4D Nanoscale Diffraction Observed by Convergent-Beam Ultrafast Electron Microscopy

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Science  30 Oct 2009:
Vol. 326, Issue 5953, pp. 708-712
DOI: 10.1126/science.1179314


Diffraction with focused electron probes is among the most powerful tools for the study of time-averaged nanoscale structures in condensed matter. Here, we report four-dimensional (4D) nanoscale diffraction, probing specific site dynamics with 10 orders of magnitude improvement in time resolution, in convergent-beam ultrafast electron microscopy (CB-UEM). As an application, we measured the change of diffraction intensities in laser-heated crystalline silicon as a function of time and fluence. The structural dynamics (change in 7.3 ± 3.5 picoseconds), the temperatures (up to 366 kelvin), and the amplitudes of atomic vibrations (up to 0.084 angstroms) are determined for atoms strictly localized within the confined probe area (10 to 300 nanometers in diameter). We anticipate a broad range of applications for CB-UEM and its variants, especially in the studies of single particles and heterogeneous structures.

In fields ranging from cell biology to materials science, structures can be imaged in real-space by using electron microscopy. Atomic-scale resolution of structures is usually available from Fourier-space diffraction (13) data, but this approach suffers from the averaging over the selected specimen area which is typically on the micrometer scale. Substantial progress in techniques has enabled localization of diffraction to nanometer- and even angstrom-sized areas by focusing a condensed electron beam onto the specimen. Parallel illumination with a single electron wave vector is reshaped to a convergent beam with a span of incident wave vectors. This method of convergent-beam electron diffraction (CBED) or electron microdiffraction (1, 4) has made possible determination of structures in three dimensions with highly precise localization to areas reaching below one unit cell. The applications have been wide-ranging, from revealing bonding charge distribution (5, 6) and local defects and strains in solids (7) to detecting local atomic vibrations and correlations (8, 9). Today, aberration-corrected, atomic-sized convergent electron beams enable analytical probing by using electron-energy-loss spectroscopy (EELS) (10) and scanning transmission electron microscopy (STEM) (11).

The above-mentioned studies all probed static, or time-averaged, structures. In order to resolve structural dynamics with appropriate spatiotemporal resolution, femtosecond (fs) and picosecond (ps) electron pulses are ideal probes because of their picometer wavelength and their large cross section, resulting from the effective Coulomb interaction with atomic nuclei and core and valence electrons of matter. Typically, ultrafast electron diffraction is achieved by initiating the physical or chemical change with a pulse of photons (pump) and observing the ensuing dynamics with electron pulses (probe) at later times. By recording sequentially delayed diffraction frames, a “movie” can be produced to reveal the temporal evolution of the transient structures involved in the processes under study. In our laboratory (1216), studies in ultrafast electron diffraction and crystallography have spanned phases ranging from isolated molecules to interfaces and macromolecular structures. Other research groups (1720) have invoked the techniques, primarily for the study of nonequilibrium melting, lattice distortion, and phonon dynamics.

The spatial resolution in all these time-resolved investigations corresponds to atomic (and subatomic) length scale, but the probed specimen area is on the order of square micrometers, as mentioned above, albeit with a substantially reduced volume when compared with x-ray diffraction [see overviews in (21, 22)]. Thus, these techniques, as with those based on parallel beam illumination, lack the capability of resolving nanometric heterogeneity, for instance in nanometer-sized particles or specific sites in an extended material. Real-space ultrafast electron microscopy (23, 24) can resolve such objects by selected-area imaging, but the diffraction from a selected area is typically averaged over micrometers or more. Accordingly, the analytical and quantitative information available from this parallel-beam diffraction within such areas is somewhat limited. It follows that nanoscale (and finer) time-resolved, convergent-beam diffraction would be of great use in materials science and biology for structural dynamics studies in heterogeneous media.

