## Abstract

We present an intuitive, simple theoretical model, coupled leaky mode theory (CLMT), to analyze the light absorption of 2D, 1D, and 0D semiconductor nanostructures. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure. Unlike conventional methods such as Mie theory that requests specific physical features of nanostructures to evaluate the absorption, the CLMT model provides an unprecedented capability to analyze the absorption using eigen values of the leaky modes. Because the eigenvalue shows very mild dependence on the physical features of nanostructures, we can generally apply one set of eigenvalues calculated using a real, constant refractive index to calculations for the absorption of various nanostructures with different sizes, different materials, and wavelength-dependent complex refractive index. This CLMT model is general, simple, yet reasonably accurate, and offers new intuitive physical insights that the light absorption of nanostructures is governed by the coupling efficiency between incident light and leaky modes of the structure.

© 2012 OSA

## 1. Introduction

Light-matter interactions at sub-wavelength structures exhibit significant resonances. [1, 2] One interesting application of the resonance is to enhance light absorption. Recent works have demonstrated strong enhancements in the absorption by leveraging on the resonant light-matter interaction at nanostructures [3, 4].This absorption enhancement is critical for the development of many high-performance absorption-based photonic devices, such as photo detectors [5], electro-optic modulators [6], and solar cells [3, 7]. In order to explore the full potential of nanostructures for absorption enhancement, it is very important to obtain better physical insights into the resonant light absorption.

Here we demonstrate an intuitive, simple theoretical model to analyze the resonant light absorption of semiconductor nanostructures, including two-dimensional (2D) planar films, one-dimensional (1D) nanowires, and zero-dimensional (0D) nanoparticles. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure, hence termed as coupled leaky mode theory (CLMT). In stark contrast to conventional methods (i.e., Mie theory [1], finite difference time domain (FDTD) [8]) that request specific physical features of the nanostructure for calculating the light absorption, this CLMT model provides an unprecedented capability to evaluate light absorption using eigenvalues of the leaky modes. Significantly, the eigenvalue only shows very mild dependence on the physical features of nanostructures. As a result, we can generally apply one set of eigenvalues calculated using a real and constant refractive index to calculations for the absorption of nanostructures with different sizes, different semiconductor materials, and wavelength-dependent complex refractive index. This CLMT model presents a general, simple, yet reasonably accurate method to analyze the light absorption of semiconductor nanostructures. It offers new intuitive physical insights that the light absorption of nanostructures is governed by the coupling efficiency between incident light and leaky modes of the structure.

## 2. General existence of leaky mode resonances in semiconductor nanostructures

TheCLMT model builds upon a framework of leaky mode resonances (LMRs) for dielectric nanostructures that we have previously developed [9]. We demonstrated that the resonant light-matter interaction at semiconductor nanowires (NWs) is rooted in the resonance of incident light with leaky modes of the nanowire. Optical responses, including light scattering and absorption, were found substantially enhanced when the incident wavelength matches one of the leaky modes supported by the NW [5, 7, 10]. All the absorption or scattering peaks in the optical spectra of NWs can be related with specific set of leaky modes. This LMRs framework establishes an intuitive connection between optical responses and intrinsic optical modes [5, 7, 10].

We find that the framework of LMRs can also apply to low-dimensional semiconductor structuresother than one dimensional (1D) wires, including two dimensional (2D) planar films and zero dimensional (0D) particles. To generalize the framework of LMRs, we define the leaky mode of a structure as a mode with propagating electromagnetic fields outside the structure. This definition is similar to the classical definition for leaky modes in optical waveguides [11].The leaky mode can be calculated by solving Maxwell’s equations in related coordinates (cartesian, cylindrical, and spherical coordinates for 2D planar films, 1D circular wires, and 0D spherical particles, respectively) and matching boundary conditions at the interface of structure/environment [9]. Without losing generality, we can set the refractive indexes of the structure and the environment to be *n* and 1, respectively. We can find that leaky modes satisfy the following equations,

*r*indicates thickness for the 2D planar film(and radius for the 1D wire and the 0D particle),

