Letter pubs.acs.org/NanoLett
Diffraction of Quantum Dots Reveals Nanoscale Ultrafast Energy Localization Giovanni M. Vanacore,† Jianbo Hu,† Wenxi Liang,† Sergio Bietti,‡ Stefano Sanguinetti,‡ and Ahmed H. Zewail*,† †
Physical Biology Center for Ultrafast Science and Technology, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125, United States ‡ L-NESS and Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, Via Cozzi 53, I-20125 Milano, Italy S Supporting Information *
ABSTRACT: Unlike in bulk materials, energy transport in low-dimensional and nanoscale systems may be governed by a coherent “ballistic” behavior of lattice vibrations, the phonons. If dominant, such behavior would determine the mechanism for transport and relaxation in various energy-conversion applications. In order to study this coherent limit, both the spatial and temporal resolutions must be sufficient for the length-time scales involved. Here, we report observation of the lattice dynamics in nanoscale quantum dots of gallium arsenide using ultrafast electron diffraction. By varying the dot size from h = 11 to 46 nm, the length scale effect was examined, together with the temporal change. When the dot size is smaller than the inelastic phonon mean-free path, the energy remains localized in high-energy acoustic modes that travel coherently within the dot. As the dot size increases, an energy dissipation toward lowenergy phonons takes place, and the transport becomes diffusive. Because ultrafast diffraction provides the atomic-scale resolution and a sufficiently high time resolution, other nanostructured materials can be studied similarly to elucidate the nature of dynamical energy localization. KEYWORDS: Femtosecond diffraction, quantum dots, energy localization, ballistic transport
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sional (nanotubes9 and nanowires10) systems. A substantial dearth of relevant experimental data has been the case for quasizero-dimensional systems, quantum dots, although they have the potential to overcome many of the limitations of 2D and 1D systems because of the expected high energy-conversion efficiency.2 Time-resolved optical methods have been used to study the dynamics in nanostructures on the time scale of their occurrence.4,7−9 These valuable studies provide the overall temporal change, however the atomic-scale lattice dynamics cannot directly be observed because of the long wavelength of the optical probe used. With ultrafast electron pulses, timeresolved diffraction enables probing of the spatiotemporal behavior of atomic lattice structures.11,12 This methodology has become the method of choice in this laboratory for the investigation of correlated atomic motions,13 bond dilation,14 and structural transformations.15−17 In other laboratories, Xrays have been used in the studies of structural and thermal dynamics, and for an overview see ref 18 and the articles by Chergui.19,20
emiconductor quantum dots (QDs), nanometer-sized systems, have been widely studied in recent years because of their appealing physical properties, which strongly depend on the interaction between the electronic and lattice degrees of freedom. Changes in the electronic and lattice structures, through variation in the length scale, significantly affect the optical absorption, the carrier mobility and the heat transport, which are crucial aspects in the design of new generation optoelectronic devices1 and thermoelectric applications.2,3 In bulk systems, heat transport is usually governed by a random walk of energy-carrying phonons (diffusive regime), and Fourier’s law of heat conduction describes the energy transfer because of the continuum nature of lattice states. However, when the size of the structure becomes smaller than the inelastic phonon mean-free path, lattice phonons can travel ballistically (no collisions)4 and a Fourier description is no longer valid.5,6 Understanding and control of the transport at the atomic-scale level is of fundamental importance, especially when the length scale reaches that of molecular structures and nanodevices.2,7,8 Experimental investigation of phonon transport in nanoscale systems is challenging. The difficulty lies with the technique used for measuring the thermal properties of the nanostructures involved and with the complexity of systematically varying their size. So far, a particular emphasis has been given to twodimensional (thin films and superlattices)4,7,8 and one-dimen© XXXX American Chemical Society
Received: June 18, 2014 Revised: July 22, 2014
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Figure 1. Diffraction, morphology, and the temporal probing scheme. (a) Representative 3D AFM images of the investigated quantum dots with the indicated dimensions. (b) Schematic layout of the photon-pump/electron-probe, UEC, experiments. (c) Diffraction pattern of GaAs dots measured with the electron beam propagating 0.7° off the [110] zone axis and with an incidence angle of 1.6°. The pattern consists of Bragg spots arranged along a characteristic chevron pattern, together with surface wave resonances (SWR) from the substrate.
both the photon energy (1.55 eV) and the band gap of the GaAs dots (1.55 eV for QD-1, 1.45 eV for QD-2, and 1.42 eV for QD-3),25,26 the excitation of charge carriers occurs locally within the dots. Figure 1a depicts the three QDs studied, which are labeled according to their width and height. The figure also shows the diffraction pattern taken with the electron beam propagating nearly along the [110] direction. The assignment of Bragg spots in Figure 1c is that of the static crystal structure, which originates from the 3D morphology of the dot. Surface wave resonance (SWR) features from the substrate are also visible, attesting to the high crystalline order of the surface.27 The diffraction pattern is not symmetric because we chose the angle between the electron beam and the zone axis to be not zero but 0.7°, so that the off-axis reflections {11l}, {22l} and {33l} can be clearly visible. Shown in Figure 2a is the observed temporal change of the diffraction intensity for the (006) Bragg spot and for the three investigated dot dimensions at 4.6 mJ/cm2. The transient for each case can be well described by a fast exponential decay with a time constant τd and a slower recovery to the equilibrium state. As displayed in Figure 2b, the intensity change when normalized to the calculated bulk value gives an enhancement factor of two to three for the smallest dot (QD-1) when compared to those of QD-2 and QD-3. This behavior is the result of quantum confinement in the investigated dots that modifies their optical absorption depending on the size. The theoretical size-dependent optical absorbance, whose calculations are detailed in the Supporting Information, is shown as a solid line in Figure 2b; only QD-1 shows the enhancement because of the confinement-caused resonant absorption at the optical wavelength of 800 nm. The change of the diffraction intensity, I(t), is proportional to the square of the structure factor, which is determined by the atomic displacements. In this case, the motions are those of optical phonons, acoustic phonons, and incoherent thermal vibrations and these occur on different time scales. From the transient behavior of I(t), we obtained the time constant τd (Figure 1), and in Figure 3, we present the dependence of τd on the QD size, excitation fluence, and the angle, δ, between the
Here, we specifically employ ultrafast electron crystallography (UEC) in the reflection geometry to map out phonon dynamics in size-controlled GaAs quantum dots. We measured the transient change of their Bragg reflections, following an ultrafast optical excitation, as a function of the dot size and excitation fluence. The results indicate a size-dependent dynamics: the phonon transport is ballistic in small dots (length scale, h = 11 nm), and becomes diffusive in large dots (h = 46 nm). This four-dimensional sampling, in space and time, demonstrates that in the ballistic regime the excitation of thermal motions is quenched and the energy remains localized in high-energy acoustic phonons that travel coherently within the dot, whereas in the diffusive regime an incoherent energy dissipation toward low-energy modes takes place. Diffraction of Size-Controlled Quantum Dots. GaAs quantum dots were grown on Al0.3Ga0.7As by droplet epitaxy.21 This method allows for the generation of strain free and stoichiometric nanostructures with accurate control of the size and density. We studied three different samples with highly homogeneous distributions of shape and size for the dots. From analysis of atomic force microscopy (AFM) images, we determine their height, h, to be 11 nm (QD-1), 22 nm (QD2), and 46 nm (QD-3) with a corresponding base width, w, of 31, 54, and 110 nm (see Figure 1a). Because the inelastic meanfree path of high-energy acoustic phonons in GaAs22,23 is Λp ∼ 26−27 nm (see also Supporting Information), different dynamical regimes of phonon transport can then be explored. The UEC geometry is schematically shown in Figure 1b for the reflection mode; more details can be found in the Appendix. The ultrashort electron pulses were focused on the sample in a grazing incidence geometry. The dynamics were initiated by ultrashort laser pulses at 800 nm (120 fs, 1 kHz), with the fluence varying between 2.3 and 4.6 mJ/cm2. The diffracted electrons were recorded in the stroboscopic mode at different delay times between the excitation laser and the electron pulse. The same transients were obtained when the experiment was performed at different positions on the sample surface, and the transient electric field24 (surface charging) effect was found to be insignificant at all fluences used. Because the band gap of the AlGaAs substrate (1.8 eV) is larger than B
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dependence on the excitation fluence and on the angle δ is observed for QD-3, whose size is larger than Λp. Slower rates were observed at higher fluences and for smaller angles δ. For example, at 4.6 mJ/cm2 the reflections (006) (δ = 0°) and (3̅35) (δ = 40.3°) give τd ≈ 12.5 and 4.1 ps, respectively; the values drop down to 5.5 and 3.1 ps at 2.3 mJ/cm2. Coherent Energy Localization. For QD-1 the reported value of τd (3.7 ps) is in good agreement with the decay time of the longitudinal optical (LO) phonon in GaAs at 300 K, τanh ≈ 3.5 ps.28−30 The generation of LO phonon occurs in ∼100 fs by electron−phonon coupling;28 a polaron picture28,31,32 has been used to describe the interaction between the electron distribution and the optical phonon. Because energy and momentum must be conserved, the LO phonon decays into the high-energy acoustic phonons by the anharmonic coupling,28,31,32 with τanh ≈ 3.5 ps. Because the diffraction intensity change is related to the mean-square amplitude of the phonons involved, we have calculated, following the approach of ref 33, for QD-1, when only the high-energy acoustic phonons are populated, an 11% decrease for the (006) reflection, which is in close agreement with the experimental result of 13% (see Figure 2a). Moreover, as shown in Figure 3b, there is no discernible fluencedependence for τd. This behavior is supported by the theoretically calculated29 dependence of τanh on lattice temperature. For the fluence range used here, a maximum change of only 0.3 ps is estimated from the calculations. The absence of additional decay components at longer times, besides the 3.7 ps component, suggests that the hot acoustic phonons are long-lived and that for QD-1 they persist on the time scale reported here; dynamically speaking, they travel unperturbed within the dot with no collisions and maintaining defined phases during the propagation, a ballistic coherent transport (see Figure 4b (right panel)). In this case, the lattice preserves its structural coherence and a breathing oscillation of the interatomic distance should be visible in the position of Bragg spots.34,35 As shown in Figure 4a, for QD-1 we observed oscillations with a frequency of 0.33 THz (period of 3 ps) for (006) reflection; the same was observed for the (2̅26) reflection. This value matches that obtained from a standingwave condition for a hemispherical nanostructure, ⟨f⟩ ≈ vhe/h ≈ 0.29−0.36 THz, when the velocity of high-energy acoustic
Figure 2. Diffraction intensity transients and optical absorption. (a) Experimentally observed temporal change of the diffraction intensity for the (006) Bragg reflection measured at 4.6 mJ/cm2 for QD-1 (blue), QD-2 (green), and QD-3 (red). (b) Intensity change for the reflections (006) (squares), (2̅26) (circles), (1̅15) (downward pointing triangles), and (3̅35) (upward pointing triangles), normalized to the calculated bulk value and plotted as a function of the dot height. The orange solid line is the calculated size-dependent optical absorbance for the investigated nanostructures (see Supporting Informations for details).
