Acoustic pulse-induced modulation

Figure 1a shows schematically the experimental arrangement used to launch acoustic strain pulses into a QCL and to observe dynamically their effect on its optical and electronic characteristics. An aluminium-film acoustic transducer was deposited on the bottom surface of the semi-insulating GaAs substrate of a 2.5‒2.75 THz QCL containing 88 periods of the active-region heterostructure32, as shown in Fig. 1b. The THz QCL device with dimensions of (2000 × 150) µm2 was mounted onto the cold finger of a cryostat, with a 1.0 × 0.3-mm2 slit aperture allowing optical access to the aluminium transducer layer. Longitudinal-acoustic pulses with a single-cycle (bipolar) form and ~20-ps33 duration were generated in the device by optically exciting the transducer using ultrafast infra-red pulses from an amplified Ti:Sapphire laser (see Section “Methods”). The optical pulses were focused using cylindrical lenses to form a beam with dimensions of ~1.1 mm × 0.3 mm at the QCL position, resulting in ~50% spatial overlap between the optical beam and the QCL device. Perturbations to the electron transport in the QCL were monitored via the device voltage, with the THz emission being simultaneously monitored on a Schottky-diode detector. The L–I–V characteristics and the band structure of the QCL device are shown in Fig. 1c, d, respectively.

Fig. 1: Experimental arrangement and QCL device structure and characteristics. a Experimental arrangement for measuring the optical and electronic perturbation of a THz QCL by laser-generated picosecond acoustic pulses. The sample is mounted on the cold finger of an optical cryostat at an operating temperature of 15 K. b Schematic diagram of the QCL device structure showing the transmitted (solid red line) and reflected (dashed red line) strain pulses. c L‒I‒V characteristics of the QCL in the absence of acoustic pulses, measured at a temperature of ~10 K and driven with 50-µs pulses at a 1-kHz repetition rate. Labels correspond to the different QCL biasing conditions used in experiments: (V 1 = 4.13 V, V A = 5.02 V, V 2 = 5.49 V and V 3 = 5.58 V). d Simplified QCL band structure showing two periods of the structure with labelled injection level (IL-|2〉), upper laser level (ULL-|11′〉) and lower lasing level (LLL). Source data for c and d are provided in ref. 50. Full size image

Figure 2a shows the time-varying THz emission, L(t), and the QCL voltage, V(t), recorded with an optical pulse energy of 10 µJ incident on the Al-film transducer and with a quiescent QCL bias of V b = 5.58 V (V 3 in Fig. 1c). The first pulse in both L(t) and V(t) appears 32 ns after the initial optical excitation (at t = 0 ns, not shown), corresponding to the transit time of the acoustic wave, propagating at the speed of longitudinal sound in GaAs (4800 ms−1), through the ~150-µm-thick substrate. Subsequent echo-pulses occur at 70-ns intervals, owing to acoustic reflections between the top metallic (Ti/Au) contact of the QCL and the bottom of the substrate. We assume that the acoustic pulse is not significantly perturbed as it propagates through the QCL structure. This is consistent with pump–probe measurements of the acoustic phonon modes of typical QCL structures34 which show phonon lifetimes less than or equal to the time for an acoustic wave to propagate once through the structure, and indicates narrow phonon stop bands in the frequency range of interest. As the acoustic wave propagates through the QCL, it perturbs the electrical transport and modulates the THz emission. Although the acoustic pulse has a duration of ~20 ps, the duration of the acoustically induced effect is determined by the round-trip transit time of the acoustic wave through the 13.9-µm-thick QCL active region, which was calculated to be ~6 ns using the speed of sound in GaAs. This is shown experimentally in Fig. 2b. For these operating conditions, with the QCL bias exceeding that required for subband alignment V A , the power modulation ΔL is negative in sign.

Fig. 2: Observation and measurement of the acoustic modulation on the QCL device. a Temporal response of the THz emission, L(t), (black trace, top) and QCL voltage, V(t), (red trace, bottom) to incident strain pulses generated with an optical pulse energy of 10 µJ at a QCL bias V b = V 3 . The initial strain pulse, arriving at time t = 32 ns, is generated by laser impact on the Al-film transducer at t = 0 and the following pulses, occurring every 70 ns, are due to multiple reflections in the device. b Shows the change in the L(t) and V(t) responses (ΔL and ΔV, respectively) to an individual acoustic pulse for the region highlighted in (a). Data have been smoothed using a 20-point rolling average filter. Source data are provided as a Source Data file. Full size image

The perturbation of the electron transport also results in a voltage modulation, ΔV(t), as shown in Fig. 2a where a positive modulation (ΔV > 0) is observed. This implies an increase in resistance (assuming constant current) caused by the acoustic wave passing through the heterostructure; this observation is consistent with predictions from a time-dependent perturbation model of the QCL under perturbation by a strain pulse (see Section “Theoretical analysis on the origin of voltage perturbations”). The fast THz power modulation is followed by a broader positive peak after the acoustically induced pulse in V(t) has ended, and coincides with the small negative ringing in V(t) that we attribute to resonance in the electrical circuit.

