Weutilize microfabricated superconducting tunnel junction STJ-based phonontransducers for the emission and detection of tunable, non-thermal and spectrallyresolved acoustic phonons, with
Spectrometer design
Device design for each spectrometer integrates two STJ phonon transducers—an emitter and a detector—mounted on opposite sides of a monolithically etched mesa on a silicon substrate (see figure 1(a)) This configuration enables direct generation and detection of phonons within a compact, mechanically stable platform The mesas are approximately 0.8 micrometers high and have widths spanning a design-dependent range, chosen to optimize phonon coupling, impedance matching, and overall device performance.
7 to 15àm, allow for the isolation of a ballistic path for phonon propagation The devices are fabricated on a 525àm thick silicon (100) wafer and the mesa sidewalls are on the Si(111)
Figure 1 (a) False-colored SEM micrograph of completed phonon spectrometer. The STJ emitter is fabricated with the tunnel junction lying mostly on the sidewall of the 0.8àm high mesa structure The width of the mesa, R=7,
The mesa structure isolates a ballistic phonon transport path between the emitter and detector, while the detector itself is a double-junction SQUID with a hot-electron finger that collects ballistically propagating phonons Finger widths Wf of 1.5, 2, 3, and 6 µm were explored to observe geometry-dependent transmission, and a magnetic field of about 1 G is applied perpendicular to the SQUID detector to suppress the Josephson current To minimize phonon backscattering from the substrate, a 0.5 µm-thick silver film is deposited on the backside of the 525 µm-thick silicon substrate The inset shows a side-view schematic of a silicon mesa with phonon transducers, and an optical microscope image reveals a 4.5 mm^2 device comprising six spectrometers Since the mean free path of phonons at the experimental temperature and frequencies is about 1 mm, the detected signal includes backscattered phonons from the bottom of the substrate, and the ballistic path along the [110] direction can be blocked by etching a trench into the mesa to quantify this contribution This measurement platform also enables monolithic integration of nanostructures into the mesa, and microfabrication makes the experiments scalable—spectrometers are fabricated in lots of 100 on 100 mm Si wafers; each 4.5 mm square chip contains up to six spectrometers, as shown in figure 1(b).
The phonon emitter is a single Al–AlOx–Al tunnel junction with most of the junction area lying on the sidewall of the mesa The aluminum films are kept thin (21e break Cooper pairs, creating fresh quasiparticles The probability that a phonon will survive traveling a distance r within the aluminum is e − r /3 ph , the mean absorption length of 3ph(ω) being dependent on phonon energy¯hωand band gap energy1e[31] If we treat the phonons as point-particles traveling ballistically within the Al, then the probability of a phonon generated at a distancez from the Al/Si interface and traveling at an angleθ to the normal, to escape into the Si before reabsorption is [31,36] e − z /(3 ph cos θ) T AlSi (θ) (2)
Here T AlSi (θ) is an acoustic-mismatch transmission factor for wave transmission from Al into
Si The films of some of our emitter STJs have lower and upper layer thicknesses of ∼20 and
∼79 nm respectively on the mesa sidewall (as determined by profilometry measurement and adjusted for sidewall angle) For simplicity, we treat all phonons as being generated within the lower layer at a spatially uniform rate We assume the phonons’ velocities are distributed uniformly in all directions, and that those entering the top layer may reflect from the Al/vacuum boundary, reenter the lower layer, and reach the Al/Si boundary For phonons to emerge and travel directly across the mesa toward the detector (an angle∼35.3 ◦ to the sidewall normal), we estimate the refraction angle within the Al using Snell’s law, assuming average wave speeds vAl=4.4×10 3 m s −1 in Al and vSi=6.6×10 3 m s −1 in Si, to be θ ∼22.7 ◦ From reported values of the acoustic impedances of Al and Si, we estimate T AlSi to be >0.9 for such an angle and to be frequency independent [6, 31] Kaplan et al [27] have calculated values for phonon decay time in Al as a function of phonon energy h¯ω and bandgap energy 1e.
We multiply these by vAl to find 3ph(100 GHz)∼=1.04àm, 3ph(400 GHz)∼=0.38àm and
These values exceed some reported experimental values for three-phonon processes in aluminum at the energy ħω, but they are comparable with measured normal-state acoustic attenuation after correction to the superconducting state By averaging equation (2) over the full aluminum layer thickness, we estimate that in the direction toward the detector, approximately 90% of phonons at the relevant frequencies escape into silicon, about 78% at a lower frequency, and about 68% at a higher frequency We apply these attenuation factors to modify the spectrum in equation (1), as shown in Figure 2c.