*k*is the wave vector in free space (

*k*= 2$\pi $/$\lambda $, $\lambda $ is the wavelength),

*J*

_{m}() and

*H*

_{m}() are the

*m*th order Bessel function and the first kind of Hankel functions, $\psi $

_{m}() and $\xi $

_{m}() are the

*m*th order Riccati-Bessel functions that can be related with spherical Bessel functions

*j*

_{m}() and spherical Hankel function

*h*

_{m}(), and the prime indicates differentiation with respect to related arguments. Even (odd) denotes leaky modes with symmetric (asymmetric) electric field distributions inside the 2D planar film. TM and TE refer to polarizations of transverse magnetic and transverse electric, respectively. For 1D wires, the TM (TE) polarization is defined as electric (magnetic) field polarized parallel to the wire axis [1]. The TM (TE) polarization for 0D particles is defined as no magnetic (electric) field in the radial direction [1]. The TM and TE polarizations are degenerate for 2D planar films, and thus their leaky modes are referred as TEM modes [12]. For simplicity, we only consider the leaky modes with zero propagation constant for both 2D and 1D structures, which corresponds to the scenarios of normal incidence.

Solving Eqs. (1)-(6) gives complex values for a normalized parameter *nkr* (*nkr* = *N*_{real}-*N*_{imag}*i*). These complex values are eigenvalues of the leaky modes. Table 1
lists the solution for typical leaky modes calculated using a constant refractive index of 4, i.e. *n* = 4. The real part of the eigenvalue*N*_{real} indicates the condition for leaky mode resonances (LMRs). For instance, we can expect to observe TM_{11} leaky mode resonance in 1D wires when the condition of *nkr* = 2.30 is satisfied. The imaginary part *N*_{imag }refers to the radiative leakage of the electromagnetic energy stored in leaky modes. For materials without intrinsic absorption loss, this imaginary part indicates spectral width of the leaky mode resonance.

The leaky mode of 2D planar films can be labeled with a mode number of *m*, as TEM_{m}. The mode number *m* corresponds to the number of half wavelength in the transverse direction of the planar film. Leaky modes in 1D wires and 0D particle scan be characterized by an azimuthal mode number, *m*, and a radial order number, *l*. Physically, the azimuthal mode number *m *indicates the number of effective wavelength around the circumference of the structure, while the radial order number *l *describes the number of radial field maxima within the structure. As a result, the modes can be termed as TM_{ml} or TE_{ml} [1, 2].

The eigenvalue of leaky modes shows interesting dependence on the subscript numbers of *m* and *l* as well as the refractive index *n*. For all the 2D, 1D and 0D structures, the real part of the eigenvalue (*N*_{real}) is always linearly dependent on the mode number *m* (upper panels in Fig. 1
) and the order number *l* (not shown). Additionally, for a given leaky mode, the real part *N*_{real}is essentially independent of the refractive index of the material (upper panels of Fig. 1). For instance, the *N*_{real }of TEM_{m} leaky modes in 2D structures is always equal to an integer number *m* of $\pi $, *N*_{real} = *m$\pi $*, *m* = 1, 2, 3 ……regardless the refractive index (the upper left panel of Fig. 1). This means that, no matter whatever the refractive index is, given LMRs always happen at fixed values of *nkr*. Changing the refractive index *n* can cause the value of *kr *for LMRs shift to keep the value of *nkr* invariant. In contrast, the imaginary part (*N*_{imag}) of the eigenvalue shows substantial dependence on both the subscript numbers and the refractive index. Interestingly, *N*_{imag} is constant for all leaky modes in the 2D structure, and increases with the refractive index increasing (the lower left panel of Fig. 1). This indicates identical radiative leakage for all the leaky modes, and the leakage is lower for materials with lower refractive index. In the 1D and0D structures, *N*_{imag} exponentially decreases with the mode number *m *and the refractive index *n* increasing (the lower middle and right panels of Fig. 1). This suggests stronger optical confinement at higher order modes and larger refractive index.

