Classification of Normal Modes - Acoustic Guided Waves in an Anisotropic Cylinder Waveguide

Chapter 4 Nano-Acoustic Guided Waves in Cylindrical Nanorods

4.5 Acoustic Guided Waves in an Anisotropic Cylinder Waveguide

4.5.3 Classification of Normal Modes

Another advantage of using a series of power of

x , y ,

and

z

as basis functions is its simplicity for classification of normal modes. According to the parity of the displacement field, guided modes can be grouped into dilatational, torsional and flexural mode in a cylindrical waveguide. The classification is defined in Table 4-1. The parity is expressed by ( , )

 

, where

  

( 1)^pand

  

( 1)^q.

Table 4-1 Classification of normal modes

Classification

u

Dilatational ( , )

 

( , )

 

( , )

 

Torsion

( , )

 

( , )

 

( , )

 

( , )

 

( , )

 

( , )

 

Flexural other Other other

Fig. 4-5 Schematic diagram of the defined parity in displacement

u and

u of

_y dilatational modes..

Based on the defined parity, the displacement components in

x y z , ,

direction of

4.5.4 Comparison with Pochhammer Chree Theory

As long as the elastic properties (stiffness constants and density) and the geometry (radius) are known, the RUS method can be used to find the dispersion relation. In this section, we use

SiO which is isotropic to test the anisotropic model – RUS method.

₂ For

SiO , the stiffness constants are :

₂

C

₁₁



78.5

GPa

C

₁₂



16.1

GPa

C

₄₄



31.2

GPa

and the density is

 

2203

kg m

/ ³. Fig. 4-6 shows the dispersion relation from Pochhammer Chree theory and RUS method. It shows good agreement between two different approaches. However, there are extra modes by using RUS method with comparison to Pochhammer Chree theory. The reason is that one method uses the x, y parity while the other uses the ,

r 

parity to specify the normal modes. This reason will be confirmed later by comparing the displacement field of every normal mode.

Dispersion Relation composed of SiO2. The red lines are from Pochhammer Chree theory, a standard solution for an isotropic waveguide. The blue circles are from RUS method.

The displacement fields of the first four modes are drawn in Fig. 4-7to Fig. 4-9. For the reason that even the same mode will show difference in mode pattern at different wave vector, and each mode will cross the other somewhere in the dispersion curve, we compare the mode pattern at a specific wave vector as

k 

0.25 10 (



⁷

m

^¹). The lowest mode, without cutoff frequency, shows the axisymmetric parity for both approaches.

But the second mode in the RUS method shows absolutely different parity in x, y directions and thus excluded from the Pochhammer Chree Theory. Mode 3 of the RUS method (Mode 2 of the Pochhammer Chree theory) shows major components in the

x, y

u u rather than u . In addition, it should be noticed that as

k 

0for this mode, the entire rod will exhibit pure vibration in the radial direction, and it is usually called

―breathing mode‖, which is most commonly excited and observed in the previous

pump-probe studies of vibration response of nano-structures [4.13]. The displacement distribution of mode 4 shows the parity like mode 2 as well, thus not appearing in the Pochhammer Chree theory again.

Displacement field of Mode1 (Pochhammer Chree Theory)

Displacement field of Mode2 (Pochhammer Chree Theory)

Fig. 4-7 Displacement fields of (a) mode 1 and (b) mode 2of Pochhammer Chree theory.

-3

Fig. 4-8 Displacement fields of (a) mode 1 and (b) mode 2 of RUS method

-3

Fig. 4-9 Displacement fields of (a) mode 3 and (b) mode 4 of RUS method

4.5.5 Dispersion Curve and Mode Distribution of Guided Waves in GaAs NRs

In our experiment, the spot size of the focused pump beam is much greater than the diameter of nanorod, thus the assumption of dilatational acoustic waves excited by the homogenous expansion of gold nanodisk is reasonable so it is the only type we interested. The dispersion curve of dilatational modes of a 340nm-diameter GaAs nanorod is given in Fig. 4-10 and compared with the isotropic model assuming uniform velocity of the longitudinal and the transverse are 4731 m/s and 3347m/s, which are the typical velocities in the (100) direction. According to the waveguide theory, it is worthy to note there are discrete modes with certain mode pattern that can propagate inside a nanorod. This is because of the confinement of the boundary, the allowed modes and its phase velocity as well as group velocity is very different from the case of a bulk crystal which can be easily described by plane waves with longitudinal and transverse velocities.