To determine the total rate of absorbed phonons, we average equation (2) over all depths and incidence angles At large values of θ the transmission coefficient T_AlSi(θ) tends to 1, independent of phonon frequency, while for angles above about 45°, T_AlSi(θ) becomes zero due to total internal reflection within the material.
Kaplan calculated the transmission coefficients for the Al–Si interface by averaging over all incidence angles and phonon polarizations, and from these results the AlSi transmission is estimated to be T_AlSi ≈ 0.44, assuming the three phonon polarizations are equally populated Consequently, for frequencies ω above the threshold ω > 21 e/ħ, the transmission remains finite.
Approximately 56% of phonons produced are reabsorbed within the aluminum (Al) To capture the frequency dependence, we multiply this baseline by the average of equation (2) taken over the full Al layer thickness and all incident angles below the critical angle As a result, the overall reabsorption fraction across all phonon directions is estimated to be about 61% at ω/2π ≈ 0 GHz, about 67% at the second frequency setting, and about 71% at the third frequency setting For each bias voltage Ve, these reabsorption fractions are applied to the spectrum in equation (1), and the total reabsorbed power is found by integrating.
Energy conservation requires that all reabsorbed power be reemitted The quasiparticles created during reabsorption relax and recombine, emitting additional phonons with frequencies lower than those originally absorbed Based on typical decay times and the geometry of our aluminum-film superconducting tunnel junction (STJ) on the mesa sidewall, the quasiparticles travel only short distances before reemission, so about 80% of the power is reemitted at the same or nearby location as the original tunneling injection.
By summing the first- and second-step relaxation—together accounting for up to about 25% of the total relaxation phonon power, including attenuation and reabsorbed/reemitted power—we find that for typical V_e values up to a few mV, the total modulated power P_tot emitted from the emitter STJ is roughly proportional to the modulated emitter current δI_e The power emitted due to recombination, however, remains fixed as V_e varies, and for large V_e it constitutes a negligibly small fraction of the total power Consequently, the total emitted differential phonon rate is approximately δI_e/e.
To obtain P_peak/P_tot at a given ω_peak, we take P(ω_peak)Δω from equation (1), attenuate this quantity for a chosen peak width Δω using equation (2), and divide by the total power P_tot calculated at V_e = (ħω_peak + 21e)/e The resulting values for typical emitter film thicknesses appear in figure 2(d) For a peak width δω/2π in GHz and a peak frequency ω/2π = 100 GHz, P_peak/P_tot is about 50%; this declines to about 32% at ω/2π = 0 GHz and continues to fall at higher peak frequencies As shown in figure 2(d), the P_peak/P_tot values from the STJ-emitted phonon spectrum compare very favorably to a Planck distribution, exceeding it by more than an order of magnitude for ω/2π > 300 GHz This analysis demonstrates that aluminum STJs made of films a few tens of nm thick will emit narrow spectral distributions of acoustic phonons into Si at frequencies up to several hundred GHz.
Phonon emission from aluminum STJs has been reported elsewhere at frequencies up to
Phonon detection with superconducting tunnel junction
Phonons incident on the detector are registered as an increase in the tunnel current through the detector junctions, yielding a measurable signal that tracks phonon absorption The STJ detector is biased with a voltage below the superconducting gap, optimizing sensitivity to phonon-induced tunneling while maintaining junction stability.
Phonons incident on the detector finger with energy equal to or above the superconducting gap (2Δ) break Cooper pairs in the aluminum detector films, and the resulting quasiparticles diffuse until some reach the detector junction and tunnel through The STJ detectors are made from aluminum films with a superconducting gap Δ such that 2Δ ≈ 360 μeV (corresponding to about 90 GHz), and these detectors essentially act as high-pass filters for acoustic phonons with a cut-off frequency around that value.
Using a lock-in detector, we isolate the modulated portion δI_d of the detector current, which corresponds to the emitter phonons that actively strike the detector Consequently, the phonon spectrum extends from roughly 90 GHz up to (eV e −21e)/h, with a sharp peak at the same upper frequency Since the modulated emitter phonon power scales with δI_e, the measured differential transfer function δI_d/δI_e directly indicates the fraction of the phonon spectrum transmitted from the emitter through the sample to the detector.
