Gigahertz Coherent Guided Acoustic Phonons in ... - ACS Publications

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Gigahertz Coherent Guided Acoustic Phonons in AlN/GaN Nanowire Superlattices Pierre-Adrien Mante,† Yueh-Chun Wu,‡ Yuan-Ting Lin,§ Cheng-Ying Ho,§ Li-Wei Tu,§ and Chi-Kuang Sun*,†,‡,⊥ †

Molecular Imaging Center, National Taiwan University, Taipei 10617, Taiwan Department of Electrical Engineering and Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan § Department of Physics and Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan ⊥ Research Center for Applied Sciences and Institute of Physics, Academia Sinica, Taipei 11529, Taiwan ‡

ABSTRACT: The generation of guided acoustic phonons in the GHz range in GaN/AlN superlattices grown atop a GaN nanowire is presented. Combined with a femtosecond laser, ultrafast pump−probe spectroscopy allows the generation and detection of guided acoustic phonons at different frequencies in the nanowire superlattices. The capability of the nanowire superlattices to be excellent detectors of acoustic phonons at specific frequencies is then used to observe the strong dispersion, as a result of nanoconfinement, of guided acoustic phonons after their propagation in the nanowire. The generation of high frequency coherent guided acoustic phonons could be useful not only to realize an acoustic transducer with a nanolateral size but also as a source to understand the thermal behavior of nanowires. KEYWORDS: Ultrafast pump−probe spectroscopy, nanowires, superlattice, guided acoustic phonons

T

is the achievable lateral resolution. The area of generation of phonons is given by the laser spot size, which means that it is limited by the diffraction effect.14 Different methods have been developed to overcome the optical limitation in the lateral resolution. For example, the use of a near-field scanning optical microscope (NSOM) to collect the light can help increase the resolution up to 200 nm.15−17 Nevertheless, the use of the NSOM only allows one to collect a small quantity of light. Another method to increase the lateral resolution is by manipulating the acoustic transducer properties.18 In this case, a piezoelectric transducer is used and, with such a transducer, the generation can be saturated. By presaturating the transducer with shaped light pulses, it was possible to generate acoustic waves with a lateral size of 100 nm. Increasing the lateral resolution should also be possible by using nanostructures. Indeed, using a nanostructure with dimensions smaller than the laser spot size will result in the generation of an acoustic wave with dimensions set by the nanostructure. Nanowires are excellent candidates for this purpose. However, up to now, only confined acoustic vibrations

o realize the imaging of microscopic objects, optical and electron microscopy are extremely efficient methods. However, these techniques have some limitations. For example, electronic microscopy is limited to the study of the surface of materials, due to the weak penetration of electrons. In a similar way, optical imaging in opaque materials can only give information on the surface. Moreover, both techniques cannot be used to examine the mechanical properties of microscopic objects. To be able to retrieve viscoelastic information, acoustic microscopy has been developed. Acoustic imaging is an efficient technique applied in a wide variety of fields. It is used for medical purposes,1 underwater imaging,2 or nondestructive testing.3 The higher the frequency of the acoustic waves, the better the in-depth resolution is. The picosecond ultrasonics technique4 takes advantage of the interaction of femtosecond light pulses with absorbing materials to generate acoustic waves in the sub-terahertz range. Using this technique, it is possible to realize measurements of the size of embedded structures.5 Thanks to the development of new transducers, such as quantum wells,6−8 superlattices,9−11 or quantum dots,12 acoustic waves with frequencies in the terahertz range can be generated, which means that even higher in-depth acoustic resolution on the order of a few nanometers is possible.13 However, this technique still has some limitations. One of these © 2013 American Chemical Society

Received: December 6, 2012 Revised: January 12, 2013 Published: February 8, 2013 1139

