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We report on the near-infrared intersubband (ISB) absorption properties of strain-free Sc_0.14Al_0.86N/GaN multiple quantum wells (MQWs) grown on c-plane GaN substrates by molecular beam epitaxy. These MQWs exhibit strong, sharp, and tunable absorption energies between 515 meV and 709 meV, for well widths ranging from 7 nm to 1.5 nm, respectively. Observation of ISB absorption in ultra-thin Sc_0.14Al_0.86N/GaN MQWs not only extends the near-infrared range accessible with Sc-containing nitrides but also highlights the challenges of growing nanometer-thick GaN quantum wells. We explore the effects of growth temperature on absorption characteristics and find that substrate temperatures above 600°C significantly enhance ISB absorption intensity but also introduce an energy redshift for the narrowest wells. The redshift is attributed to increased interface roughness due to ScAlN surface morphology degradation at higher temperatures. Additionally, a comparison of experimental results with simulated band-structures indicates that the magnitude of net polarization rises faster with Sc-composition than previously suggested by theoretical calculations. This study advances the prospects of ScAlN/GaN heterostructures for novel photonic devices in the technologically important near-infrared range. 

Using comprehensive x-ray reciprocal space mapping, we establish the precise lattice-matching composition for wurtzite ScxAl1−xN layers on (0001) GaN to be x = 0.14 ± 0.01. 100 nm thick ScxAl1−xN films (x = 0.09–0.19) were grown in small composition increments on c-plane GaN templates by plasma-assisted molecular beam epitaxy. The alloy composition was estimated from the fit of the (0002) x-ray peak positions, assuming the c-lattice parameter of ScAlN films coherently strained on GaN increases linearly with Sc-content determined independently by Rutherford backscattering spectrometry [Dzuba et al., J. Appl. Phys. 132, 175701 (2022)]. Reciprocal space maps obtained from high-resolution x-ray diffraction measurements of the (101¯5) reflection reveal that ScxAl1−xN films with x = 0.14 ± 0.01 are coherently strained with the GaN substrate, while the other compositions show evidence of relaxation. The in-plane lattice-matching with GaN is further confirmed for a 300 nm thick Sc0.14Al0.86N layer. The full-width-at-half-maximum of the (0002) reflection rocking curve for this Sc0.14Al0.86N film is 106 arc sec and corresponds to the lowest value reported in the literature for wurtzite ScAlN films. 

Wurtzite ScxAl1−xN/GaN (x = 0.13–0.18) multi-quantum wells grown by molecular beam epitaxy on c-plane GaN are found to exhibit remarkably strong and narrow near-infrared intersubband absorption in the technologically important 1.8–2.4 μm range. Band structure simulations reveal that, for GaN wells wider than 3 nm, the quantized energies are set by the steep triangular profile of the conduction band caused by intrinsic polarization fields. As a result, the intersubband transition energies provide unique and direct access to essential ScAlN polarization parameters. Measured infrared absorption indicates that the spontaneous polarization difference of the presumed lattice-matched Sc0.18Al0.82N/GaN heterostructure is smaller than the theoretically calculated value. The intersubband transition energies are relatively insensitive to the barrier alloy composition indicating negligible variation of the net polarization field in the probed 0.13–0.18 Sc composition range. 

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also show a novel protocol. 1. We prove that secret-key cryptography exists if O˜(n‾√)-approximate SVP is hard for 2εn-time algorithms. I.e., we extend to our setting (Micciancio and Regev's improved version of) Ajtai's celebrated polynomial-time worst-case to average-case reduction from O˜(n)-approximate SVP to SIS. 2. We prove that public-key cryptography exists if O˜(n)-approximate SVP is hard for 2εn-time algorithms. This extends to our setting Regev's celebrated polynomial-time worst-case to average-case reduction from O˜(n1.5)-approximate SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical, generalizing Peikert's polynomial-time classical reduction from O˜(n2)-approximate SVP. 3. We show a 2εn-time coAM protocol for O(1)-approximate CVP, generalizing the celebrated polynomial-time protocol for O(n/logn‾‾‾‾‾‾‾√)-CVP due to Goldreich and Goldwasser. These results show complexity-theoretic barriers to extending the recent line of fine-grained hardness results for CVP and SVP to larger approximation factors. (This result also extends to arbitrary norms.) 4. We show a 2εn-time co-non-deterministic protocol for O(logn‾‾‾‾‾√)-approximate SVP, generalizing the (also celebrated!) polynomial-time protocol for O(n‾√)-CVP due to Aharonov and Regev. 5. We give a novel coMA protocol for O(1)-approximate CVP with a 2εn-time verifier. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs.