shown above traces to the right in Fig. 3) from the absorbed FIR pump powers by dividing by 0.15 (that fraction of the zero-order propagating light reaching the bolometer) and by 0.016 , the FIR absorbtance of the bolometer. The latter was calculated using the theory of electromagnetic wave propagation in stratified dielectric and metallic media 33 , using the low-temperature FIR index of refraction of silicon 34 , 3.318, and the resistivity, 235 µΩ -cm, of the biased bolometer. Estimates of the acoustic pulse power within the doping DSL result from dividing the bolometer's absorbed acoustic power by 0.30, the estimated acoustic wave absorptance of the bolometer. The latter is consistent with dielectric crystal/metal interface studies 35 . Figure 4 displays separately, the LA and TA phonon spectra as obtained by Si:B pieozophonon spectroscopy. The averaged (10 3 single shots) peak of a bolometer response to a phonon pulse, for a time gate of width ∆t positioned at the arrival time for the mode in question, is plotted versus the Si:B resonant scattering energy E.
Here the peak value of each bolometer phonon signal, prior to averaging, was divided by a the peak value of a separately detected portion of the FIR pump radiation that reflected from a wiremesh beamsplitter and focused onto a pyroelectric detector. As a result, the averaged bolometer responses shown are in arbitrary units, and the corresponding pump power was not estimated from the bolometer response in this case. The phonon energies for both LA and TA signals (the drop in the signals) are identified as 4.3 meV, corresponding to 1.04 THz, in excellent agreement with the presumed AO D4WM process. On average, the TA signal is weaker (× 1 4 ) . It is well known from phonon focusing studies in silicon 36 using incoher- ent sources that TA phonon signals are stronger than LA signals along <111>. Since our data shown in Figs. 3 and 4 indicates the opposite, we have further evidence for coherent, and not incoherent, phonon generation. Figure 5 shows a plot of the estimated LA power versus the 1.04 THz estimated pump power. Here, we have divided the full incident LA power by 4 to account for power assumed to emanate from four <111> equivalent directions. For the pump power P ∼ 9.4 mW and estimated <111> LA power ∼ 18 µ W, we find a conversion efficiency: η ∼ 2 × 10 −3 . The propagating acoustic pulse, contains 9 × 10 −14 J in a volume V = v LA ∆tA/ √ 3 = 1.4 × 10 −11 cm 3 . The estimated phonon occupation number in this volume is n = 1.4 × 10 8 . The zero point amplitude 37 is given by ζ zp = √ ρV * ω ∼ 3.2×10 −18 m , where we choose for the initial coherence volume V * ∼ 1.4×10 −15 m 3 , a cylinder formed by the circular projected bolometer area A/ √ 3 , and a coherence length Λ LA /2 (due to elastic scattering at B and Sb impurities). The avalanche of stimulated LA phonons than corresponds to a classical acoustic wave of amplitude, ζ = ζ zp √ n + 1 ∼ 3.7 × 10 −14 m. This amplitude corresponds to 2.7×10 −4 of the distance between an impurity and the nearest Si atom, or strain, S ∼ 3×10 −4 . Picosecond acoustic solitons formed in Si have reported strain levels of this order of magnitude 38 . For an active medium length of L = 30 × d × √ 3 = 0.42 µm, and assuming a unity zero-point fluctuation, we find a large exponential gain coefficient: g LA = ln(n/1)/L 4 × 10 5 cm −1 .
In summary, we report evidence for a self-starting MAPPO operating at 1.04 THz. A MAPPO may prove useful as a novel source of coherent acoustic phonons with both compressional and shear waves available for probing material properties at the nanoscale. In principle, a MAPPO might also be operated in reverse 18 as an acoustic wave detector; coherent incident acoustic waves could excite coherent FIR radiation which could be coupled out using a grating or a prism. Given that stimulated optical 3WM and 4WM have been used to study quantum corre-lations between photon beams 9,39,40 , nonlinear MAPPOs could also provide a testbed for analogous studies of nonclassical acoustic phonon fields.