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have been observed.19−22 Therefore, a transducer for acoustic waves propagating inside nanostructures has to be designed. The observation of traveling acoustic waves in nanowires could also assist the study of acoustic phonon propagation behavior inside the nanostructures, thus leading to quantitative investigation of different thermal properties of such structures. In particular, it could lead to a better understanding of the origin of the low thermal conductivity of nanowires, as well as the diameter and temperature dependence,23,24 which are much needed for the manipulation of thermal properties and the creation of new devices.25−27 In this letter, we propose and demonstrate that the use of superlattices grown atop the nanowires enables the generation of multiple frequency gigahertz coherent guided acoustic phonons (CGAPs). We first derive the model to calculate the dispersion relation of acoustic phonons in a nanowire and in a nanowire superlattice (NWSL). We then investigate the acoustic behavior of the NWSLs using ultrafast pump−probe spectroscopy. We analyze the generation and propagation of acoustic waves inside the nanowire and compare the results with the calculated dispersion relation. We then examine the dispersion of CGAPs in the NWSL. Our observation of gigahertz guided acoustic phonons in the proposed NWSL is the first step toward high lateral and depth resolution acoustic imaging. Propagating acoustic waves in nanowires are responsible for heat transport, and their analysis is much needed. Up to now, the experimental investigation of the acoustic behavior of nanowires has been limited to confined acoustic modes.19−22 To enable the generation of guided acoustic waves, a transducer has to be deposited on top of the nanowires. Superlattices are efficient acoustic transducers, which also allow detection of acoustic waves at specific frequencies determined by the periodicity of the superlattice. Our purpose is to design a superlattice to allow the generation and detection of CGAPs at multiple frequencies so that the dispersion can be analyzed. To do so, we need the dispersion relations of acoustic waves in nanowires and NWSL. The propagation of acoustic waves in a cylinder has been studied independently by Pochhammer28 and Chree.29 A semianalytical model allowing the description of the vibrational behavior of structures in any type of geometry has been developed by Visscher et al.30 This model has already been derived for nanowires,31,32 and we applied it to wurtzite nanowires in the case of a circular cross section. Considering the theory of elasticity, we can write the lagrangian of the system as follows: 1 2

L=

⎡

[ω 2E − Γ]a = 0

j

l

⎦

Eiα , lβ = δi , l

∑ aiα Φα α

Γiα , lβ =

⎛ x ⎞m⎛ y ⎞n iqz ⎜ ⎟ ⎜ ⎟ e ⎝r ⎠ ⎝r ⎠

∫V Φ*βΦα dV

(5)

Cijkl V

∫V

∂Φ*β ∂Φα dV ∂xj ∂xk

(6)

Using this algorithm, we are able to calculate the dispersion relation of GaN and AlN nanowires. Due to the generation method, the laser spot size is bigger than the rod diameter and only dilatational modes can be generated. Flexural or torsional modes are not calculated in the following. The dispersion relations of dilatational phonon modes of a 75 nm diameter GaN nanorod with a circular cross section are reproduced in Figure 1a.

Figure 1. (a) Dispersion relations of dilatational phonon modes of circular GaN nanowire with a diameter of 75 nm. (b) Dispersion relations of phonon modes in an AlN/GaN superlattice in the case of a linear dispersion relation (without nanowire, black curve) or for guided acoustic phonons (with nanowire, red curve). The dispersion relations of higher order guided acoustic phonons are not represented.

Now that we have obtained the dispersion relation for GaN and AlN nanowires, we combined them to obtain the dispersion relation of a NWSL. In order to do so, we used Rytov’s equation:33

(1)

cos(Qd) = cos(q1d1) cos(q2d 2) − sin(q2d 2)

1 + E2 sin(q1d1) 2E (7)

where di is the thickness of medium i, d is the period of the superlattice, qi is the wave vector in medium i, Q is the wave vector of the superlattice, and E is the ratio between the acoustic impedance of media 1 and 2. In Figure 1b, we present the phonon dispersion relations of an AlN/GaN superlattice with a GaN thickness of 56 and 42 nm for AlN, in the case of a linear dispersion relation based on

(2)

Using the method developed by Visscher et al.,30 we chose a basis function which is a power of the directions x, y, and z Φα =

ρ V

and

where ui is the lattice displacement component, Cijkl is the elastic stiffness tensor, ρ is the mass density, and ω is the angular frequency. We then expand the displacement fields in terms of a complete set of function Φα as

ui =

(4)

with

∂u ∂u ⎤

∫V ⎢⎢ρω2uiui − Cijkl ∂xi ∂xk ⎥⎥ dV ⎣

where r is the radius of the nanowire and q is the longitudinal wave vector of acoustic phonon modes. The index α = {m, n} is a set of two non-negative integers satisfying the condition m + n ≤ N, where N is set to obtain convergence of the results. In our case, no differences were observed for N ≥ 12; the following results are therefore obtained for N = 12. By applying the expansion of eq 3 to eq 1 and taking the derivative with respect to the expansion coefficients, we can reformulate the problem into an eigenvalue problem:

(3) 1140

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Figure 2. Scanning electron microscope and transmission electron microscope images of the NWSL.

equilibrium position, thus launching acoustic waves. This generation mechanism is often referred to as the deformation potential mechanism.35−37 Another kind of acoustic generation can occur through the thermo-elastic mechanism. The excitation of electrons increases their temperature, and the electrons then give their excess energy to the lattice by electron−phonon collisions.4,38 However, with our experimental conditions, this second generation mechanism is negligible, about 138 times smaller, compared to the deformation potential.39 The time-delayed probe beam then monitors the acoustic phonon induced reflectivity changes. The obtained traces are then averaged to increase the signal-to-noise ratio. The transient reflectivity of the sample obtained with UV or IR probe is presented in Figure 3.

their bulk properties (black lines), i.e., for a superlattice grown on a bulk sample and for the first order of guided acoustic phonons (red lines). A consequence of the periodicity of the superlattice can be seen for both curvesthe dispersion relations are folded into a mini-Brillouin zone. The gap opening at the zone-center and the zone-edges, except for the first zoneedge gap, occurs at lower frequency for the guided acoustic phonons. This is the result of the confinement: the velocity of phonons is reduced, creating a lower frequency gap opening. To verify the generation and propagation of CGAP in a NWSL, we grew GaN nanowires along the surface normal of a Si substrate by plasma-assisted molecular beam epitaxy. Then, on top of the GaN nanowires, a five period AlN/GaN superlattice is grown. The average nanowire diameter is 75 ± 15 nm, the total length is 1400 ± 50 nm, one period of the superlattice is 98 nm thick with a 42 nm AlN layer and 56 nm for GaN. The average length and width are obtained by performing the statistical average above a large number of measurements on SEM images. To observe the dispersion of CGAP, we chose to grow relatively long nanowires, nevertheless to reduce the effect of attenuation,34 the period of the superlattice was chosen to be large in order to generate lower frequencies. Images of the nanowires obtained using scanning electron microscope and transmission electron microscope can be seen in Figure 2. Ultrafast pump−probe spectroscopy4 experiments were performed using a tunable Ti:sapphire oscillator in a conventional reflectivity scheme. The laser produces 100 fs optical pulses at a repetition rate of 76 MHz centered at a wavelength adjustable between 700 and 950 nm. In order to photoexcite carriers in GaN, the photon energy of the laser pulse needs to be higher than the band gap. We therefore use a 360 nm wavelength pump beam obtained by second harmonic generation in a β-BaB2O4 (BBO) crystal. The wavelength of the probe beam is either 360 or 720 nm, and the spot size is slightly larger than 30 μm, much larger than the wire diameter. We are thus sensitive to the average wire dimensions, and the spread in these dimensions will cause inhomogeneous broadening on the measured signals. When the pump beam is absorbed in GaN, electrons are excited from the valence to the conduction band, the binding between atoms is modified, and atoms move out of their

Figure 3. (a) Transient reflectivity obtained for a probe wavelength of 360 (blue line) or 720 nm (red line). One can observe the generation of CGAP and their echoes after a round trip in the NWSL.

We observe multiple contributions on these signals. Between 0 and 100 ps, damped oscillations can be seen. They can be detected with both probe wavelengths. Due to the absorption of light in GaN, we can assume that the excitation took place at the top of the nanowire, i.e., in the superlattice. This first feature corresponds to the generation and detection of CGAP in the superlattice. At a longer time scale, we noticed other broader features, denoted echoes on Figure 3 and corresponding to the round-trip of CGAP in the NWSL. For both probe 1141