scattering vector, s, and the [001] axis (see inset in Figure 3c). Because local absorption occurs only within the dot, the created phonons transfer their energy to the substrate along the perpendicular to the interface direction since transverse dissipation of energy in the dot is not feasible. Thus, in this case the height (h) of the dot defines the length scale of phonon transport. For QD-1, whose size is smaller than the inelastic mean-free path, Λp, τd ≈ 3.7 ps for all the Bragg spots studied and for all the fluences used. In contrast, a strong
Figure 3. Size, fluence, and angle dependencies. (a) Size and fluence dependence of the experimental (full symbols) and simulated (solid lines) time constant, τd, for the (006) Bragg spot at 4.6 mJ/cm2 (black), 3.4 mJ/cm2 (gray), and 2.3 mJ/cm2 (light gray). Λp is the phonon inelastic mean-free path in GaAs. (b) Experimental (open circles) and simulated (solid lines) τd as a function of the excitation fluence for the (006) spot in QD-1 (blue) and QD-3 (red). (c) Experimental (open circles) and simulated (solid lines) τd as a function of the angle δ (see inset) at 4.6 mJ/cm2 for QD-1 (blue) and QD-3 (red). The time constant for different δ’s is obtained by monitoring the behavior of different Bragg reflections: δ = 0° for (006), δ = 11.4° for (1̅17), δ = 15.8° for (1̅15), δ = 25.2° for (2̅26), and δ = 40.3° for (3̅35). C
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4c (right panel)), leading to a progressive randomization of the phases. The consequence of this incoherent diffusive transport is the loss of coherence in structural dynamics and, hence, the absence of breathing oscillations as observed experimentally; see Figure 4a. When the transport is diffusive, the diffusion time scales with the square of the diffusion length. Because the investigated dots have a “dome-like” shape, a spatial dependence of the time constant τd should be expected. This dependence was probed by monitoring the change of τd with the angle δ. As shown in Figure 3c, no discernible dependence is observed for QD-1, whereas in QD-3 larger values of τd are measured at smaller δ angles, confirming the validity of the diffusive picture in large dots and its absence in small dots. Additional observations support this picture. Our results for large dots are consistent with previous ultrafast diffraction measurements on GaAs bulklike structures.37 A time constant of 12 ps was measured for the (008) spot at 10.9 mJ/cm2, which correlates well with the value (12.5 ps) measured for the (006) spot in QD-3 at high fluence. Moreover, because phonon−phonon scattering randomizes the distribution of acoustic modes, besides the intensity change one also expects a broadening of the Bragg peaks.37 For QD-3, the transient increase of the width for the observed Bragg spots is about a factor of four larger than that measured for QD-1. Structural Dynamics. To elucidate the different regimes of energy localization we have calculated for the recorded Bragg reflections the transient behavior of the structure factor (and diffraction intensity) when a size-dependent phonon transport is considered. The model, which is detailed in the Appendix and the Supporting Information, separately incorporates the ballistic, diffusive, and the dissipative contributions to the total heat flux. The results of the simulations for the time scale of the structural dynamics are plotted as solid lines in Figure 3, and a satisfactory agreement with the experimental results is obtained. The observed dependence of τd on the angle δ is also reproduced in the calculated space-resolved maps of the time constant, shown in Figure 4b,c (left panels). For QD-1, the calculated τd is 3.6−3.7 ps and uniformly distributed inside the dot, whereas in QD-3 the calculated τd is maximum (11 ps) at the center and exhibits a progressive decrease (down to 5 ps) toward the sides of the dot. Besides the time constant, when the intensity change is monitored as a function of the angle δ, the spatial distribution of the mean-square displacement of the atoms, ⟨u2⟩, can be explored. In bulk systems, ⟨u2⟩ has an isotropic distribution. In such case, ⟨u2⟩ is the same for all Bragg reflections37,38 and can be obtained through the Debye− Waller relation: ⟨u2⟩ ∝ W = (1/s2)ln(I0/I). In Figure 5a the Debye−Waller factor, W, is plotted for the (006), (22̅ 6) and (3̅35) reflections as measured for QD-1, and in Figure 5b W is shown as a function of the angle δ. The existence of an anisotropic structural deformation along the [001] direction is evident in the progressive decrease of W for larger δ. This is quantitatively confirmed by the simulated map of the local atomic displacement, displayed in Figure 5c for QD-1, and by the calculated change of W with δ, as shown in Figure 5a,b. The implications of the results obtained from this fourdimensional sampling are worth mentioning. Because theoretical calculations4,29 have shown that most of the heat is conducted by low-energy phonons, we expect the small dots, where only high-energy phonons are excited, to exhibit a very low thermal conductivity. This makes them particularly suitable for thermoelectric and heat management applications, where
Figure 4. Coherent oscillations and phonon transport regimes. (a) Transient behavior of the position change of the (006) Bragg spot for QD-1 (left) and QD-3 (right). For clarity, the background due to incoherent signal was subtracted. For QD-1, coherent oscillations are observed with a frequency of 0.33 THz (period of 3 ps). (b, left) Space-resolved map of the calculated time constant τd for QD-1. (b, right) Schematic representation of ballistic coherent phonon transport across the small-sized dot. The blue spark represents the energy transfer from the laser to the acoustic phonon population. (c, left) Space-resolved map of the calculated time constant τd for QD-3. (c, right) Schematic representation of the diffusive, incoherent phonon transport across the large-sized dot. The small red sparks represent phonon−phonon scattering processes within the acoustic population.