To illustrate the general existence of LMRs in semiconductor nanostructures, we calculate spectral light absorption for the nanostructures with well-established analytical techniques, for example, using the Lorenz-Mie formalism for 1D and 0D nanostructures. Without losing generality, we use silicon as an example [13]. Figure 2
shows the calculated absorption spectra for a 100 nm thick 2D thin film (left panel), a 1D wire (middle panel) and a 0D particle (right panel) both in radius of 100 nm. For the 1D wire, only the calculation for a TM-polarized incidence is given and for the 0D particle only for a TE-polarized incidence. We compare the absorption peaks with the real eigenvalue (*N*_{real}) of leaky modes (given as blue ticks above the absorption spectra in Fig. 2). For the convenience of comparison, the calculated absorption spectra are plotted as a function of the normalized parameter *n*_{real }*kr,* where *n*_{real }is the real part of the refractive index of silicon. We can find that the absorption peaks of all the nanostructures show very good consistence with the eigenvalues of leaky modes. This suggests that the resonant light absorption of 2D, 1D, and 0D semiconductor nanostructures can all be correlated to the resonance of incident light with leaky modes of the structures.

## 3. Coupled leaky mode theory

The framework of LMRs indicates that the light absorption of nanostructures is governed by the resonance of incident light with leaky modes of the nanostructure. As a further step, we develop a model of coupled leaky mode theory (CLMT) to quantitatively evaluate the light absorption from the perspective of LMRs. In contrast to typical rigorous methods (i.e., Mie theory, FDTD) for analyzing the light absorption that requests pecitic physical features of nanostructures, this CLMT model provides a capability of analyzing the light absorption only using eigenvalues of the leaky modes.

The CLMT considers nanostructures as low-quality-factor resonators, and models the optical response of nanostructures as resulting from the coupling of incident light with leaky modes of the nanostructure [7, 9].The absorbed power (*P*_{abs}) in a nanostructure can be modeled as the absorption loss of electromagnetic energy (Ι*a*Ι^{2}) stored in the nanostructure resonator,

_{abs}indicates the intrinsic absorption loss in the material. The stored energyΙ

*a*Ι

^{2}is dictated by the coupling between leaky modes of the nanostructure and incident light, and can be derived using a formalism of couple-mode theory (CMT) [14–16].where

*a*is the amplitude of electromagnetic fields and is normalized to reflect the energy stored in the mode, $\omega $

_{0}and $\gamma $

_{rad }are the resonant frequency and the radiative loss of the leaky mode, κ corresponds to the coupling between the leaky mode and incident light

*W*

_{i}. By applying energy conservation and time-reversal symmetry, we can get $k=\sqrt{C{\gamma}_{rad}}$,

*C*is a constant equal to 1 for 2D structure (standing wave resonator), and 2 for 1D and 0D structures (traveling wave resonator). For an incident frequency

*ω*, we may have

*N*

_{real}) and imaginary (

*N*

_{imag})parts of the eigenvalue as ${\omega}_{0}$ =

*c*.

*N*

_{real}/(

*n*.

*r*), and ${\gamma}_{rad}$ =

*c*.

*N*

_{imag}/(

*n*.

*r*),

*c*is speed of light. Substituting these expressions into Eq. (7) and (9), the absorption can be related with the leaky mode as

*q*

_{abs}and

*q*

_{rad}are the quality factors of the nanostructure resonator due to absorption loss and radiation loss, respectively.

*q*

_{abs}can be derived from the intrinsic complex refractive index (

*n*=

*n*

_{real}-

*n*

_{imag}*i) of the material,

*q*

_{abs}=

*n*

_{real}/(2*

*n*

_{imag}), and

*q*

_{rad}is related with the eigenvalue of leaky modes,

*q*

_{rad}=

*N*

_{real}/(2*

*N*

_{imag}).