Here, we report the development of convergent-beam ultrafast electron microscopy (CB-UEM) with applications in the study of nanoscale, site-selected structural dynamics initiated by ultrafast laser heating (1014 K s–1). Because of the femtosecond pulsed-electron capability, the time resolution is 10 orders of magnitude improved from that of conventional TEM, which is milliseconds; and, because of beam convergence, high-angle Bragg scatterings are visible with their positions and intensities being very sensitive to both the three-dimensional (3D) structural changes and amplitudes of atomic vibrations. The CB-UEM configuration is shown in Fig. 1; our chosen specimen is a crystalline silicon slab, a prototype material for such investigations. From these experiments, we found that the structural change within the locally probed site occurs with a time constant of 7.3 ± 3.5 ps, which is on the time scale of the rise of lattice temperature known for bulk silicon from the detailed spectroscopic studies of Sjodin et al. (25). For these local sites, the temperatures measured at different laser fluences range from 299 to 366 K, corresponding to vibrational amplitude changes from 0.077 to 0.084 Å, respectively.

Fig. 1

Schematic of the CB-UEM setup (top) and observed low-angle diffraction discs. Femtosecond electron pulses are focused on the specimen to form a nanometer-sized electron beam. Structural dynamics are determined by initiating a change with a laser pulse and then observing the consequences by using electron packets delayed in time. Insets (right) show the CB-UEM patterns taken along the Si [011] zone axis at different magnifications. At the high camera length used, only the ZOLZ discs indexed in the figure are visible; the kinematically forbidden 200 disc appears as a result of dynamic scattering. In the reciprocal space representation of the diffraction process (bottom), the Ewald sphere has an effective thickness of 2α, the convergence angle of the electron beam. The diamond structure of Si forbids any reflections from odd-numbered Laue planes when the zone axis is [011].

The electron microscope is integrated with a fs oscillator and amplifier laser system in California Institute of Technology’s (Caltech’s) UEM-2. The fundamental mode of the laser at 1036 nm was split into two beams: The first was frequency doubled to 518 nm and used to initiate the heating of the specimen, whereas the second, which was frequency tripled, was directed to the microscope for extracting electrons from the cathode. The time delay between pump and probe was adjusted by changing the relative optical path lengths of these two pulses. The pulses were sufficiently separated in time (5 μs) to allow for cooling of the specimen.

The electron packets were accelerated to 200 keV (corresponding to a de Broglie wave vector of 39.9 Å–1), demagnified, and lastly focused (with a 6-mrad convergence angle) to an area of 10- to 300-nm diameter on the wedge-shaped specimen, as shown in Fig. 1. A wide range of thicknesses, starting from ~2 nm, was accessible simply by moving the electron beam laterally. The silicon specimen was prepared by mechanical polishing of a wafer along the (011) planes, followed by Ar ion milling for final thinning and smoothing; the wedge angle was 2°. In the microscope, Kikuchi lines were observed and used as a guide to orient the specimen with the [011] zone axis being either parallel or tilted relative to the incident electron beam direction.

In Fig. 1, we display typical high-magnification (high-value camera length) CB-UEM patterns of Si obtained when the specimen is unexcited and the zone axis is very close to [011]; the relative magnification can be seen by comparing the disc length scale in Fig. 1 and circle radius in Fig. 2. Unlike parallel-beam diffraction, which yields spots, convergent-beam diffraction produces discs in reciprocal space (back focal plane of the objective lens) with their diameter given by the convergence angle (2α) of the electron pulses. These discs form the zero-order Laue zone (ZOLZ) of the pattern; they show white contrast with thin specimens and exhibit the interference patterns displayed in Fig. 1 when the thickness is increased, as expected (1, 4, 5).

