Abstract
Crystal defects affect the thermal and heat-transport properties of materials by scattering phonons and modifying phonon spectra1,2,3,4,5,6,7,8. To appreciate how imperfections in solids influence thermal conductivity and diffusivity, it is thus essential to understand phonon–defect interactions. Sophisticated theories are available to explore such interactions, but experimental validation is limited because most phonon-detecting spectroscopic methods do not reach the high spatial resolution needed to resolve local vibrational spectra near individual defects. Here we demonstrate that space- and angle-resolved vibrational spectroscopy in a transmission electron microscope makes it possible to map the vibrational spectra of individual crystal defects. We detect a red shift of several millielectronvolts in the energy of acoustic vibration modes near a single stacking fault in cubic silicon carbide, together with substantial changes in their intensity, and find that these changes are confined to within a few nanometres of the stacking fault. These observations illustrate that the capabilities of a state-of-the-art transmission electron microscope open the door to the direct mapping of phonon propagation around defects, which is expected to provide useful guidance for engineering the thermal properties of materials.
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The datasets generated during and/or analysed during the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
The experimental work was supported by the Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under grant DESC0014430. DFT studies were supported by the US Department of Energy (grant no. DE-FG02-05ER46237) and the National Energy Research Scientific Computing Center (NERSC). This research was partially supported by the NSF through the University of California-Irvine Materials Research Science and Engineering Center (grant no. DMR-2011967). We thank Z. Cheng and S. Graham (Georgia Institute of Technology) and Q. Ramasse (SuperSTEM Laboratory) for their constructive suggestions. TEM experiments were conducted using the facilities in the Irvine Materials Research Institute (IMRI) at the University of California, Irvine.
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X.Y., R.W. and X.P. conceived this project and designed the studies. X.Y. performed STEM-EELS experiments. X.Y. and C.A.G. analysed EELS data. C.L., L.G. and R.W. designed and performed the theoretical simulations. T.A., T.C.L., N.D. and O.L.K. set up the angular resolved condition with high-energy-resolution EELS. The SiC films were provided by D.G.S. All authors discussed and commented on the results. The manuscript was prepared by X.Y., C.L., C.A.G., O.L.K., R.W. and X.P., with contributions from all other co-authors.
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T.C.L., N.D. and O.L.K. declare a financial interest in Nion Company.
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Extended data figures and tables
Extended Data Fig. 1 STEM observations of 3C-phase SiC showing stacking faults.
The SiC was grown on a Si(001) wafer. a, b, High-angle (a) and medium-angle (b) ADF-STEM of SiC along [01\(\bar{1}\)]. The medium-angle condition is more sensitive to the defect structure, owing to the contribution of diffraction contrast, and hence the stacking faults and other defects have a stronger contrast in b. The SiC film was epitaxially grown on an (001) plane of the Si wafer. The stacking faults form along the <111> direction with angles of 54.7° with respect to the interface between SiC and Si. c, Atomic-resolution high-angle ADF-STEM image of the SiC–Si interface along [01\(\bar{1}\)]. Owing to the large lattice mismatch of 24.5% between film and substrate (aSiC = 0.543 nm and aSi = 0.436 nm), many structural defects, including multiple stacking faults (arrowed), are generated from the interface. d, Low-magnification medium-angle ADF-STEM image with marked electron beam locations for acquiring the aloof, SiC and SiC-SF vibrational spectra shown in main-text Fig. 2a. e, HAADF STEM image of one stacking fault with a large field of view acquired with a convergence semi-angle of 33 mrad and an ADF detector collection semi-angle of 70–210 mrad. f, Vertically integrated intensity profile along the blue box in e. The grey dots are average intensity and used in the line profile of the ADF intensity in main-text Fig. 3b. The average intensities of the atomic columns are uniform except the region near the stacking fault.