*α*is the ratio between the incident frequency

*ω*and the eigenfrequency

*ω*

_{0},

*α*=

*ω*/

*ω*

_{0}or

*n*

_{real}

*kr*/

*N*

_{real}. Equation (10) can be rewritten as

*I*

_{0}and the dimensionality of the nanostructure. Arbitrary incident wave

*E*

_{i}can always be expanded into a series of harmonic terms, and $\left|{W}_{i}\right|$ represents the power carried by each of the harmonic terms. For simplicity, we consider a plane wave as the incidence,

*E*

_{i}=

*E*

_{0}

*exp*[

*i*(

*kx*-

*ωt*)]. For 1D wires, the incident wave can be expanded in cylindrical coordinate (

*r*,

*ϕ*,

*z*) as,

*exp*(-i

*ωt*)), respectively. The power carried in the

*m*th order incoming wave can be calculated

*I*

_{0},

*λ*/2π

*I*

_{0}, and (2

*m*+ 1)

*λ*

^{2}/2π

*I*

_{0}for 2D, 1D and 0D nanostructures, respectively.

Substituting the expressions of $\left|{W}_{i}\right|$ into Eq. (11), we can find out the light absorption cross section *C*_{abs }using *C*_{abs} = *P*_{abs}/*I*_{0}. Subsequently, we can derive the absorption efficiency *Q*_{abs} as *Q*_{abs} = *C*_{abs}/*G*.*G* is the geometrical cross section of nanostructures, which is unity, 2*r* and π*r*^{2} for 2D, 1D and 0D structures, respectively. Expressions for the absorption efficiency *Q*_{abs }can be written as

*n*

_{real}and

*n*

_{imag}), the absorption of nanostructures for a given frequency

*ω*is dictated by the eigenvalue of leaky modes,

*N*

_{real}and

*N*

_{imag}. These equations calculate the absorption efficiency contributed by one single leaky mode. For nanostructures that typically involve multiple leaky modes, we need sum up the absorption efficiency

*Q*of each leaky mode to get the total absorption efficiency ${Q}_{abs}^{T}$,To minimize the interference between different leaky modes, the absorption efficiency for a single mode

_{abs,ml }*Q*needs be corrected from the value calculated using Eqs. (15-17) by ${Q}_{abs,ml}={Q}_{abs}/[1+2{({n}_{real}kr-{N}_{real})}^{2}]$. This essentially limits every leaky mode can only interact with incident wavelengths at the proximity of each resonant wavelength.

_{abs,ml}Equation (18) can nicely reproduce the absorption efficiency calculated using conventional rigorous methods, such as Lorentz-Mie theory for 1D and 0D structures [1]. Figure 3 shows calculated absorption spectra of 2D (left panel), 1D (middle panel) and 0D (right panel) of silicon nanostructures using the conventional analytical methods (blue curve) and the CLMT model (red curve). Again, without losing generality, the thickness of the 2D thin film is set 100 nm, and the radii of the 1D wire and the 0D particle both are 100 nm. For simplicity, we only calculate the absorption for normal incidence of plane waves with linear polarization (TM polarization for 1D wire, and TE polarization for 0D particle). In the calculations using the CLMT model, we use the eigenvalues of leaky modes listed in Table 1, which are substituted into Eq. (18) along with the intrinsic refractive index of silicon. We can find from Fig. 3 that the CLMT calculations for all the structures are reasonably consistent with the rigorous calculations.

Notably, while the refractive index of semiconductor materials typically shows substantial wavelength dependence, the CLMT model can reproduce the rigorous solution using the eigenvalue of leaky modes that are calculated with a constant refractive index. For instance, the CLMT calculations shown in Fig. 3 use the eigenvalues of leaky modes (*N*_{real} and *N*_{imag}) calculated with a constant refractive index of 4.As a reference, the refractive index of silicon materials varies in a range of 4.6-3.5 in this same spectral range [13].We can get similar CLMT calculations by assuming the refractive index as other constants, for example, 3.5, or 5 when calculating the eigenvalue. This robustness of the CLMT calculation over the refractive index can be understood from Eqs. (15)-(17). The absorption of nanostructures of given materials (*n*_{imag}, *n*_{real} are fixed) is essentially dictated by *N*_{imag} and *N*_{real}. We have demonstrated in Fig. 2 that *N*_{real }of a given leaky mode is approximately independent of the refractive index of materials. This leaves *N*_{imag }as the only variables that could vary with the refractive index. The leaky modes involved in the light absorption of nanostructures are typically low order modes (*m*< 3 and *l*< 3), and the *N*_{imag }of these leaky modes show moderate variation for different refractive indexes. For instance, *N*_{imag} of the TM_{11} leaky mode in 1D wires can be found equal to 0.163 and 0.113 for a refractive index of 4 and 5, respectively. Such moderate variation in *N*_{imag }can causes only minor change in the overall absorption efficiency.

