spie proceedings [spie photonics and micro- and nano- structured materials 2011 - yerevan, armenia...

Optics of anisotropic nanostructures (nanowires and nanorods)

Alexander Shik* and Harry E. Ruda

Centre for Advanced Nanotechnology, University of Toronto, 170 College St., Toronto M5S 3E3, Canada

ABSTRACT

A survey of polarization-dependent optical phenomena in semiconductor and metal nanowires and nanorods is presented. Due to the large dielectric constant mismatch between nanostructures and their environment, the amplitude of the optical electric field inside the former depends dramatically on the angle between the direction of light polarization and the nanostructure axis. As a result, optical absorption, photoconductivity, and nonlinear photoresponse in semiconductor structures are strongly anisotropic, with the maximal value for the parallel light polarization. In metal structures, absorption anisotropy depends on the light frequency ω, and for ω close to the transverse plasmon frequency is maximal for the perpendicular light polarization. Luminescence emitted by semiconductor nanowires and nanorods is strongly polarized along their axis. Joint action of polarization effects in absorption and luminescence results in the polarization memory, when luminescence of a random ensemble of nanorods is polarized in the same direction as the exciting light. Keywords: nanowires, nanorods, light absorption, luminescence, anisotropy

1. INTRODUCTION Our goal is to present a large group of optical phenomena in anisotropic nanostructures, nanowires (NWs) and nanorods (NRs) placed in environment with the dielectric constant ε0 different from that of the nanostructure ε, and caused by polarization effects (image forces). Due to the dielectric mismatch, the intensity of the light electric field inside nanostructures, which, in turn, determines all light-induced phenomena, strongly depends on the orientation of NWs and NRs related to the light polarization. Similarly, light emitted by such nanostructure appears to be strongly polarized.

2. POLARIZATION OF ABSORPTION If a cylindrical NW is placed in external electric field E0 its parallel component, E||, being continuous at the interface, remains the same inside a wire: E|| = E0||. At the same time, the component normal to the NW axis, E⊥, is modified inside the NW:

.2

00

0⊥⊥ ε+ε

ε= EE (1)

If, instead of a long NW, we consider a NR, which can be considered as a prolate ellipsoid of revolution with semi-axes a and l/2 (a < l/2), the internal field is expressed in terms of the so-called depolarization factors n||,⊥ depending only on the ratio a/l:1

.)( ||,0

||,00

0||, ⊥

⊥⊥ ε−ε+ε

ε= E

nE (2)

Though the given expressions are derived for static electric fields, they remain valid also for high frequency fields, as long as the NW radius a or the both NR dimensions a, l remain much less than the electromagnetic wavelength. For ε > ε0, (1) and (2) show that the amplitude of the high-frequency electric field and, hence, the probability of optical transitions in a nanostructure depend dramatically on the light polarization, acquiring maximal values for light with polarization parallel to its axis. This results in a strong dependence of the absorption coefficient α on the light polarization. The ratio of α in NWs for the two light polarizations is: *[email protected]; phone 1 416 946-7898; fax 1416 978-3801

Plenary Paper

Photonics and Micro- and Nano-structured Materials 2011, edited by Rafael Kh. Drampyan, Proc. of SPIE Vol. 8414, 841408 · © 2012 SPIE · CCC code: 0277-786X/12/$18 · doi: 10.1117/12.923133

Proc. of SPIE Vol. 8414 841408-1

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.2

2

0

0||

εε+ε

=α

α

⊥ (3)

For most semiconductors this factor exceeds 30 for free standing NWs (ε0 = 1). Note that so far we have considered classical electrodynamic effects caused exclusively by the difference of the refractive indices for nanostructures and environment. In NRs with noticeable size quantization, the optical matrix element becomes anisotropic, which may modify the described polarization effects so that (3) acquires an additional factor equal to the square of the ratio of the matrix elements for the two perpendicular directions. The effect in its pure form was first observed and explained in 2. The authors saw photoconductivity in InP NWs increasing up to 49 times when the polarization of the exciting light changed from perpendicular to parallel. This number exactly corresponds to Eq.(3). Similar effects with a slightly lesser degree of polarization (maybe, owing to the misorientation of the NWs, the role of which is discussed elsewhere 3) were also observed in photoluminescence of Si NWs 4 and photoconductivity of ZnO and GaN NWs 5,6.