Fig. 2

Temporal frames of CB-UEM. (A) High-angle SOLZ ring obtained for a tilt angle of 5.15° from the [011] zone axis. Besides SOLZ, Kikuchi lines and periodic bands (resulting from atomic correlations) are visible. The ZOLZ discs are blocked (top left) to enhance the dynamic range in the area of interest, indicated by the curved box; the disc of the direct beam (the center one in Fig. 1 discs) is indicated by a circle. The intensity scale is logarithmic. (B to D) Time frames of the SOLZ ring are shown by color mapping for visualization of dynamics. The intensity of the ring changes within picoseconds, but the surrounding background remains at the same level.

In the reciprocal space, the effective thickness of the Ewald sphere is 2α (bottom image of Fig. 1), giving rise to multiple spheres that can intersect with higher-order Laue zone (HOLZ) reflections, the focus of this study (Fig. 2) and the key to 3D structural information; the first and second zones, FOLZ and SOLZ, are examples of such zones or rings. The interference patterns in the discs are the result of dynamical scattering in silicon (1, 4, 5) and are reproduced in our CB-UEM patterns (Fig. 1).

The scattering vectors of HOLZ rings (R) are related to the interzone spacing in the reciprocal space (hz in Å–1) by the tilt angle from the zone axis (η) and by the magnitude of the incident electron’s wave vector (k0). In the plane of the detector and for our tilt geometry, the HOLZ ring scattering vector is given byR[k02sin2(η)+2k0hz]12k0sin(η)(1)where, for our case of the [011] zone axis, hz=n/(a2); n=1,2,3... for the different Laue zones with a being the lattice constant. Additionally, for this zone axis, k+l=n, where (hkl) are the Miller indices of the reciprocal space. When k+l=1 for FOLZ, k and l must have different parity, which is forbidden by the symmetry of the diamond Si structure. Therefore, the FOLZ along the [011] zone axis should be absent, and the first visible ring should belong to SOLZ; in general, all odd numbered zones will be forbidden. Here, HOLZ indexing is defined according to the face-centered cubic unit cell and not to the primitive one (1).

The HOLZ ring taken with the CB-UEM is presented in Fig. 2. In order to reduce the strong on-zone-axis dynamic scattering (and to bring the high scattering angles into the range of the recording camera), the slab was tilted 5.15° away from the [011] zone axis, along the [022¯] direction. The scattering vector of the Bragg points of the ring, from the direct beam position, was measured to be 2.2 Å–1, close to the value of 2.22 Å–1 obtained by using Eq. 1 forn=2, which identifies the spots shown as part of the SOLZ. From this value, the known lattice separation of 5.4 Å was obtained for silicon.

In addition to the SOLZ ring, Kikuchi lines and some oscillatory bands are also visible in the CB-UEM, as seen in Fig. 2A. Kikuchi lines arise from elastic scatterings of the inelastically scattered electrons, whereas the oscillatory bands in the thermal diffuse scattering (TDS) background result from correlations between the atoms (9). We also observed deficit HOLZ lines and interference fringes in ZOLZ discs (1, 4). Although the dynamics in these features may reveal additional intriguing behavior, this report is primarily concerned with the relative intensity change of the SOLZ spots of the ring.

The temporal behavior is displayed in Fig. 2, with three CB-UEM frames taken at time delays of t = –14.8 ps, +5.2 ps, and +38.2 ps, together with a static image; the zero of time is defined by the coincidence of the pump and probe pulses in space and time. The frame at negative time has a higher ring intensity than that observed at +38.2 ps, whereas the +5.2 ps frame shows an intermediate intensity value. The results indicate that the intensity change is visible within the first 5 ps of the structural dynamics. For quantification, the intensities in each frame were normalized to the area of azimuthally integrated background. The normalization of the HOLZ ring intensities to the TDS background makes the atomic vibration estimations insensitive to the thickness changes of the probed area (8), which may result from slight beam jittering.