The CLMT model can generally apply to 2D, 1D, and 0D nanostructures of all kinds of semiconductor materials. Most semiconductor materials have a refractive index in the range of 3~5. We find that the eigenvalue of leaky modes calculated with a constant refractive index of 4 can reasonably reproduce the spectra absorption of a wide range of semiconductor nanostructures. Figure 4 show the calculated absorption spectral of 1D nanostructures of a variety of materials, including amorphous silicon (a-Si), gallium arsenide (GaAs), germanium (Ge), and copper indium gallium selenide (CuInGaSe), using rigorous analytical methods (blue curve) and the CLMT model (red curve). The CLMT calculations use the eigenvalues of leaky modes for a constant refractive index of 4 (listed in Table 1). To illustrate the robustness of the CLMT model, the CLMT calculations using the eigenvalues of leaky modes for a constant refractive index of 3 are also given for a-Si and CuInGaSe (black dashed line). We can find that the CLMT calculations generally show reasonable consistence with the results calculated from rigorous analytical methods. For a-Si, both CLMT calculations with n = 4 (red solid curve) and n = 3 (black dashed line) are reasonable approximations for the Mie calculation. However, for CuInGaSe, the CLMT calculation with n = 3 shows a better approximation for the Mie calculation than the one with n = 4. This is because the real part of the refractive index of CuInGaSe is close to 3 across the whole spectrum.

## 4. Conclusion

We demonstrate anew theoretical model, coupled leaky mode theory (CLMT), to analyze the light absorption in 2D, 1D, and 0D semiconductor nanostructures. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure. The CLMT model provides a capability of evaluating the light absorption of nanostructures using the eigenvalue of leaky modes, instead of specific physical features of the nanostructure as conventional analytical methods. Significantly, the eigenvalue only shows mild dependence on the physical features of nanostructures. As a result, we can generally apply one set of eigenvalues calculated using a real and constant refractive index to calculations for the absorption of nanostructures with different sizes, different semiconductor materials, and wavelength-dependent complex refractive index. This CLMT model provides a general, simple and reasonably accurate approach for the analysis of light absorption in nanostructures as an alternative to existing methods that are typically computation intensive.

More importantly, the CLMT provides new physical insights into the light absorption that cannot be obtained from existing analytical methods. It reveals that the light absorption of nanostructures is determined by the coupling between incident light and leaky modes of the structures. This insight opens a new door for the development of high-performance absorption-based photonic devices. For instance, we can see from Eqs. (15)-(17) that, upon resonances (*n*_{real}*kr*/*N*_{real}-1 = 0), the absorption efficiency is dictated by (*N*_{imag}/*N*_{real}).(*n*_{imag}/*n*_{real})/(*N*_{imag}/*N*_{real} + *n*_{imag}/*n*_{real})^{2}. This absorption can be maximized by tuning the radiative quality factor *q*_{rad} = *N*_{real}/(2**N*_{imag})of leaky modes equal to the absorption quality factor *q*_{abs} = *n*_{real}/(2**n*_{imag}). Regardless the intrinsic absorption of materials for specific incidence, properly nanostructuring the materials in nanostructures can always maximize the absorption efficiency to the same level as ½, 1/(2*kr*), and (2m + 1)/[2(kr)^{2}] for 2D, 1D and 0D structures, respectively. Therefore, to design high-performance nanostructure photodetectors for a specific wavelength, we need tune the wavelength close to a leaky mode resonance with a radiative quality factor *q*_{rad }comparable to the intrisinc absorption quality factor *q*_{abs} of the materials at this wavelength.

## Acknowledgments

This work has been supported by start-up fund from North Carolina State University. L.C. acknowledges the Ralph E. Powe Junior Faculty Enhancement Award.

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