The quasistatic formulae (1)-(3) are adequate only for thin nanostructures with a much less than the light wavelength. To obtain the expression valid for arbitrary a, we must find the electric field distribution in NWs of different orientation by solving the wave, rather than the Laplace, equation. It was done in 7 and the results are presented in Fig.1, which shows the spectral dependence of the ratio )/()( |||| ⊥⊥ +−= ααααaP characterizing the polarization dependence of absorbed

light intensity. For ωa/c → 0, this ratio tends to the value (ε2+2εε0–3ε02)/(ε2+2εε0+5ε0

2) corresponding to (3). However, at higher light frequencies (or thicker NWs) we can no longer claim that light with parallel polarization has much higher absorption coefficient since α|| and α⊥, as well as their ratio, demonstrate strong frequency dependence, especially dramatic for α⊥, which tends to infinity in a series of critical points corresponding to Pa = – 1 in Fig.1. Considering a cylindrical NW as an optical fiber, we can express each of critical points as the cutoff of a fiber mode LP1m (m = 1, 2, 3...). At the cutoff, these modes become purely transverse and excited by an incident wave. The described frequency dependence of the polarization-sensitive absorption has not yet been studied experimentally, except for the observation 5 that two different frequencies of excited light caused different degree of photoconductivity anisotropy.

Figure 1. Spectral dependence of the polarization ratio )/()( |||| ⊥⊥ +−= ααααaP in a NW with ε = 9, ε0 = 1.

Returning to thin NWs, we note that (3) applied to both metallic and semiconductor nanostructures. In semiconductors the degree of polarization does not explicitly depend on ω. In metals, on the contrary, ε has a strong frequency dispersion caused by the electron plasma, which makes the polarization characteristics frequency-dependent even in the quasistatic limit of small nanostructures. For analytical description of these effects, we will use for ε the simplest Drude expression:

)](/[1 2 νωωωε ip −−= where ωp is the plasma frequency in a bulk metal and ν is the scattering rate. Substituting this



expression into (2), we see that at small ν (weak electron scattering), optical electric field inside NRs increases dramatically at the frequency

⊥⊥

⊥⊥ +−ε

ω=ω

||,||,0

||,||, )1( nn

np (4)

called the localized plasmon frequency. Optical absorption spectrum has a strong maximum near the plasmon frequency. Light polarized along the NR axis generates plasmons with the frequency ω|| called longitudinal plasmons, while light with the perpendicular polarization generates transverse plasmons with the frequency ω⊥ > ω||. For long NWs ω|| → 0, in other words, longitudinal plasmons cease to exist. Formally, it is due to the fact that in infinitely long cylinders the parallel component of electric field must be continuous at the interface and hence is uniform, without any structure characterizing plasma oscillations. If we substitute the Drude expression for ε(ω) into (3), we see that anisotropy of optical absorption in metallic NWs has an interesting spectral dependence. Far below the plasmon frequency, |ε(ω)| is very large, and the situation is similar to that in semiconductor NWs where absorption is much higher for the light with parallel polarization. However, near the plasmon frequency absorption of the perpendicularly polarized light increases drastically (the so-called plasmon amplification), changing the sign of anisotropy.

3. POLARIZATION OF LUMINESCENCE In the previous section we considered physical phenomena related to strong polarization of the external non-polarized light inside a NW with high refractive index. Now we discuss the problem of polarization of light emitted by a NW. As a first step, we solve the auxiliary problem of finding the electric field created by an electric dipole placed at the axis of a cylinder with radius a and dielectric constant ε, in environment with ε0. Details of these calculations can be found in 3, and here we present only the final results. For the dipole moment d0 parallel to the cylinder axis (z-axis), the electric field of the radiation far from the NW, has the same amplitude and configuration as would be created by a dipole with the moment d0 in free space. For the normal orientation of d0, the corresponding effective moment is given by d = 2ε0d0/(ε+ε0). As a result, even for NWs luminescence acquires a strong polarization under the influence of image forces. Now we may consider the net polarization of luminescence from a line x,y = 0 containing a mixture of randomly oriented emitting dipoles. We take some point z along the wire axis containing three mutually perpendicular dipoles dx, dy, dz with frequency ω and calculate the resulting electric field far from the NW at some point (x0,0,0) using the standard formula for the electric field of an emitting dipole and adding contributions of all line segments by integrating over z. The resulting Poynting vector S is directed along the x-axis and for the two different light polarizations is:

( ) ( ),26

22

032

0

4

|| zxx ddxc

S +πεηω

= (5)

( ) 2

032

0

4

2 yx dxc

Sπεηω

=⊥ (6)

where η is the density of dipoles per unit NW length. By substituting dx,y = 2ε0d0x,y/(ε+ε0); dz = d0z into (5),(6), we get the resulting polarization of light emitted by a NW. The intensity ratio for different light polarizations is

.6

323

220

200

2

2

22||

εεεεε ++

=+

=⊥ x

zxe

e

ddd

II (7)

If the bare dipole moments d0x,y,z do not coincide due to size quantization or/and crystal anisotropy, this modifies polarization character introducing anisotropy even in the absence of image forces, similar to the case of absorption. It is seen from (7) that light emitted by a NW with a dielectric constant higher than that of its environment, is strongly polarized in the NW direction, which was clearly demonstrated experimentally 2,4. The effect is qualitatively similar to that for absorption considered in Sec.2 but by comparing (3) with (7), one can see that the degree of polarization for luminescence is less than for absorption. It is caused by the fact that, according to (5), light with parallel polarization is generated not only by dz dipole moments but also by dx which are weakened by image forces. For InP NWs (ε = 12.7)



where, as was mentioned above, α||/α⊥ ≅ 49, (7) gives 30/|| ≅⊥ee II , in good agreement with the experimentally observed

value 24/|| ≅⊥ee II 2.

All the results obtained above for semiconductor NWs remain valid also for NRs, except for the numerical values of polarization coefficients, which for NRs depend not only on ε/ε0 but also on the depolarization coefficients, or, in other words, on the NR aspect ratio. Now we consider the polarization properties of luminescence from thick NWs where the electric field cannot be assumed uniform. Using the formulae 8 for the field created by an emitting dipole embedded into a dielectric cylinder, we get 7 the expressions

,)()()()(

)()()()()1(

100)1(

01

)1(10

)1(01

0 kaHkaJkaHkaJkaHkaJkaHkaJ

dd zzεε

ε−

−= (8)

ck

kaHkaJkaHkaJkaHkaJkaHkaJ

dd xxεω

εεε =

−

−= ,

)()()()()()()()(

)'1(10

)1(0

'10

)'1(10

)1(0

'1

0, (9)

(Jn and Hn(1) are the Bessel and Hankel functions) and generalizing the formulae for dx,z to the case of arbitrary a.

Figure 2. (a) – spectral dependence of the dipoles dx (1) and dz (2) for a NW with ε = 9, ε0 = 1; (b) – spectral

dependence of the polarization ratio Pa for the same NW.

Fig.2a shows the frequency dependence of the effective dipoles dx and dz. At ω → 0 they acquire the static values dz = d0z, dx = 2ε0d0x/(ε+ε0), and then demonstrate strong oscillations. At larger ω (or larger a), the phases of oscillations close in, dz approaches dx, so that the radiation becomes almost unpolarized. So far we have analyzed the emission characteristics of one single effective dipole in a NW. To obtain radiation characteristics of the whole NW, we must perform integration over its length assuming the dipoles distributed uniformly along z, exactly as it was done in the previous section. Substituting (8),(9) into the first equality of (7) and assuming isotropic internal emission (d0z = d0x), we obtain the spectral dependence of the resulting polarization ratio

( ) ( )eeeee IIIIP ⊥⊥ +−= |||| / presented at Fig.2b. The latter is seen to have a strong oscillatory character changing sign at some critical frequencies differing from those describing polarization dependence of absorption (Fig.1). It is evident from Fig.2b that, if the luminescence spectrum of a NW material contains several lines with noticeably different frequencies, the polarization of these lines can be essentially different and even have opposite sign. It was confirmed by experiments 9 in ZnO NWs. As seen from Fig.3, this material contains two strong luminescence lines with



frequencies differed by 1.3 times and these lines have opposite angular dependences caused by opposite signs of luminescence anisotropy.