Figure 3 depicts the transient behavior of the SOLZ ring intensity for two different laser power, 10 mW and 107 mW, corresponding to pulse fluence of 1.2 and 13 mJ cm–2, respectively, after correcting for power losses; the heating laser beam diameter on the specimen is 60 μm (23). The intensities were normalized to the average value obtained at negative times. Whereas the intensity change is essentially absent in the 10-mW data, the results for the 107-mW set show a transient behavior with a characteristic time of 7.3 ± 3.5 ps obtained from the monoexponential fit shown in red in the figure. The temporal response of UEM-2 is on the fs time scale, as obtained by EELS, and it is much shorter than the 7 ps reported here (26, 27).

Fig. 3

Diffraction intensities at different times and fluences. Normalized, azimuthally integrated intensity changes of the SOLZ ring are shown, with time ranging from –20 ps to +100 ps, for two different laser powers. Whereas the 10-mW response does not show noticeable dynamics, the 107-mW transient has an intensity change with a characteristic time (τ) of 7.3 ± 3.5 ps. The range of fluences studied was 1.2 to 15 mJ cm–2 (Fig. 4). The red curve is a monoexponential fit based on the Debye-Waller effect, as discussed in the text. The red dashed line through the 10 mW data is an average of the points after +20 ps. The dependence on fluence is given in Fig. 4.

The local heating of the lattice is responsible for the SOLZ intensity change with time. A pump laser, in our case at 518 nm (2.4 eV), excites the valance electrons of Si to the conduction band; one-photon absorption occurs through the indirect bandgap at 1.1 eV, and multiphoton absorption excites electron-hole pairs through the direct gap. The excited carriers thermalize within 100 fs (28), via carrier-carrier scatterings, and then electron cooling takes place in ~1 ps, by electron-phonon coupling. During this time, lattice heating occurs through increased atomic vibration, reducing SOLZ intensity. The effective lattice temperature is ultimately established with a time constant of a few picoseconds depending on density of carriers or fluence (25). However, in CB-UEM measurements the lattice-temperature rise could be slower than in bulk depending on the dimension of the specimen relative to the mean free path of electrons in the solid.

The dynamical change can be quantified by considering a time-dependent Debye-Waller factor with an effective temperature describing the decrease in the Bragg spot intensity with time. If the root mean square (rms) displacement of the atoms, ux212, along one of the three principle axes is denoted by ux for simplicity and the scattering vector by s, then the HOLZ ring intensity can be expressed asIRingF(t)=I0(t)exp[4π2s2ux2(t)](2)where IRingF(t) is the measured intensity for a given fluence, F, and the vibrational amplitude is now time dependent. Note that ux2 is one-third of the total, utotal2 (29).

In the Einstein model of atomic vibrations, which has been used successfully for silicon (8, 30), the atoms are treated as independent harmonic oscillators, with the three orthogonal components of the vibrations decoupled. As a result, a single frequency (ω) is sufficient to specify the energy eigenstates of the oscillators. The relationship of the vibrational amplitude to temperature can be established by simply considering the Boltzmann average over the populated eigenstates. Consequently, the probability distribution of atomic displacements is derived to be of Gaussian form, with a standard deviation corresponding to the rms (ux) of the vibration involved (8, 30)ux=[(/2ωm)coth(ω/2kΒTeff)]12(3)where is Planck’s constant, kB the Boltzmann constant, Teff in our case the effective temperature, and m the mass of the oscillator. In the high-temperature limit, that is, when ω/2kΒT<<1, Eq. 3 simplifies to mω2ux2=kΒT, which is the classical limit for a harmonic oscillator; the zero-point energy, which contributes almost half of the mean vibration amplitude at room temperature, is included in Eq. 3. The value of ħω is 25.3 meV. Despite its simplicity, the Einstein model in Eq. 3 was remarkably successful in predicting the HOLZ rings and TDS intensities by multislice simulations (8, 30), although it was not as successful in predicting the oscillatory bands in the background, which are due to correlations (9).