phonons, vhe ≈ 3.2−4.0 km/s,22,29 and h = 11 nm is considered (see Supporting Information). As shown in Figure 3a,b, for QD-3 the time constant τd is longer than the value observed for smaller dots, indicating that, besides the LO phonon decay, an additional mechanism of energy exchange is present. If we consider that only high-energy acoustic phonons are populated, a 2% decrease in intensity is calculated33 for the (006) reflection, contrary to our result of 7% change (see Figure 2a). This calculated value of 2% differs from that of QD-1 (11%) because of the difference in the dot’s optical absorption. When we allow for the relaxation of the hot phonons to the low-energy ones, which may take place via anharmonic scattering, an 8% drop is predicted because the diffraction intensity is inversely proportional to the square of the phonon frequency. The 8% is indeed close to the measured value. Furthermore, the strong fluence-dependence of τd observed for QD-3 (see Figure 3b) is typical of an energy equilibration process whose time scale for the relaxation is predicted to be proportional to the extent of the external excitation.36 In our case, this translates into a time constant that lengthens when the fluence increases. These observations suggest that in QD-3 phonon−phonon scattering dominates. The hot acoustic phonons experience “collisions” during their propagation within the dot (see Figure D
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(equivalent pressure ∼5 × 10−5 Torr) was irradiated on the surface to crystallize the Ga droplets into GaAs dots, while the substrate temperature was reduced down to 150 °C. After the growth, the samples were capped with a low-temperature amorphous As layer able to prevent the formation of native oxide due to air exposure during the following transfer to the UEC setup. Prior to the UEC experiments, the As capping layer was removed by annealing the samples in situ at 250 °C for 20 min and at 320 °C for 30 min, producing a clean and atomically flat surface between the dots. The size distributions of the dots have been obtained by atomic force microscopy. The mean values and the standard deviations (in parentheses) of their height and base width are as follows: the height, h, is 46(5.5) nm, 22(2.5) nm, and 11(2.1) nm for QD-3, QD-2, and QD-1, respectively; the width, w, is 110(3.4) nm, 54(3.2) nm, and 31(3) nm for QD-3, QD-2, and QD-1, respectively. UEC Experiment
The setup for UEC experiments at Caltech has already been described in details elsewhere.27,37 Briefly, electron pulses with energy per particle of 30 keV and sub-picosecond pulse width are generated in a photoelectron gun (Kimball Physics, Inc.) after irradiation of a LaB6 photocathode with 120 fs UV laser pulses (λ = 266 nm). The electron beam is focused at grazing incidence (0.5−2.5°) on the sample surface. The sample is mounted on a 5-axis goniometer, allowing for simultaneous adjustment of the incidence and the azimuthal angles. The electron beam diffracted from the surface is recorded on a phosphor-screen/MCP/CCD assembly. The ultrafast dynamics are initiated by femtosecond laser pulses (1 kHz, 120 fs, 800 nm) focused in normal incidence to the sample surface. The laser-induced change of the diffraction pattern is monitored in the stroboscopic mode by varying the delay time between the electron and the laser pulses. The velocity mismatch and the non-coaxial geometry between electrons and photons are responsible for a different arrival time of the electrons at different regions of the sample surface illuminated by the pump pulse. This effect is compensated for by tilting the wavefront of the optical pulse with respect to its propagation direction.14,40 The fluence of the excitation laser at the sample position was calibrated by scanning a knife edge across the laser profile and recording the passing residual pulse energy. In the grazing-incidence geometry, transient electric field (TEF) may contribute to temporal changes of the detected diffraction features (see ref 24 and references therein). However, here control experiments were carried out, supporting that TEF effect is negligible in the fluence range employed. First, we monitored the time-dependent angular deflection of the direct electron beam (ΔϑDB) tangential to the sample surface, which provides a quantitative measure of the field-induced deflection.24 At the highest fluence used (4.6 mJ/ cm2), ΔϑDB was found to be 21 μrad. This value should be compared with the typical diffraction angle, ϑdiff, and angular width, ϑw, of the investigated Bragg reflections; for the (006) spot, ϑdiff = 39.6 mrad and ϑw = 7.9 mrad, orders of magnitude larger than ΔϑDB. Second, we found that Δϑ scales linearly with the diffraction angle for the (00n) Bragg spots: Δϑ/ϑ is 1.9 × 10−3 for the (004), 2 × 10−3 for the (006), and 1.8 × 10−3 for the (008) (with an error of ±0.1 × 10−3) for the case of QD-1. Because the field-induced deflection is independent of the diffraction angle,24 this observed scaling behavior further confirms that the TEF is negligible.
Figure 5. Structural dynamics and its anisotropy. (a) Experimental (left) and calculated (right) transient behavior of the Debye−Waller factor, W, derived from the reflections (006) (dark blue), (2̅26) (blue), and (3̅35) (light blue) in QD-1. The inset is a schematic representation of the scattering vectors of the shown Bragg reflections plotted in the [110] plane. (b) Experimental (open circles) and calculated (solid lines) Debye−Waller factors as a function of the angle δ for QD-1 (blue), QD-2 (green), and QD-3 (red). (c) Simulated space-resolved map of the local atomic displacements, represented in the [110] plane at a delay time of 12 ps.
the energy-conversion efficiency is inversely proportional to the thermal conductivity. Spatiotemporal visualization of structural dynamics in GaAs quantum dots reveals the atomic-scale details of the sizedependent energy localization in nanoscale systems. Structural changes are examined by observing the transient behavior of their electron diffraction following an ultrafast optical excitation. For dots smaller than the phonon inelastic meanfree path (ballistic regime), the excitation of thermal motions is quenched and structural dynamics is mainly controlled by the anharmonic decay of the optical phonon formed by the unit cell atoms. In this case, the energy remains localized in high-energy acoustic modes that “travel” coherently within the dot. As the dot size increases, slower phonon−phonon scattering within the acoustic population takes place, and the transport becomes diffusive in nature. These results are fundamental for the understanding of energy conversion in nanoscale materials, and for the control of properties involving thermal conductivity and optical design.
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APPENDIX
Quantum Dots Growth
The investigated samples were grown by droplet epitaxy (DE) technique in a Gen II molecular beam epitaxy (MBE) system equipped with As, Ga, and Al cells. In situ electron diffraction is used to monitor the growth process.39 The substrate is n-type doped GaAs(001). After the oxide removal, a 250 nm layer of GaAs and a 90 nm layer of Al0.3Ga0.7As were grown at 580 °C. The formation of GaAs quantum dots is obtained with a twostep procedure. First, Ga droplets are created on the surface by the deposition of gallium: 2, 1, and 0.25 monolayers (MLs) are deposed at 350 °C with a flux of 0.05 ML/s for the samples QD-3, QD-2, and QD-1, respectively. Then, a flux of As4 E
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Structural Dynamics Simulations
Author Contributions
We carried out simulations of structural dynamics within the framework of a phenomenological model that includes electron−phonon interaction and non-Fourier phonon transport. The main equations are reported in the Supporting Information. Here, we briefly describe the main ideas. The optical excitation creates electron-hole carriers with a carrier density Nex that depends on the absorbed fluence. Such excitation induces a weakening of the interatomic bonds by transferring electrons from bonding (valence band) to antibonding (conduction band) states. This leads to an instantaneous stress, electronic pressure Pe, which for GaAs can be written as34 Pe = Nexadef, where adef is the deformation potential. The excitation of acoustic phonons from the decay of optical phonons is able to drive an additional stress on the lattice, quantitatively described by the phonon pressure Pph = C· α(TA(t) − Tref), where C is the stiffness tensor, α is the linear expansion coefficient, Tref = 300 K, and TA is the effective temperature of the acoustic phonon population. The total stress within the dot induced by the optical excitation is given by the combination of the electron and phonon pressures: σ0(t) = Pe + Pph. The acoustic phonon temperature TA can be calculated by invoking a three-temperature model (3TM),41,42 where the carriers, the optical phonons, and the acoustic phonons subsystems are each in a local state defined by the temperatures Te, TO, and TA. The size-dependent acoustic phonon transport is included in the TA-equation of the 3TM by separately incorporating the diffusive, the dissipative, and the ballistic contributions. The diffusive component is given by the Fourier’s law of heat conduction. The dissipative contribution, which takes into account the thermalization of the hot phonons, is included using the Cattaneo correction43 to a pure Fourier equation. The ballistic transport is modelled by introducing an additional component, whose weight depends on the ratio Λp/h, which compensates for the over-prediction of the heat flux by the diffusive-dissipative term when h < Λp. Its analytical expression is derived from two similar approaches, giving equivalent results: (i) the ballistic-diffusive model by Chen,44 and (ii) the multiphysics model of Siemens et al.5 (see also Supporting Information for further details). With the boundary conditions defined by the dot geometry, we numerically solved the modified-3TM model coupled with the stress-strain equations using σ0 as initial stress.45 The temporal behavior of the atomic displacement field, u, is then calculated, from which the transient change of the scattering factor (and the diffraction intensity) for the observed Bragg spots can be finally obtained.46
G.M.V. and A.H.Z. designed the experiments; S.B. and S.S. provided the samples; G.M.V., W.L., and J.H. performed the measurements; G.M.V. and J.H. analyzed the data and performed the simulations; G.M.V. and A.H.Z. wrote the paper with the support of J.H., W.L., S.S., and S.B.