Extended Data Fig. 2 Relationship of the diffraction pattern and the FBZ in defect-free SiC and in a region containing a stacking fault.
a, b, Convergent beam diffraction patterns of defect-free SiC (a) and a stacking-fault region (b) taken with a 3 mrad convergence semi-angle. The additional diffuse scattering signal marked by the blue arrow in b is correlated with the stacking-fault structure and local symmetry breaking. c, d, Reciprocal space diagrams, including the diffraction pattern and the location of the FBZ (blue contours), corresponding to a and b, respectively. For the stacking fault, a 2H phase is used to simulate the phonon dispersion relations in d. Although the X point in 3C phase is outside the FBZ of 2H phase, an equivalent path to Γ–X is indicated by the red lines along Γ–P–X′, in which X′ is at the edge of the FBZ of the 2H-SiC structure and has coordinates of (1/2, 0, 2/3) in fractions of reciprocal lattice vectors (g1, g2, g3) of 2H phase. g1 and g3 are in-plane reciprocal lattice vectors and indicated as two arrows, while g2 is out-of-plane and not indicated here.
Extended Data Fig. 3 Interaction between the energy–momentum relation for fast electrons and the phonon dispersion curves of SiC.
a, The energy–momentum (E–q) relation for fast electrons (red line) along with the calculated phonon dispersion curves of 3C-phase SiC (black lines), which are duplicated from main-text Fig. 2b. The E–q relation for fast electrons follows E = ħvq in the long-wavelength range, where ħ is the reduced Planck constant and v = 1.34 × 108 m s−1 is the velocity of fast electrons for an accelerating voltage of 60 kV. Because the red line is near-vertical, we show in b and c enlarged sections of a for clarity. b, Enlarged plot to show intersections between the E–q relation for fast electrons and the optical branches. [00q] denotes the direction towards the X point and parallel to [001] with a momentum of q. c, Enlarged plot of the corner of the Γ point with very small energy and momentum. To further clarify the E–q relation for fast electrons at this small momentum, the energy is divided by 1,000 as indicated by the red dashed line. The polariton mode occurs when the E–q relation of the fast electrons intersects the phonon dispersion curves53,54. Owing to the high speed of fast electrons, their E–q relation is very close to the vertical line at the Γ point, and intersects with LO and TO branches (shown in b), but does not have any non-zero intersections with either LA or TA branches (as shown in c) because of their slow velocities (9.5 × 103 m s−1 and 4.1 × 103 m s−1 for LA and TA in 3C-phase SiC, respectively).
Extended Data Fig. 4 Vibrational spectra of defect-free SiC and of a region containing a stacking fault obtained using a convergence semi-angle (α) of 10 mrad.
a, Diffraction pattern of defect-free SiC along the [01\(\bar{1}\)] direction with α = 10 mrad. The blue contour is the FBZ. 200, 111 and 1\(\bar{1}\bar{1}\) diffraction spots are marked by orange circles with a radius of 10 mrad. The small red circle indicates the location of the EELS entrance aperture with a collection semi-angle of 2.5 mrad, which is at the X point to avoid the polariton mode. b, Vibrational spectra of defect-free SiC (black curve) and a stacking-fault-containing region (SiC-SF, red curve) with α = 10 mrad. Background-subtracted vibrational spectra in b are almost identical to the data obtained using 33 mrad in Fig. 2a. The TA signal (at 30–50 meV) in the spectrum of stacking-fault region is slightly stronger than that in the spectrum of the defect-free region, which is also consistent with the observed discrepancy using 33 mrad in Fig. 2a.
Extended Data Fig. 5 Spatially resolved simulation of projected PDOS across a stacking fault.
a, Eleven-layer atomic structure for the simulation, comprising 3C-phase SiC with a two-layer stacking fault (purple) in the middle. The number of each layer is given on the left-hand side. b, Projected PDOS plots corresponding to individual layers in the atomic structure. The two simulated spectra in main-text Fig. 2c are from the stacking fault (purple curve) and from the defect-free region (green curve). These results even reproduced the 4-meV red shift of the TA mode centre in the stacking fault. c, Reciprocal space diagram illustrating the location of the FBZ of the 11-layer defect structure in a (orange contour) along with FBZs of the 3C-phase structure (light blue contour) and the 2H-phase structure (blue contour). The X′′ point is equivalent to the X point of the 3C structure and positioned at the edge of the new FBZ with coordinates (1/2, 0, 1/6) in fractions of reciprocal lattice vectors (g1, g2, g3′) of the 11-layer defect structure. g1 and g3′ are in-plane reciprocal lattice vectors as indicated by two arrows, while g2 is out-of-plane and not indicated here. The X′′ point was further used to calculate the angle-resolved vibrational spectra. d, Integrated signal of the TA mode in the projected PDOS spectra in the energy range 30–40 meV.