Modeling the detector behavior
Quasiparticle–phonon interactions provide a robust framework for modeling and quantifying phonon detector behavior For the differential rate ˙n_ph, d of phonons of frequency ω striking the detector finger, the average differential rate of phonon-induced quasiparticle generation ˙n_QP,ph is determined by a coupling relation that links the incoming phonon flux to quasiparticle production This relation captures how the detector’s response scales with phonon input across frequencies and underpins quantitative predictions of sensitivity and timing By analyzing the ω-dependent transfer from phonons to quasiparticles, we can optimize detector design and interpretation of the detected phonon signal.
In equations (3), T SiAl is the acoustic transmission factor for phonons transiting from Si into
Al, which we estimate from acoustic impedances to be>0.9 over all incidence angles [33] The fraction of phononsαabs(ω)absorbed in the finger will be approximatelyαabs(ω)=1−e 2d /3 ph (ω)
For detector fingers with thickness d in the phonon-incidence direction of 140–205 nm, the absorption coefficient αabs(ω) is expected to be at least 0.2 at ω/2π ≈ 0 GHz and to reach at least 0.8 near ω/2π ≈ 700 GHz The observed decrease of Ppeak/Ptot as the peak frequency increases (Fig 2d) motivates treating αabs as independent of peak frequency and taking αabs ≈ 0.25 In a typical spectrometer signal transmitted through bulk silicon, the resulting modulated response is consistent with this αabs value and with the detector response described by Equations (3).
To extract the quasiparticle population n_QP,ph and the phonon arrival rate ṅ_ph,d from the measured differential detector tunnel current δI_d, one must account for quasiparticle loss processes inside the detector The leading loss channel is diffusion of quasiparticles into the attached wiring leads, followed by recombination into Cooper pairs Using conventional tunneling-rate and quasiparticle-recombination theory, the nondimensional efficiency factor Eff can be defined as Eff = δI_d / (e ṅ_QP,ph) for each detector (see Appendix B) [31,40,42].
Equation (4) is 2e^2 R_n N_0 W_tr d_tr, where R_n denotes the normal-state tunneling resistance of the junction, N_0 is the normal density of states at the Fermi level (approximately 1.75 × 10^10 eV^−1 cm^−3 in aluminum) [31], and W_tr and d_tr are, respectively, the average total width and thickness of the wiring trace connected to the detector STJ.
D∼ cm 2 s −1 is the diffusion constant for quasiparticles in Al, andτrec∼30às is the average quasiparticle recombination time in Al at a temperature of 0.3 K [30, 31, 41, 43, 44] In our detectors{Eff}is typically∼0.1.
Fabrication techniques and challenges 10 4 Instrumentation, measurement technique and characterization of spectrometer 13 4.1 Low temperature apparatus
Dc characterization of superconducting tunnel junctions emitters and detectors 14 4.3 Josephson current suppression
DC characteristics of emitter and detector tunnel junctions were determined from current-biased I–V measurements at approximately 0.3 K The superconducting gap is estimated from the I–V behavior (Fig 6a), and the normal-state resistance Rn of the junctions is calculated from the same I–V curves Figure 6b displays the current-biased I–V curves in the subgap regime for four SQUID detectors with the current normalized by their normal-state resistances to enable direct comparison, with the red plot showing pronounced rounding due to poor filtering on that signal line that allows stray voltage noise to perturb the junction voltage Figure 6c presents the resistance-normalized I–V curves for four emitters with normal-state resistance values of 212, 935, 2250 and 5559 Ω.
Figure 6 presents two key measurements on SQUID-based detectors: in panel (a) a typical current-biased I–V trace of an emitter is shown, with the junction 21e’s band gap estimated at roughly 400 eV; in panel (b) the I–V curves of four detectors are shown without suppression of the Josephson current, with a focus on the subgap regime All devices are superconducting quantum interference devices (SQUIDs), and their detector resistances range from 167 Ω (black trace) upward.
213 (magenta), 849 (red), to 817 (blue) Poorly filtered lines lead to rounding-off of gap rise as shown in the red plot (c) I×R n –V curves of four
Figure 6 continues the analysis of emitters in the subgap region, showing emitter resistances of 935 (black), 2250 (magenta), 212 (red), and 5556 (blue) The plot demonstrates how to identify common emitter problems, with the magenta curve indicating a partly shorted device, and the red curve revealing severe anomalies.
