Chapter 8 Photonic Nanodevices
8.1
Goals
• Introduce basic concepts of optics • Plasmonics • Metamaterials Read: • Bruchez et al., Science 281, 2013 (1998). Semiconductor Nanocrystals as Fluorescent Biological Labels.
8.2
Background
Photonics, the application of optics, is now an integral part of our technological world, side by side with electronics. Its biggest societal impact so far is in optical communication which is at the heart of the internet revolution. Some believe photonics will eventually all but replace electronics. A few of the advantages of light over electric current are: speed, higher bandwidth, and less interference. The present chapter serves as an introduction to photonics in nanotechnology so that many important topics such as optical fibers will not be considered. A recent survey of the field can be found in the book by Prasad [21]. 95
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CHAPTER 8. PHOTONIC NANODEVICES
8.3
Optics
• Wave-particle duality
E = hν =
• Reflection
θi = θ r .
• Refraction
n1 sin θ1 = n2 sin θ2 .
• Diffraction
8.4
h hc ,p = . λ λ
2d sin θ = mλ.
Nanophotonics
• Light interaction with nanomaterials. • Nanoscale confinement of light.
8.5 8.5.1
Light Interaction with Nanomaterials Fluorescent quantum dots
Fluorescence is an imaging technique that is useful for studying cellular structure and functions. A fluorophore is a material that absorb light at one wavelength and re-emits at another. Semiconductor quantum dots (typically CdSe/ZnS core-shell of size of a few nm) have been demonstrated to act as luminescent labels that are 20 times brighter, 100 times more stable against photobleaching, and have 3 times sharper emission spectra than conventional dyes such as rhodamine. [22] Invitrogen is one of the leading companies making semiconductor quantum dots for biological applications (Fig. 8.2). The increased efficiency is in part due to the relatively larger surface area of nanoparticles. The sharper spectra is due to the quantization of the energy states of the QD’s into discrete atomic-like levels. In addition, the emission energies can be tuned by using QD’s of different shapes and sizes. The emitted photon energy is approximately given by E0 , (8.1) R2 where Eg is the bulk band gap of CdSe, Ee and Eh are the confinement energies of electrons and holes, and R is the radius of the QD. E ∼ E g + Ee + Eh ∼ Eg +
8.5. LIGHT INTERACTION WITH NANOMATERIALS
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Figure 8.1: (A) Size- and material-dependent emission spectra of several surfactant-coated semiconductor nanocrystals in a variety of sizes. The blue series represents different sizes of CdSe nanocrystals (16) with diameters of 2.1, 2.4, 3.1, 3.6, and 4.6 nm (from right to left). The green series is of InP nanocrystals (26) with diameters of 3.0, 3.5, and 4.6 nm. The red series is of InAs nanocrystals (16) with diameters of 2.8, 3.6, 4.6, and 6.0 nm. (B) A truecolor image of a series of silica-coated core (CdSe)-shell (ZnS or CdS) nanocrystal probes in aqueous buffer, all illuminated simultaneously with a handheld ultraviolet lamp. [22]
8.5.2
History
“Seminal developments in the story of nanocrystal technology emerged in the early 1980s from the labs of Louis Brus at Bell Laboratories and of Alexander Efros and A.I. Ekimov of the Yoffe Institute in St. Petersburg (then Leningrad) in the former Soviet Union. Dr. Brus and his collaborators experimented with nanocrystal semiconductor materials and observed solutions of strikingly different colors made from the same substance. This work contributed to the understanding of the quantum confinement effect that explains the correlation between size and color for these nanocrystals. Two scientists from Bell Labs – Dr. Moungi Bawendi and Dr. Paul Alivisatos – moved to MIT and UC Berkeley, respectively, and continued investigating quantum dot optical properties. These researchers found ways to make the quantum dots water soluble. They also discovered that adding a passivating inorganic ”shell” around the nanocrystals, and then shining blue light on them, caused the quantum dots to light up brightly.” From the web site of Invitrogen (www.qdots.com)
8.5.3
Metallic nanoparticles
Metals have a lot of mobile electrons. Under the action of an external electric field, they respond very fast to screen it. Moving over a positive background, their mutual Coulomb interaction results in them moving collectively like a wave. Such a collective excitation is
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CHAPTER 8. PHOTONIC NANODEVICES
Figure 8.2: Schematic of a QdotTM quantum dot, size-dependent emission, and fluorescence from mouse cerebellum (www.qdots.com).