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Science Foundation and the Air Force Office of Scientific Research in the Center for Physical Biology at Caltech supported by the Gordon and Betty Moore Foundation. We thank Dr. Paolo Biagioni for the finite difference time domain (FDTD) simulations and Dr. Sang-Tae Park for useful discussions.
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ASSOCIATED CONTENT
S Supporting Information *
A detailed description of the optical absorption in quantum dots, the analytical formulation of the structural dynamics calculations, and additional experimental data are presented. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
(1) Heiss, M.; Y. Fontana, Y.; Gustafsson, A.; Wüst, G.; Magen, C.; O’Regan, D. D.; Luo, J. W.; Ketterer, B.; Conesa-Boj, S.; Kuhlmann, A. V.; Houel, J.; Russo-Averchi, E.; Morante, J. R.; Cantoni, M.; Marzari, N.; Arbiol, J.; Zunger, A.; Warburton, R. J.; Fontcuberta i Morral, A. Nat. Mater. 2013, 12, 439−444. (2) Maldovan, M. Nature 2013, 503, 209−217. (3) Cahill, D. G.; Braun, P. V.; Chen, G.; Clarke, D. R.; Fan, S.; Goodson, K. E.; Keblinski, P.; King, W. P.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Phillpot, S. R.; Pop, E.; Shi, L. Appl. Phys. Rev. 2014, 1, 011305. (4) Luckyanova, M. N.; Garg, J.; Esfarjani, K.; Jandl, A.; Bulsara, M. T.; Schmidt, A. J.; Minnich, A. J.; Chen, S.; Dresselhaus, M. S.; Ren, Z.; Fitzgerald, E. A.; Chen, G. Science 2012, 338, 936−939. (5) Siemens, M. E.; Li, Q.; Yang, R.; Nelson, K. A.; Anderson, E. H.; Murnane, M. M.; Kapteyn, H. C. Nat. Mater. 2010, 9, 26−30. (6) Minnich, A. J.; Johnson, J. A.; Schmidt, A. J.; Esfarjani, K.; Dresselhaus, M. S.; Nelson, K. A.; Chen, G. Phys. Rev. Lett. 2011, 107, 095901. (7) Pernot, G.; Stoffel, M.; Savic, I.; Pezzoli, F.; Chen, P.; Savelli, G.; Jacquot, A.; Schumann, J.; Denker, U.; Mönch, I.; Deneke, Ch.; Schmidt, O. G.; Rampnoux, J. M.; Wang, S.; Plissonnier, M.; Rastelli, A.; Dilhaire, S.; Mingo, N. Nat. Mater. 2010, 9, 491−495. (8) Yu, J.-K.; Mitrovic, S.; Tham, D.; Varghese, J.; Heath, J. R. Nat. Nanotechnol. 2010, 5, 718−721. (9) Chang, C. W.; Okawa, D.; Garcia, H.; Majumdar, A.; Zettl, A. Phys. Rev. Lett. 2008, 101, 075903. (10) Hochbaum, A. I.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. Nature 2008, 451, 163− 167. (11) 4D Electron Microscopy: Imaging in Space and Time; Zewail, A. H., Thomas, J. M., Eds.; Imperial College Press: London, 2010. (12) Gao, M.; Lu, C.; Jean-Ruel, H.; Liu, L. C.; Marx, A.; Onda, K.; Koshihara, S.; Nakano, Y.; Shao, X.; Hiramatsu, T.; Saito, G.; Yamochi, H.; Cooney, R. R.; Moriena, G.; Sciaini, G.; Miller, R. J. D. Nature 2013, 496, 343−346. (13) Yang, D. S.; Lao, C.; Zewail, A. H. Science 2008, 321, 1660− 1664. (14) Baum, P.; Yang, D. S.; Zewail, A. H. Science 2007, 318, 788− 792. (15) Ruan, C.-Y.; Lobastov, V. A.; Vigliotti, F.; Chen, S.; Zewail, A. H. Science 2004, 304, 80−84. (16) Gedik, N.; Yang, D.-S.; Logvenov, G.; Bozovic, I.; Zewail, A. H. Science 2007, 316, 425−429. (17) Carbone, F.; Baum, P.; Rudolf, P.; Zewail, A. H. Phys. Rev. Lett. 2008, 100, 035501. (18) Highland, M.; Gundrum, B. C.; Koh, Y. K.; Averback, R. S.; Cahill, D. G.; Elarde, V. C.; Coleman, J. J.; Walko, D. A.; Landahl, E. C. Phys. Rev. B 2007, 76, 075337.
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. F
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(19) Mansart, B.; Cottet, M. J. G.; Mancini, G. F.; Jarlborg, T.; Dugdale, S. B.; Johnson, S. L.; Mariager, S. O.; Milne, C. J.; Beaud, P.; Grübel, S.; Johnson, J. A.; Kubacka, T.; Ingold, G.; Prsa, K.; Rønnow, H. M.; Conder, K.; Pomjakushina, E.; Chergui, M.; Carbone, F. Phys. Rev. B 2013, 88, 054507. (20) Milne, C. J.; Penfold, T. J.; Chergui, M. Coord. Chem. Rev. 2014, DOI: doi.org/10.1016/j.ccr.2014.02.013. (21) Reyes, K.; Smereka, P.; Nothern, D.; Millunchick, J. M.; Bietti, S.; Somaschini, C.; Sanguinetti, S.; Frigeri, C. Phys. Rev. B 2013, 87, 165406. (22) Blakemore, J. S. J. Appl. Phys. 1982, 53, R123−R181. (23) Maris, H. J.; Tamura, S. Phys. Rev. B 2012, 85, 054304. (24) Schäfer, S.; Liang, W.; Zewail, A. H. Chem. Phys. Lett. 2010, 493, 11−18. (25) Marzin, J.-Y.; Bastard, G. Solid State Commun. 1994, 92, 437− 442. (26) Simulation software NEXTNANO 3 (www.nextnano.de/ nextnano3/), (accessed Nov 30, 2013). (27) Schäfer, S.; Liang, W.; Zewail, A. H. J. Chem. Phys. 2011, 135, 214201. (28) Machinikowski, P.; Jacak, L. Phys. Rev. B 2005, 71, 115309. (29) Luo, T.; Garg, J.; Shiomi, J.; Esfarjani, K.; Chen, G. Europhys. Lett. 2013, 101, 16001. (30) Li, X. Q.; Nakayama, H.; Arakawa, Y. Phys. Rev. B 1999, 59, 5069. (31) Zibik, E. A.; Grange, T.; Carpenter, B. A.; Porter, N. E.; Ferreira, R.; Bastard, G.; Stehr, D.; Winnerl, S.; Helm, M.; Liu, H. Y.; Skolnick, M. S.; Wilson, L. R. Nat. Mater. 2009, 8, 803−807. (32) Grange, T.; Ferreira, R.; Bastard, G. Phys. Rev. B 2007, 76, 241304(R). (33) Schäfer, S.; Liang, W.; Zewail, A. H. New J. Phys. 2011, 13, 063030. (34) Bargheer, M.; Zhavoronkov, N.; Gritsai, Y.; Woo, J. C.; Kim, D. S.; Woerner, M.; Elsaesser, T. Science 2004, 306, 1771−1773. (35) Liang, W.; Vanacore, G. M.; Zewail, A. H. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 5491−5496. (36) Schäfer, S.; Liang, W.; Zewail, A. H. Chem. Phys. Lett. 2011, 515, 278−282. (37) Yang, D.-S.; Gedik, N.; Zewail, A. H. J. Phys. Chem. C 2007, 111, 4889−4919. (38) Raman, R. K.; Murdick, R. A.; Worhatch, R. J.; Murooka, Y.; Mahanti, S. D.; Han, T.-R. T.; Ruan, C.-Y. Phys. Rev. Lett. 2010, 104, 123401. (39) Koguchi, N.; Ishige, K.; Takahashi, S. J. Vac. Sci. Technol., B 1993, 11, 787−790. (40) Baum, P.; Zewail, A. H. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16105−16110. (41) Carbone, F.; Gedik, N.; Lorenzana, J.; Zewail, A. H. Adv. Condens. Matter Phys. 2010, 2010, 958618. (42) Tao, Z.; Han, T. T.; Ruan, C.-Y Phys. Rev. B 2013, 87, 235124. (43) Joseph, D. D.; Preziosi, L. Rev. Mod. Phys. 1990, 62, 375−391. (44) Chen, G. J. Heat Trans. 2002, 124, 320−328. (45) Yurtsever, A.; Schafer, S.; Zewail, A. H. Nano Lett. 2012, 12, 3772−3777. (46) X-Ray Diffraction; Warren, B. E., Eds., Dover Publications Inc.: New York, 1990.