Extended Data Fig. 6 Angle-resolved vibrational spectra along X→Γ′ and phonon dispersions of all phonon branches in the defect-free SiC as well as aloof spectra of SiC film using a convergence semi-angle of 3 mrad.
a, Diffraction pattern of SiC with annotations including the FBZ (solid blue contour) and selected momentum positions of the EELS entrance aperture (red circles) along X→Γ′, where Γ′ is the centre of the adjacent BZ (dashed blue contour). b, Angle-resolved vibrational spectra along X→Γ′. The TA (red), LA (blue), TO (green) and LO (orange) modes are separated after multi-Gaussian peak fitting. Four dashed lines are guides to the eye to show the shift of corresponding modes at different momentum positions. The value of momentum q is the reciprocal distance from the selected position to Γ′. c, Experimental phonon dispersion relation of the phonon energy (red points) from b, superimposed with the simulated phonon dispersion curves (black curves). In order to exclude the surface phonon polariton signal, the angle-resolved line-scan was acquired along X→Γ′ in the adjacent BZ rather than the FBZ. On the basis of the equivalence of each BZ, the angle-resolved phonon information along X→Γ′ should be identical to that along X→Γ. At the Γ′ point (q = 0 Å−1), the LO and TO modes can be distinguished in the raw spectrum without the presence of the polariton signal, which is due to the long wavelength dipole scattering. TA and LA modes can also be detected at most momentum positions as shown in b, but were not easily separable at q = 0.36 Å−1 because of their energy difference being smaller than the energy resolution. The experimental peak locations of all four branches are consistent with theoretical phonon dispersion curves as shown in c. The discrepancy between experiment and theory is much reduced compared to previous work30, thanks to the improved energy resolution. d, Aloof spectra of SiC film with different momentum exchanges by angle-resolved vibrational spectroscopy. The convergence semi-angle was 3 mrad and spectra were obtained at the Γ point (centre of FBZ, red curve) and the X point (black curve) of the SiC film. Spectra were acquired when the electron beam was 20 nm from the surface of the SiC sample. The intensity of the X point spectrum is multiplied by 1,000. The polariton mode only appears in the aloof spectrum of the Γ point, which is consistent with the aloof spectrum acquired at large convergence semi-angle in main-text Fig. 2a.
Extended Data Fig. 7 Contribution of local atomic vibrations at a stacking fault along three orthogonal directions.
a, Atomic structure of an 11-layer defect structure, showing x, y and z directions. The electron beam (e-beam) direction is parallel to the x direction in our experimental setup. b–e, Bubble plots of phonon dispersion of the structure shown in a along the Γ to X″ direction with the contribution of local atomic vibrations at the stacking fault (inside purple box) in all directions (b, black dots), and along x (c, red dots), y (d, blue dots) and z (e, green dots) directions. The X″ point is the same as shown in Extended Data Fig. 5c. The size of each dot (the ‘bubble’) represents the local vibrational density of states at the stacking-fault region. The density of states was calculated by summing the square of an eigenvector’s amplitude for selected phonon modes and directions. In b–e, the calculated phonon dispersion curves are added as dashed curves. Vibrations along the electron beam direction (the x direction in a and c) cannot be detected in our experimental setup44. As a result, the vibrational spectrum at the stacking fault should comprise vibrations moving along both y and z directions in d and e.