Figure 3. Luminescence spectrum of ZnO NW (left) with two emission lines and angular dependence of the

luminescence intensity (right) in these lines 9.

4. NONLINEAR PHENOMENA Polarization dependence of the internal optical field in NWs and NRs must result in anisotropy not only of linear but also of nonlinear optical phenomena, such as, e.g., the second harmonic generation (SHG). Moreover, one may expect the anisotropy of nonlinear effects to be larger than of linear, since the former are proportional to a higher degree of electric field. However, the analysis of SHG in NWs and NRs is less straightforward than that of linear effects since the effect itself is absent in uniform bulk materials characterized by inversion symmetry and appears only in nanostructures with ε ≠ ε0. The theory of SHG in NWs was developed in 10 using the approach elaborated earlier 11 for spherical nanocrystals. It was shown that SHG amplitude depends not only on the mutual orientation of the NW axis and light polarization E, as in linear phenomena, but also on the direction of the light wave vector k. For the three possible cases: (i) longitudinal polarization, (ii) transverse polarization at k parallel to the NW axis, and (iii) transverse polarization at k perpendicular to the NW axis, the ratio of SHG amplitudes is (ε/ε0)3 : 36 : 4.

Figure 4. The SHG image of a set of three NWs recorded at two perpendicular directions of the laser polarization 12 indicated with a double arrow. The grey scale numbers represent photon counts per pixel.

Experimental investigations of nonlinear phenomena 12,13 was performed on ZnSe NWs with length 8-10 micron and diameters 80-100 nm, excited by a laser at 1029 nm wavelength, that is satisfying the quasi-static condition ka << 1. A typical SHG image of a set of three NWs is shown in Fig.4. The image demonstrates a dramatic dependence of the signal intensity on the exciting light polarization. For a quantitative study, we took one NW and measured the angular dependence of the integral SHG signal ISH(θ) measured by rotating the sample with respect to the excitation beam polarization. It is presented by a number of experimental points at Fig.5. It is seen that this dependence has a very strong



character with ISH(0) (parallel polarization) exceeding ISH(90o) (perpendicular polarization) by more than an order of magnitude. Similar giant angular oscillations were observed also for two other nonlinear phenomena: third harmonic generation and two-photon luminescence.

Figure 5. Polarization dependence of the SHG intensity 12. The dots represent experimentally measured values, the solid

line corresponds to the theoretical calculations for 2)(δθ = 0.2.

For particular conditions of the experiment, the theory 10 predicts the following angular dependence of ISH:

.)1()1(

sin)1(3.10cos24

424

⎥⎦

⎤⎢⎣

⎡

++−

+∝εεθεθSHI (10)

With ε of ZnSe, (10) gives the amplitude of angular oscillations several times larger than the experimental one. The most evident explanation is attributed to deviations from an ideal cylindrical NW geometry assumed in theoretical calculations. Free-standing NWs are inevitably bent, so that the local value of θ changes along the NW. The best fitting

of results was obtained for the value of average NW corrugation 2)(δθ ≅ 0.2. The corresponding theoretical curve is given by the solid line in Fig.5.

