NANO LETTERS
Optical Interference from Pairs and Arrays of Nanowires
2006 Vol. 6, No. 4 862-865
Monica Lexholm, Dan Hessman,* and Lars Samuelson Solid State Physics/The Nanometer Structure Consortium, Lund UniVersity, Box 118, S-221 00 Lund, Sweden Received December 14, 2005; Revised Manuscript Received February 14, 2006
ABSTRACT The angle dependence of the scattered light from pairs and one-dimensional arrays of nanowires was studied. The intensity of the scattered light varied distinctly during rotation of the structure. The results could be theoretically modeled by treating a pair of nanowires as a double slit and an array of nanowires as a grating. The correspondence between theory and experimental results conclusively proves that the variations are due to the proposed interference effects.
For the past decade there has been a revolution in onedimensional nanostructures such as carbon nanotubes1 and semiconductor nanowires.2 Nanowires have a promising future as building blocks in electronics and photonics2-4 as well as for life-science applications.5 Today, not only the dimensions and chemical composition of the nanowires can be controlled but also their location.6-8 The ability to accurately position nanowires in arrays opens the door to new applications. For electronics, an array of single nanowire devices could be connected to form complex networks, and for life-science applications, an array of pillars could function as a size filter for molecules.9 For photonics, a twodimensional array of nanowires creates a periodicity of the dielectric constant provoking a range of forbidden photon energies, a photonic band gap.10-12 The band gap can be tuned by changing the periodicity, hereby providing a way to controllably modify how nanowires couple to an external electromagnetic field. In the present work, we have studied the scattered light from pairs and arrays of nanowires looking for interference patterns and compared the results with a model where a pair of nanowires is modeled as a double slit and an array of nanowires as a grating. The nanowires were grown with chemical beam epitaxy, CBE, using a gold particle to catalyze growth.13,14,7 The diameter and position of each nanowire were chosen by controlling the size and position of the gold particle through electron beam lithography, EBL.6,7 InAs nanowires, in the 〈111〉 direction, with a diameter of 100 nm, a length of 2-3 µm, and a surface roughness on the order of a few atomic diameters were grown on InAs (111)B substrates. Details on the fabrication of the samples can be found in ref 7. The band gap of InAs is about 0.4 eV, considerably smaller than * Corresponding author. E-mail:
[email protected]. 10.1021/nl052467j CCC: $33.50 Published on Web 03/11/2006
© 2006 American Chemical Society
Figure 1. Schematic view of the setup. The slightly focused laser beam hits the structure at an angle γ and the scattered light is detected at an angle θ. The laser beam is fixed at an angle V relative to the objective throughout the measurement. Only the sample is rotated, hereby changing both θ and γ ) V - θ. The intensity of the scattered light as a function of θ is obtained by analyzing the CCD images for different θ.
the photon energy used in these experiments. The nanowires are therefore not transparent, and any photoluminescence that might be emitted would not be detectable with the camera used in this work. The studied structures were pairs and arrays of 10 nanowires with spacings ranging from 650 to 2400 nm. The fabricated samples were studied in a scanning electron microscope, SEM, in order to measure the exact dimensions of each structure. For the light scattering experiments a laser,15 with a typical power of 0.2 mW, was slightly focused to yield a spot diameter of about 100 µm centered on a pair or an array of nanowires. Light scattered by the nanowires was collected with an optical microscope at a fixed angle, V, relative to the incoming laser beam and the intensity was measured with a charge coupled deVice, CCD, camera. V was measured for each measurement series with a typical value of 80°. The sample was then rotated manually changing both the detec-
interference by using two slits as wave sources was first proposed by Young in 1804.17 If the finite width of a slit is considered, then the interference pattern from N number of slits is described by18
{(
I ) I0
) } {( ) }
sin(NR) sin(R)
2
‚
sin(β) β
2
(1)
where
Figure 2. Model of the scattered light at a point P at an angle θ from the optical axis of an inverted double slit illuminated by a laser beam at an angle γ. In our experiments the angle V between the laser beam and observation point is fixed and the sample is rotated, changing both θ and γ. The cross section of the nanowires is taken to be quadratic in order to simplify the model.
tion angle, θ, and the angle at which the laser illuminated the structure, γ, see Figure 1. For each angle θ a CCD image was captured and the intensity of the area corresponding to the structure under study was measured. There are two reasons for using an optical microscope; one is to distinguish the studied structure from other structures and particles on the sample, and the other is to collect enough light from the structure. High spatial resolution and efficient light collection are, however, obtained at the expense of angular resolution. That is, light from a range of angles, given by the numerical aperture of the objective, contributes to the measured intensity at each angle θ. The experimental approach described above will thus introduce a broadening of the interference pattern. To optimize the tradeoff between resolution and a small range of angles, a variable aperture was put in front of the objective. The aperture was made just large enough, about 1 mm, for the signal to be detectable and for distinguishing one structure from the rest. More details on the setup can be found in ref 16. Huygen’s principle gives that when illuminated by light an opening of infinitesimal width and infinite length, an ideal slit, can be treated as a line source. The idea of studying
R)
π a(sin(θ) - sin(γ)) λ
β)
π b(sin(θ) - sin(γ)) λ
θ is the detection angle, γ is the angle of the incoming wave, b is the slit width, a is the slit distance, and I0 is the incoming irradiance. The first part of eq 1 describes interference from N ideal slits, and the second part describes diffraction from a single slit. The finite width of the slits is thus considered by modulating the ideal interference pattern with a single slit diffraction envelope. For the inverted case, see Figure 2, where the scattered light from a pair of nanowires is detected instead of the transmitted light from a double slit, the same equation should be valid for the interference pattern. One interesting aspect of using nanowires as slits is that their diameter can be made much smaller (down to 20 nm) than the ordinary laser wavelength (400-700 nm). This means that they appear to have almost infinitesimal width and in this sense approach the idea of an ideal double slit where the envelope function factor in eq 1 becomes negligible. First we look at how different spacing between the nanowires affects the angle dependence of the scattered light. Figure 3a shows that the experimental maxima are more separated for smaller distances between the nanowires, just as theory states, see eq 1.