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Diffraction of Quantum Dots Reveals Nanoscale Ultrafast Energy Localization Giovanni M. Vanacore,† Jianbo Hu,† Wenxi Liang,† Sergio Bietti,‡ Stefano Sanguinetti,‡ and Ahmed H. Zewail*,† †
Physical Biology Center for Ultrafast Science and Technology, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125, United States ‡ L-NESS and Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, Via Cozzi 53, I-20125 Milano, Italy S Supporting Information *
ABSTRACT: Unlike in bulk materials, energy transport in low-dimensional and nanoscale systems may be governed by a coherent “ballistic” behavior of lattice vibrations, the phonons. If dominant, such behavior would determine the mechanism for transport and relaxation in various energy-conversion applications. In order to study this coherent limit, both the spatial and temporal resolutions must be sufficient for the length-time scales involved. Here, we report observation of the lattice dynamics in nanoscale quantum dots of gallium arsenide using ultrafast electron diffraction. By varying the dot size from h = 11 to 46 nm, the length scale effect was examined, together with the temporal change. When the dot size is smaller than the inelastic phonon mean-free path, the energy remains localized in high-energy acoustic modes that travel coherently within the dot. As the dot size increases, an energy dissipation toward lowenergy phonons takes place, and the transport becomes diffusive. Because ultrafast diffraction provides the atomic-scale resolution and a sufficiently high time resolution, other nanostructured materials can be studied similarly to elucidate the nature of dynamical energy localization. KEYWORDS: Femtosecond diffraction, quantum dots, energy localization, ballistic transport
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sional (nanotubes9 and nanowires10) systems. A substantial dearth of relevant experimental data has been the case for quasizero-dimensional systems, quantum dots, although they have the potential to overcome many of the limitations of 2D and 1D systems because of the expected high energy-conversion efficiency.2 Time-resolved optical methods have been used to study the dynamics in nanostructures on the time scale of their occurrence.4,7−9 These valuable studies provide the overall temporal change, however the atomic-scale lattice dynamics cannot directly be observed because of the long wavelength of the optical probe used. With ultrafast electron pulses, timeresolved diffraction enables probing of the spatiotemporal behavior of atomic lattice structures.11,12 This methodology has become the method of choice in this laboratory for the investigation of correlated atomic motions,13 bond dilation,14 and structural transformations.15−17 In other laboratories, Xrays have been used in the studies of structural and thermal dynamics, and for an overview see ref 18 and the articles by Chergui.19,20
emiconductor quantum dots (QDs), nanometer-sized systems, have been widely studied in recent years because of their appealing physical properties, which strongly depend on the interaction between the electronic and lattice degrees of freedom. Changes in the electronic and lattice structures, through variation in the length scale, significantly affect the optical absorption, the carrier mobility and the heat transport, which are crucial aspects in the design of new generation optoelectronic devices1 and thermoelectric applications.2,3 In bulk systems, heat transport is usually governed by a random walk of energy-carrying phonons (diffusive regime), and Fourier’s law of heat conduction describes the energy transfer because of the continuum nature of lattice states. However, when the size of the structure becomes smaller than the inelastic phonon mean-free path, lattice phonons can travel ballistically (no collisions)4 and a Fourier description is no longer valid.5,6 Understanding and control of the transport at the atomic-scale level is of fundamental importance, especially when the length scale reaches that of molecular structures and nanodevices.2,7,8 Experimental investigation of phonon transport in nanoscale systems is challenging. The difficulty lies with the technique used for measuring the thermal properties of the nanostructures involved and with the complexity of systematically varying their size. So far, a particular emphasis has been given to twodimensional (thin films and superlattices)4,7,8 and one-dimen© XXXX American Chemical Society
Received: June 18, 2014 Revised: July 22, 2014
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Figure 1. Diffraction, morphology, and the temporal probing scheme. (a) Representative 3D AFM images of the investigated quantum dots with the indicated dimensions. (b) Schematic layout of the photon-pump/electron-probe, UEC, experiments. (c) Diffraction pattern of GaAs dots measured with the electron beam propagating 0.7° off the [110] zone axis and with an incidence angle of 1.6°. The pattern consists of Bragg spots arranged along a characteristic chevron pattern, together with surface wave resonances (SWR) from the substrate.
both the photon energy (1.55 eV) and the band gap of the GaAs dots (1.55 eV for QD-1, 1.45 eV for QD-2, and 1.42 eV for QD-3),25,26 the excitation of charge carriers occurs locally within the dots. Figure 1a depicts the three QDs studied, which are labeled according to their width and height. The figure also shows the diffraction pattern taken with the electron beam propagating nearly along the [110] direction. The assignment of Bragg spots in Figure 1c is that of the static crystal structure, which originates from the 3D morphology of the dot. Surface wave resonance (SWR) features from the substrate are also visible, attesting to the high crystalline order of the surface.27 The diffraction pattern is not symmetric because we chose the angle between the electron beam and the zone axis to be not zero but 0.7°, so that the off-axis reflections {11l}, {22l} and {33l} can be clearly visible. Shown in Figure 2a is the observed temporal change of the diffraction intensity for the (006) Bragg spot and for the three investigated dot dimensions at 4.6 mJ/cm2. The transient for each case can be well described by a fast exponential decay with a time constant τd and a slower recovery to the equilibrium state. As displayed in Figure 2b, the intensity change when normalized to the calculated bulk value gives an enhancement factor of two to three for the smallest dot (QD-1) when compared to those of QD-2 and QD-3. This behavior is the result of quantum confinement in the investigated dots that modifies their optical absorption depending on the size. The theoretical size-dependent optical absorbance, whose calculations are detailed in the Supporting Information, is shown as a solid line in Figure 2b; only QD-1 shows the enhancement because of the confinement-caused resonant absorption at the optical wavelength of 800 nm. The change of the diffraction intensity, I(t), is proportional to the square of the structure factor, which is determined by the atomic displacements. In this case, the motions are those of optical phonons, acoustic phonons, and incoherent thermal vibrations and these occur on different time scales. From the transient behavior of I(t), we obtained the time constant τd (Figure 1), and in Figure 3, we present the dependence of τd on the QD size, excitation fluence, and the angle, δ, between the
Here, we specifically employ ultrafast electron crystallography (UEC) in the reflection geometry to map out phonon dynamics in size-controlled GaAs quantum dots. We measured the transient change of their Bragg reflections, following an ultrafast optical excitation, as a function of the dot size and excitation fluence. The results indicate a size-dependent dynamics: the phonon transport is ballistic in small dots (length scale, h = 11 nm), and becomes diffusive in large dots (h = 46 nm). This four-dimensional sampling, in space and time, demonstrates that in the ballistic regime the excitation of thermal motions is quenched and the energy remains localized in high-energy acoustic phonons that travel coherently within the dot, whereas in the diffusive regime an incoherent energy dissipation toward low-energy modes takes place. Diffraction of Size-Controlled Quantum Dots. GaAs quantum dots were grown on Al0.3Ga0.7As by droplet epitaxy.21 This method allows for the generation of strain free and stoichiometric nanostructures with accurate control of the size and density. We studied three different samples with highly homogeneous distributions of shape and size for the dots. From analysis of atomic force microscopy (AFM) images, we determine their height, h, to be 11 nm (QD-1), 22 nm (QD2), and 46 nm (QD-3) with a corresponding base width, w, of 31, 54, and 110 nm (see Figure 1a). Because the inelastic meanfree path of high-energy acoustic phonons in GaAs22,23 is Λp ∼ 26−27 nm (see also Supporting Information), different dynamical regimes of phonon transport can then be explored. The UEC geometry is schematically shown in Figure 1b for the reflection mode; more details can be found in the Appendix. The ultrashort electron pulses were focused on the sample in a grazing incidence geometry. The dynamics were initiated by ultrashort laser pulses at 800 nm (120 fs, 1 kHz), with the fluence varying between 2.3 and 4.6 mJ/cm2. The diffracted electrons were recorded in the stroboscopic mode at different delay times between the excitation laser and the electron pulse. The same transients were obtained when the experiment was performed at different positions on the sample surface, and the transient electric field24 (surface charging) effect was found to be insignificant at all fluences used. Because the band gap of the AlGaAs substrate (1.8 eV) is larger than B
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dependence on the excitation fluence and on the angle δ is observed for QD-3, whose size is larger than Λp. Slower rates were observed at higher fluences and for smaller angles δ. For example, at 4.6 mJ/cm2 the reflections (006) (δ = 0°) and (3̅35) (δ = 40.3°) give τd ≈ 12.5 and 4.1 ps, respectively; the values drop down to 5.5 and 3.1 ps at 2.3 mJ/cm2. Coherent Energy Localization. For QD-1 the reported value of τd (3.7 ps) is in good agreement with the decay time of the longitudinal optical (LO) phonon in GaAs at 300 K, τanh ≈ 3.5 ps.28−30 The generation of LO phonon occurs in ∼100 fs by electron−phonon coupling;28 a polaron picture28,31,32 has been used to describe the interaction between the electron distribution and the optical phonon. Because energy and momentum must be conserved, the LO phonon decays into the high-energy acoustic phonons by the anharmonic coupling,28,31,32 with τanh ≈ 3.5 ps. Because the diffraction intensity change is related to the mean-square amplitude of the phonons involved, we have calculated, following the approach of ref 33, for QD-1, when only the high-energy acoustic phonons are populated, an 11% decrease for the (006) reflection, which is in close agreement with the experimental result of 13% (see Figure 2a). Moreover, as shown in Figure 3b, there is no discernible fluencedependence for τd. This behavior is supported by the theoretically calculated29 dependence of τanh on lattice temperature. For the fluence range used here, a maximum change of only 0.3 ps is estimated from the calculations. The absence of additional decay components at longer times, besides the 3.7 ps component, suggests that the hot acoustic phonons are long-lived and that for QD-1 they persist on the time scale reported here; dynamically speaking, they travel unperturbed within the dot with no collisions and maintaining defined phases during the propagation, a ballistic coherent transport (see Figure 4b (right panel)). In this case, the lattice preserves its structural coherence and a breathing oscillation of the interatomic distance should be visible in the position of Bragg spots.34,35 As shown in Figure 4a, for QD-1 we observed oscillations with a frequency of 0.33 THz (period of 3 ps) for (006) reflection; the same was observed for the (2̅26) reflection. This value matches that obtained from a standingwave condition for a hemispherical nanostructure, ⟨f⟩ ≈ vhe/h ≈ 0.29−0.36 THz, when the velocity of high-energy acoustic
Figure 2. Diffraction intensity transients and optical absorption. (a) Experimentally observed temporal change of the diffraction intensity for the (006) Bragg reflection measured at 4.6 mJ/cm2 for QD-1 (blue), QD-2 (green), and QD-3 (red). (b) Intensity change for the reflections (006) (squares), (2̅26) (circles), (1̅15) (downward pointing triangles), and (3̅35) (upward pointing triangles), normalized to the calculated bulk value and plotted as a function of the dot height. The orange solid line is the calculated size-dependent optical absorbance for the investigated nanostructures (see Supporting Informations for details).