Closer inspection of the voltage pulses (Fig. 2b) reveals they comprise an initial ~3-ns pulse, followed by a ~3-ns-long shoulder owing to reflection of the strain pulse in the opposite direction. The reflected pulse amplitude is ~40% that of the forward pulse, which is consistent with the amplitude reflection coefficient for the strain pulse at the GaAs/Au interface, r = (Z Au – Z GaAs )/(Z Au + Z GaAs ), where Z Au = 63 × 106 kg m2 s−1 and Z GaAs = 25 × 106 kg m2s−1 are the acoustic impedances of Au and GaAs, respectively.

It is important to note that, owing to the short (~ps) carrier lifetimes in QCLs, a strain-induced band structure deformation will perturb the carrier transport on ultrafast timescales. The rise time of the THz power modulation in our device will therefore be limited fundamentally by the acoustic-pulse duration (~20 ps) and its transit time through an individual injection region of the heterostructure stack (see Section “Theoretical analysis”). Nevertheless, the perturbation in V(t), shown in Fig. 2a, b, exhibits a rise time that is limited to ~800 ps by parasitic device impedance (see Section “Methods”). The rise time of L(t) is similarly limited by the Schottky-detector response, rather than the timescales of the underlying acoustoelectric processes.

Figure 3 shows the effect of the initial acoustic pulse on L(t) and V(t) (3a and 3b, respectively), recorded with the QCL bias, V b , set either below or above the subband alignment voltage, V A . The acoustic pulse induces an increase in V(t) in all cases, whereas the sign of ΔL depends on the QCL bias: below subband alignment, ΔL > 0, and above subband alignment, ΔL < 0. Furthermore, for measurements performed with V b = V A (not shown) the THz-power modulation was indistinguishable from noise (ΔL ≈ 0). These observations can be reconciled using a quantitative phenomenological analysis based on the measured relationship between the bias voltage and the unperturbed THz power (see Section “Relation between voltage and THz power modulation”). It is important to note that both the voltage and power modulations are unipolar in nature, although distortion by ringing due to resonance in the electrical circuit does cause the modulation in L(t) to appear bipolar at certain biasing conditions. However, as is evident in Fig. 3a, b (in which the shaded areas indicate the times at which the acoustic pulse will be acting on the QCL active region), it can be seen that these ringing effects occur after the acoustic pulse has left the QCL ridge. As such, these effects are attributed to the active region relaxing to its unperturbed state, and not attributed directly to the modulation due to the passage of the acoustic pulse through the active region.

Fig. 3: Measurement and prediction of effect of the acoustic modulation. Temporal responses of the QCL to the first two strain pulses at three different QCL operating biases V b = V 1 to V 3 : a THz power modulation, L(t) (solid lines) and b QCL voltage perturbation, ΔV(t). Predicted power modulation waveforms are shown in (a) (dashed lines), with amplitudes normalised to the experimental data by factors of 75, 3 and 4 for V 1 , V 2 and V 3 , respectively. Highlighted regions indicate the temporal position of the acoustic modulations. Incident strain pulses were generated with an optical pulse energy of 10 µJ. Data have been smoothed using a 20-point rolling average filter. Source data are provided as a Source Data file. Full size image

Figure 4 shows the amplitude of ΔL and ΔV as a function of the optical pulse energy measured outside the cryostat. Since the amplitude of the strain pulse is proportional to the laser fluence incident on the Al-film transducer31, we conclude that both perturbations increase linearly with strain amplitude. No temporal shift relative to the optical excitation pulse was observed in either the ΔV or ΔL responses in any of these measurements, indicating that the strain amplitude was insufficient to cause nonlinear acoustic propagation or shock wave formation. The maximum THz power modulation in Fig. 3a, ΔL = −0.25 mW, was observed for QCL bias V 3 , where the unperturbed THz power was L ≈4 mW, giving a fractional change |ΔL/L| ≈6%. This modulation depth was measured using a 10-µJ optical pulse, although it was found that the optical pulse energy could be increased to at least 24 µJ without damage to the Al transducer layer. As there is no evidence of nonlinear acoustic effects, it is reasonable to assume, via extrapolation, that >15% THz modulation depth could be achievable.

Fig. 4: Amplitude of acoustic modulation measured as a function of optical pulse energy. Amplitude of voltage ΔV (red squares) and power ΔL (black circles) perturbation signals for the first strain-induced pulse, measured at a bias voltage V b = 5.38 V. Error bars represent the measurement errors on the pulse energy and amplitudes. The lines are linear fits to the data. Source data are provided as a Source Data file. Full size image

Theoretical analysis on the origin of voltage perturbations

QCLs are resonant-tunnelling devices, in which quantised electron states localised in adjacent periods are brought into resonance at the alignment bias, V A , as shown in Fig. 1d. Inter-period transport is dominated by resonant tunnelling across an injection barrier, and models of the transport and band structure typically exploit the structural periodicity of the device35. However, the acoustic-strain pulses induce a localised perturbation, which causes electric-field-domain formation36 and breaks this periodicity. The ~20-ps acoustic pulse duration is comparable to the state lifetimes within the QCL, and therefore a time-dependent perturbation (TDP) model37 is needed to calculate the transition probabilities for resonant tunnelling between two periods of the QCL.

The effect of the acoustic pulses on the inter-period tunnelling rate between an injector state (IL) in the first period, and the upper laser level (ULL) in the second period, denoted |2〉 and |11′〉, respectively, was calculated using a TDP approximation (see Section “Methods”). The amplitude of the perturbation caused by the acoustic wave is typically ~0.01–1 meV38, corresponding to strain amplitudes in the range 10−6–10−4 (using the deformation potential of GaAs is ~10 eV per unit strain), which is consistent with piezospectroscopic measurements using metal transducers on GaAs38. Although the amplitude of the perturbation may be comparable with the anti-crossing energy (~0.4 meV), our TDP model gives a simpler and more intuitive analytical approach than other methods such as Landau–Zener theory39,40 or a full time-dependent Schrödinger solution.

Figure 5 shows the calculated time-dependent net tunnelling probabilities between states |2〉 and |11′〉 arising due to propagation of the acoustic wave through the heterostructure, under the assumption of equal initial state populations. Device operation was considered at three bias points: below the alignment bias, V b < V A ; at alignment, V b = V A and above alignment, V b > V A . In all three cases the duration of the simulated perturbation is commensurate with the ~20 ps acoustic pulse duration and causes a greater increase in reverse tunnelling than in forward tunnelling, leading to a net reduction in forward-tunnelling probability. This is consistent with the experimentally observed increase in device voltage V(t) above the laser threshold for a constant driving current. The time-dependent net tunnelling probabilities between the injector and upper lasing states were found to be perturbed by up to ~40% by the presence of the acoustic wave; for comparison, the same calculation between the upper and lower lasing states yielded a ~0.001% effect.

Fig. 5: Net tunnelling probabilities for the perturbed QCL device using a time-dependent model. Time-dependent net tunnelling probabilities between the injection level (|2〉) and the upper laser level (|11'〉) due to transit of the acoustic wave, calculated using a time-dependent perturbation model for QCL biases below subband alignment (V b < V A , V b = 3.53 kV cm−1), at alignment (V b = V A , V b = 3.63 kV cm−1) and above alignment (V b > V A , V b = 3.85 kV cm−1). The timescale of the simulation is set so that the pulse can propagate over two adjacent periods. Source data are provided as a Source Data file. Full size image

It is important to note that the model considers carrier injection into the ULL only and not into parasitic current pathways (i.e., states not involved in photon emission). Experimentally, the net current injection is constant, as the perturbed region has a much lower impedance than the source and the rest of the device. The corresponding voltage change across the perturbed region modulates the state populations and hence photon generation in the device. This implies a qualitative explanation for the observed polarity of the THz-power modulation: below V A , an increase in voltage causes an increase in THz power, whereas the converse applies above V A .

Relation between voltage and THz power modulation

The interaction between acoustic waves and the QCL band structure at the quantum level is complex, with multiple subbands perturbed by the propagating strain wave. A direct quantum analysis of the THz power perturbation is extremely difficult. However, a quantitative phenomenological analysis can be obtained from the measured relationship between the bias voltage, V b , and the unperturbed THz power, L (derived from Fig. 1c), as this encapsulates the underlying quantum phenomena within the single observable parameter, L. A linear perturbation approximation then allows a direct prediction of the change in THz power as the voltage varies dynamically through a small perturbation around the bias point.

Owing to the different spatial localisations of the static voltage (L–V) measurements and the temporal voltage responses V(t), it was necessary to equate them by converting the voltage to the internal electric field for both data sets. For the unperturbed L–V data measurements of Fig. 1c, the voltage is dropped across the QCL ridge of height D = 13.9 µm, assuming negligible contact resistance. This gives an internal field of F 0 = V b /D throughout the device, assuming a single electric-field domain. For the temporal responses in Fig. 3b, the voltage perturbation is assumed to correspond to the average internal field perturbation, localised over the spatial extent of the acoustic wave, d = 96 nm (determined from the pulse duration and the speed of sound in GaAs). The spatially averaged local field perturbation is then ΔF(t) = ΔV(t)/d.

In this way, the measured voltage perturbation across the QCL can be linked directly to a perturbation in THz emission power according to the relationship ΔL(t) = (dL/dF 0 )ΔF(t), where dL/dF 0 is the slope efficiency per period of the active region heterostructure.