Back bending of the gap rise, driven by overinjection and local suppression of the superconducting gap, is commonly observed at low emitter resistance The gap-rise width, illustrated by the black and blue plots, signals inhomogeneity in the superconducting gap that limits the energy resolution for phonon spectroscopy.
This plot highlights several potential problems in emitter performance In the 212-emitter (red plot), we observe a back bending of the gap-rise step at V_e ≈ Δ/e, a signature of quasiparticle overinjection This behavior appears consistently in emitter STJs with R_n < 700 Ω and leads to local suppression of the superconducting gap and degraded phonon energy resolution.
In the 2250 junction (magenta plot), the I–V curve shows a signature of a partially shorted device, likely formed during fabrication or processing, which adds an uncontrolled thermal phonon population to the junction’s emission The black and blue curves indicate a limitation on emitter energy resolution For an ideal STJ, the gap-rise step at V ≈ Δ/e would be infinitely sharp, but in practice we observe a broadening of about 60–80 μV (≈ 15–20 GHz) This broadening most likely indicates that the superconducting gap Δ varies within the junction.
∼60–80àeV (corresponding to a∼15–20 GHz imprecision in emitted phonon frequency).
To suppress the Josephson current in the detector so it can be voltage-biased and its quasiparticle tunneling current clearly distinguished, we apply a magnetic field perpendicular to the SQUID loop using a small superconducting coil mounted as close as possible to the top of the chip to minimize vibration-coupled flux noise For the coil geometry shown in Figure 5(c), the axial magnetic field is calculated via the Biot–Savart law to be 1.27 G mA−1 The heat load from typical coil current is ≤2 μW The maximum supercurrent in the SQUID detector junction, assuming perfect symmetry, follows Ic(Φ) = 2 Ic(0) cos(π Φ/Φ0).
Φ0 = 2.07×10^-15 Wb is the flux quantum, Φ denotes the applied magnetic flux, and Ic(0) is the critical current at zero magnetic field When a magnetic flux equal to an odd integer multiple of Φ0/2 is applied (Φ = nΦ0/2 with n odd), the supercurrent is fully suppressed.
To minimize flux trapping, we typically operate at the minimum effective flux (equivalent to 1) In practice, the supercurrent is not always fully suppressed, probably due to asymmetry between the two junctions Figure 7(a) shows our method for determining the detector bias point for phonon transport studies: the detector voltage is swept in the subgap regime from roughly −300 μV to +300 μV, with the coil current swept at each voltage step.
The tunnel current is measured in steps from 0 to 2 mA, and Figure 7(a) presents a three-dimensional plot of the current as a function of detector bias voltage and coil current We set the detector bias point to approximately 1 d/e (about 180 mV) and the coil current to roughly 1 mA, where the minimum critical current is obtained For the detector (R_n ≈ 6), the measured zero-voltage and zero-field supercurrent is about 1.2 µA along the z-axis, closely matching the Ambegaokar–Baratoff expression for T ≈ 0 K, Ic0 = 2e R π1 n [51] By applying a magnetic field (~1 G) at this bias point, the supercurrent is suppressed to about 1 nA.
Figure 7 shows detector Josephson current suppression: the subgap tunnel current, measured before Josephson current suppression, is recorded for coil currents from 0 to 2 mA as the bias voltage sweeps from −300 to +300 μV In spectrometer operation, the detector is typically biased in the subgap region (about 180 μV) and at an external magnetic field of roughly 1 G (corresponding to about 1.27 G mA−1), a condition that largely suppresses the critical current The data also highlight the periodic nature of the critical current with the applied magnetic field Panel (b) shows a plot of βL.
Figure 8 shows the ratio of the minimum suppressed critical current to the calculated critical current (I_c(min)/I_c0) at 10 mK for several SQUID designs Junctions formed on a flat surface are represented by solid symbols, while junctions formed on the sidewall are represented by open symbols Loop areas range from ≤ 2 μm^2 (represented by squares) to larger values, illustrating how geometry and junction placement affect the suppression ratio.
Three device areas were studied: approximately 10 nm^2 (circles), 120 nm^2 (triangles), and 180 nm^2 (diamonds) In panel (c), the temperature dependence of the subgap tunnel current after Josephson current suppression was measured at a drain voltage of 180 mV The results are shown as a red plot versus temperature, while the blue plot represents the tunnel current predicted by BCS theory.
