Abstract
Nanoscale structures can produce extreme strain that enables unprecedented material properties, such as tailored electronic bandgap1,2,3,4,5, elevated superconducting temperature6,7 and enhanced electrocatalytic activity8,9. While uniform strains are known to elicit limited effects on heat flow10,11,12,13,14,15, the impact of inhomogeneous strains has remained elusive owing to the coexistence of interfaces16,17,18,19,20 and defects21,22,23. Here we address this gap by introducing inhomogeneous strain through bending individual silicon nanoribbons on a custom-fabricated microdevice and measuring its effect on thermal transport while characterizing the strain-dependent vibrational spectra with sub-nanometre resolution. Our results show that a strain gradient of 0.112% per nanometre could lead to a drastic thermal conductivity reduction of 34 ± 5%, in clear contrast to the nearly constant values measured under uniform strains10,12,14,15. We further map the local lattice vibrational spectra using electron energy-loss spectroscopy, which reveals phonon peak shifts of several millielectron-volts along the strain gradient. This unique phonon spectra broadening effect intensifies phonon scattering and substantially impedes thermal transport, as evidenced by first-principles calculations. Our work uncovers a crucial piece of the long-standing puzzle of lattice dynamics under inhomogeneous strain, which is absent under uniform strain and eludes conventional understanding.
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Codes central to the main work described in this manuscript are deposited in Code Ocean (https://codeocean.com/capsule/9335891/tree). Source data are provided with this paper.
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Acknowledgements
We thank the College of Engineering, Peking University, for its support. P.G. and J.D. acknowledge support from National Natural Science Foundation of China (52125307, 12004010) and the National Key R&D Program of China (2019YFA0708200, 2021YFB3501500). P.G. acknowledges support from the New Cornerstone Science Foundation through the XPLORER PRIZE. We thank B. Liao for the helpful discussion regarding the theoretical modelling on the effects of inhomogeneous strain. We thank R. Shi and R. Qi for helping with EELS data processing. We thank X. Sun for helping with schematic drawing. We acknowledge the Electron Microscopy Laboratory of Peking University, China, for the use of Cs corrected Nion U-HERMES200 scanning transmission electron microscopy.
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L.Y. proposed and directed the research. L.Y. fabricated the Si nanoribbons and performed thermal conductivity measurements while working in D.L.’s lab. J.D. performed EELS measurements and analysis, electron diffraction, and GPA analysis under the direction of P.G. S.Q. carried out the preparation of TEM samples and helped with data processing. H.L. and Z.D. conducted strain distribution analyses. S.Y., Y.T. and L.Y. performed theoretical modelling. The manuscript was prepared and revised by L.Y., J.D., S.Y., D.L. and P.G. with input from all co-authors.
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Extended data figures and tables
Extended Data Fig. 1 HRTEM examination showing the single crystalline nature of the bent SiNR.
(a) TEM image of a bent SiNR. (a1-a7) HRTEM images covering the full width of the examined region for the bent SiNR in (a), and the blue dashed rectangles in (a) mark the different locations along the length of the bent SiNR for HRTEM tests.
Extended Data Fig. 2 SAED patterns of a bent SiNR.
(a1-a5) SAED patterns of the bent SiNR, where the inserts show the corresponding portion of the sample in Extended Data Fig. 1 within the selected aperture with diameter of ~180 nm.
Extended Data Fig. 3 Stress-free sample.
(a) SEM image of the stress-free straight Si nanoribbon sample, where the ends are treated with EBID Pt/C composite to minimize the effects of contact thermal resistance. (b) SEM image of the stress-free kinked Si nanoribbon sample. Note that the kinked Si nanoribbon are patterned with 90° kink angle, and thus the straight dashed line delineating on the ribbon edge as well as 90° kink angle formed between the two kink arms suggest this kinked ribbon is in stress-free condition.
Extended Data Fig. 4 Measured EELS map of the transverse acoustic mode under bending.
(a) HAADF image for the bent Si nanoribbon without kink, and the calculated strain contour is overlaid on top to visualize the strain distribution. The yellow rectangle represents the area where the EELS signals are acquired by summing each column spectra of the three-dimension dataset along the axial direction (perpendicular to the strain gradient direction) to enhance the signal-to-noise ratio. (b) The vibrational spectra map for the transverse acoustic (TA) mode along the beam shift direction. (c) Measured EELS profiles for the TA phonon mode along the strain gradient, which shows a peak shift from 25.8 meV to 28.3 meV as the strain changes from -2.78% to 2.65%.
Extended Data Fig. 5 Calculated phonon density of states under elastic strain.
(a) Calculated phonon density of states (PDOS) of Si atoms as the elastic strain changes from 4.77% to −4.77%. (b) Extracted TO peak position for the measured bent SiNR with kink in Fig. 3b–d as a function of elastic strain, where the theoretically calculated peak position for the TO mode is also plotted for comparison. Error bars represent the peak fitting error. Note that the slight discrepancy between the experimental results and theoretical calculation could be caused by local uniform strain approximations in DFT to reproduce the inhomogeneous strain distribution in bent Si nanoribbons, and measurement uncertainties in EELS due to the limited energy resolutions, as well as errors introduced during background subtraction and phonon peak fitting.
Extended Data Fig. 6 Bent Si nanoribbon sample preparation procedures.
(a-b) A Si nanoribbon initially rested on a PDMS substrate and was picked up by a microprobe through van der Waals interaction. (c) The ribbon sample was placed on the suspended microdevice to bridge the gap of the two suspended membranes. (d) The nanoribbon was inverted to present its thickness side upward using the microprobe. (e) One end of the ribbon adhered well with the underlying electrode via van der Waals interaction, and the other end was moved by the microprobe to achieve the desired bending configuration. (f) The two ends of the bent ribbon were fixed with the underlying electrode through focused electron beam induced deposition (EBID) of Pt/C composite.
Extended Data Fig. 7 Contact thermal resistance characterization.
SEM images of the 34 nm thick, 85 nm wide kinked SiNR with suspended length, L, of (a) 3.8 μm, and (b) 10.1 μm, respectively. (c) SEM image of the (34, 85) kinked SiNR with suspended length of 4.7 μm. The electron beam induced deposition (EBID) of Pt/C composite is performed at the two ends. (d) Measured thermal conductivity of the three samples in (a-c) in the temperature range from 50–400 K, where the close thermal conductivity indicates the negligible contribution from the contact thermal resistance.
Extended Data Fig. 8 Phonon frequency broadening as a function of elastic strain difference.
Calculated absolute value of phonon frequency broadening for TA, LA, and TO (LO) modes as a function of strain difference at the selected q-points of (a) (0.5, 0.5, 0.0) and (b) (0.75, 0.5, 0.25), respectively. As the elastic strain is linearly distributed along the thickness direction for a bent Si nanoribbon (Supplementary Fig. 3), and the phonon frequency shift is linearly dependent on elastic strain, this in turn validates our assumption that the broadened phonon spectra are linearly distributed in real space.
Supplementary information
Supplementary Information
This file contains Supplementary Figs 1–12, Supplementary Sections 1–6 and Supplementary References.
Supplementary Video 1
Bent SiNR sample preparation procedures. An illustration video showing the bent Si nanoribbon sample preparation process on the suspended microdevice.
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Yang, L., Yue, S., Tao, Y. et al. Suppressed thermal transport in silicon nanoribbons by inhomogeneous strain. Nature (2024). https://doi.org/10.1038/s41586-024-07390-4
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DOI: https://doi.org/10.1038/s41586-024-07390-4
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