The actual displacement field of excited guided modes will be given in the later chapter for discussion. However, it can be found that the distribution of displacement field of the first four modes of anisotropic case is nearly the same as the isotropic one at low wavenumber region; however, significant difference in dispersion relation at higher wavenumber region will make the mode pattern more complicate and different.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10⁷ 0

0.5 1 1.5 2 2.5x 10¹⁰

Wavenumber (m^-1)

Frequency (Hz)

Fig. 4-10 Calculated dispersion relations of a 340nm-diameter GaAs nanorod. The red dot is calculated from Pochhammer Chree equation, and the solid blue line is calculated from RUS method.

Reference

[4.1] C. Kittel, Introduction to Solid State Physics. 8^th edition. New York: Wiley (2005).

[4.2] Sebastain G. Volz and G. Chen, ―Molecular dynamics simulation of thermal conductivity of silicon nanowires,‖ Appl. Phys. Lett., 75, 2056 (1999).

[4.3] S.J.A. Koh, H.P. Lee, C. Lu and Q.H. Cheng, ―Molecular dynamics simulation of a solid platinum nanowire under uniaxial tensile strain: Temperature and strain rate effects,‖ Phys. Rev. B., 72, 085414 (2005).

[4.4] E. Dieulesaint and D. Royer, Elastic waves in solids. Springer, (1995).

[4.5] Anthony D. Puckeet, M.L. Peterson, ―A semi-analytical model for predicting multiple propagating axially symmetric modes in cylindrical waveguides,‖

Ultrasonic, 43, 197 (2005).

[4.6] Mirsky. ―Axisymmeteric vibration of orthotropic cylinders,‖ J. Acoust. Soc. Am.

36, 2106 (1964).

[4.7] A.-C. Hladky-Hennion. ―Finite element analysis of the propagation of acoustic waves in waveguides,‖ J. Sound. Vib. 194(2), 119 (1996).

[4.8] Marzani, E. Viola, I. Bartoli, F.L. di Scalea, P. Rizzo. ―A semi-analytical finite element formulation for modeling stress wave propagation in axisymmeteric damped waveguided.‖ J. Sound. Vib. 318, 488 (2006).

[4.9] Migliori, J.L. Sarrao, William M. Visscher, T.M. Bell, Ming Lei, Z. Fisk and R.G.

Leisure. ―Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli od solids.‖ Physica B. 183, 1 (1993)

[4.10] William M. Visscher, A. Migliori, T. M. Bell, and R.A. Reinert. ―On the normal modes of free vibration oh inhomogeneous and anisotropic elastic objects.‖

J. Acous. Soc. Am., 90(4), 2154 (1991).

[4.11] N. Nishiguchi, Y. Ando and M. N. Wybourne. ―Acoustic phonon modes of retangular quantum wires.‖ J. Phys.: Condens. Matter. 9, 5751-5764 (1997).

[4.12] R.G. Leisure and F.A. Willis. ―Resonant ultrasound spectroscopy.‖ J. Phys.:

Condens. Matter, 9, 6001-6029 (1997).

[4.13] M. Hu, X. Wang, G. Hartland, P. Mulvaney, J. Juste, and J. Saders,

―Vibrational response of nanorods to ultrafast laser induced heating: Theoretical and experimental analysis.‖ J. Am. Chem. Soc, 125, 14925 (2003).

49

Chapter 5 Experimental Results and Discussion

5.1 Experiment Results (probe 880nm, pump 440nm)

As we mentioned in chapter 2, two-color pump-probe scheme was used for a much better signal-to-noise ratio (SNR). Since the modulation of optical constants perturbed by the acoustic disturbance is more effective near the band gap of GaAs (

E

 1.43 eV

, 873nm), the probe wavelength was first chosen to be 880nm (from mode-locked Ti-sapphire laser) and thus the pump wavelength was 440nm (frequency doubled from Ti-sapphire laser).

0 200 400 600 800 1000

0.2 0.4 0.6

-500 0 500 1000 1500 2000

-1.0 -0.5 0.0 0.5

 R (a .u .)

Time (ps)

The inset of Fig. 5-1 shows the typical transient reflectivity change of pump-probe measurement of Au-attached GaAs nanorods. We divided the signals into two parts: (1) the carrier dynamic, which lead to the main character of the signal, i.e. the sharp change

Fig. 5-1 Transient change in reflectivity of the sample induced by the pump beam (880nm). A small oscillatory signal that is attributed to coherent phonon generation of the sample is highlighted in the figure. The inset shows the full measured trace.

and the following rapid exponential decay. The sharp change is induced by the arrival of pump pulse which excites sample’s electronic configuration and thus perturbs the reflectivity of the following probe. After the excitation, the electronic configuration would rapidly release the excess energy and return to steady state, and this lead to the following exponential decay character.

However, carrier dynamics studies in the designed samples about size and optical wavelength dependencies are beyond this thesis scope so the comparison and the discussion about the difference of this main character of the signal are ignored. The only thing we interested is the second part: damped oscillations, which are highlighted in Fig.

5-1, accompanying with the carrier dynamic response after the excitation, and it is attributed to the modulation due to coherent phonon generation.

For the purpose of studying the characters of acoustic responses of Au-attached GaAs nanorods, we first fitted the function of exponential decay to remove carrier dynamic background. The frequencies of acoustic responses were identified by performing fast Fourier transform (FFT) to the oscillatory signals. The background-subtraction oscillations and the frequency spectra of 340nm-diameter samples are shown in Fig. 5-2. Since the frequency peaks that below the noise level is meaningless, and the possible artifacts which may result from the fitting and FFT procedures (the picket-fence effect and the leakage effect), we only discussed the most dominating frequency. In the 340nm-diameter case, the frequency is around 9.3GHz. In other cases of smaller diameter, the measured frequencies show dependency on the diameter. Here, according to the possible acoustic dynamic response as we discussed in Chapter 3, we can reasonably conclude that the frequency was contributed from the radial breathing mode vibration of individual GaAs nanorod. The interpretation can be supported by the following arguments.

Based on the previous investigation done by Yi-Hsin Chen (a former member of our group) in which he showed that for the samples with Au attached and without Au attached the dominating frequencies are almost the same under this pump-probe wavelength setup (probe: 880nm, pump: 440nm). In spite of the elastic anisotropism of GaAs, both results are in good agreement with the predicted breathing mode frequency of simplified isotropic theory. Second, by analyzing the phase of oscillation, the signals in both cases behave like cosinusoidal oscillation, i.e. the phase is around

0

^oat time zero. The phase was believed to be one of the characters of the signal from structure vibration for a step-like excitation [5.1].

To further confirm the validity of this conclusion, the size dependency was compared to formula (2.1), where

V

_L is 4731m/s, the typical longitudinal velocity in [001] direction, and the parameter is 2.14 for Poisson ratio used is 0.312. As shown in Fig. 5-3, the frequencies show the inverse proportionality to the diameter, and this is a typical character for radial vibration, i.e. the smaller object has the higher vibration frequencies. All the measured results exhibit good agreement to the breathing mode theory.

0 5 10 15 20

100 150 200 250 300 350 400 450 500 550

Fig. 5-2 Frequency spectra of 340nm-diameter nanorod samples.

(black line: the sample with Au attached, red line: without Au attached)

2.14 f V

 D



5.2 Experimental Result (probe 1120nm, pump 390nm)

Although in the former scheme (probe 880nm, pump 440nm) the signal of breathing vibration of individual rod is dominating and overwhelming, the other acoustic dynamic responses may play a role for different probe wavelength since the detection mechanism of the each response depends on the probing optical wavelength, such as backward Brillouin oscillations and surface acoustic waves (SAWs). So in this section, we changed the probe wavelength from near infrared (880nm) to infrared (1120nm) region to study the acoustic responses of the structure. The wavelength was settled on 1120nm due to the power optimization for the OPO that was pumped by 780nm pulses which are the strongest power output from the Ti-sapphire laser. Here, although the pump wavelength is different (440nm vs. 390nm), the excitation can be still regarded as equivalent under these two schemes due to the same optical response of Au and GaAs in this ultraviolet regime.

0 500 1000 1500 2000

-1.5 -1.0 -0.5 0.0 0.5

 R (a .r. b .)