The extent to which the supercurrent in the SQUID detectors may be suppressed is dependent on two geometric properties: self-induced flux and junction symmetry The self- induced flux is proportional to the self-inductance, L, of the SQUID loop, which we estimate based on the inductance of a rectangular loop [52] The more closely identical the two junctions are, the more closely the current flowing through them may be made to cancel In figure7(b), we plot the ratio of the minimum obtainable critical current to the maximum zero voltage critical current (I c (min)/I c0 )versus the parameter β L = 2L I 8 0 c , the ratio of the self-induced flux to the flux quantum Each symbol in figure 7(b) represents a unique SQUID design based on the location of the junction and the loop area: junctions formed on a flat surface are represented by solid symbols, while the open symbols represent junctions formed on the sidewall; loop areas vary from ⩽2àm 2 (squares), to ∼10àm 2 (circles), to ∼120àm 2 (triangles) and to
Smaller loop areas and larger junction resistances reduce βL and generally improve supercurrent suppression in these diamond-based SQUID detectors However, detectors formed on the sidewall show large suppression variation for devices with similar βL, a behavior likely driven by junction asymmetry In contrast, devices fabricated on the flat (100) surface exhibit more consistent supercurrent suppression, achieving more than three orders of magnitude reduction when βL is below a certain threshold.
2×10 −3 , indicating more symmetric junction formation We also note a tradeoff in detector design: while Josephson critical current scales inversely with normal-state tunnel resistance
I c0 = 2e R π1 n , detector efficiency (equation (4)) also scales inversely with R n In practice we find that a loop area of∼2àm 2 and detector resistance R n ∼200–300enable both suppression of
I c to levels smaller than thermal quasiparticle tunneling current, as well as detector efficiencies of∼0.1 that permit readily measurable spectrometer signals.
Modulated phonon transport measurements
Figure 8(a) shows the schematic of our phonon transport experiments For phonon emission (V_e ≥ 21 e/e), the emitter is current-biased by applying a dc bias, V_b = V_R e n R_b, through a bias resistor R_n ≈ 500 kΩ, where V_e = I_e R_n is the voltage across the emitter junction and R_n is the emitter’s normal-state resistance All device wiring uses filtered twisted-pair lines, and shielded coaxial cables are used for all connections The dc current through the emitter junction is stepped from I_e ≈ 0.35–2 µA, corresponding to emitter voltages in that range.
V_e is in the range of about 0.35 to 2 mV for a junction with a normal-state resistance R_n of 1 kΩ In addition to the DC bias current applied to the junction, an AC modulation current δI_e of approximately 20 nA rms is introduced by superimposing an AC modulation δV_b onto the DC bias level V_b through a unity-gain isolation amplifier, such as the Burr-Brown ISO124P.
100× voltage divider; the output is independent of frequency between 4 and 1000 Hz and exhibits noise of ∼10 − 6 V Hz −1 / 2 The typical modulation frequencies for our measurements range between 7 and 11 Hz.
Phonon detection is achieved by biasing the detector in the subgap regime (V_d ∼ 1d/e), which suppresses the Josephson current The detector signal consists of a steady-state component and a modulated component, as shown in figure 8(a) The steady-state dc detector current I_d is about 1–2.5 nA for emitter voltages V_e in the range 0.35–5 mV, as illustrated in figure 8(b) For dc detector tunnel currents I_d up to 1.5 times the unperturbed (thermal) level of the steady-state detector current, we treat τ_rec
Figure 8 encapsulates the core of the phonon spectroscopy measurements: a schematic of the setup, the steady-state detector current Id, and the differential transfer function δId/δIe, which represents the fraction of emitted phonon flux that reaches the detector The emitter tunnel junction becomes active above a certain bias Ve and emits detectable relaxation phonons only beyond that threshold, with a higher bias enabling emitted phonons to break multiple Cooper pairs in the detector A peak near 4 mV is attributed to resonant backscattering from oxygen impurities in silicon, a feature typically observed around 870 GHz Panel (d) shows the voltage-biased detector I–V curves under varying conditions.
Figure 8 continues with emitter voltages and partially suppressed Josephson current, and the differential conductance is calculated from the I–V measurements in (d) with colors matching those in (d) The detector is modeled as a current source in the equivalent circuit, where the dc current Id and the modulated current δId follow the incident flux of phonons We treated Id as effectively constant with {Eff} fixed, and validated this by raising the device temperature until Id increased by a factor of three, observing only a small change in the differential transfer function δId/δIe Consequently, for Id < 1.5 times its thermal level, the detector response remains linear with the incident phonon flux, while for Id > 1.5 times the thermal level nonlinearities may arise In these devices, the recovery time τrec may be limited by magnetic flux trapped in the Al detector film as well as by the quasiparticle population.