Figure 3. (a) Intensity of the scattered light from two pairs of nanowires spaced by 650 and 1450 nm, respectively, using a laser wavelength of 532 nm. (b) Intensity of the scattered light from a pair of nanowires, spaced by 1450 nm, illuminated by a laser beam of wavelength 532 and 677 nm, respectively. Nano Lett., Vol. 6, No. 4, 2006
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Figure 4. (a) Intensity of the scattered light from an array of 10 nanowires spaced by 1000 nm collected with a 0.15 NA microscope objective (dashed line) and reduced aperture size (solid line). (b) Intensity of the scattered light from one single nanowire.
The role played by the laser wavelength can be studied by comparing the results from two measurements on the same structure using different wavelengths, see Figure 3b. According to theory, see eq 1, longer wavelength should give a larger separation between the interference maxima, as is the case experimentally. The effect of detecting a range of angles is seen clearly in a comparison of the results with different aperture size, see Figure 4a. A larger range of angles, due to larger aperture size, broadens the peaks. Nanowires grown in the 〈111〉 direction are known to have a hexagonal cross section.19 The angle dependence of the scattered light from one single nanowire was detected in order to study how the facets might influence the interference pattern. The result, see Figure 4b, shows variations in the intensity of about 20%, which is much less than the modulation depth in the other measurements. The main reason for the fluctuations is probably unintentional translation of the structure during the rotation. Such translation within an uneven laser spot can easily give rise to 20% variations in the light intensity. The 3D shape of a nanowire seems to be negligible, which could be expected because its dimensions are much smaller than the laser wavelength. To compare the results more specifically with theory, see Figure 5, a computer simulation program based on eq 1 was used. The broadening effect due to the optical microscope was included by simply adding the theoretical values of the intensity within a range of angles. This is not the complete picture but serves as a first approximation. The size of the range of angles used in the simulations was fitted to the experimental results. The range of angles derived from the fitting was 14° for the 0.15 NA objective and 4-5° for the case of an additional aperture, in good agreement with the actual dimensions. The change in detection angle, θ, was well defined, but its absolute value was difficult to measure. The experimental results were therefore adjusted to the theoretical simulations by shifting the detection angle, the x axis, within the margins of its estimated uncertainty of about 5°. The absolute value of the experimental amplitude did not contain any useful information because different laser 864
Figure 5. Intensity of the scattered light (solid lines) from a pair of nanowires spaced by (a) 650 nm, (b) 1450 nm, and an array of 10 nanowires spaced by (c) 1000 nm and (d) 2400 nm compared to theoretical simulations (dashed lines).
intensities were used in different measurements. For simplicity, the simulations were therefore multiplied by a factor so that the theoretical maximum value matches the experimental one. The experimental results agree surprisingly well with the simulations given the simplicity of the model. In general, the scattering from a nanowire should depend on the observation angle with the details given by parameters such as wire dimensions, refractive index, and absorption coefficient. However, in the experiments described in this paper, the observation angle is kept fixed relative to the incoming light while the sample is rotated. As long as the nanowires are rotationally symmetric, the influence of the nanowire details can therefore be reduced to diffraction because of the nonzero width and a multiplicative factor describing the scattering efficiency in a fixed direction. Nano Lett., Vol. 6, No. 4, 2006
The interference pattern from a pair of nanowires might provide a way to study the mechanics of nanowires experimentally. A static displacement relative to each other, caused by, for example, a charging of the nanowires, would change the phase periodicity in the interference pattern, whereas an oscillating pair would broaden the peaks with a maximum broadening at their resonance frequency. Analyzing the width of the interference peaks as a function of driving frequency could therefore be a way of measuring resonance frequency. Detecting the shift in resonance frequency due to mass adsorption is a well-known technique for measuring mass.20 The small dimensions of a nanowire imply a large relative mass change, making the nanowire an ideal candidate for a cantilever in the quest for a single molecule detector.21 In summary, scattered light from pairs and arrays of nanowires showed distinct interference patterns. The results could be described adequately by modeling a pair of nanowires as a double slit and an array of nanowires as a grating. It is expected that interference effects can be used for sensor applications. Acknowledgment. We thank M. T. Bjo¨rk, K. A. Dick, A. I. Persson, and L. E. Fro¨berg for growing the nanowires and M. T. Bjo¨rk, S. Ghatnekar-Nilsson, K. A. Dick, and I. Maximov for discussions and help with EBL and SEM. This work was performed within the Nanometer Structure Consortium at Lund University and supported by the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation.
Nano Lett., Vol. 6, No. 4, 2006
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