scattering vector, s, and the [001] axis (see inset in Figure 3c). Because local absorption occurs only within the dot, the created phonons transfer their energy to the substrate along the perpendicular to the interface direction since transverse dissipation of energy in the dot is not feasible. Thus, in this case the height (h) of the dot defines the length scale of phonon transport. For QD-1, whose size is smaller than the inelastic mean-free path, Λp, τd ≈ 3.7 ps for all the Bragg spots studied and for all the fluences used. In contrast, a strong
Figure 3. Size, fluence, and angle dependencies. (a) Size and fluence dependence of the experimental (full symbols) and simulated (solid lines) time constant, τd, for the (006) Bragg spot at 4.6 mJ/cm2 (black), 3.4 mJ/cm2 (gray), and 2.3 mJ/cm2 (light gray). Λp is the phonon inelastic mean-free path in GaAs. (b) Experimental (open circles) and simulated (solid lines) τd as a function of the excitation fluence for the (006) spot in QD-1 (blue) and QD-3 (red). (c) Experimental (open circles) and simulated (solid lines) τd as a function of the angle δ (see inset) at 4.6 mJ/cm2 for QD-1 (blue) and QD-3 (red). The time constant for different δ’s is obtained by monitoring the behavior of different Bragg reflections: δ = 0° for (006), δ = 11.4° for (1̅17), δ = 15.8° for (1̅15), δ = 25.2° for (2̅26), and δ = 40.3° for (3̅35). C
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4c (right panel)), leading to a progressive randomization of the phases. The consequence of this incoherent diffusive transport is the loss of coherence in structural dynamics and, hence, the absence of breathing oscillations as observed experimentally; see Figure 4a. When the transport is diffusive, the diffusion time scales with the square of the diffusion length. Because the investigated dots have a “dome-like” shape, a spatial dependence of the time constant τd should be expected. This dependence was probed by monitoring the change of τd with the angle δ. As shown in Figure 3c, no discernible dependence is observed for QD-1, whereas in QD-3 larger values of τd are measured at smaller δ angles, confirming the validity of the diffusive picture in large dots and its absence in small dots. Additional observations support this picture. Our results for large dots are consistent with previous ultrafast diffraction measurements on GaAs bulklike structures.37 A time constant of 12 ps was measured for the (008) spot at 10.9 mJ/cm2, which correlates well with the value (12.5 ps) measured for the (006) spot in QD-3 at high fluence. Moreover, because phonon−phonon scattering randomizes the distribution of acoustic modes, besides the intensity change one also expects a broadening of the Bragg peaks.37 For QD-3, the transient increase of the width for the observed Bragg spots is about a factor of four larger than that measured for QD-1. Structural Dynamics. To elucidate the different regimes of energy localization we have calculated for the recorded Bragg reflections the transient behavior of the structure factor (and diffraction intensity) when a size-dependent phonon transport is considered. The model, which is detailed in the Appendix and the Supporting Information, separately incorporates the ballistic, diffusive, and the dissipative contributions to the total heat flux. The results of the simulations for the time scale of the structural dynamics are plotted as solid lines in Figure 3, and a satisfactory agreement with the experimental results is obtained. The observed dependence of τd on the angle δ is also reproduced in the calculated space-resolved maps of the time constant, shown in Figure 4b,c (left panels). For QD-1, the calculated τd is 3.6−3.7 ps and uniformly distributed inside the dot, whereas in QD-3 the calculated τd is maximum (11 ps) at the center and exhibits a progressive decrease (down to 5 ps) toward the sides of the dot. Besides the time constant, when the intensity change is monitored as a function of the angle δ, the spatial distribution of the mean-square displacement of the atoms, ⟨u2⟩, can be explored. In bulk systems, ⟨u2⟩ has an isotropic distribution. In such case, ⟨u2⟩ is the same for all Bragg reflections37,38 and can be obtained through the Debye− Waller relation: ⟨u2⟩ ∝ W = (1/s2)ln(I0/I). In Figure 5a the Debye−Waller factor, W, is plotted for the (006), (22̅ 6) and (3̅35) reflections as measured for QD-1, and in Figure 5b W is shown as a function of the angle δ. The existence of an anisotropic structural deformation along the [001] direction is evident in the progressive decrease of W for larger δ. This is quantitatively confirmed by the simulated map of the local atomic displacement, displayed in Figure 5c for QD-1, and by the calculated change of W with δ, as shown in Figure 5a,b. The implications of the results obtained from this fourdimensional sampling are worth mentioning. Because theoretical calculations4,29 have shown that most of the heat is conducted by low-energy phonons, we expect the small dots, where only high-energy phonons are excited, to exhibit a very low thermal conductivity. This makes them particularly suitable for thermoelectric and heat management applications, where
Figure 4. Coherent oscillations and phonon transport regimes. (a) Transient behavior of the position change of the (006) Bragg spot for QD-1 (left) and QD-3 (right). For clarity, the background due to incoherent signal was subtracted. For QD-1, coherent oscillations are observed with a frequency of 0.33 THz (period of 3 ps). (b, left) Space-resolved map of the calculated time constant τd for QD-1. (b, right) Schematic representation of ballistic coherent phonon transport across the small-sized dot. The blue spark represents the energy transfer from the laser to the acoustic phonon population. (c, left) Space-resolved map of the calculated time constant τd for QD-3. (c, right) Schematic representation of the diffusive, incoherent phonon transport across the large-sized dot. The small red sparks represent phonon−phonon scattering processes within the acoustic population.