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dispersion relation of an AlN/GaN superlattice but with a linear dispersion relation for each material. We used the properties of bulk AlN and GaN,41 and the thicknesses of each layer were similar to the ones in our nanowires. The comparison between this dispersion relation and the one obtained for CGAPs is presented in Figure 1b. For a linear dispersion, the frequency of the first gap opening was located at 90 GHz, which is different from the 70 GHz we observed experimentally for CGAPs. Now, if we consider that the observed 70 GHz frequency is a result of the mechanical changes due to size reduction, it would mean that the average sound velocity in the superlattice is 25% smaller compared to bulk materials. However, we recently demonstrated, through the observation of confined acoustic phonons, that the mechanical properties of GaN nanowires were unchanged for similar dimensions.19,20 We therefore attribute this discrepancy in frequency to the generation of CGAPs. The main difference between the signal obtained with a UV probe and the one obtained with an IR probe is the appearance of two low frequency peaks at 25 and 60 GHz. In superlattices, the detection of acoustic waves is enhanced when the wave vector of the phonons is equal to 2 times the wave vector of light, qphonons = 2klight.40 In the case of our nanowires, we need to obtain the effective refractive index. We realized the reflectivity spectrum of our sample, and by fitting this spectrum in a similar way as in ref 19, we obtained the refractive index of the nanowire array, and thus the wave vector of light. The horizontal red line in the lower part of Figure 4 corresponds to 2 times the wave vectors of the IR probe, while the wave vector of the UV probe is out of the mini-Brillouin zone. One can see that this line intersects the phonon dispersion relation at frequencies close to the experimental ones. We now analyze the second contribution: the echoes. As we have described previously, these are guided acoustic phonons that went all the way down to the nanowire−substrate interface, got reflected, and traveled back to the surface. The broad echoes that can be observed in Figure 3 correspond to the envelope of the acoustic waves. The width of the echo is given by the size of the superlattice,40 and since the superlattice is relatively large, 490 nm, this envelope is low frequency. However, upon this envelope, the frequencies generated by the superlattice are superimposed. By filtering out the frequencies

wavelengths, we had a structure appearing around 350 ps, and for the IR probe, a second one at 700 ps. We are now going to discuss both contributions. First, we focus on the generation. To understand the origin of these oscillations, we need to clarify the properties of superlattices. It has been demonstrated that the generation of acoustic waves by superlattices is extremely efficient at frequency corresponding to the zone-center of the mini-Brillouin zone.40 In our experimental traces, oscillations with different frequencies were observed. To obtain these frequencies, we realized the Fourier transform of both signals. The normalized spectra for both probe wavelengths are presented in the lower part of Figure 4.

Figure 4. Top: Dispersion relation of the first and third order CGAPs in the NWSL. The red line corresponds to 2 times the wave vector of the 720 nm probe. Bottom: Fourier spectra of the signals obtained with UV and IR probe.

As expected from the time domain data, multiple frequencies are present. Most of these frequencies are common to both signals. By solving eq 4 for a 75 nm diameter nanowire for GaN and AlN and by injecting the obtained dispersion relations in eq 7, we obtained the dispersion relations for CGAPs in the superlattice. These dispersion relations are reproduced in the upper part of Figure 4. We remark that, as expected, the experimental frequencies correspond to the zone-center phonons of the mini-Brillouin zone. It is interesting to note that we are able to generate multiple modes in this NWSL. The first common frequency, pointed to in Figure 4 by an arrow, is located around 70 GHz. In order to demonstrate the effect of the confinement on phonons, we calculated the

Figure 5. (a) Top: Transient reflectivity obtained with a 720 nm probe wavelength with the envelope of the acoustic pulse filtered out (black curve), with a filter centered at 25 GHz (blue curve) and a filter at 60 GHz (red curve). Bottom: Time frequency analysis of the transient reflectivity with the envelope filtered out. (b) Schematic representation of the paths followed by the 25 GHz (1 and 2) and 60 GHz phonons (3). (c) Phonon group velocity in GaN and AlN 75 nm diameter nanowires. 1142