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8.6. METAMATERIALS
known as a plasmon and the latter has a resonant frequency of oscillation. This is described by a frequency-dependent dielectric constant known as the Drude dielectric function: ε(ω) = 1 −
ωp2 , ω 2 + iγω
(8.2)
where ωp2 = ne2 /(ε0 m∗ ) is the plasma frequency (squared) and γ is a damping constant due to the scattering of the electrons. In a nanoparticle, this includes boundary scattering which then introduces the size dependence in the dielectric function. For Au spherical nanoparticle of size 9 nm, the peak absorption is around 500 nm and it redshifts for larger nanoparticles. Fabrication and optical properties of metallic nanoparticles is reviewed in Ref. [23]. The peculiar optical properties of metallic nanoparticles have led to their use for local field enhancement for apertureless near-field microscopy and for electromagnetic waveguiding using nanoparticles smaller than the wavelength. Diffraction limits the dimensions of optical waveguides in the visible to λ/(2η) or a few 100 nm. It has been found that, not only can an array of nanoparticles guide electromagnetic fields when they are near-field coupled, but it is also possible to carry the energy around sharp bends. For a planar geometry consisting of a metal in contact with a dielectric, the plasmons are localized near the interface and are known as surface plasmons (Fig. 8.3). The electric field z Dielectric Electric field E
_
+
_
+
_
+
Metal
Figure 8.3: Electric field distribution of a plasmon oscillation at the interface of a metal and a dielectric. of surface plasmons are in both layers (though mostly in the dielectric), hence the wavelength of a surface plasmon depends upon the dielectric function of both layers [24, 25]: λsp = λ
s
εm + ε d , εm εd
(8.3)
If εm ∼ −εd , then the plasmon wavelength can be very small and of the order of the nanosizes; this would allow strong coupling.
8.6
Metamaterials
Metamaterials are artificial inhomogeneous systems of materials and are expected to have properties quite different from those of the constituents. The simplest example is a dielectric
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CHAPTER 8. PHOTONIC NANODEVICES
waveguide. An example of the latter is a slab of glass or an optical fiber. In optics, we know that light can be internally reflected if there is a change in the dielectric constant (i.e., refractive index). This acts as an optical confinement analogous to the confinement of electrons in a potential well. For optics, length scale for confinement is of the order of micrometer.
8.6.1
Maxwell’s equations and scaling properties
There is no length scale; hence, electromagnetic properties are similar at all length scales. Mathematically, rewriting Maxwell’s equations as (for the H field) #
"
1 ∇ × H(r) = ∇× ε(r)
!
ω2 H(r). c2
(8.4)
Now rescale the dielectric regions (Fig. 8.4): ε0 (r) = ε(r/s).
(8.5)
This is equivalent to rescaling the position vector
6
ε(r) ε0 (r)
−2 −1
1
2
-
−2
2
r r0 = sr
Figure 8.4: Length scaling of inhomogeneous system with s = 2. r0 = sr, giving
"
#
1 s∇0 × H(r0 /s) = s∇ × ε(r0 /s) 0
!
ω2 H(r0 /s). c2
101
8.6. METAMATERIALS But ε(r0 /s) = ε0 (r0 ); thus, #
"
1 ∇ × 0 0 ∇0 × H(r0 /s) = ε (r ) 0
!
ω2 H(r0 /s), s2 c2
i.e., similar solutions. The new mode is H0 (r0 ) = H(r0 /s),
(8.6)
and the new frequencies are
ω , s i.e., there is no need to solve the problem again at a new length scale. ω0 =
8.6.2
(8.7)
Photonic crystals
We have seen in the previous chapter that a periodic arrangement of atoms leads to the creation of band gaps in the electronic spectrum. It is, therefore, not surprising that a periodic arrangement of inclusions or holes of different refractive indices in a transparent matrix can also lead to “photonic band gaps” (PBG). The significance of the latter is that light of energies falling within those gaps cannot propagate freely. The dimensions of the structures is obviously of the order of the wavelength of the wave that is involved. This was first proposed by Yablonovitch in 1987 [26]. A schematic of such inhomogeneous dielectric crystals is shown in Fig. 8.5.