phonons, vhe ≈ 3.2−4.0 km/s,22,29 and h = 11 nm is considered (see Supporting Information). As shown in Figure 3a,b, for QD-3 the time constant τd is longer than the value observed for smaller dots, indicating that, besides the LO phonon decay, an additional mechanism of energy exchange is present. If we consider that only high-energy acoustic phonons are populated, a 2% decrease in intensity is calculated33 for the (006) reflection, contrary to our result of 7% change (see Figure 2a). This calculated value of 2% differs from that of QD-1 (11%) because of the difference in the dot’s optical absorption. When we allow for the relaxation of the hot phonons to the low-energy ones, which may take place via anharmonic scattering, an 8% drop is predicted because the diffraction intensity is inversely proportional to the square of the phonon frequency. The 8% is indeed close to the measured value. Furthermore, the strong fluence-dependence of τd observed for QD-3 (see Figure 3b) is typical of an energy equilibration process whose time scale for the relaxation is predicted to be proportional to the extent of the external excitation.36 In our case, this translates into a time constant that lengthens when the fluence increases. These observations suggest that in QD-3 phonon−phonon scattering dominates. The hot acoustic phonons experience “collisions” during their propagation within the dot (see Figure D
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(equivalent pressure ∼5 × 10−5 Torr) was irradiated on the surface to crystallize the Ga droplets into GaAs dots, while the substrate temperature was reduced down to 150 °C. After the growth, the samples were capped with a low-temperature amorphous As layer able to prevent the formation of native oxide due to air exposure during the following transfer to the UEC setup. Prior to the UEC experiments, the As capping layer was removed by annealing the samples in situ at 250 °C for 20 min and at 320 °C for 30 min, producing a clean and atomically flat surface between the dots. The size distributions of the dots have been obtained by atomic force microscopy. The mean values and the standard deviations (in parentheses) of their height and base width are as follows: the height, h, is 46(5.5) nm, 22(2.5) nm, and 11(2.1) nm for QD-3, QD-2, and QD-1, respectively; the width, w, is 110(3.4) nm, 54(3.2) nm, and 31(3) nm for QD-3, QD-2, and QD-1, respectively. UEC Experiment
The setup for UEC experiments at Caltech has already been described in details elsewhere.27,37 Briefly, electron pulses with energy per particle of 30 keV and sub-picosecond pulse width are generated in a photoelectron gun (Kimball Physics, Inc.) after irradiation of a LaB6 photocathode with 120 fs UV laser pulses (λ = 266 nm). The electron beam is focused at grazing incidence (0.5−2.5°) on the sample surface. The sample is mounted on a 5-axis goniometer, allowing for simultaneous adjustment of the incidence and the azimuthal angles. The electron beam diffracted from the surface is recorded on a phosphor-screen/MCP/CCD assembly. The ultrafast dynamics are initiated by femtosecond laser pulses (1 kHz, 120 fs, 800 nm) focused in normal incidence to the sample surface. The laser-induced change of the diffraction pattern is monitored in the stroboscopic mode by varying the delay time between the electron and the laser pulses. The velocity mismatch and the non-coaxial geometry between electrons and photons are responsible for a different arrival time of the electrons at different regions of the sample surface illuminated by the pump pulse. This effect is compensated for by tilting the wavefront of the optical pulse with respect to its propagation direction.14,40 The fluence of the excitation laser at the sample position was calibrated by scanning a knife edge across the laser profile and recording the passing residual pulse energy. In the grazing-incidence geometry, transient electric field (TEF) may contribute to temporal changes of the detected diffraction features (see ref 24 and references therein). However, here control experiments were carried out, supporting that TEF effect is negligible in the fluence range employed. First, we monitored the time-dependent angular deflection of the direct electron beam (ΔϑDB) tangential to the sample surface, which provides a quantitative measure of the field-induced deflection.24 At the highest fluence used (4.6 mJ/ cm2), ΔϑDB was found to be 21 μrad. This value should be compared with the typical diffraction angle, ϑdiff, and angular width, ϑw, of the investigated Bragg reflections; for the (006) spot, ϑdiff = 39.6 mrad and ϑw = 7.9 mrad, orders of magnitude larger than ΔϑDB. Second, we found that Δϑ scales linearly with the diffraction angle for the (00n) Bragg spots: Δϑ/ϑ is 1.9 × 10−3 for the (004), 2 × 10−3 for the (006), and 1.8 × 10−3 for the (008) (with an error of ±0.1 × 10−3) for the case of QD-1. Because the field-induced deflection is independent of the diffraction angle,24 this observed scaling behavior further confirms that the TEF is negligible.
Figure 5. Structural dynamics and its anisotropy. (a) Experimental (left) and calculated (right) transient behavior of the Debye−Waller factor, W, derived from the reflections (006) (dark blue), (2̅26) (blue), and (3̅35) (light blue) in QD-1. The inset is a schematic representation of the scattering vectors of the shown Bragg reflections plotted in the [110] plane. (b) Experimental (open circles) and calculated (solid lines) Debye−Waller factors as a function of the angle δ for QD-1 (blue), QD-2 (green), and QD-3 (red). (c) Simulated space-resolved map of the local atomic displacements, represented in the [110] plane at a delay time of 12 ps.
the energy-conversion efficiency is inversely proportional to the thermal conductivity. Spatiotemporal visualization of structural dynamics in GaAs quantum dots reveals the atomic-scale details of the sizedependent energy localization in nanoscale systems. Structural changes are examined by observing the transient behavior of their electron diffraction following an ultrafast optical excitation. For dots smaller than the phonon inelastic meanfree path (ballistic regime), the excitation of thermal motions is quenched and structural dynamics is mainly controlled by the anharmonic decay of the optical phonon formed by the unit cell atoms. In this case, the energy remains localized in high-energy acoustic modes that “travel” coherently within the dot. As the dot size increases, slower phonon−phonon scattering within the acoustic population takes place, and the transport becomes diffusive in nature. These results are fundamental for the understanding of energy conversion in nanoscale materials, and for the control of properties involving thermal conductivity and optical design.
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APPENDIX
Quantum Dots Growth
The investigated samples were grown by droplet epitaxy (DE) technique in a Gen II molecular beam epitaxy (MBE) system equipped with As, Ga, and Al cells. In situ electron diffraction is used to monitor the growth process.39 The substrate is n-type doped GaAs(001). After the oxide removal, a 250 nm layer of GaAs and a 90 nm layer of Al0.3Ga0.7As were grown at 580 °C. The formation of GaAs quantum dots is obtained with a twostep procedure. First, Ga droplets are created on the surface by the deposition of gallium: 2, 1, and 0.25 monolayers (MLs) are deposed at 350 °C with a flux of 0.05 ML/s for the samples QD-3, QD-2, and QD-1, respectively. Then, a flux of As4 E
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Structural Dynamics Simulations
Author Contributions
We carried out simulations of structural dynamics within the framework of a phenomenological model that includes electron−phonon interaction and non-Fourier phonon transport. The main equations are reported in the Supporting Information. Here, we briefly describe the main ideas. The optical excitation creates electron-hole carriers with a carrier density Nex that depends on the absorbed fluence. Such excitation induces a weakening of the interatomic bonds by transferring electrons from bonding (valence band) to antibonding (conduction band) states. This leads to an instantaneous stress, electronic pressure Pe, which for GaAs can be written as34 Pe = Nexadef, where adef is the deformation potential. The excitation of acoustic phonons from the decay of optical phonons is able to drive an additional stress on the lattice, quantitatively described by the phonon pressure Pph = C· α(TA(t) − Tref), where C is the stiffness tensor, α is the linear expansion coefficient, Tref = 300 K, and TA is the effective temperature of the acoustic phonon population. The total stress within the dot induced by the optical excitation is given by the combination of the electron and phonon pressures: σ0(t) = Pe + Pph. The acoustic phonon temperature TA can be calculated by invoking a three-temperature model (3TM),41,42 where the carriers, the optical phonons, and the acoustic phonons subsystems are each in a local state defined by the temperatures Te, TO, and TA. The size-dependent acoustic phonon transport is included in the TA-equation of the 3TM by separately incorporating the diffusive, the dissipative, and the ballistic contributions. The diffusive component is given by the Fourier’s law of heat conduction. The dissipative contribution, which takes into account the thermalization of the hot phonons, is included using the Cattaneo correction43 to a pure Fourier equation. The ballistic transport is modelled by introducing an additional component, whose weight depends on the ratio Λp/h, which compensates for the over-prediction of the heat flux by the diffusive-dissipative term when h < Λp. Its analytical expression is derived from two similar approaches, giving equivalent results: (i) the ballistic-diffusive model by Chen,44 and (ii) the multiphysics model of Siemens et al.5 (see also Supporting Information for further details). With the boundary conditions defined by the dot geometry, we numerically solved the modified-3TM model coupled with the stress-strain equations using σ0 as initial stress.45 The temporal behavior of the atomic displacement field, u, is then calculated, from which the transient change of the scattering factor (and the diffraction intensity) for the observed Bragg spots can be finally obtained.46
G.M.V. and A.H.Z. designed the experiments; S.B. and S.S. provided the samples; G.M.V., W.L., and J.H. performed the measurements; G.M.V. and J.H. analyzed the data and performed the simulations; G.M.V. and A.H.Z. wrote the paper with the support of J.H., W.L., S.S., and S.B.