In Fig. 4, we present the change in the asymptotic intensity with fluence (inset) and the derived vibrational amplitudes for the different temperatures. The amplitudes are directly obtained from Eq. 2 because s is experimentally measured. The relative temperature change (from t to t+) is then derived from Eq. 3, taking the value of ux at room temperature (297 K) to be 0.076 Å [from both CBED (30) and x-ray (31) measurements] for the unexcited specimen. The amplitude of atomic vibrations, and hence the temperature, increases as the fluence of the initiating pulse increases. Although the trend is expected for an increased ux with temperature, the absolute values, from 0.077 to 0.084 Å, correspond to a large 3.2% to 3.6% change in nearest neighbor separation; these values are still well below the 15% criterion for a melting phase transition.

Fig. 4

Amplitudes of atomic vibrations (rms) plotted against the observed intensity change at different fluences. (Inset) The monoexponential temporal behavior, with the asymptotes highlighted (circles) for their values at different fluences. The fluence was varied from 1.2 to 15 mJ cm–2. The configuration of the laser excitation is similar to that given in (23). This comparative study of the effect of the fluence was performed at a slightly different sample tilt (corresponding to s = 2.7 Å–1), corresponding to a thickness of ~80 nm. For each fluence, the temperature represents the effective value for the lattice structural change. The error bars given were obtained from the fits at the asymptotes shown in the inset, and they are determined by the noise level of temporal scans.

The linear thermal expansion coefficient has been accurately determined for silicon (32), and for a value of 2.6 × 10–6 K–1 at room temperature the vibrational amplitudes reported here are much higher than the equilibrium thermal values at the same temperature. This is because the effective temperature applies to a lattice arrested in a picosecond time window; at longer times, the vibrations equilibrate to a lower temperature (33). As such, measuring nanoscale local temperatures on the ultrashort time scale enhances the sensitivity of the probe thermometer by orders of magnitude. Moreover, the excitation per site is substantially enhanced. For a single-photon absorption at the fluence used, we estimate, for a 60-nm-thick specimen, the number of absorbed photons per Si atom (for the fs pulse used) to be ~0.01, as opposed to 10−9 photons per atom if the experiments were conducted in the time-averaged mode.

The achievement of nanoscale diffraction with CB-UEM opens the door to exploration of different structural, morphological, and electronic phenomena. The spatially focused and timed electron packets enable studies of single particles and structures of heterogeneous media. Extending the methodology reported here to other variants, such as EELS (26), STEM (11), and nanotomography (34), promises possibilities for mapping individual unit cells and atoms on the ultrashort time scale of structural dynamics.

References and Notes

  1. For this study, we used 500 to 1000 electrons per pulse, and care was taken to account for any instrumental broadening in fitting the transient response by proper convolution with the two pulse-widths involved. By measuring the energy spread of the photoelectron packets with EELS in the same microscope (UEM-2), we obtained a maximum temporal width of 0.7 ps. The path difference caused by the condenser lens system, at 10 mrad convergence angle, is 0.3 ps.
  2. In the standard formula for the Debye-Waller effect, three components of vibrational amplitudes are considered, and the exponent becomes exp[(4/3)π2s2utotal2]. All three components in the Einstein model are equal, and hence ux2=1/3utotal2. From an experimental point of view, the intensities could be expressed as IRingF(t)~exp[4π2s2ux2(t)]/{1exp[4π2s2ux2(t)]} where the denominator represents the TDS background (30). However, under our experimental conditions, this background is insignificant in the narrow angular range studied. Moreover, we note that TDS and other background contributions, such as inelastic plasmon scattering, which can be filtered out (30), are not of concern here because we are monitoring the temporal change in reference to frames recorded at negative delay times.
  3. This work was supported by the NSF and the Air Force Office of Scientific Research in the Gordon and Betty Moore Center for Physical Biology at CalTech. We thank O.-H. Kwon and H. S. Park for the experimental aid in UEM-2 laboratory and acknowledge the helpful use of the Cornell TEM sample preparation facility. Caltech has filed a provisional patent application for variants of 4D microscopy.
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