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500 ps for 60 GHz, which is in good agreement with what we observe experimentally. Due to promising future applications,25−27 studies on the thermal conductivity of nanowires have recently attracted much interest.23,24 Low thermal conductivity compared to bulk and deviation from the T3 dependence for small diameter have been observed and tentatively attributed to phonon-boundary scattering.23 It has also been proposed that the modification of the phonon velocity could play a major role.24 Unfortunately, the measurement of the thermal conductivity cannot give direct evidence of the relative influence of each mechanism. Therefore, no clear understanding can be achieved without direct observation of phonon behaviors. Our observation of the dispersion of CGAPs shows that the modification of the phonon velocity cannot be discarded, and it also enables the separate and controlled studies of these different mechanisms, which will lead to a better understanding of phonon transport and thermal properties in nanowires. In summary, we have shown that, thanks to the use of a superlattice, we were able to realize the first ever observation of guided acoustic phonons in a nanowire. The analysis of the generation and the assignment of each observed frequency have been performed. By taking advantage of the capability of superlattices to detect acoustic waves at specific frequencies, we studied the propagation of guided acoustic phonons in the nanowire, revealing a strong dispersion phenomenon. The generation and observation of the propagating guided acoustic phonons show the potential of such a system for low lateral size acoustic transducer and provide a new way to investigate phonon transport and thermal properties of nanowires.

below 10 GHz, we can get rid of the envelope and observe these specific frequency components as displayed on the black curve of the upper part of Figure 5a. On this signal, three features are present. The first one appears between 0 and 200 ps, which corresponds to the previously analyzed generation signal. Then, from 250 to 400 ps, a first group of oscillations can be observed. Finally, from 450 to 550 ps, higher frequency components, around 60 GHz, appear. By performing the time frequency analysis of the 10 GHz high-pass filtered curve, reproduced in the lower part of Figure 5a, we observed these features. Between 0 and 200 ps, the spectrum of the signal is broad, as we have seen previously when analyzing the generation. From 250 to 400 ps, the spectrum of the signal is centered at 25 GHz, which is the frequency of the first intersection between the wave vector of light and the phonon dispersion relation in Figure 4. In this part of the signal, lower frequencies are detected at a shorter time delay than higher frequencies. Due to dispersion, the waveform is gradually distorted as it propagates through the nanorod, thus leading to this trend. From 450 to 550 ps, a signal around 60 GHz is detected, which corresponds to the second intersection between the wave vector of light and the phonon dispersion in Figure 4. The posterior arrival of these phonons is due to the strong dispersion in the nanowires. To further analyze the dispersion, we realized frequency filters to isolate the 25 and 60 GHz components. The filtered signals are reproduced in the upper part of Figure 5a. The difference of time arrival for the echo of both contributions is striking. We also notice a difference in the shape of the generation signal for both frequencies. The generation signal of the 25 GHz component is first getting stronger and then decreases, while, for the 60 GHz component, it is only decreasing. These trends provide information on the location of the generation of both frequencies. The detection of the 25 GHz signal is enhanced in the superlattice, and the increase of the amplitude of the signal implies that the generation takes place in the GaN nanowires below the superlattices. As a result, the CGAPs corresponding to this signal have followed the path noted 1 in Figure 5b. CGAPs generated in the GaN nanowires are either traveling toward the surface or the substrate. The maximum amplitude of the 25 GHz echo corresponds to CGAPs having followed path 2 of Figure 5b, as they suffer less attenuation losses and are only reflected one time compared to the CGAPs that traveled through paths 1 and 2. From these considerations, we can conclude that the time between the maximum amplitude of the generation and of the echo of the signal filtered around 25 GHz corresponds to a round-trip in the GaN nanowire. The decrease of the 60 GHz generation signal means that these CGAPs are generated in the superlattice. Since the superlattice only has five periods, the generated zone-center phonons have a broad spectrum,42 that extends to 60 GHz. The detected 60 GHz echo is thus resulting from the round-trip of these CGAPs in the superlattice/ nanowire system, as depicted in path 3 of Figure 5b. Inside the nanowire, phonons travel at the group velocity. Figure 5c shows the group velocity in GaN and AlN obtained by differentiating the dispersion relation of CGAPs. We obtain the following velocities: vgGaN(25 GHz) = 7.45 nm·ps−1, vgAlN(25 GHz) = 10.4 nm·ps−1, vgGaN(60 GHz) = 5 nm·ps−1, and vgAlN(60 GHz) = 9.8 nm·ps−1. From the velocities and the thicknesses of each layer, we can estimate the expected arrival time of each echo. We obtain an estimated arrival time of 300 ps for 25 GHz and

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work was sponsored by the National Science Council of Taiwan, R.O.C., under Grant No. 101-2120-M-002-005.

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