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Science Foundation and the Air Force Office of Scientific Research in the Center for Physical Biology at Caltech supported by the Gordon and Betty Moore Foundation. We thank Dr. Paolo Biagioni for the finite difference time domain (FDTD) simulations and Dr. Sang-Tae Park for useful discussions.
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ASSOCIATED CONTENT
S Supporting Information *
A detailed description of the optical absorption in quantum dots, the analytical formulation of the structural dynamics calculations, and additional experimental data are presented. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
(1) Heiss, M.; Y. Fontana, Y.; Gustafsson, A.; Wüst, G.; Magen, C.; O’Regan, D. D.; Luo, J. W.; Ketterer, B.; Conesa-Boj, S.; Kuhlmann, A. V.; Houel, J.; Russo-Averchi, E.; Morante, J. R.; Cantoni, M.; Marzari, N.; Arbiol, J.; Zunger, A.; Warburton, R. J.; Fontcuberta i Morral, A. Nat. Mater. 2013, 12, 439−444. (2) Maldovan, M. Nature 2013, 503, 209−217. (3) Cahill, D. G.; Braun, P. V.; Chen, G.; Clarke, D. R.; Fan, S.; Goodson, K. E.; Keblinski, P.; King, W. P.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Phillpot, S. R.; Pop, E.; Shi, L. Appl. Phys. Rev. 2014, 1, 011305. (4) Luckyanova, M. N.; Garg, J.; Esfarjani, K.; Jandl, A.; Bulsara, M. T.; Schmidt, A. J.; Minnich, A. J.; Chen, S.; Dresselhaus, M. S.; Ren, Z.; Fitzgerald, E. A.; Chen, G. Science 2012, 338, 936−939. (5) Siemens, M. E.; Li, Q.; Yang, R.; Nelson, K. A.; Anderson, E. H.; Murnane, M. M.; Kapteyn, H. C. Nat. Mater. 2010, 9, 26−30. (6) Minnich, A. J.; Johnson, J. A.; Schmidt, A. J.; Esfarjani, K.; Dresselhaus, M. S.; Nelson, K. A.; Chen, G. Phys. Rev. Lett. 2011, 107, 095901. (7) Pernot, G.; Stoffel, M.; Savic, I.; Pezzoli, F.; Chen, P.; Savelli, G.; Jacquot, A.; Schumann, J.; Denker, U.; Mönch, I.; Deneke, Ch.; Schmidt, O. G.; Rampnoux, J. M.; Wang, S.; Plissonnier, M.; Rastelli, A.; Dilhaire, S.; Mingo, N. Nat. Mater. 2010, 9, 491−495. (8) Yu, J.-K.; Mitrovic, S.; Tham, D.; Varghese, J.; Heath, J. R. Nat. Nanotechnol. 2010, 5, 718−721. (9) Chang, C. W.; Okawa, D.; Garcia, H.; Majumdar, A.; Zettl, A. Phys. Rev. Lett. 2008, 101, 075903. (10) Hochbaum, A. I.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. Nature 2008, 451, 163− 167. (11) 4D Electron Microscopy: Imaging in Space and Time; Zewail, A. H., Thomas, J. M., Eds.; Imperial College Press: London, 2010. (12) Gao, M.; Lu, C.; Jean-Ruel, H.; Liu, L. C.; Marx, A.; Onda, K.; Koshihara, S.; Nakano, Y.; Shao, X.; Hiramatsu, T.; Saito, G.; Yamochi, H.; Cooney, R. R.; Moriena, G.; Sciaini, G.; Miller, R. J. D. Nature 2013, 496, 343−346. (13) Yang, D. S.; Lao, C.; Zewail, A. H. Science 2008, 321, 1660− 1664. (14) Baum, P.; Yang, D. S.; Zewail, A. H. Science 2007, 318, 788− 792. (15) Ruan, C.-Y.; Lobastov, V. A.; Vigliotti, F.; Chen, S.; Zewail, A. H. Science 2004, 304, 80−84. (16) Gedik, N.; Yang, D.-S.; Logvenov, G.; Bozovic, I.; Zewail, A. H. Science 2007, 316, 425−429. (17) Carbone, F.; Baum, P.; Rudolf, P.; Zewail, A. H. Phys. Rev. Lett. 2008, 100, 035501. (18) Highland, M.; Gundrum, B. C.; Koh, Y. K.; Averback, R. S.; Cahill, D. G.; Elarde, V. C.; Coleman, J. J.; Walko, D. A.; Landahl, E. C. Phys. Rev. B 2007, 76, 075337.
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. F
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(19) Mansart, B.; Cottet, M. J. G.; Mancini, G. F.; Jarlborg, T.; Dugdale, S. B.; Johnson, S. L.; Mariager, S. O.; Milne, C. J.; Beaud, P.; Grübel, S.; Johnson, J. A.; Kubacka, T.; Ingold, G.; Prsa, K.; Rønnow, H. M.; Conder, K.; Pomjakushina, E.; Chergui, M.; Carbone, F. Phys. Rev. B 2013, 88, 054507. (20) Milne, C. J.; Penfold, T. J.; Chergui, M. Coord. Chem. Rev. 2014, DOI: doi.org/10.1016/j.ccr.2014.02.013. (21) Reyes, K.; Smereka, P.; Nothern, D.; Millunchick, J. M.; Bietti, S.; Somaschini, C.; Sanguinetti, S.; Frigeri, C. Phys. Rev. B 2013, 87, 165406. (22) Blakemore, J. S. J. Appl. Phys. 1982, 53, R123−R181. (23) Maris, H. J.; Tamura, S. Phys. Rev. B 2012, 85, 054304. (24) Schäfer, S.; Liang, W.; Zewail, A. H. Chem. Phys. Lett. 2010, 493, 11−18. (25) Marzin, J.-Y.; Bastard, G. Solid State Commun. 1994, 92, 437− 442. (26) Simulation software NEXTNANO 3 (www.nextnano.de/ nextnano3/), (accessed Nov 30, 2013). (27) Schäfer, S.; Liang, W.; Zewail, A. H. J. Chem. Phys. 2011, 135, 214201. (28) Machinikowski, P.; Jacak, L. Phys. Rev. B 2005, 71, 115309. (29) Luo, T.; Garg, J.; Shiomi, J.; Esfarjani, K.; Chen, G. Europhys. Lett. 2013, 101, 16001. (30) Li, X. Q.; Nakayama, H.; Arakawa, Y. Phys. Rev. B 1999, 59, 5069. (31) Zibik, E. A.; Grange, T.; Carpenter, B. A.; Porter, N. E.; Ferreira, R.; Bastard, G.; Stehr, D.; Winnerl, S.; Helm, M.; Liu, H. Y.; Skolnick, M. S.; Wilson, L. R. Nat. Mater. 2009, 8, 803−807. (32) Grange, T.; Ferreira, R.; Bastard, G. Phys. Rev. B 2007, 76, 241304(R). (33) Schäfer, S.; Liang, W.; Zewail, A. H. New J. Phys. 2011, 13, 063030. (34) Bargheer, M.; Zhavoronkov, N.; Gritsai, Y.; Woo, J. C.; Kim, D. S.; Woerner, M.; Elsaesser, T. Science 2004, 306, 1771−1773. (35) Liang, W.; Vanacore, G. M.; Zewail, A. H. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 5491−5496. (36) Schäfer, S.; Liang, W.; Zewail, A. H. Chem. Phys. Lett. 2011, 515, 278−282. (37) Yang, D.-S.; Gedik, N.; Zewail, A. H. J. Phys. Chem. C 2007, 111, 4889−4919. (38) Raman, R. K.; Murdick, R. A.; Worhatch, R. J.; Murooka, Y.; Mahanti, S. D.; Han, T.-R. T.; Ruan, C.-Y. Phys. Rev. Lett. 2010, 104, 123401. (39) Koguchi, N.; Ishige, K.; Takahashi, S. J. Vac. Sci. Technol., B 1993, 11, 787−790. (40) Baum, P.; Zewail, A. H. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16105−16110. (41) Carbone, F.; Gedik, N.; Lorenzana, J.; Zewail, A. H. Adv. Condens. Matter Phys. 2010, 2010, 958618. (42) Tao, Z.; Han, T. T.; Ruan, C.-Y Phys. Rev. B 2013, 87, 235124. (43) Joseph, D. D.; Preziosi, L. Rev. Mod. Phys. 1990, 62, 375−391. (44) Chen, G. J. Heat Trans. 2002, 124, 320−328. (45) Yurtsever, A.; Schafer, S.; Zewail, A. H. Nano Lett. 2012, 12, 3772−3777. (46) X-Ray Diffraction; Warren, B. E., Eds., Dover Publications Inc.: New York, 1990.
G
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