5. POLARIZATION MEMORY The previous sections were devoted to optical properties of individual NWs or NRs and now we will discuss similar effects in random NW arrays (say, in a polymer matrix or in solution). Such a system has macroscopically isotropic optical properties but simultaneously must possess a very interesting property of polarization memory 3,14. If we excite photoluminescence in the system using polarized light, in accordance with Sec.2, non-equilibrium carriers will be generated mostly in the NWs that are oriented close to parallel to the light polarization. According to Sec.3, light emitted by these NWs will have preferable polarization parallel to them, or in other words, parallel to the polarization of the exciting radiation. To calculate this effect in a random system of thin NWs, we take some NW with orientation characterized by the spherical angles (θ,α) where the z-axis is chosen along the polarization of the exciting light. Considering NW as a line of emitting dipoles with components d0 along the NW axis and 2d0ε0/(ε+ε0) in two other directions and using the ideas presented in Sec.3, we calculate the relative intensity of emission for light polarized parallel and perpendicular to the z-axis. Finally, we average the result over θ and α with the weight (ε+ε0)2cos2θ + 4ε0sin2θ, which, according to (3), is the



relative light intensity absorbed in a given NW. The result gives the total intensities I||,⊥ of light emission with polarizations parallel and perpendicular to the polarization of exciting light:

.168)(63]2)[(44)](2)[(3336)(21]2)[(20)](2)[(9

20

20

20

20

20

20

20

20

20

20

20

20

20

20||

εεεεεεεεεεεεεεεεεεεεεε

++++++++++++++++++

=⊥I

I (11)

(11) shows that maximal possible polarization, reached at |ε/ε0| >> 1, is equal to I||/I⊥ = 3. In terms of the polarization ratio ( ) ( )⊥⊥ +−= IIIIP |||| / it corresponds to P = 0.5. For InP NWs in air we get P = 0.44. It is worth noting that at

ε < ε0 (e.g., in metallic NWs near the plasmon resonance), when electric field in NWs has a preferably perpendicular orientation, the system also has a polarization memory. This effect is weaker than at large ε, with the maximal P = 6/68 = 0.088. The effect of polarization memory was first observed in porous Si 15 which has only a distant resemblance to a system of NWs or NRs, so that for demonstation of its main experimental features we have chosen subsequent more detailed measurements in CdSe/ZnS NRs 16 dissolved in a liquid and hence having a randomly distributed orientation. Fig.6 shows the luminescence spectrum consisting of two peaks corresponding to interband transitions between different size-quantized states. Polarization properties of the emitted light are presented in the form of luminescence spectra for different angular positions of the analyzer related to the direction of exciting light polarization E0. The luminescence is seen to be polarized mostly parallel to E0, in qualitative agreement with theoretical predictions.

Figure 6. Photoluminescence spectrum of the NR solution for excitation by linearly polarized light 16. Different curves

correspond to the luminescence components with polarization in the direction forming the angle θ with the polarization of excitation. θ increases from the upper to the lower curve with the step 10o. The amplitude of luminescence at λ < 520 nm is shown with magnification factor 10.

Quantitative treatment of the results meets some difficulties. According to (11), the amplitude of polarization memory in small nanostructures has some universal value depending only on ε/ε0. However, in the experiment this parameters differs noticeably for two different spectral lines in the same NRs. Besides, for the short wavelength peak the amplitude of polarization memory exceeded the theoretically predicted one, while for the long wavelength peak was always less. These results could be explained if the matrix element responsible for the ground state 573 nm long-wavelength peak is larger for perpendicular light polarization (it is really the case for NRs with a small aspect ratio 17) while that at 470 nm is either more isotropic or larger for parallel polarization. This is in agreement with the theoretical 18 and experimental 19 conclusions of essentially different polarization properties of different optical transitions in NRs.



6. CONCLUSIONS We have described a large variety of polarization-sensitive optical phenomena in nanostructures of anisotropic shape, such as NWs and NRs. The phenomena have a universal character being due exclusively to the difference in dielectric constants between nanostructures and their environment. They include strong dependence of linear (absorption, photoluminescence, photoconductivity, etc.) and nonlinear optical properties on polarization of the exciting light, as well as high degree of polarization of luminescence emitted by these systems. Joint action of the absorption and emission anisotropy results in polarization memory in random arrays of NWs and NRs. In metallic nanostructures all the mentioned phenomena demonstrate dramatic dependence (including the change of sign for some of them) on the light frequency, related to plasmon effects. All the described effects are provided with proper theoretical description and illustrated by experimental observations performed in the recent years.

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