The modulated AC detector current, i.e., the differential response or differential transfer function, of our detector (Figure 8(c)) represents the modulated portion of the incident phonons and is isolated with a low-noise current pre-amplifier (DL 1211) and a lock-in amplifier (SRS 830) over a range from 0 to approximately 1 pA rms As shown in Figure 8(c), the emitter tunnel junction turns on once the emitter voltage exceeds Ve, and a step in the detector response appears at that voltage This step arises because the emitted relaxation phonons with energy eVe become energetic enough to break Cooper pairs in the detector, i.e., to overcome the detector’s superconducting gap Δd.
At around 90 GHz, when V_e = (21e+41d)/e, a further change in the detected signal level occurs as the emitted relaxation phonons gain enough energy to break multiple Cooper pairs in the detector (consistent with equation (3)) We also evaluate the impact of microwave Josephson radiation on the detector signal In one spectrometer, biasing the emitter at V_e = 0 V and modulating the Josephson branch of the emitter I–V curve results in a zero detector response.
We conclude that our measurement is not influenced by Josephson radiation or inductive coupling of the emitter Josephson current into the detector.
The peak frequency of the emitted relaxation phonon distribution is related to the emitter bias voltage via the relation (eV e −21e)/h A feature near Ve ≈ 4 mV in figure 8(c) is attributed to backscattering by oxygen impurities in silicon This peak was observed at approximately 870 GHz in prior studies of STJ phonon spectroscopy Taken together, these observations confirm that the aluminum STJ-based spectrometer emits a strong, tunable signal well above 800 GHz However, at such high frequencies (figure 2(d)), only about 20% of the total phonon power resides at the peak frequency (eV e −21e)/h.
In figure 8(d), voltage-biased I–V curves of the detector are shown as the emitter voltage Ve is swept from 0 to about 5 mV, noting that Josephson current could not be suppressed below roughly 5 nA At Ve = 0, the subgap current at Vd ≈ Δ/e (about 180 μV) matches the curve shown in figure 7(c) at a temperature of roughly 313 mK As the phonon flux reaching the detector increases, the total quasiparticle density in the detector rises well above the thermal level, driving the detector current higher In figure 8(e), the differential conductance dI/dV is calculated from the subgap regime.
From the I–V measurements in figure8(d), the detector’s conductance remains essentially constant as the emitter voltage is varied At the typical bias point Vd = 1 d/e, the conductance G stays fixed at approximately 5×10^−6 S, and the only difference is in the total current level.
In the simplified STJ phonon detector model (Figure 8f), the detector is represented as a current source in parallel with a resistance 1/G, with the dc current Id and the modulated current δId following the incident phonon flux The detector sits in series with the current amplifier (input impedance RAMP) and the line resistance RLINE, and its bias point is set by an isolated voltage source (Stanford Research SIM928) across the entire network, fed through a 10^5 voltage divider Typical values are RLINE ≈ 70 Ω and RAMP ≈ 2 kΩ (RAMP per the manufacturer’s specification) This model, together with the measurements in Figures 8d and 8e, shows that the STJ maintains a steady bias throughout the measurement range—even if Id rises by 1 nA, the bias across the STJ changes by only a few μV—and the current through the amplifier accurately reflects the modulated current δId The modulated amplifier current δIAMP equals δId /(1 + G(RLINE + RAMP)), which differs from δId by only about 1% for typical values of RLINE, RAMP, and G.
Results of phonon spectroscopy measurements 21 1 Energy resolution and sensitivity
Ballistic phonon propagation
Evidence for the ballistic nature of phonon transport is shown by comparing the differential detector response δI_d/δI_e across spectrometers with varying mesa widths, detector finger widths, blocked ballistic paths, and offset line-of-sight between emitters and detectors (Figs 10a–d) To enable cross-detector comparisons, each measured δI_d/δI_e value is divided by Eff for that detector to produce the phonon transmission signal Per equations (3) and (4), the resulting scaled value is expected to equal T_SiAl αabs.
2e δ n ˙ ph,d δ I e for 21d⩽¯hω