Time (ps)

Diameter:179nm Period:300nm Diameter:256nm Period:355nm Diameter:340nm Period:500nm

Fig. 5-4 Transient reflectivity change traces of different sample of different diameter and period under 1120nm probe used

Fig. 5-4 shows the experimental results for three different size cases under the infrared probe: (1)340nm diameter (period 500nm) (2) 256nm diameter (period 355nm) and (3) 179nm diameter (period 355nm). All of these samples are with around 720nm height.

Similarly, we neglected the difference of carrier dynamics and mainly focused to the damped oscillations observed. As one can see, there are many interesting and surprising things under this scheme: first, relatively huge oscillatory signal detected in small-diameter cases (< ~300nm) for 1120nm probe used. The amplitude of the oscillatory signal is quite different as compared to the results of near infrared probe (880nm). Fig. 5-5 shows the comparison between the results of two different probe wavelength but of identical sample (diameter: 231nm), one can easily observe that the sensitivity of detected oscillatory signal of these two are very different (although the probe. (each trace was normalized by its minimum value)

(a) (b)

noted as the second interesting things that there are echo-like signals at a later point in time. These phenomena observed are quite different to those observed in that of 880nm probe used. In Fig. 5-6, the overall signal can be roughly identified that there is

5.2.1 Frequency Domain Analysis

Since the appearance of the echo-like signals states the possibility that different types of oscillations may occur at different timeframe, FFT was employed to the different parts of overall oscillatory signal. One was chosen for all the recorded time (around 2000ps), which possess the better resolution in the frequency domain, another was chosen for the initial oscillations, and the other was chosen to the echo-like oscillations. From Fig. 5-7, it is clear that dominating frequency at different time are nearly the same. It means that the oscillations of proceeding one and the later one come from the same physical mechanism. This fact strikes out the possibility of different probe sensitivity in the rod and substrate region which make the detected signal appear once again when acoustic waves traveling into the substrate [5.2].

However, it is most important and surprising that the dominating oscillatory signal here is not the breathing mode of GaAs nanorods since the frequencies are absolutely signal of the sample of 273nm diameter and 355nm period. Note: Zero padding was used to make the length of each data equal to fairly compare

0 2 4 6 8 10 12 14 16 18 20

Initial oscillation (before 400ps) and zero padding (401-2000ps)

Echo-like oscillation (401+2000ps) (with zero padding 400ps) 6.4GHz

probe. The frequencies observed are absolutely different for different probe in these cases. (One would expect that frequency of vibration modes are unrelated to the optical probing wavelength used).

Fig. 5-8 Frequency spectra of the signal of the sample of 180nm diameter for 880nm probe (black-square line) and 1120nm probe (red-circle line).

The dominating peak in red-circle line represents the radial breathing mode of GaAs nanorods.

Fig. 5-9 Frequency spectra of the signal of the sample of 250nm diameter for each 880nm probe (black-square line) and 1120nm probe (red-circle line). The dominating peak in red-circle line represents the radial breathing mode of GaAs nanorods.

We conducted a size-dependency experiment by testing a series of different samples. Similarly, the frequencies measured show the diameter dependency. Therefore, it also reveals that the signal should be another vibration modes of the structure but not the breathing mode of GaAs nanorods. The frequency versus the rod diameter under the infrared probe (1120nm) is shown in Fig. 5-10. The blue line is the predicted breathing mode frequency as we stated in previous chapter. The red line represents the lateral vibration of a free Au nanodisk. In the figure, the measured dominating frequencies match the red line (lateral vibration of a free Au nanodisk) much better than the blue line (breathing mode of GaAs nanorods). A slight discrepancy between the red line and the measured results should result from the boundary condition assumption. In reality, since the Au is deposited on the GaAs nanorod, the free boundary condition assumed somehow is not valid. The vibration frequency would become higher because of boundary effect. The experimental results are in agreement with the predicted trend.

100 150 200 250 300 350 400 450 500

Lateral vibration of a free Au disk Breathing mode of GaAs nanorods

Fig. 5-10 Dominating frequency versus the diameter of the rod under 1120nm probe.

Therefore we suggested that the signal is induced by the vibration of the Au nanodisk for infrared probe (1120nm). There are more arguments to further support the above statement: a high frequency signal with around 52GHz just after the time zero can be observed in this scheme, but it occurs nowhere in the cases of 880nm probe used.