Extended Data Fig. 8 Atomic structure of simulated models and corresponding phonon dispersion curves.
a, 4 × 4 × 4 supercell of 3C SiC. b, 3 × 3 × 3 supercell of 2H SiC. c, Simulated phonon dispersion curves of 2H SiC (red) superimposed with those of 3C phase (black) in the Γ–X direction. The 2H phase was used to mimic the stacking-fault structure, and the Γ–P–X′ path in the 2H phase is identical to the Γ–X path in 3C phase, as mentioned in Extended Data Fig. 2. d, Energy positions of simulated phonon modes of the stacking fault (SiC-SF) and defect-free 3C-phase SiC in c at the X point. e, Vibrational spectra of defect-free SiC (black) and the stacking-fault region (red) with a convergence semi-angle of 3 mrad at the Δ point, which is the midpoint of the Γ–X line. The location of the Δ point is marked by a violet bar in c.
Extended Data Fig. 9 Spatial distribution of local phonon resonances.
a, Top panel, the TA (red dots) and LA (blue dots) signals were integrated in the respective energy ranges 45–50 meV and 70–75 meV using the spectrum image data in main-text Fig. 4b, and overlaid with Gaussian-type fitting curves. ‘Nor.’ means normalized with respect to the height of the ZLP in each spectrum. Bottom panel, ADF intensity of the STEM image in main-text Fig. 4a. b, Fitting of the signal intensity of TA as a function of the distance to the stacking fault. Red dots are duplicated from the positive x-axis data of the integrated intensity in the energy range 45–50 meV in a. The fitting function is indicated near the red dashed fitting curve, where y is the signal intensity, and x is the distance to the stacking fault. We estimated the decay length of the defect phonon mode near the stacking fault according to the one-dimensional Schrödinger equation for the quasiparticle (the defect phonon at the X point). The wavefunction (ψ) of the defect mode should decay exponentially outside the fault, following ψ(x) = Aexp(−x/ξ), where x is the distance away from the stacking fault and A is a coefficient. The localization length, ξ, depends on the energy difference between the defect mode and corresponding TA mode in the defect-free region, ΔEPh (3.8 meV in this case), and can be estimated from \(1/\xi =\sqrt{2{m}^{* }\Delta {E}_{{\rm{Ph}}}}/\hbar \), where m* is the effective mass of the defect mode and ħ is the reduced Planck constant. Taking the effective mass from the phonon dispersion curve of 3C-phase SiC at the X point (m* ≈ 2.03 × 10−28 kg, only absolute values in the square root), ξ is about 2.7 nm. This is close to the experimental observation, 4.6 nm, obtained through a fitting as shown in b. It appears that the relationship \(1/\xi =\sqrt{2{m}^{* }\Delta {E}_{{\rm{Ph}}}}/\hbar \) holds well for estimating the decay length of local phonon modes.
Extended Data Fig. 10 Local PDOS of the defect structure calculated by the real-space Green’s function approach.
a, Projected PDOS plots corresponding to individual layers in a 42-layer structure at the X point. This structure was constructed by inserting a 2-layer stacking-fault structure into a periodic 3C-bulk SiC of 40 layers (each layer was designed as in Extended Data Fig. 5a, and the thickness of the 42-layer structure is about 10.5 nm). The stacking-fault structure was inserted at layers L21 and L22, marked on the right-hand side. Then, the retarded Green’s function was calculated at the specific momentum, and simulation details followed the procedure in the literature55. The simulated results were broadened with a Gaussian peak of 8 meV FWHM, resulting in all curves as shown. b, Peak centre of the 40–50 meV TA mode as a function of distance to the stacking fault. This result predicted a 4-meV red shift of TA modes at 40–50 meV in a. Also, the variation extended to a distance of 1.125 nm around the stacking fault and affects a region 2.25 nm wide in total, which is much broader than the size of the stacking fault (0.25 nm). This broadening is closer to our experimental observation than the DFT results in Extended Data Fig. 5b.
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Yan, X., Liu, C., Gadre, C.A. et al. Single-defect phonons imaged by electron microscopy. Nature 589, 65–69 (2021). https://doi.org/10.1038/s41586-020-03049-y
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DOI: https://doi.org/10.1038/s41586-020-03049-y
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