Figure 8.5: Photonic crystals. The physics behind the phenomenon is the same as for electrons in a crystal whereby the interference of multiple traveling waves scattered off the microstructures can be constructive (band formation) or destructive (band gaps). Such wave scattering is also known as Bragg scattering. Using 2ηd sin θ = mλ, (8.8) we find that Bragg scattering is strong if d∼
λ , 2η
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CHAPTER 8. PHOTONIC NANODEVICES
Figure 8.6: An electronic structure and a photonic band structure. which is of the order of 200 nm for visible light. An example of a photonic band structure is shown in Fig. 8.6. Note that the group velocity, defined by vg =
∂ω , ∂k
is not constant and can reach zero. Thus light propagation can be slowed down in a PBG crystal; this could be useful if one wishes to increase interaction time. Creating a defect in the structure has a similar effect as for electrons, whereby an electromagnetic mode becomes allowed inside the gap. Thus the overall structure would act as a waveguide similar to an optical fiber. A significant difference is that, if the photonic crystal is bent, then the radiation is expected to follow the waveguide even for sharp bends. This contrast to optical fibers where a sharp bend would allow radiation to leak out of the fiber due to internal reflection not being satisfied.
8.6.3
Left-handed materials (LHM)
In 1967–8, Veselago [27, 28] theoreticized what kind of electromagnetic waves would be sustained in materials with a negative refractive index. His main conclusion is that the material would act like one with left-handedness. This would reverse a number of wellknown physical phenomena. For example, an approaching light source would not be Doppler blue shifted but rather red shifted. In 1998 [29, 30], such a metamaterial was constructed at the micron scale and demonstrated for microwaves (Fig. 8.7). A LHM is one whereby the E, H, and k vectors form a left-handed set. Note, however, that the E, H, and S vectors would still form a RH set. This subtlety can be illustrated
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8.7. SUMMARY
Figure 8.7: The original left-handed (http://physics.ucsd.edu/lhmedia). n>0
material
at
microwave
frequency
n>0 S2* k 2 *
S1
Hk H2 HH H 66
k1
Y H
-
x
S H2
j H
k1t k2t n<0
Figure 8.8: Refraction at a RHM/RHM and a RHM/LHM interface [31]. using Fig. 8.8 [31]. The two solutions depicted on the RHS of the interface are both allowed in the sense of conserving the transverse momentum of the incident wave. In addition, one requires the Poynting vector to have a positive x component. As an illustration of the peculiar properties of a LHM, consider a flat slab sandwiched between two RHM, in effect an extension of Fig. 8.8. This is shown in Fig. 8.9. Among the more interesting properties predicted in the optical regime is negative refraction (Fig. 8.10) and cloaking (Fig. 8.11).
8.7
Summary
• Increasing the degree of confinement (from bulk to quantum dots) should lead to semiconductor lasers with lower threshold on the basis of a density-of-states argument.
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CHAPTER 8. PHOTONIC NANODEVICES n>0
H
HH
object
n<0
Y HH * Hj H
HH
H H
HH
n>0
HH HH
HH
H H
* Hj H
-
x
image
Figure 8.9: Focussing using a RHM/LHM/RHM slab geometry [31].
Figure 8.10: Illustration of negative refraction. The length scale is ∼ 10 nm. The other important consequence of nanostructuring is the control of the transition energy. • The optical properties of metallic nanoparticles are size and shape dependent and this is a classical electrodynamics effect. Light absorption can create surface plasmons of a shorter wavelength allowing subwavelength phenomena. • The dispersion relation of light can be modified by using an inhomogeneous medium, including the creation of photonic band gaps. The length scale is of the order of the λ/(2η).
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8.7. SUMMARY
Figure 8.11: Illustration of electromagnetic cloaking. √ • Both solutions η = ± µ are allowed. This leads to left-handed propagation. A consequence is flat-lens focussing.