The fact is illustrated in Fig. 5-11. (The signal do occur in all the cases of small diameter (<300nm) under 1120nm probe.) Such a high frequency signal in our designed samples could only come from the thickness vibration of the gold nanodisk. By substituting the longitudinal velocity of 3240m/s in Au [5.3] into formula (2.5), the estimated thickness is around 30nm, which is roughly the same as the actual size of deposited Au nanodisk that is identified from the SEM image. (As shown in Fig. 5-12)

0 100 200 300 400 500 600

-0.2 0.0

 R (a .u .)

Time (ps)

Diameter:256nm Period:355nm Probe:1120nm

11ps 31ps

50ps

Fig. 5-11 The measured trace of transient change in reflectivity of the sample of 256nm diameter and 355nm period.(The frequency of the oscillation highlighted is around 52GHz)

These experimental results are in agreement with the intuition of the mechanical vibration: for a disk, a mostly possible mode excited by homogeneous thermal expansion is radial vibration rather than thickness vibration. This is because the expansion coordinates projects relative huge components on radial mode rather than the other. Another point is the nature that a mode with lower frequency are easily excited and survived. So the radial mode is the major one under this optical excitation is not surprising. By deducing from the above discussion, we concluded that not only thickness vibration but also the radial vibration (major one) are observed for infrared probe used, so it is quite reasonably to recognize the oscillatory signals are mainly induced by the Au nanodisk vibration.

However, for another cases of 340nm diameter and 440nm diameter, the signals observed return to that of breathing mode of GaAs nanorods, as shown in Fig. 5-10.

Additionally, as shown in Fig. 5-4, the sensitivity of detected oscillatory signal is relative small as compared to the other cases for the infrared probe (1120nm). In fact,

Lateral vibration Thickness vibration

Lateral vibration

Fig. 5-12 (a) SEM image of the Au-attached GaAs nanorods. In this figure the thickness of metal layer can be identified as ~27nm. The graph is taken by Hung-Pin Chen who is a member of our group. (b) Schematic diagram of lateral and thickness vibration of a nanodisk.

(a) (b)

the sensitivity of these two is more like those of 880nm probe used.

5.2.2 Detection Mechanism for the Infrared Probe (1120nm)

Now we have to figure out the problems about why different type of oscillatory signals dominate in two different probe wavelength schemes (for the samples of diameter < 300nm), and why the signal alters as the rod size is bigger than around 300nm (for the same probe of 1120nm)?

In fact, all of these phenomena discussed above reveal the implicit answer of the intervention of localized surface plasmon resonance (LSPR) occurred in the deposited Au nanodisks. As we mentioned in Chapter 3, by choosing a suitable optical wavelength, most of researchers made use of the LSPR effect to enhance the optical detection of acoustic modes in metal nanoparticles. That is because the resonance frequency of collective electrons in metal nanoparticles is highly sensitive to geometry and the environment surrounded. Acoustic modes permit a way to modulate the volume and the shape thus change the corresponding optical resonance wavelength. It causes severe change of optical response of metal nanoparticles and thus reflects onto the transient change of reflectivity and transmissivity of the probe that is located near the plasmon band.

Thanks to the abundant LSPR studies on 2D-periodical metal nanostructures [5.4], we know that the resonance of metal nanodisk array depends on size, geometry, periodicity, and permittivities of the metal and the environment. We suggested this is the reason that why we can see strong sensitivity on detection of vibration modes of Au nanodisk in infrared probe (1120nm), but it disappeared in that of 880nm probe. Also, it can explain why we see the abrupt change of signal as the rod become greater than

~300nm under the 1120nm probe since the plasmon resonant band would be redshifted

from the used probe (1120nm). The transmission and reflection spectra of the samples was preliminary examined to support the argument. The results exhibited that the transmission has a valley at around 1.1

 m

for the sample of 273nm diameter, and the valley is blueshifted for the sample of small diameter (with the same period of 355nm).

The character is corresponding to that of LSPR in metal nanodisks. However, the further investigation about the characters including the resonance field distribution in space and the width of the resonance band is beyond this thesis scope, so the discussion is left here.

5.2.3 Time Domain Analysis

Here, as we have proposed plasmon resonance would play an important role for infrared probe, we know that the gold nanodisk not only serves to be an opto-acoustic

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